64c3814
==== TABLE OF CONTENTS ====
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//bibliography.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//main.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/charts.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/discs.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/dynamics.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/ellipsoids.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/flow.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/introduction.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/preliminaries.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/uniformization.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/variation.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//unicode-map.tex
===========================
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//bibliography.tex ====
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==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//bibliography.tex ====
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\title{Uniformization of complete K\"ahler manifolds\protect\\
with positive bisectional curvature}
\author{OpenAI}
\date{September 23, 2026}
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\begin{document}
\maketitle
\begin{abstract}
We prove that every complete connected noncompact K\"ahler manifold with
strictly positive holomorphic bisectional curvature is biholomorphic to
complex Euclidean space. This resolves Yau's uniformization conjecture
positively in every complex dimension.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/flow}
\input{sections/discs}
\input{sections/variation}
\input{sections/ellipsoids}
\input{sections/charts}
\input{sections/dynamics}
\input{sections/uniformization}
\input{bibliography}
\end{document}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//main.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/charts.tex ====
\section{Harnack inequalities and shrinking holomorphic charts}
\label{sec:charts}
The joint positivity of the cotangent norm will now give an exhausting
system of inverse charts with quantitative transition estimates.
We first prove positive-time completeness and construct uniform local
charts. These charts may have several sheets, so continuing their
inverse germs across compact sets also requires a topological step:
we will contract loops before constructing single-valued inverse branches.
For the quantitative estimates, volume loss must force contraction in
every direction, with an exponent common to all compact sets.
The local chart and path-lifting constructions have antecedents in Chau--Tam
\cite[Section~2]{ChauTamSimply}\cite[Sections~2--3]{ChauTamStein}.
Throughout this Section, a metric on a cotangent space is the Hermitian
dual metric, including the transpose in its matrix representation.
\subsection{From joint positivity to completeness}
The differential Harnack inequalities for K\"ahler--Ricci flow have
their classical antecedent in Cao's work \cite{CaoHarnack}.
Here we derive the needed inequalities from joint positivity before
using them to prove completeness of the evolving metric.
\begin{proposition}\label{prop:harnack-completeness}
The flow of Theorem~\ref{thm:localized-flow} has nonnegative holomorphic
bisectional curvature and is complete at every positive time. Its
Ricci tensor is positive definite. Writing $A=\Ric(h_t)$ and
$R=\tr_{h_t}A$, one has, in fixed holomorphic coordinates,
\begin{align}
A_t+A h_t^{-1}A+t^{-1}A&\ge0,
\label{chart:matrix-harnack}\\
R_t+t^{-1}R+2\operatorname{Re}\partial_XR+A(X,\overline X)&\ge0
\qquad(X\in T^{1,0}M).
\label{chart:scalar-harnack}
\end{align}
In particular $h_{t'}\le h_t$ for $t'\ge t$, $R(o,t)\le C$ for
$t\ge0$, and
\begin{equation}\label{chart:integrated-harnack}
R(x,t)\le R(y,t')\frac{t'}t
\exp\!\left(\frac{C\dist_{h_t}(x,y)^2}{t'-t}\right)
\qquad(0<t<t').
\end{equation}
On each fixed compact set, $h_t\le C_Kt^{-a_K}g$ for $t\ge1$, with
$a_K>0$.
\end{proposition}
\begin{proof}
By Theorem~\ref{thm:joint-positivity}, the dual squared norm of the
tangent metric pulled back to $(x,w)$, where $t=e^{w+\bar w}$, is
plurisubharmonic. Completing the square in the free first derivative
of a cotangent section identifies this assertion with Griffiths
nonnegativity of the pulled-back tangent metric. At a spatially
K\"ahler-normal point its curvature blocks are
\[
\Theta_{i\bar j\,\alpha\bar\beta}
=\Rm_{i\bar j\alpha\bar\beta},\qquad
\Theta_{w\bar j\,\alpha\bar\beta}
=t\nabla_{\bar j}A_{\alpha\bar\beta},\qquad
\Theta_{w\bar w}=tA+t^2(A_t+A h_t^{-1}A).
\]
The spatial block gives nonnegative bisectional curvature and hence
$A\ge0$ and metric monotonicity. The last block gives
\eqref{chart:matrix-harnack}. Trace the curvature in its fiber
indices and evaluate on the spacetime direction $(X,t^{-1})$.
The identities
\[
\tr_{h_t}A_t=R_t-|A|_{h_t}^2,
\qquad \tr_{h_t}(A h_t^{-1}A)=|A|_{h_t}^2,
\qquad \tr_{h_t}\nabla_X A=\partial_XR
\]
give \eqref{chart:scalar-harnack}.
At a fixed point choose a moving unitary frame satisfying
$E_t=\tfrac12 A^{\#}E$, where $A^{\#}=h_t^{-1}A$.
Differentiation shows that
\[
\frac{d}{dt}\bigl[tA(E,\overline E)\bigr]
=t\bigl(A_t+A h_t^{-1}A+t^{-1}A\bigr)(E,\overline E)\ge0.
\]
Initial strict bisectional positivity and local smooth convergence
give $A>0$ at sufficiently small positive times on any prescribed
compact set. The preceding inequality propagates this positivity
and gives $tA\ge a_Kh_t$ there for $t\ge1$, after changing the compact
constant. Integrating $\partial_t h_t=-A$ proves the asserted
power contraction. Applied to the trace, the same inequality gives
monotonicity of $tR(o,t)$. The linear bound for $\rho(o,t)$ in
Theorem~\ref{thm:localized-flow} implies, for $t\ge1$,
\[
tR(o,t)\log2
\le\int_t^{2t}R(o,v)\,dv
=\rho(o,2t)-\rho(o,t)\le C(1+2t).
\]
Smoothness on compact time intervals supplies the remaining
basepoint bound.
For clarity, the next integration precedes the completeness proof.
Choose a piecewise smooth path from $x$ to $y$ whose $h_t$ length is
within $\eps$ of their distance, and run it at constant $h_t$ speed
on $[t,t']$. Such paths exist by the definition of the distance.
Along this path, \eqref{chart:scalar-harnack}, divided by $R>0$,
and $A\le Rh_v$ give
\[
\frac{d}{dv}\log R(\gamma(v),v)
\ge-\frac1v-C|\dot\gamma(v)|_{h_v}^2.
\]
Since $h_v\le h_t$, its energy is at most
$(\dist_{h_t}(x,y)+\eps)^2/(t'-t)$. Integrating and then letting
$\eps\downarrow0$ proves \eqref{chart:integrated-harnack} without
using a minimizing geodesic or completeness of $h_t$.
Fix $T<\infty$. Put $d=\dist_{h_t}(x,o)$ and apply
\eqref{chart:integrated-harnack} with $y=o$ and
$t'=t+1+d^2$. The basepoint bound gives
\begin{equation}\label{chart:quadratic-curvature}
R(x,t)\le\frac{C_T(1+d^2)}t,
\qquad 0<t\le T.
\end{equation}
Let $D_L=B_g(o,L)$, and let $l_L(t)$ be the infimum of lengths of
paths from $o$ to first exit from $D_L$, measured in $h_t$.
The initial metric is complete, so $\overline D_L$ is compact.
There is a minimizing segment realizing $l_L(t)$, contained in
$\overline D_L$, and every initial part of it minimizes to its
endpoint. On this segment the distance in
\eqref{chart:quadratic-curvature} is at most $l_L(t)$.
Write $l=l_L(t)$. In the summed real index forms use perpendicular
parallel fields times a cutoff which rises from zero to one on
an endpoint interval of width
\[
a=\min\left(\frac l2,\frac{\sqrt t}{1+l}\right).
\]
If $l\ge2\sqrt t/(1+l)$, the index inequality and the two endpoint
curvature bounds imply
\[
\int_0^l\Ric_{h_t}(\dot\gamma,\dot\gamma)\,dr
\le \frac C a+\frac{C_T(1+l^2)a}{t}
\le\frac{C_T(1+l)}{\sqrt t}.
\]
Here and below fixed real-versus-complex normalization factors are
included in $C_T$. If $l<2\sqrt t/(1+l)$, use
\eqref{chart:quadratic-curvature} on the whole segment to obtain the
same bound. The function $l_L$ is locally Lipschitz in time, since
the metrics are smoothly equivalent on the compact closure. At a
differentiability time, compare a current minimizing segment with
its length at a slightly earlier time. The metric variation formula
therefore gives
\[
l_L'(t)\ge-\frac{C_T(1+l_L(t))}{\sqrt t}
\quad\text{for almost every }t\in(0,T].
\]
Consequently
\begin{equation}\label{chart:exit-distance}
1+l_L(t)\ge(1+L)e^{-2C_T\sqrt t}.
\end{equation}
As $L\to\infty$, every divergent path has infinite $h_t$ length.
Equivalently, each finite $h_t$ ball is trapped in a compact initial
ball; its closure is compact by local equivalence of the smooth
metrics. This proves completeness.
\end{proof}
\begin{lemma}[Uniform immersed charts]\label{chart:local-charts}
For every fixed compact $K\subset M$, there are $r_K,c_K,C_K>0$
such that, for sufficiently large $t$ and every $p\in K$, there is
a holomorphic local biholomorphism
\[
\Phi:B_{\C^n}(r_K)\longrightarrow M,\qquad
\Phi(0)=p,\qquad d\Phi(0)\text{ unitary for }h_t,
\]
with $c_Kg_{\rm eucl}\le\Phi^*h_t\le C_Kg_{\rm eucl}$.
The pullback metrics have bounds of every fixed derivative order
on smaller balls. The corresponding local smooth compactness
statement holds for rescaled flows whenever curvature is uniformly
bounded on fixed-radius moving balls during a fixed two-sided time
interval.
\end{lemma}
\begin{proof}
For $t\ge1$, \eqref{chart:integrated-harnack}, with a fixed future
time increment, bounds $R$ on every fixed-radius $h_t$ ball about
$o$. Nonnegative bisectional curvature gives $|\Rm|\le C_nR$.
Centers in $K$ are handled by a compact family of paths from $o$:
their $h_t$ lengths are bounded, so balls about these centers lie
in a bounded-radius ball about $o$.
To apply local curvature derivative estimates, take a ball for the
earliest time of a short time interval. It remains inside the
corresponding larger later-time balls because metrics decrease.
The curvature bound gives metric equivalence on this fixed region
throughout the interval. A path measured in the later metric
cannot first exit the earlier ball at arbitrarily small length;
thus a smaller later-time ball is contained in this region.
Local Ricci-flow derivative estimates
\cite{Shi}\cite[Theorem~1.4]{Kotschwar} now give all curvature
derivatives on smaller balls and inner time intervals.
Here the estimate is applied on shifted sufficiently short cylinders,
with a time buffer before each observation slice. Metric equivalence
keeps the initial balls required by the local estimate compactly
inside the fixed region. This construction applies verbatim after
a fixed rescaling.
Here is why no injectivity-radius hypothesis enters the chart
statement; compare \cite[Lemma~2.2]{ChauTamSimply}.
Pull back by the exponential map on a small tangent ball. The
curvature bound excludes conjugate points at this scale, although
the exponential map may identify distinct points. Jacobi-field
estimates and their differentiated equations give uniformly
controlled pullback metrics and parallel complex structures, with
a fixed Euclidean lower metric bound. In particular any sequence
has a smoothly convergent subsequence on a smaller tangent ball.
The identities $J^2=-\Id$ and $N_J=0$ pass to the smooth limit.
The Newlander--Nirenberg theorem supplies local holomorphic coordinates
for the limit complex structure \cite{NewlanderNirenberg}
\cite[Theorem~1.1]{HillTaylor}. In the smooth case their components
satisfy the smooth elliptic equation $\bar\partial_J^*\bar\partial_J f=0$,
so interior elliptic regularity makes these coordinates smooth.
Restrict them to a still smaller ball. Their squared radius is uniformly strictly
plurisubharmonic for all sufficiently close complex structures.
Solve the small $\dbar$ errors of these coordinate functions on
a smaller strictly pseudoconvex coordinate ball. One can use the
weighted $L^2$ estimate \cite{Hormander,Demailly}, viewing scalar
functions as top forms with values in the anticanonical bundle.
Give this bundle the metric induced by the original uniformly
controlled pullback metric. A fixed large multiple of the squared
radius pays its uniformly
bounded curvature, and the strictly plurisubharmonic exhaustion
of the ball supplies an auxiliary complete K\"ahler metric.
The source errors tend to zero smoothly, so the correcting
functions tend to zero in local $L^2$. We first justify smoothness of
each correction, then obtain uniform estimates on a fixed smaller
real ball. Write the corrections as $u_j$
and their equations as $\dbar_{J_j}u_j=\eta_j$.
For each index separately, the existing complex structure $J_j$
has smooth holomorphic coordinates, with no uniform radius asserted.
In those coordinates,
\[
\Delta_{\mathrm{eucl}}u_j
=4\sum_a\partial_{z_a}\eta_{j,a}
\]
is an identity of distributions with smooth right side. A compactly
supported smooth extension of the right side, convolved with the
Euclidean fundamental solution, gives a smooth particular solution
on a smaller ball.
The remaining harmonic distribution is smooth: its mollifications
satisfy the mean-value identity against a fixed smooth radial
unit-mass kernel, and passage to the distributional limit identifies
the distribution with its smooth convolution by that kernel.
Thus each $u_j$ is qualitatively smooth.
For uniform estimates return to the fixed real ball and use
$P_j=\dbar_{J_j}^*\dbar_{J_j}$ with the original controlled
pullback K\"ahler metric, rather than the auxiliary weighted metric.
The scalar K\"ahler identity identifies $P_j$, up to a fixed sign
and normalization, with the real Laplace--Beltrami operator.
Its coefficients have uniform bounds of every fixed order and a
uniform ellipticity lower bound. Moreover
$P_ju_j=\dbar_{J_j}^*\eta_j=:F_j$ tends smoothly to zero.
Testing this equation with $\chi^2\overline{u_j}$ and absorbing
the cross term gives the Caccioppoli estimate
\[
\int\chi^2|\nabla u_j|^2
\le C\int_{\supp\chi}(|u_j|^2+|F_j|^2)
\]
for a fixed interior cutoff. The qualitative smoothness just
proved legitimizes this test. Uniform metric equivalence gives a
$W^{1,2}$ bound in the fixed real coordinates. Interior elliptic
estimates \cite[Theorems~8.8 and~8.10]{GilbargTrudinger}, on
successively smaller fixed balls, now yield
\[
\|u_j\|_{H^{m+2}(B')}
\le C_m\bigl(\|u_j\|_{L^2(B)}+\|F_j\|_{H^m(B)}\bigr).
\]
Choose $m+2>k+n$ and use Sobolev embedding in real dimension $2n$
\cite[Section~5.6.3, Theorem~6]{EvansPDE}. This proves convergence
to zero in every fixed $C^k$ norm on smaller balls. The corrected
coordinate maps are therefore $C^1$ close to the limiting coordinate
map. On a sufficiently small fixed ball their derivatives are
uniformly close to its invertible center derivative. They are
injective on that ball, and their images contain a common ball about
their center values, by the quantitative inverse function theorem.
Subtract the coordinate center values and invert on this common ball.
Composing with the exponential parametrization and making a bounded
linear normalization, with one further fixed reduction of radius,
gives the required maps into $M$ with unitary center derivative.
If no common radius and bounds existed for the original family, a
sequence witnessing their failure would have a smoothly convergent
subsequence of pullbacks. The construction just given supplies common
charts on that subsequence, a contradiction. This proves the uniform
chart assertion.
For a flow, use the exponential parametrization at its central
time as a fixed parametrization. The curvature bound controls
the time derivative of the metric; the first Ricci derivative
controls that of its connection. Induction using the higher
covariant derivative estimates gives ordinary space-time
derivative bounds for the metrics and their parallel complex
structures. Arzel\`a--Ascoli on smaller balls gives the claimed
local smooth limits.
\end{proof}
\subsection{Total distortion in the initial metric}
In an $h_t$-unitary chart at $x$, the product of the initial-metric
singular lengths is $e^{\rho(x,t)/2}$. This determinant identity alone
allows the growth to occur in only some directions. The next lemma
bounds the greatest singular length by a fixed power of the least,
so volume loss will force contraction in every direction.
\begin{lemma}[Static distortion]\label{lem:static-distortion}
Fix $r,c>0$ and a compact set $K\subset M$. Suppose
$\Phi:B_{\C^n}(r)\to M$ is a holomorphic local biholomorphism,
$\Phi(0)\in K$, and $\widetilde g=\Phi^*g\ge c g_{\rm eucl}$.
Let $A$ and $B$ be the least and greatest singular lengths of
$d\Phi(0)$, using the Euclidean metric in the domain and $g$ in
the target. There are constants depending only on $K,r,c$ and
the initial geometry on a compact neighborhood of $K$ such that
\begin{equation}\label{chart:static-distortion}
B\le C(1+A)^C.
\end{equation}
In particular these constants are independent of the particular
chart and its total distortion.
\end{lemma}
\begin{proof}
We construct a plurisubharmonic weight whose upper bound is of order
$\log(1+A)$ and whose Levi form is strict near the center.
Its moving-ball definition will also give a nonnegative lower bound
on the support of the weighted $\dbar$ construction. These bounds
make the extension estimate independent of the total distortion.
We then extend a local function whose differential detects $B$;
its $L^2$ bound gives the required polynomial bound.
Choose a constant holomorphic covector $\alpha=df$ with
$|\alpha|_{\rm eucl}=A$ and $|\alpha|_{\widetilde g}(0)=1$.
The logarithmic dual norm
\[
\psi=\log|\alpha|_{\widetilde g}^2
\quad\text{satisfies}\quad
\psi(0)=0,\qquad \psi\le C+2\log A.
\]
For a tangent $X$, the curvature formula for this logarithmic norm is
\[
\dd\psi(X,\overline X)
=\frac{\Rm(X,\overline X,v,\overline v)}{|v|^2}
+\frac{|D'_X\alpha|^2|\alpha|^2
-|\langle D'_X\alpha,\alpha\rangle|^2}{|\alpha|^4},
\]
where $v$ is the metric dual of $\alpha$ and all norms in this
formula use $\widetilde g$. The second summand is nonnegative.
Initial bisectional positivity therefore makes $\psi$
plurisubharmonic. On any region whose target image
lies in a prescribed compact neighborhood $K^+$ of $K$, it gives
\begin{equation}\label{chart:strict-covector}
\dd\psi\ge\kappa\widetilde g
\end{equation}
with $\kappa>0$ uniform. The normalization of $\alpha$ cancels
from this estimate.
Shrink the Euclidean domain by a fixed factor. Choose $r_0>0$
so small that any path of $\widetilde g$ length at most $3r_0$
starting in this smaller domain stays in $B(r)$; the lower
metric bound makes this choice uniform. The small closed metric
balls used below are compact there and their distances are
attained by minimizing segments. Put
\begin{equation}\label{chart:metric-ball-envelope}
\Psi(z)=\max_{\dist_{\widetilde g}(z,y)\le r_0}\psi(y).
\end{equation}
This function is continuous. Compactness gives upper
semicontinuity. For lower semicontinuity, move a maximizing
endpoint a little towards the center along a minimizing segment;
the resulting endpoint is admissible for all nearby centers and
its value tends to the maximum.
We verify the Levi inequality for this moving-ball maximum.
Let $\varphi$ be a smooth upper test at $z$, let $y$ realize the
maximum, and prescribe a nonzero complex tangent $V$ at $z$.
Choose a holomorphic center germ with this derivative. If
$\dist_{\widetilde g}(z,y)<r_0$, any holomorphic endpoint germ
with derivative equal to the parallel translate of $V$ along a
minimizing segment stays admissible for a sufficiently small
parameter. Suppose instead that the constraint is active.
For two real parameter directions corresponding to $V$ and $JV$,
choose trial paths whose first variation fields are the parallel
translates along this segment. Their first length variations
vanish. The trace of their second length variations is
\[
-\int_0^{r_0}
\bigl[\Rm(T,V^{\parallel},V^{\parallel},T)
+\Rm(T,JV^{\parallel},JV^{\parallel},T)\bigr]\,dr<0.
\]
Tangential components make no curvature contribution, and
holomorphic endpoint germs have zero trace covariant
acceleration, so there are no omitted trace boundary terms.
The strict sign is precisely the initial bisectional sign.
The trace inequality alone is insufficient to preserve a
two-parameter length constraint. At the far endpoint we may
prescribe its holomorphic second jet freely while retaining
its first derivative. Changing that jet by a real tangent $W$
(regarded as a complex tangent by $J$) changes the real length
Hessian by
\[
\begin{pmatrix}
\langle W,T\rangle&\langle JW,T\rangle\\
\langle JW,T\rangle&-\langle W,T\rangle
\end{pmatrix}
\]
at the final endpoint. Vectors in the real span of $T,JT$
realize every trace-free symmetric matrix of size two.
Choose the jet to remove the trace-free part of the length
quadratic. The trace remains
strictly negative; hence the full quadratic is negative definite.
Smooth trial paths joining these holomorphic endpoint germs
then have length less than $r_0$ for every sufficiently small
nonzero parameter. The actual distance is no greater than this
trial length. This proves admissibility also at an active
constraint, without differentiating a possibly nonsmooth
distance function.
On either kind of admissible endpoint germ $y(a)$,
$\varphi(z(a))\ge\Psi(z(a))\ge\psi(y(a))$, with equality at
$a=0$. Taking the parameter Levi derivative proves
$\dd\Psi\ge0$ in upper-test sense, and thus in the
plurisubharmonic sense. If
$\dist_{\widetilde g}(0,z)<r_0/4$, all the relevant paths and
endpoints map into a fixed compact $K^+$; applying
\eqref{chart:strict-covector} to the transported tangent gives
\begin{equation}\label{chart:strict-envelope}
\dd\Psi\ge\kappa\widetilde g
\quad\text{on }B_{\widetilde g}(0,r_0/4),
\qquad \Psi\ge0\quad\text{on }B_{\widetilde g}(0,r_0).
\end{equation}
The latter inequality uses the admissible endpoint $0$.
Throughout the fixed Euclidean domain,
$\Psi\le C+2\log A$.
We next specify the weighted extension, including its support.
A fixed small initial holomorphic coordinate neighborhood $U$
of $p=\Phi(0)$ lifts to a single sheet through $0$. Indeed its
short radial paths and their short homotopies lift by local
inversion of $\Phi$, and their $\widetilde g$ lengths equal
their initial target lengths. The lower metric bound prevents
escape from the coordinate domain. Uniqueness of local lifts
and the short homotopies make this an inverse branch on $U$.
Choose $U$ uniformly over $p\in K$, with the whole sheet inside
$B_{\widetilde g}(0,r_0/4)$.
In these initial target coordinates $w$, centered at $p$, choose
a linear holomorphic function $\ell$ whose differential has
target norm one and whose pullback differential at $0$ has
Euclidean norm $B$. Let $\chi$ be a fixed cutoff on this sheet,
equal to one near $0$ and zero near its boundary. The function
$a=\chi\ell$, extended by zero outside the sheet, is smooth and
uniformly bounded. Its source $\beta=\dbar a$ is supported in
a fixed target coordinate annulus and satisfies
\[
|\beta|_{\widetilde g^{-1}}\le C,
\qquad
\int_{\supp\beta}dV_{\rm eucl}
\le c^{-n}\int_{\supp\beta}dV_{\widetilde g}
\le C.
\]
The last integral is the volume of one initial target annulus,
since the chosen sheet is injective. These constants do not
contain $B$.
On the same sheet choose a nonpositive logarithmic-pole weight
$\lambda$, equal to $(n+1)\log(|w|^2/\delta^2)$ near $0$ and
cut off to zero within the strict region. Its negative Hessian
is bounded by $C\widetilde g$ on the cutoff annulus, uniformly
also for smooth pole regularizations. Choose a fixed large
$k$ so that the weight
\[
\varphi_*=k\Psi+\lambda+|z|^2
\]
is plurisubharmonic and its Levi form dominates a positive
multiple of $\widetilde g$ on $\supp\beta$.
On that support, $\Psi\ge0$ and $\lambda$ is bounded below.
The scalar weighted $\dbar$ estimate on the fixed Euclidean
ball \cite{Hormander}\cite[Theorem~5.1 and Remark~4.5]{Demailly} gives
$\dbar v=\beta$ with
\begin{equation}\label{chart:weighted-extension}
\int |v|^2e^{-\varphi_*}\,dV_{\rm eucl}
\le\int |\beta|^2_{(\dd\varphi_*)^{-1}}
e^{-\varphi_*}\,dV_{\rm eucl}\le C.
\end{equation}
For a smooth-weight justification, first convolve $\Psi$ on
a slightly larger Euclidean region and regularize the pole.
For each particular smooth $\widetilde g$, the convolution
scale can be chosen small enough to retain half of
\eqref{chart:strict-envelope} on the compact support regions.
The estimates just obtained are independent of that scale.
Weak $L^2$ limits and Fatou's lemma give
\eqref{chart:weighted-extension} for the singular weight.
The correction $v$ is holomorphic near $0$. Integrability with
the pole $|w|^{-2(n+1)}$ forces its constant and linear terms
to vanish. Thus $a-v$ is holomorphic on the Euclidean ball
and has the prescribed differential of norm $B$ at $0$.
Since $\lambda\le0$ and
$\sup\Psi\le C+2\log A$, dropping the weight in
\eqref{chart:weighted-extension} gives
\[
\|a-v\|_{L^2(dV_{\rm eucl})}^2\le C(1+A)^{2k}.
\]
Interior differentiation of a holomorphic function bounds its
center derivative by this norm, proving
\eqref{chart:static-distortion}.
\end{proof}
\subsection{The volume clock and selected Ricci windows}
The letter $s$ now denotes the volume clock, rather than the time
variable used in the disc envelope. Set
\begin{equation}\label{chart:clocks}
\tau=\log t,\qquad s=\rho(o,t),\qquad
q(s)=\frac{ds}{d\tau}=tR(o,t).
\end{equation}
For all sufficiently large times this is a smooth increasing change
of clock.
We need two kinds of control. Estimates valid at every sufficiently
late clock will govern the chart transitions and their polynomial
models. We will also select late times at which the rescaled Ricci
tensor is uniformly pinched at $o$ and the scalar Harnack inequality
approaches equality. The surrounding intervals provide compactness
for the limits used to contract loops.
\begin{proposition}\label{prop:ricci-windows}
The function $q$ is nondecreasing, $q\ge c>0$ for large $s$, and
\begin{equation}\label{chart:q-growth}
s\longrightarrow\infty,\qquad
q\le Ce^\tau,\qquad
\liminf_{s\to\infty}\frac{q(s)}s\le2.
\end{equation}
Uniformly on every fixed compact set, $\rho(x,t)/s\to1$, and
there is $c_K>0$ such that, eventually on that compact set,
\begin{equation}\label{chart:metric-decay}
e^{-2s}g\le h_t\le e^{-c_Ks}g.
\end{equation}
There are fixed $a_0\in(0,1)$ and $b_0>0$ such that the least
eigenvalue at $o$ of $t\Ric(h_t)$ relative to $h_t$ satisfies
\begin{equation}\label{chart:ricci-lower}
\lambda_{\min}(s)\ge b_0q(a_0s)
\quad\text{for all sufficiently large }s.
\end{equation}
Moreover there are $S_i\to\infty$ and $K_0<\infty$ with
\begin{equation}\label{chart:good-windows}
q(2S_i)\le K_0q(a_0S_i).
\end{equation}
One can choose $s_i\in[5S_i/4,7S_i/4]$ so that
$(\log q)'(s_i)\to0$. If $t_i$ corresponds to $s_i$ and
$R_i=R(o,t_i)$, the rescaled flows
\begin{equation}\label{chart:rescaled-flows}
g_i(z)=R_i h_{t_i+z/R_i}
\end{equation}
have uniform curvature and derivative bounds on each fixed-radius
moving ball about $o$ in a fixed small two-sided time interval.
At $(o,0)$ their scalar curvature is one and their Ricci
eigenvalues have a fixed positive lower bound.
\end{proposition}
\begin{proof}
Proposition~\ref{prop:harnack-completeness} gives monotonicity and
the positive lower bound for $q$, together with $q\le Ce^\tau$.
Hence $s\to\infty$. If $q(s)>2s$ for every sufficiently large
$s$, integration of $ds/d\tau>2s$ would give
$s\ge ce^{2\tau}$, contradicting the linear-in-$t$ bound for
$\rho(o,t)$. This proves \eqref{chart:q-growth}.
For a compact family of initial paths from $o$, the power
contraction in Proposition~\ref{prop:harnack-completeness} makes
their $h_t$ lengths tend uniformly to zero. The factor on the
right of \eqref{chart:integrated-harnack}, with time increment
one, therefore tends uniformly to one on the compact set. Apply
the inequality in both spatial directions and integrate in time.
More explicitly, for every $\eps>0$ and all sufficiently large
$v$, uniformly on the compact set,
\[
R(x,v)\le(1+\eps)R(o,v+1),\qquad
R(o,v)\le(1+\eps)R(x,v+1).
\]
The first inequality also bounds $R(x,v)$ there. Shifting either
integral endpoint by one costs a bounded amount. After division
by $s\to\infty$, and then letting $\eps\downarrow0$, the two
inequalities give $\rho(x,t)/s\to1$. No integrability of the
error factor is required.
Let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $h_t$
relative to $g$ at $x$. They lie in $(0,1]$, and their product
is $e^{-\rho(x,t)}$. Thus every $\lambda_j\ge e^{-\rho(x,t)}$,
which proves the lower bound of \eqref{chart:metric-decay}.
Choose the instantaneous unitary chart of
Lemma~\ref{chart:local-charts} at $x$. Its initial-metric
singular lengths are $\lambda_j^{-1/2}$, whose product is
$e^{\rho(x,t)/2}$. Lemma~\ref{lem:static-distortion} bounds
their largest value by a fixed power of their smallest, uniformly
for $x$ in the compact set. Since all these lengths are at least
one, their product forces the smallest to be at least
$C^{-1}e^{c\rho(x,t)}$. This proves the upper bound of
\eqref{chart:metric-decay}, after reducing $c_K$ and increasing
the starting time.
For each fixed nonzero covector $\xi\in T_o^{*(1,0)}M$, the
function
\[
L_\xi(\tau)=\log|\xi|_{h_{e^\tau}^{-1}}^2
\]
is convex. Indeed the logarithmic norm of a holomorphic dual
section is plurisubharmonic for a Griffiths-nonnegative tangent
metric, and its restriction to complexified $\tau$ is independent
of the imaginary part. Its derivative is the Rayleigh quotient
of $t\Ric$ on the dual unit vector. Normalize $|\xi|_{g^{-1}}=1$.
At $o$, \eqref{chart:metric-decay} gives
\[
L_\xi(\tau(s))-L_\xi(\tau(a_0s))
\ge(c_o-2a_0)s.
\]
Choose $a_0<c_o/4$, reducing it further to be less than one if
needed. Monotonicity of $q$ gives
\[
\tau(s)-\tau(a_0s)
=\int_{a_0s}^s\frac{dv}{q(v)}
\le\frac{s}{q(a_0s)}.
\]
The endpoint derivative of a convex function dominates its secant
slope. Taking the minimum over all covectors proves
\eqref{chart:ricci-lower}.
The estimates obtained so far hold at every sufficiently late clock.
We now select intervals on which the lower Ricci bound is comparable
to the scalar curvature. Put $L=2/a_0>1$ and choose $K_0>L^2$. If
\eqref{chart:good-windows} failed for all large $S$, then
$q(Lr)>K_0q(r)$ for all large $r$. Iteration and monotonicity
would give $q(r)\ge cr^{\log K_0/\log L}$ for every large $r$,
contradicting \eqref{chart:q-growth}. Thus such windows occur
arbitrarily far out. On their middle halves,
\[
\int_{5S_i/4}^{7S_i/4}(\log q)'(s)\,ds\le\log K_0.
\]
Choose $s_i$ there with
$(\log q)'(s_i)\le2\log K_0/S_i$.
Write $q_i=q(s_i)=t_iR_i$. On the rescaled clock of
\eqref{chart:rescaled-flows},
\begin{equation}\label{chart:clock-rescaling}
\frac{ds}{dz}=\frac{q(s)}{q_i+z},\qquad
\widehat R_i(o,z)=\frac{R(o,t_i+z/R_i)}{R_i}
=\frac{q(s)}{q_i+z}.
\end{equation}
Since $q_i\ge c>0$, choose $\delta<c/4$. As long as $s$ lies
in the good window, $|ds/dz|\le2K_0$ for $|z|\le\delta$.
The chosen $s_i$ lie a distance at least $S_i/4$ from its ends,
so this estimate keeps the whole fixed time interval in the
window for large $i$. Formula~\eqref{chart:clock-rescaling}
then bounds the rescaled basepoint scalar curvature above and
below by fixed positive constants. Apply the rescaled
\eqref{chart:integrated-harnack}, using a fixed future-time
buffer, to get a uniform scalar bound on every fixed-radius
moving ball in a smaller interval. Bisectional nonnegativity
controls $|\Rm|$, and Lemma~\ref{chart:local-charts} supplies
the local higher bounds. Finally
\eqref{chart:ricci-lower} and \eqref{chart:good-windows} give
at the central point
\[
\lambda_{\min}(\Ric(g_i))
\ge\frac{b_0q(a_0s_i)}{q_i}\ge\frac{b_0}{K_0},
\qquad \widehat R_i(o,0)=1.
\]
\end{proof}
\subsection{Harnack equality and contraction of loops}
The uniform charts are still only local biholomorphisms. To continue
their inverse germs to single-valued maps on compact subsets of $M$,
we must remove possible monodromy. The selected windows do this by
producing equality in a limiting Harnack inequality. We will show
that this equality forces real strict concavity of the scalar
curvature, then use its gradient flow to contract loops in the
corresponding small intrinsic balls.
\begin{proposition}\label{prop:simple-connectivity}
The manifold $M$ is simply connected.
\end{proposition}
\begin{proof}
Take the rescaled flows in Proposition~\ref{prop:ricci-windows}.
Their pointed pullbacks on tangent balls have a smooth local
subsequential limit, denoted $g(z)$. Pass to a further subsequence
so that
\[
k_i(z)=\frac1{q_i+z}\longrightarrow k(z),\qquad k_z=-k^2.
\]
This includes $k=0$ when $q_i\to\infty$. Denote the limit
scalar curvature by $R$ and its Ricci form by $A$. At the
central spacetime point, $R=1$ and $A\ge b\,g$ with $b>0$.
Formula~\eqref{chart:clock-rescaling} gives the exact identity
\[
\left[\partial_z\widehat R_i+k_i\widehat R_i\right]_{(o,0)}
=(\log q)'(s_i)\longrightarrow0.
\]
The limit therefore has $S:=R_z+kR=0$ at the central point.
The matrix Harnack inequality has the rescaled form
$A_z+A g^{-1}A+kA\ge0$. Its trace is $S$, because
$R_z=\tr_g A_z+|A|_g^2$. Thus this nonnegative matrix vanishes
at the point in question. Also $A_z=\dd R$ under K\"ahler--Ricci
flow. If $H_{j\bar l}=\nabla_j\nabla_{\bar l}R$, then
\begin{equation}\label{chart:equality-mixed-hessian}
H=-A^2-kA
\end{equation}
there, writing endomorphism products in a unitary frame.
On a small spacetime neighborhood, $A$ is positive definite.
Minimizing the scalar Harnack inequality over $X$ gives
\begin{equation}\label{chart:equality-Z}
Z=S-Q\ge0,\qquad Q=|\partial R|_{A^{-1}}^2.
\end{equation}
At the equality point $S=0$, so $\partial R=0$ and $Z=0$.
We give the scalar Hessian calculation, since a vanishing
Harnack quantity by itself does not supply real strict concavity.
Use $R_z=\Delta R+|A|^2$, $A_z=\dd R$, and $k_z=-k^2$.
The derivative of the Laplacian on a scalar is
$(\partial_z\Delta)R=\langle A,\dd R\rangle$, while
\[
\partial_z|A|^2=2\langle A,\dd R\rangle+2\tr(A^3).
\]
Consequently
\begin{equation}\label{chart:equality-evolution}
(\partial_z-\Delta)S
=3\langle A,\dd R\rangle+2\tr(A^3)
+k|A|^2-k^2R.
\end{equation}
At the equality point, substitution of
\eqref{chart:equality-mixed-hessian} reduces this to
$-\tr\bigl(A(A+k\Id)^2\bigr)$.
Diagonalize $A$ there, with eigenvalues $\lambda_j>0$.
Since $\partial R=0$, derivatives of the coefficients of
$A^{-1}$ make no contribution to $\Delta Q$ or $Q_z$, and
\[
Q_z=0,\qquad
\Delta Q=\sum_{j,l}\lambda_j^{-1}
\left(|R_{j\bar l}|^2+|R_{jl}|^2\right).
\]
The first square sum is exactly
$\tr\bigl(A(A+k\Id)^2\bigr)$ by
\eqref{chart:equality-mixed-hessian}. Hence at this point
\[
(\partial_z-\Delta)Z
=\sum_{j,l}\lambda_j^{-1}|R_{jl}|^2\ge0.
\]
On the other hand $Z\ge0$ has an interior spacetime zero, so
$Z_z=0$ and $\Delta Z\ge0$ there. The displayed sum must
vanish. Thus the pure complex Hessian $R_{jl}$ is zero, while
the mixed Hessian is negative definite by
\eqref{chart:equality-mixed-hessian}. The real Hessian of $R$
is therefore negative definite at the center, with a definite
bound furnished by the positive lower bound for $A$.
This conclusion holds uniformly on actual small metric balls
in the original rescaled manifolds, not only on their immersed
parametrizations. Smooth convergence gives the center Hessian
bound and $|\nabla\widehat R_i|(o,0)\to0$. The intrinsic
third-derivative estimates from
Proposition~\ref{prop:ricci-windows} allow one to parallel
transport the Hessian along any minimizing radial geodesic.
Shrinking a fixed radius $r_*>0$, independently of $i$, gives
constants $a>0$ with
\begin{equation}\label{chart:intrinsic-concavity}
\operatorname{Hess}_{g_i(0)}\widehat R_i
\le-a g_i(0)
\quad\text{on }B_{g_i(0)}(o,2r_*),\qquad
|\nabla\widehat R_i|(o,0)\longrightarrow0.
\end{equation}
Indeed the change in the transported Hessian is bounded by
$Cr_*$, so one chooses $r_*$ after the uniform center bound.
This argument uses intrinsic derivative estimates along actual
geodesics and is unaffected by multiple sheets of a local chart.
Hold the complete metric $g_i(0)$ fixed and flow by the real
vector field $Y_i=\nabla\widehat R_i(\cdot,0)$. Along a
minimizing radial geodesic of length $r\le2r_*$,
\[
\langle Y_i,\partial_r\rangle
\le |Y_i(o)|-ar.
\]
For large $i$ this is negative on $r=r_*$. At cut points,
first variation along any minimizing radial segment gives the
same upper bound for the upper directional derivative of
distance. Thus the closed ball of radius $r_*$ traps trajectories
starting inside it. Completeness makes this closed ball compact,
so these trajectories exist for all positive flow time.
For a carried tangent vector $V$, \eqref{chart:intrinsic-concavity}
gives
\[
\frac{d}{dv}|V|_{g_i(0)}^2
=2\operatorname{Hess}\widehat R_i(V,V)
\le-2a|V|_{g_i(0)}^2.
\]
Every loop contained well inside the trapping ball consequently
has carried length tending exponentially to zero.
Let a fixed piecewise smooth loop in $M$ be given. Its image
and a finite collection of paths from $o$ lie in a fixed compact
set. Their $h_{t_i}$ lengths tend to zero by
\eqref{chart:metric-decay}, and $R_i\le C$ by
Proposition~\ref{prop:harnack-completeness}. Thus this loop
lies inside $B_{g_i(0)}(o,r_*/4)$ for sufficiently large $i$.
Fix one such $i$. The compact trapping ball has a positive
minimum injectivity radius for this one smooth metric. A
sufficiently short carried loop lies in a normal ball and is
contractible. The gradient flow up to that finite time is a
homotopy from the original loop to it. Approximation of
continuous loops by piecewise smooth loops finishes the proof.
\end{proof}
\subsection{Inverse charts and one contraction exponent}
Simple connectivity permits continuation of the inverse germs, but
their first contraction estimates will depend on the compact set.
The following Liouville principle, applied to limits of logarithmic
norms, will make the exponent common to all compact sets. We state it
in the required regularity class; it is also a special case of
\cite[Theorem~9]{LiuThreeCircle}.
\begin{lemma}\label{chart:psh-liouville}
Every bounded-above plurisubharmonic function on $M$ is constant,
including a function which is not assumed continuous.
\end{lemma}
\begin{proof}
Allow the constant $-\infty$, and otherwise let $v$ be such a
function. The compact-ball maximum
$m(r)=\max_{\overline B_g(o,r)}v$ is finite and nondecreasing.
It is convex as a function of $\log r$. To see this, fix
$0<r_1<r_2$ and compare on the closed annulus with the
logarithmic chord
\[
H(r)=m(r_1)
+\frac{m(r_2)-m(r_1)}{\log r_2-\log r_1}
(\log r-\log r_1).
\]
If its slope is zero the desired comparison is immediate from
the definition of $m(r_2)$. For positive slope, a positive
maximum of $v-H(\dist_g(o,\cdot))$ would occur in the annulus
interior. At such a point choose a minimizing radial segment.
The length index form with linear parallel test fields in the
$J$-radial direction gives an upper Hessian bound for distance.
More explicitly, let $T$ be the final radial unit vector, let
$T(\ell)$ denote its parallel extension, and let $\bar r$ be
the length upper support obtained from these trial paths. At
the endpoint of a segment of length $r$ the real Hessian satisfies
\[
\operatorname{Hess}\log\bar r(T,T)
+\operatorname{Hess}\log\bar r(JT,JT)
\le-\frac1{r^3}\int_0^r
\ell^2\Rm(T(\ell),JT(\ell),JT(\ell),T(\ell))\,d\ell<0.
\]
Indeed the radial term is $-1/r^2$, while the linear test field
in the $JT$ direction gives $1/r^2$ minus the displayed
curvature integral. This is the strictly negative Levi
derivative on the radial complex line.
This support argument applies also at cut points: smoothly vary
the endpoint and use trial paths along the chosen minimizing
segment. Their lengths give a smooth upper support with the
same index-form bound. Since $H$ is increasing, it produces
a smooth upper test for $v$ at the hypothetical positive
maximum, with a negative Levi derivative. A plurisubharmonic
function admits no such upper test, whether or not it is
continuous. Hence $v\le H$ on the annulus, proving the
claimed convexity. A nondecreasing convex function of
$\log r\in\R$ which is bounded above must be constant.
Upper semicontinuity implies
$\lim_{r\downarrow0}m(r)=v(o)$, so $v$ attains its global
maximum at $o$. The plurisubharmonic maximum principle on
connected $M$ shows that $v$ is constant.
\end{proof}
Write $J_dF$ for the Taylor jet of $F$ through total degree $d$ at zero,
and fix a coefficient norm on each finite-dimensional jet space.
\begin{proposition}\label{prop:shrinking-charts}
For every sufficiently large volume clock $s$, there is a
holomorphic local biholomorphism
$\Phi_s:B_{\C^n}(r_0)\to M$ centered at $o$, with unitary
center derivative for the metric at clock $s$, and
\begin{equation}\label{chart:uniform-chart-metric}
C^{-1}g_{\rm eucl}\le\Phi_s^*h_{t(s)}\le Cg_{\rm eucl}.
\end{equation}
For fixed $0<r_1<r_0$, chosen sufficiently small, the transition
germs $F_{s,u}=\Phi_u^{-1}\Phi_s$, $u\ge s$, extend to
nonsingular holomorphic maps on $B(r_1)$ with a common bound.
They fix $0$, compose as germs, and satisfy
\begin{equation}\label{chart:center-transitions}
\|F_{s,u}'(0)\|\le1,\qquad
|\det F_{s,u}'(0)|=e^{-(u-s)/2},\qquad
\|F_{s,u}'(0)^{-1}\|\le e^{(u-s)/2}.
\end{equation}
There are coherent inverse branches $f_s$ on neighborhoods of
any fixed compact exhaustion set for all sufficiently large
$s$, satisfying $f_s(o)=0$ and $\Phi_sf_s=\Id$. A constant
$\sigma>0$, common to all compact sets, satisfies
\begin{equation}\label{chart:common-exponent}
\sup_K|f_s|\le e^{-\sigma s}
\quad\text{for every fixed compact }K
\text{ and all sufficiently large }s.
\end{equation}
For every fixed jet order $d$ there is $C_d$ such that
\begin{equation}\label{chart:transition-jets}
|J_dF_{s,u}|\le C_d\min\{1,e^{C_ds-\sigma u}\}
\qquad(u\ge s\text{ sufficiently large}).
\end{equation}
There are $c,\gamma_0>0$ such that, for each fixed
$0<\gamma\le\gamma_0$, one has
\begin{equation}\label{chart:transition-small-balls}
\sup_{|z|\le e^{-\gamma s}}|F_{s,u}(z)|
\le e^{-c\gamma u}
\qquad\text{for every }u\ge s\ge s_\gamma.
\end{equation}
The constants in the last two estimates are independent of the
later endpoint $u$; the starting threshold in
\eqref{chart:common-exponent} may depend on $K$.
\end{proposition}
\begin{proof}
Take the charts of Lemma~\ref{chart:local-charts} at $o$.
We use the elementary short-path lifting property of these
possibly noninjective charts; compare
\cite[Lemmas~2.2--2.3 and Corollary~2.2]{ChauTamStein}.
A path beginning at the center and having sufficiently small
current-metric length lifts from $0$ by local inversion.
The lower bound in \eqref{chart:uniform-chart-metric} bounds
the Euclidean length of its lift. If the given length is
smaller than the fixed distance to the chart boundary, the
lift cannot escape, and hence extends over the whole path.
The same reasoning applies to a homotopy all of whose paths
obey this length bound. Local uniqueness of lifts supplies
uniqueness throughout the homotopy.
For $u\ge s$, the image under $\Phi_s$ of a short radial path
has $h_{t(u)}$ length at most its $h_{t(s)}$ length. Lift it
through $\Phi_u$ from $0$. Short radial homotopies give a
well-defined map on a fixed ball $B(r_1)$; locally this map
is $\Phi_u^{-1}\Phi_s$, so it is holomorphic with nonsingular
derivative. Its lift has a fixed Euclidean length bound, and
therefore the maps are uniformly bounded on $B(r_1)$.
They compose wherever both sides are defined near $0$ by
uniqueness of local inverses. This is the local transition
construction of \cite[Lemmas~3.1--3.2]{ChauTamStein} with the
metric bounds already established here.
Unitary center normalization and $h_{t(u)}\le h_{t(s)}$
give the first estimate in \eqref{chart:center-transitions}.
Taking determinants of these normalizations and using
$\det h_{t(s)}(o)=e^{-s}\det g(o)$ gives the determinant
identity. All singular values are at most one and their
product is $e^{-(u-s)/2}$, so the least is at least this
product. This proves the inverse estimate.
We now use simple connectivity to continue the inverse germs.
Fix a connected relatively compact neighborhood of a compact
set and $o$. Cover its closure by finitely many small convex
initial target neighborhoods with connected pairwise
intersections, enlarging the connected neighborhood if
necessary to include fixed paths from $o$ to their centers.
Choose one such path to each center. On each target
neighborhood it extends the inverse germ at $o$ by path
lifting. For each nonempty overlap, use one point in the
overlap to form the two-path loop. By
Proposition~\ref{prop:simple-connectivity} this loop bounds
a fixed piecewise smooth homotopy. There are only finitely
many paths and homotopies; their images lie in a fixed
compact set and their initial lengths have a uniform bound.
Equation~\eqref{chart:metric-decay} makes every path in these
homotopies sufficiently short in the current metric when
$s$ is large. All lifts stay inside the chart. Thus the
two inverse germs agree at the chosen overlap point, and
then throughout the connected overlap by analytic
continuation. They patch to an inverse $f_s$ on a
neighborhood of the compact set.
Choose nested connected relatively compact open exhaustion
sets $D_m$, and choose nondecreasing thresholds $T_m\ge m$
so that all shortness conditions for $D_m$ hold whenever
$s\ge T_m$. At clock $s$, use the construction on
$D_{\max\{m:T_m\le s\}}$. On a smaller set, the branches
obtained from two larger constructions agree by continuation
from $o$. This gives the stated coherent branches with
domains exhausting $M$. The length estimate for a lift,
together with \eqref{chart:metric-decay} on the finite path
compact, gives for each fixed $K$
\begin{equation}\label{chart:local-exponent}
\sup_K|f_s|\le C_Ke^{-a_Ks}
\end{equation}
with $a_K>0$. It also gives
$f_u=F_{s,u}\circ f_s$ on any fixed compact set whenever
$s$ is sufficiently large and $u\ge s$: both sides are
the same continued inverse, and
\eqref{chart:local-exponent} keeps $f_s$ in the transition
domain.
It remains to make the exponent independent of the compact.
The functions $v_s=s^{-1}\log|f_s|$ are plurisubharmonic on
their exhausting domains and locally bounded above.
For any sequence $s\to\infty$,
Lemma~\ref{lem:psh-compactness-bridge} and diagonal extraction
give either convergence to
$-\infty$ locally uniformly from above, or local $L^1$
convergence to a plurisubharmonic function on $M$.
In the second case the upper-regularized limit is at most
zero everywhere by \eqref{chart:local-exponent}.
Lemma~\ref{chart:psh-liouville} makes it constant. On one
fixed neighborhood of $o$, \eqref{chart:local-exponent}
bounds this constant by a fixed negative number, say
$-a_*$. The constant-limit conclusion of
Lemma~\ref{lem:psh-compactness-bridge} then gives
$\limsup\sup_Kv_s\le-a_*$ on every fixed compact along
this subsequence. The collapsing case gives the same
conclusion with $-\infty$. A sequence violating
\eqref{chart:common-exponent}, with $\sigma=a_*/2$, would
contradict these alternatives. This proves a common
exponent on the full family of clocks.
For the two remaining transition estimates, transfer the common
exponent from a fixed initial neighborhood back to a small ball in
the chart domain. Choose a fixed compact initial neighborhood $K_0$
of $o$ with positive initial distance from $o$ to its
boundary. Until a path $\Phi_s(rz)$ first exits $K_0$,
\eqref{chart:metric-decay} and
\eqref{chart:uniform-chart-metric} bound its initial length
by $Ce^s|z|$. It follows that, for a fixed $C_0>1$ and
all large $s$,
\[
\Phi_s\bigl(B(e^{-C_0s})\bigr)\subset K_0.
\]
For every $u\ge s$, the map $f_u\Phi_s$ on this ball is
the same local inverse composition as $F_{s,u}$ near zero,
and hence everywhere there. Equation~\eqref{chart:common-exponent}
therefore gives the bound
\begin{equation}\label{chart:transition-inner-bound}
\sup_{B(e^{-C_0s})}|F_{s,u}|\le e^{-\sigma u}
\qquad\text{for all }u\ge s\text{ and large }s.
\end{equation}
Cauchy's estimates on this inner ball give the factor
$e^{dC_0s-\sigma u}$ for derivatives of order at most $d$.
Cauchy's estimates on a fixed transition ball give a uniform
bound. Combining the two proves
\eqref{chart:transition-jets}.
For completeness, fix an outer radius $R<r_1$ and a common
bound $B_*$ for the transitions there. Euclidean
three-circle interpolation, applied to their scalar linear
projections and then taking the supremum, gives at radius
$e^{-\gamma s}$, for fixed small $\gamma>0$,
\[
\log\sup_{B(e^{-\gamma s})}|F_{s,u}|
\le-\theta_s\sigma u+(1-\theta_s)\log B_*,
\qquad
\theta_s=\frac{\log(R/e^{-\gamma s})}
{\log(R/e^{-C_0s})}
\ge\frac{\gamma}{2C_0}
\]
for all sufficiently large $s$. In this formula choose
$\gamma_0<C_0$, so the middle radius lies between the
inner and outer radii. Since $u\ge s$, the fixed term
$\log B_*$ is absorbed uniformly for every later endpoint,
giving \eqref{chart:transition-small-balls} with, for
example, $c=\sigma/(4C_0)$ and a threshold depending on
$\gamma$. This proves all the stated uniformities.
\end{proof}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/charts.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/discs.tex ====
\section{The cotangent envelope: compactness and frames}
\label{sec:discs}
We study the flow through boundary averages of its dual squared norm on
holomorphic cotangent discs. Let $\Delta=\{z\in\C:|z|<1\}$, let $dA$
denote Euclidean area on $\Delta$, and write
$\pi:Y=T^{*(1,0)}M\to M$. Put
\[
N(x,\xi,t)=|\xi|^2_{h_t^{-1}}.
\]
Thus $N(\cdot,t)\ge N(\cdot,0)$ by
Theorem~\ref{thm:localized-flow}. The function $N(\cdot,0)$ is
plurisubharmonic: the squared norm on the total space of a Griffiths
nonpositive bundle is plurisubharmonic, and the dual of $T^{1,0}M$ has
this curvature sign.
Fix $T<\infty$ and denote physical time at a disc center by $s$.
If $y\in Y$ and $0<s<T$, let $v^{\rm raw}(y,s)$ be the infimum of
\begin{equation}
\left\langle N\bigl(f(\zeta),s e^{w(\zeta)+\overline{w(\zeta)}}\bigr)
\right\rangle,
\qquad (f(0),w(0))=(y,0),
\label{var:envelope}
\end{equation}
over holomorphic discs $f:\overline\Delta\to Y$ and
$w:\overline\Delta\to\C$, smooth on the closed unit disc, with
$s e^{w+\bar w}<T$ there. Brackets denote normalized circle average.
Let $v$ be the lower semicontinuous regularization in $(y,s)$.
Constant discs, the plurisubharmonicity of $N(\cdot,0)$, and
$N(\cdot,t)\ge N(\cdot,0)$ give
\begin{equation}
N(y,0)\le v(y,s)\le v^{\rm raw}(y,s)\le N(y,s).
\label{var:sandwich}
\end{equation}
Multiplying the fiber component of every competing disc by a nonzero
complex number shows that $v$ is homogeneous of degree two in the fiber.
The assertion at the zero section follows from \eqref{var:sandwich}.
Minimizing boundary averages over discs with a prescribed center is the
Poisson-envelope construction in the theory of Poletsky and Rosay
\cite{Poletsky1991,Rosay2003}; see also
\cite[Sections~1--2]{LarussonSigurdsson} for the manifold framework.
Here we prove the differential inequality needed for the evolving norm
directly, with the compactness and quantitative deformation estimates
specified below.
Our goal is to prove $v=N$. This equality will give the submean
inequality for $N(x,\xi,e^{w+\bar w})$, hence its joint
plurisubharmonicity. The competitors in \eqref{var:envelope} have no
common boundary compact or area bound. In Section~\ref{sec:variation}
we will localize almost minimizing discs by boundary walls and a positive
area penalty. The compact containing their full images may then depend
on the penalty. With centers in a fixed compact coordinate region, the
frame estimates used in the differential inequality must depend on
boundary and center data, independently of the area bound and the
compact containing the interiors. This Section establishes these two
kinds of control, together with the cotangent evolution identity used
in that estimate. All compactness arguments use the complete initial
metric $g$.
\subsection{The symplectic Hessian operator}
The canonical holomorphic bivector on $Y$ is
\[
\mathcal J=\sum_{i=1}^n
\frac{\partial}{\partial x_i}\wedge
\frac{\partial}{\partial\xi_i}.
\]
We use column matrices for Hermitian forms, so a form with matrix $W$
has quadratic value $v^*Wv$. In a frame $e_1,\ldots,e_{2n}$, write
$\mathcal J=\sum_{A<B}J^{AB}e_A\wedge e_B$ and define
\begin{equation}\label{disc:p-definition}
p(W,\mathcal J)=
\sum_{A<B,\,C<D}\overline{J^{AB}}J^{CD}
(W_{AC}W_{BD}-W_{AD}W_{BC}).
\end{equation}
This is the induced exterior-square squared norm when $W$ is positive,
and is a real polynomial for every Hermitian $W$. The definition is
independent of frame. In canonical cotangent coordinates we abbreviate
it to $p(W)$; if
$W=\left(\begin{smallmatrix}A&B\\B^*&C\end{smallmatrix}\right)$, then
\begin{equation}\label{disc:p-block}
p(W)=\tr(AC^{\mathsf T})-
\sum_{i,j}B_{ij}\overline{B_{ji}}.
\end{equation}
Here and below the transpose records the tangent--cotangent pairing.
In particular, if tangent coefficients have metric matrix $h$, the
column matrix of the dual norm is $b=h^{-\mathsf T}$.
\begin{lemma}[Symplectic evolution]\label{lem:symplectic-evolution}
For the flow of Theorem~\ref{thm:localized-flow},
\begin{equation}\label{disc:symplectic-evolution}
\partial_tN=p(\dd_YN).
\end{equation}
For every real smooth base function $d=d(x,t)$, with $t$ held fixed in
the spatial Hessian,
\begin{equation}\label{disc:base-addition}
p(\dd_Y(N+d))=\partial_tN+\Delta_{h_t}d.
\end{equation}
These identities do not assume that $h_t$ is complete or has
nonnegative curvature at positive time.
\end{lemma}
\begin{proof}
Fix $(x,t)$ and use K\"ahler-normal coordinates for $h_t$ there.
At that point $h_t=b_t=I$, all first spatial derivatives of $b_t$
vanish, and the Hessian of $N$ has blocks
\[
\dd_YN=\begin{pmatrix}\mathcal R_\xi&0\\0&I\end{pmatrix}.
\]
The horizontal form $\mathcal R_\xi$ is the bisectional curvature form
of $h_t$ with one pair of indices contracted against the metric dual
of $\xi$. Indeed differentiating $b=h^{-\mathsf T}$ twice at a normal
point gives the negative second derivatives of $h$, transposed, which
are precisely those curvature components. The K\"ahler symmetries give
\[
\tr\mathcal R_\xi=\xi^*\Ric(h_t)^{\mathsf T}\xi.
\]
On the other hand $\partial_th=-\Ric(h)$ implies
$\partial_tb=h^{-\mathsf T}\Ric(h)^{\mathsf T}h^{-\mathsf T}$.
Equation~\eqref{disc:p-block} now proves
\eqref{disc:symplectic-evolution}. Adding a base Hessian changes only
the horizontal block. Its contribution to $p$ is
$\tr((\dd_xd)b^{\mathsf T})=\tr(h_t^{-1}\dd_xd)$, proving
\eqref{disc:base-addition}. The calculation is tensorial, so the
normal-coordinate verification proves both identities everywhere.
\end{proof}
\subsection{Compact boundary and bounded area}
\begin{proposition}[Compactness of discs]\label{prop:disc-compactness}
For each compact $K\subset M$ and each $A_0<\infty$, there is a
compact $K'\subset M$ such that every holomorphic disc
$f:\overline\Delta\to M$, smooth up to the circle, satisfying
\[
f(\partial\Delta)\subset K,\qquad
\int_\Delta f^*\omega_g\le A_0
\]
has $f(\overline\Delta)\subset K'$. Every sequence of such discs
has a subsequence converging holomorphically on compact subsets of
$\Delta$.
The same statements hold for discs in $Y$ with compact boundary in
$Y$ and with $\int_\Delta(\pi\circ f)^*\omega_g\le A_0$.
The resulting compact may depend on $K$ and $A_0$.
\end{proposition}
The area bound first places part of each disc in a fixed compact.
An inverse-Jacobian estimate for the global injective immersion then
propagates this control away from isolated possible singularities of a
limit; properness of the immersion is not assumed. The following
extra-integrability lemma removes those punctures. A second lemma
records the subharmonic compactness needed to propagate the initial
control.
The puncture estimate assumes no extension of the bundle or its metric
across the puncture.
\begin{lemma}[Extra integrability]\label{lem:extra-integrability}
Let $(E,k)$ be a smooth Hermitian holomorphic vector bundle on
$\Delta^*=\{0<|z|<1\}$, with Griffiths nonnegative curvature.
Suppose that $A:E\to\Delta^*\times\C^q$ is holomorphic, pointwise
injective, and has bounded operator norm. If $a$ is a holomorphic
section with $\int_{\Delta^*}|a|_k^2\,dA<\infty$, then there are
$r_0>0$ and $\sigma>0$ such that
\begin{equation}\label{disc:extra-integrability}
\int_{0<|z|<r_0}|a|_k^2|z|^{-\sigma}\,dA<\infty.
\end{equation}
The exponent may depend on the section and the bundle data.
\end{lemma}
\begin{proof}
Choose a fixed smooth function $\chi:\R\to[0,1]$ which is zero on
$(-\infty,-2]$ and one on $[-1,\infty)$, and set
\[
\chi_m(z)=\chi(m^{-1}\log|z|),\qquad
\phi_m(z)=\frac2m\log|z|+|z|^{1/m}.
\]
Writing $r=|z|$, one has
\[
|\dbar\chi_m|\le\frac{C}{mr},\qquad
(\phi_m)_{z\bar z}=\frac1{4m^2}r^{1/m-2}.
\]
On a complex curve Griffiths and Nakano nonnegativity coincide. Apply
the bundle-valued H\"ormander estimate to $E$-valued $(1,0)$-forms,
with the metric $ke^{-\phi_m}$ and source
$a\,\dbar\chi_m\wedge dz$; see \cite{Hormander} and
\cite[Theorem 5.1 and Remark 4.5]{Demailly}.
One may use any complete auxiliary K\"ahler metric on $\Delta^*$.
In dimension one its conformal factor cancels both from the norm of
the unknown top form times volume and from the curvature-inverse
source pairing times volume. Dropping the nonnegative curvature of
$k$ therefore gives a solution $v_m$ of
$\dbar v_m=a\,\dbar\chi_m$ satisfying
\begin{align}
\int_{\Delta^*}|v_m|_k^2e^{-\phi_m}\,dA
&\le C\int_{e^{-2m}<r<e^{-m}}
|a|_k^2r^{-3/m}e^{-r^{1/m}}\,dA,\label{disc:weighted-solve}\\
\int_{\Delta^*}|v_m|_k^2r^{-2/m}\,dA
&\le C\int_{e^{-2m}<r<e^{-m}}|a|_k^2\,dA\longrightarrow0.
\label{disc:correction-small}
\end{align}
For the second estimate we used $r^{-3/m}\le e^6$ on the source
annulus and $e^{-1}\le e^{-r^{1/m}}\le1$. The constants are
independent of $m$. The estimate concerns smooth metrics on the
punctured domain; it does not assert a curvature extension at zero.
The sections $a_m=\chi_ma-v_m$ are holomorphic and converge to $a$
in unweighted $L^2$. Near zero they equal $-v_m$, so each has a
positive extra integrability exponent, namely $2/m$. Since $A$ is
bounded, $Aa_m$ and $Aa$ are ordinary $L^2$ holomorphic vector
functions on the punctured disc. Their negative Laurent coefficients
vanish by $L^2$ integrability; hence they extend across zero.
Their $L^2$ convergence and the interior Cauchy estimates give
convergence of every finite jet at zero.
Here is the closure argument that recovers a positive exponent for
$a$ itself. Let $R=\mathcal O_{\C,0}$, with maximal ideal
$\mathfrak m=(z)$, and let $S\subset R^q$ consist of all germs $Ab$
where $b$ is holomorphic on some punctured neighborhood and
\[
\int |b|_k^2r^{-\varepsilon}\,dA<\infty
\quad\hbox{for some }\varepsilon>0.
\]
These images extend across zero by the same $L^2$ argument. The set
$S$ is an $R$-submodule: for a sum take the smaller of the two positive
exponents, and multiplication by a holomorphic germ is bounded after
shrinking the neighborhood. The ring $R$ is Noetherian, so $S$ is a
submodule of a finite free module and $Q=R^q/S$ is a finite module.
For each $k$, the image of $S$ in $R^q/\mathfrak m^kR^q$ is a
finite-dimensional vector subspace and is closed. Jet convergence of
$Aa_m\in S$ consequently gives
\[
Aa\in S+\mathfrak m^kR^q\qquad\hbox{for every }k.
\]
Krull's intersection Theorem for the finite local module $Q$ gives
$\bigcap_k\mathfrak m^kQ=0$ \cite[Lemma 10.51.4]{StacksKrull}.
Thus $Aa\in S$. By definition it is the image of one section $b$
with some positive extra exponent. Pointwise injectivity of $A$
implies $a=b$ on a sufficiently small punctured neighborhood, proving
\eqref{disc:extra-integrability}. No common exponent for $a_m$ was
used.
\end{proof}
Let $F=(F_1,\ldots,F_N):M\to\C^N$ be the global injective
holomorphic immersion of Proposition~\ref{prop:initial-data}. Set
\[
\psi=\log\sum_{|I|=n}|dF_I|_g^2,
\qquad dF_I=dF_{i_1}\wedge\cdots\wedge dF_{i_n}.
\]
Immersivity makes $\psi$ smooth. Along any holomorphic disc,
$\psi$ is subharmonic: each $dF_I$ is a holomorphic canonical form,
whose logarithmic squared norm is plurisubharmonic under the curvature
hypothesis, and the log-sum has the same property. Similarly
$|dF|_g^2=\sum_i|dF_i|_g^2$ is plurisubharmonic.
\begin{lemma}[Compactness with a lower anchor]\label{lem:psh-compactness-bridge}
Let $\Omega_j$ be increasing open subsets exhausting a connected
complex manifold $X$, and let $u_j$ be plurisubharmonic on $\Omega_j$.
Assume that for each compact $K\subset X$, the functions $u_j$ are
uniformly bounded above on $K$ once $K\subset\Omega_j$.
Every sequence has a subsequence for which either $u_j$ tends to
$-\infty$ locally uniformly from above, or $u_j$ converges in
$L^1_{\mathrm{loc}}$ to a plurisubharmonic function.
A lower bound $u_j(z_j)\ge-A$ at points $z_j$ in a fixed compact set
excludes the first alternative along that sequence. In particular,
lower bounds on moving subsets of a fixed compact set of uniformly
positive measure suffice. If the $L^1_{\mathrm{loc}}$ limit is the
constant $c$, then
\[
\limsup_j\sup_K u_j\le c
\]
for every compact $K\subset X$.
\end{lemma}
\begin{proof}
We explain the lower-anchor step rather than building it into the
compactness theorem. Pass to a subsequence with $z_j\to z_*$ and
work in a coordinate ball about $z_*$. On a slightly larger ball
choose a common upper bound $C$ and set $w_j=C-u_j$.
These functions are nonnegative and superharmonic. The supermean
inequality at $z_j$ bounds their integrals on a ball of fixed radius
about $z_j$ by its volume times $C+A$. A fixed smaller ball about
$z_*$ is eventually contained in those balls, so the $L^1$ norms
there are bounded.
Such a bound propagates along finite chains of overlapping coordinate
balls. In a fixed overlap of positive measure, the integral bound
supplies a point at which $w_j$ is bounded by its average over the
overlap. Applying the supermean inequality on a larger ball centered
at that point, still contained in the coordinate neighborhood, bounds
the integral on the next smaller ball. On each fixed compact overlap, the Euclidean volume measures in
the two coordinate charts are uniformly comparable. Increasing the
local upper bound $C$ when moving between charts only adds a fixed constant.
Connectedness and a finite covering therefore give $L^1$ bounds on
every fixed compact subset of $X$.
Families of subharmonic functions bounded in $L^1_{\mathrm{loc}}$
are relatively compact in that topology \cite[Chapter~I, Proposition~4.21]{DemaillyBook}.
Apply this result in coordinate balls and take a diagonal subsequence.
Positivity of the complex Hessians passes to the distributional
limit, so its upper semicontinuous representative is
plurisubharmonic. If no subsequence has a lower anchor in any fixed
compact set, then the suprema on each member of a compact exhaustion
tend to $-\infty$; this is the other alternative. This also proves
the assertion for exhausting domains, since each finite chain of
balls is contained in every sufficiently late $\Omega_j$.
Finally suppose the limit is the constant $c$. In a coordinate ball
whose concentric enlargement is relatively compact, apply the
submean inequality on a fixed-radius ball centered at each point of
the smaller ball. It bounds $u_j-c$ from above by a fixed constant
times the $L^1$ norm of $u_j-c$ on the enlargement. This norm tends
to zero. A finite covering proves the asserted upper bound on $K$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:disc-compactness}]
We first control centers of discs $f_l$ in $M$. The maximum principle
gives uniform bounds
\begin{equation}\label{disc:boundary-upper}
|F\circ f_l|\le C_K,\qquad
|dF|_g\circ f_l\le C_K,\qquad
\psi_l:=\psi\circ f_l\le C_K.
\end{equation}
The area bound controls the energy, with a fixed normalization
constant. In particular, the functions
\[
E_l(\theta)=\int_{1/2}^1
|\partial_rf_l(re^{i\theta})|_g^2r\,dr
\]
have uniformly bounded angular integrals. On a set of angles of
measure bounded below, $E_l(\theta)\le C(A_0+1)$. For each such
angle and every $r\in[1/2,3/4]$, Cauchy--Schwarz gives
\[
\dist_g(f_l(re^{i\theta}),K)
\le E_l(\theta)^{1/2}
\left(\int_{1/2}^1\frac{dr}{r}\right)^{1/2}.
\]
Completeness and Hopf--Rinow put these images in a fixed compact.
Thus on a subset of the fixed annulus of area bounded below,
$\psi_l$ has a fixed lower bound. Lemma~\ref{lem:psh-compactness-bridge}, together with
\eqref{disc:boundary-upper}, now gives a subsequence converging in
$L^1_{\mathrm{loc}}(\Delta)$ to a subharmonic function
$\psi_\infty$; the lower bounds on the anchor subsets exclude
collapse to $-\infty$. We retain their positive-area property for
the later step that avoids neighborhoods of the possible atoms. Let
$\mu_l=(2\pi)^{-1}\Delta_{\mathrm{eucl}}\psi_l$ and
$\mu=(2\pi)^{-1}\Delta_{\mathrm{eucl}}\psi_\infty$.
Then $\mu_l\to\mu$ weakly on compact subsets.
We spell out the required inverse-Jacobian estimate. Let
$Z=\{z:\mu(\{z\})\ge1\}$, a locally finite subset of $\Delta$.
Near any point outside $Z$, choose a small disc $D$ whose boundary
has zero $\mu$-measure and for which $\mu(D)<b<4/3$.
For large $l$, $\mu_l(D)<b$. The Riesz decomposition on $D$
\cite[Chapter~I, Proposition~4.22]{DemaillyBook} is
\[
\psi_l=h_l+\int_D\log|z-\zeta|\,d\mu_l(\zeta).
\]
On each smaller concentric disc the harmonic terms $h_l$ are bounded
in absolute value: their $L^1$ norms are bounded by $L^1$ convergence
of $\psi_l$, the bounded masses, and local integrability of the
logarithmic kernel; harmonic interior estimates then apply.
For a positive measure $\nu$ of mass $b_l\le b$, Jensen's inequality
gives, with the zero-mass case interpreted separately,
\[
\exp\left(-p\int_D\log|z-\zeta|\,d\nu(\zeta)\right)
\le\frac1{b_l}\int_D|z-\zeta|^{-pb_l}\,d\nu(\zeta),
\qquad p=\tfrac32.
\]
Integrating in $z$ gives a uniform bound since $pb<2$. Therefore
$e^{-\psi_l}$ is locally bounded in $L^{3/2}(\Delta\setminus Z)$.
Let $s_1\le\cdots\le s_n$ be the singular values of $dF$ measured
from $g$ to the Euclidean metric. Cauchy--Binet and
\eqref{disc:boundary-upper} imply
\[
e^{\psi_l}=\prod_i s_i(f_l)^2,\qquad
s_1(f_l)^{-2}\le C_K e^{-\psi_l}.
\]
Consequently
\begin{equation}\label{disc:inverse-jacobian}
|f_l'|_g^2\le C_K e^{-\psi_l}|(F\circ f_l)'|^2.
\end{equation}
The bounded holomorphic maps $F\circ f_l$ have uniformly bounded
derivatives on smaller discs. Hence $df_l$ is locally bounded in
$L^3$ off $Z$. For $p\in M$ the function
$v_{l,p}(z)=\dist_g(f_l(z),p)$ has weak gradient bounded by
$|df_l|_g$. Morrey's local gradient-only estimate in real dimension
two, with exponent $1-2/3=1/3$, therefore applies
\cite[Section~5.6.2, Theorem~4 and the Remark on p.~283]{EvansPDE}.
Its constant depends on the nested domain discs, not on $p$;
no bound on the values of $v_{l,p}$ is needed.
Taking $p=f_l(w)$ gives the uniform local metric H\"older estimate.
These local estimates propagate the compact anchors. Indeed, remove
small neighborhoods of the finitely many points of $Z$ in the anchor
annulus, with total area smaller than half the anchor lower bound.
Choose anchor points in the remaining compact set and pass to a
convergent subsequence of their domain positions. The local
H\"older estimate puts their neighboring images in a fixed bounded
$g$-neighborhood of the anchor compact. Any compact subset of the
connected set $\Delta\setminus Z$ can be reached by finitely many
overlapping small discs of this kind. Completeness makes each
successive bounded neighborhood compact. Arzel\`a--Ascoli, local
holomorphic coordinates, and diagonal extraction give a holomorphic
limit $f:\Delta\setminus Z\to M$. Its area is finite by lower
semicontinuity, and $dF$ remains uniformly bounded along it.
Fix a puncture of this limit and move its domain coordinate to zero.
The bundle $E=f^*T^{1,0}M$ has Griffiths nonnegative curvature,
$A=dF:E\to\C^N$ is bounded and injective, and the holomorphic
section $a=f'$ has finite $L^2$ norm. Lemma~\ref{lem:extra-integrability}
applies. Fubini then gives an angle $\theta$ with
\[
\int_0^{r_0}|a(re^{i\theta})|_g^2r^{1-\sigma}\,dr<\infty.
\]
Since $\int_0^{r_0}r^{\sigma-1}\,dr<\infty$, the corresponding
radial path has finite length and converges to a point $p\in M$.
The bounded map $F\circ f$ extends holomorphically to zero and its
value there is $F(p)$. Choose a relatively compact coordinate
neighborhood $U$ of $p$ with compact boundary. Injectivity of $F$
gives
\[
\dist_{\mathrm{eucl}}(F(p),F(\partial U))>0.
\]
For small enough $r$, $F\circ f$ on $0<|z|<r$ avoids
$F(\partial U)$, and the radial path supplies a point with image in
$U$. Connectedness traps the entire punctured image in $U$.
Bounded coordinate functions extend, so $f$ extends holomorphically.
This argument uses compactness of a local chart boundary and global
injectivity of $F$; it does not require $F$ to be proper.
Off the puncture, $\psi_\infty=\psi\circ f$ almost everywhere.
The latter now extends smoothly across it, so the two functions
define the same distribution there and $\mu$ has no atom at the
puncture. Thus every alleged point of $Z$ is removable and has
zero atomic mass. The preceding exponential-integrability and
Morrey estimates apply across it as well, and the original sequence
converges on all compact subsets of $\Delta$. Since this applies to
every sequence, the family of centers is relatively compact.
If complete images could escape every compact, choose an escaping
interior point on each disc and precompose with a disc automorphism
sending zero to that point. Both the boundary compact and the area
bound are unchanged, contradicting the center conclusion. This
proves the first assertion.
For the cotangent assertion take global holomorphic fields
$X_1,\ldots,X_m$ spanning $T^{1,0}M$, as supplied by
Proposition~\ref{prop:initial-data}. The base images are compact by
the first part. The functions $\xi(X_i)$ along each disc are
bounded by their boundary values. Over the resulting compact base
the evaluation map $\xi\mapsto(\xi(X_i))_i$ has a uniform positive
least singular value. Its bounded values therefore bound the
cotangent vectors and put the images in a compact subset of $Y$.
Local holomorphic normal-family compactness proves the stated
subsequence conclusion.
\end{proof}
\subsection{Boundary factorization and holomorphic frames}
We now construct frames normalized on the boundary of each disc.
Their symplectic coefficients and their values at a fixed compact set
of centers must remain controlled when the disc area increases.
Fix the injective immersion
\begin{equation}\label{disc:immersion-y}
G:Y\longrightarrow\C^q,\qquad
G(x,\xi)=\big(F(x),\xi(X_1(x)),\ldots,\xi(X_m(x))\big).
\end{equation}
It is injective because its base entries separate base points and
the fiber evaluations separate covectors. It is immersive because
$dF$ detects base tangent vectors and the evaluations detect vertical
ones. There is also a finite family of global holomorphic fields
spanning $TY$: take the cotangent lifts
\[
\widetilde X_i=\sum_aX_i^a\partial_{x_a}
-\sum_{a,b}\xi_b(\partial_{x_a}X_i^b)\partial_{\xi_a}
\]
and the vertical translations by $dF_j$.
The former span after projection to $TM$, and the latter span the
vertical tangent. Smooth coefficients on any fixed compact subset
of $Y$ can be represented by smooth functions of $G$ there, by local
embedded coordinate neighborhoods and a finite partition of unity.
For a vector or matrix function $a$ on the circle, write
\begin{equation}\label{disc:half-sobolev}
D(a)^2=\sum_{k\in\mathbb Z}|k|\,\|a_k\|_F^2
=\frac1{(2\pi)^2}\int_0^{2\pi}\!\int_0^{2\pi}
\frac{\|a(e^{i(t+\theta)})-a(e^{it})\|_F^2}
{|e^{i\theta}-1|^2}\,dt\,d\theta,
\end{equation}
where $a_k$ are Fourier coefficients and $\|\cdot\|_F$ is the
Frobenius norm. For holomorphic $a$, this is
$\pi^{-1}\int_\Delta\|a'(z)\|_F^2\,dA$.
\begin{lemma}[Disc frames and Gauss area]\label{lem:disc-frames}
Let $f$ be a holomorphic disc in $Y$ extending past the unit circle.
There is a holomorphic frame $E_f$ of $f^*TY$, smooth on the closed
disc, for which $U=dG\,E_f$ satisfies
\begin{equation}\label{disc:frame-unitary}
U^*U=I\quad\hbox{on }\partial\Delta,\qquad
\sup_\Delta\|U\|_{\mathrm{op}}\le1.
\end{equation}
Let $\mathfrak g:Y\to\Gr(2n,q)$ send $y$ to $dG(T_yY)$, and
normalize the Pl\"ucker form by
$\omega_{\Gr}=i\partial\bar\partial\log\det(V^*V)$ for a local
holomorphic frame $V$ of its tautological plane. Then
\begin{equation}\label{disc:gauss-area}
D(U)^2=\frac1{2\pi}\int_\Delta
(\mathfrak g\circ f)^*\omega_{\Gr}.
\end{equation}
If $f(\partial\Delta)\subset K$ for a fixed compact $K\subset Y$,
the holomorphic component matrix $J_{E_f}$ of $\mathcal J$ in this
frame is bounded throughout $\Delta$ by a constant depending only
on $K$ and $G$. If also $f(0)$ ranges in a compact subset $K_0$
of a fixed coordinate chart, the frame matrix at zero and its
inverse are bounded by constants depending only on $K$, $K_0$, $G$
and that chart. These constants do not depend on the disc area or
on an interior compact containing its image.
\end{lemma}
We will first trivialize $f^*TY$ and then normalize the boundary Gram
matrix of that frame. For the normalization we use the smooth
Wiener--Masani factorization \cite{WienerMasani1957} in the Hardy-space
form described by Berndtsson--Rosay
\cite[Theorem~2.1 and Section~6]{BerndtssonRosay}.
We include its construction and the boundary half-Sobolev estimates
that will control the later deformation error.
\begin{lemma}[Boundary factorization]\label{lem:boundary-factorization}
Let $L$ be an $r\times r$ smooth Hermitian matrix on the circle with
$mI\le L\le MI$, where $0<m\le M<\infty$. There is a holomorphic
invertible matrix $H$ on $\Delta$, smooth with smooth inverse on
$\overline\Delta$, such that $L=H^*H$ on the circle. It satisfies
\begin{align}
\|H\|_{\infty,\mathrm{op}}&\le\sqrt M,&
\|H^{-1}\|_{\infty,\mathrm{op}}&\le m^{-1/2},\label{disc:factor-sup}\\
D(H)^2&\le\frac{M}{2m^2}D(L)^2,&
D(H^{-1})^2&\le\frac1{2m^3}D(L)^2.
\label{disc:factor-seminorm}
\end{align}
In particular the seminorm constants depend only on the boundary
spectral bounds, independently of higher derivatives of $L$.
\end{lemma}
\begin{proof}
Equip the vector Hardy space with
$\|v\|_L^2=\langle v^*Lv\rangle$, where brackets denote normalized
circle average. This norm is equivalent to the ordinary Hardy norm.
The shift $v\mapsto zv$ is an isometry, its range has codimension
$r$, and $\bigcap_{k\ge0}z^kH^2=\{0\}$. An orthonormal basis
$p_1,\ldots,p_r$ of the orthogonal complement of the range, followed
by all its successive shifts, is therefore an orthonormal basis of
the weighted space. Indeed the orthogonal remainder after the first
$k$ shift levels is $z^kH^2$, whose intersection is zero.
Let $P$ be the holomorphic matrix with columns $p_i$. Orthogonality
of all their shifts says that every Fourier coefficient of
$P^*LP-I$ vanishes, so $P^*LP=I$ almost everywhere on the circle.
Thus $P$ is bounded. Multiplication by $P$ maps the ordinary Hardy
space isometrically onto the weighted space by the basis property.
Apply this to constant target columns to obtain a holomorphic matrix
$Q$ of Hardy functions with $PQ=I$. Consequently $P$ is invertible
everywhere, $Q=P^{-1}$, and the boundary identity gives $L=Q^*Q$.
It also bounds both $P$ and $Q$ in $H^\infty$ by their boundary
values. Taking $H=Q$ proves \eqref{disc:factor-sup}.
For the seminorm estimate put
$L_\theta(z)=L(e^{i\theta}z)$ and
$A_\theta(z)=H(e^{i\theta}z)H(z)^{-1}$. This holomorphic matrix
has $A_\theta(0)=I$. Its mean is therefore $I$, and
\[
E_\theta:=\langle\|A_\theta-I\|_F^2\rangle
=\langle\tr(L_\theta L^{-1}-I)\rangle.
\]
Changing variables in the expression for $E_{-\theta}$ gives
\begin{align*}
E_\theta+E_{-\theta}
&=\left\langle\tr\big((L_\theta-L)L^{-1}
(L_\theta-L)L_\theta^{-1}\big)\right\rangle\\
&\le m^{-2}\langle\|L_\theta-L\|_F^2\rangle.
\end{align*}
The identities
\[
H_\theta-H=(A_\theta-I)H,\qquad
H_\theta^{-1}-H^{-1}=H_\theta^{-1}(I-A_\theta)
\]
bound their squared differences by $ME_\theta$ and
$m^{-1}E_\theta$, respectively. Integrating against the symmetric
kernel in \eqref{disc:half-sobolev} proves
\eqref{disc:factor-seminorm}.
It remains to justify smooth boundary regularity. The same difference
inequality, divided by $\theta^2$, first gives one full $L^2$
tangential derivative of $H$ and $H^{-1}$. With prime denoting that
derivative, set $B=H'H^{-1}$; this is the boundary value of
$izH_zH^{-1}$ and has zero constant Fourier coefficient. Differentiating
$L=H^*H$ gives
\[
H^{-*}L'H^{-1}=B+B^*.
\]
The strictly positive Fourier projection recovers $B$. The
one-dimensional Sobolev algebra property now bootstraps: if
$H,H^{-1}\in W^{k,2}$, the displayed expression is in $W^{k,2}$,
hence so is $B$, and $H'=BH$ gives $H\in W^{k+1,2}$.
The inverse derivative formula gives the same for $H^{-1}$.
For this one-dimensional algebra assertion, Cauchy--Schwarz on
Fourier coefficients and $\sum_k(1+k^2)^{-1}<\infty$ give
$W^{1,2}(S^1)\subset L^\infty(S^1)$. In each Leibniz term of total
order at most $k\ge1$, place a derivative of order at most $k-1$
in $L^\infty$ and the other factor in $L^2$; this gives the
$W^{k,2}$ product estimate. The positive-frequency projection has
norm at most one in every $W^{k,2}$. Iterating the preceding
identities therefore gives all Sobolev orders, and the same Fourier
estimate for derivatives proves smoothness on the closure.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:disc-frames}]
We first construct a preliminary frame on a neighborhood of the closed
disc. The bundle $E=f^*TY$ is embedded by $dG$ in a trivial bundle
and is generated by the pullbacks of the finite global spanning
fields. Start on a disc of radius greater than one and choose a
generator not identically zero. On a slightly smaller disc, divide
it by the finitely many common zero factors of its ambient
components, using at each zero the minimum component vanishing
order. The resulting section $e$ is holomorphic, remains in $E$ by
continuity, and is nowhere zero.
Write its ambient components as $e_1,\ldots,e_q$ and put
$\ell_j=\overline{e_j}/|e|^2$. Thus $\sum_j e_j\ell_j=1$.
The smooth $(0,1)$-forms
\[
b_{jk}=\ell_j\dbar\ell_k-\ell_k\dbar\ell_j
\]
are antisymmetric and satisfy
$\sum_j e_jb_{jk}=\dbar\ell_k$.
Solve $\dbar c_{jk}=b_{jk}$ with $c_{jk}=-c_{kj}$ on a further
slightly smaller disc. This scalar step follows directly by
multiplying the coefficient of each $(0,1)$-form by one common smooth
compactly supported cutoff equal to one on that disc and convolving
on $\C$ with $1/(\pi z)$;
the distributional identity
$\dbar(1/(\pi z))=\delta_0$ gives a smooth solution for the smooth
source. It does not use a frame of $E$.
The functions
\[
a_k=\ell_k-\sum_j e_jc_{jk}
\]
are holomorphic and obey $\sum_k e_ka_k=1$.
The ambient row $a$ restricts to a holomorphic dual of $e$ on $E$,
so $E=\C e\oplus\ker a$. If $s_i$ are the current generators,
then $s_i-e\,a(s_i)$ generate $\ker a$.
Thus the kernel is again an ambiently embedded, finitely generated
holomorphic bundle, now of rank one less. Repeat the construction
on a finite nested sequence of discs whose radii remain greater
than one. Induction gives a holomorphic frame of $f^*TY$ on a
neighborhood of $\overline\Delta$.
Choose this preliminary frame and apply
Lemma~\ref{lem:boundary-factorization} to its $dG$ Gram matrix on
the circle. Multiplication of the frame by $H^{-1}$ yields $E_f$.
The first identity in \eqref{disc:frame-unitary} follows immediately.
For each constant column $v$, the maximum principle applied to the
holomorphic vector $Uv$ gives $|Uv|\le|v|$, proving the second.
Put $Q=U^*U$. Since $Q=I$ on the circle,
\[
\partial_r\tr Q=\tr(\partial_rQ)
=\partial_r\log\det Q
\quad\hbox{on }\partial\Delta.
\]
Green's formula and holomorphicity give
\[
4\pi D(U)^2
=\int_\Delta\Delta_{\mathrm{eucl}}\tr Q\,dA
=\int_\Delta\Delta_{\mathrm{eucl}}\log\det Q\,dA
=2\int_\Delta(\mathfrak g\circ f)^*\omega_{\Gr},
\]
which proves \eqref{disc:gauss-area}.
On the boundary, $E_f$ is orthonormal for the metric $G^*g_{\rm eucl}$.
Thus the components of $\mathcal J$ there are bounded by its norm
in that metric on $K$. They are holomorphic functions in the
frame, so the maximum principle gives the same bound inside.
For the last assertion, write $E_f=\partial S$ in the fixed chart
at the center, and let $J$ be the invertible coordinate matrix of
$\mathcal J$. Then
\[
J_{E_f}=S^{-1}JS^{-\mathsf T},\qquad
S^{-1}=J_{E_f}S^{\mathsf T}J^{-1}.
\]
Let $\gamma>0$ be the infimum on $K_0$ of the least singular value
of $dG$ in the coordinate Euclidean metric. By
\eqref{disc:frame-unitary}, $\|S(0)\|_{\mathrm{op}}\le\gamma^{-1}$.
The preceding identity gives
\[
\|S(0)^{-1}\|_{\mathrm{op}}
\le \sup_\Delta\|J_{E_f}\|_{\mathrm{op}}\,
\gamma^{-1}\sup_{K_0}\|J^{-1}\|_{\mathrm{op}}.
\]
Every quantity on the right has the stated dependence.
\end{proof}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/discs.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/dynamics.tex ====
\section{Polynomial models with nonuniform contraction}
\label{sec:dynamics}
We retain the volume clock $s=\rho(o,t)$, its logarithmic-time speed
$q(s)=ds/d\tau=tR(o,t)$, and the charts of
Proposition~\ref{prop:shrinking-charts}.
Polynomial normalization and inverse iteration are classical tools
for attracting basins, notably in Rosay--Rudin's work
\cite[Appendix]{RosayRudin1988}; Chau--Tam brought this strategy into
K\"ahler uniformization \cite{ChauTamComplex2006,ChauTamStein,ChauTamSimply}.
The estimates proved here address the nonuniform contraction permitted
by Proposition~\ref{prop:ricci-windows}. For sequences of holomorphic
automorphisms of $\C^n$, $n\ge2$, fixing zero and satisfying
$A|z|\le|f_j(z)|\le B|z|$ on one fixed ball about zero, with
$0<A<B<1$ independent of $j$, Bera--Verma prove that the attracting
basin is biholomorphic to $\C^n$ \cite[Theorem~1.1]{BeraVerma2024}.
The chart transitions here require the variable-rate estimates below.
In particular, we use only
\begin{equation}
q\text{ nondecreasing},\qquad q\ge c>0,\qquad
\liminf_{s\to\infty}\frac{q(s)}s\le2,\qquad
\lambda_{\min}(t\Ric)(o)\ge b_0q(a_0s),
\label{eq:dyn:rate-input}
\end{equation}
where $0<a_0<1$ and $b_0>0$ are fixed.
We continue to use $J_d$ and the fixed coefficient norms introduced in
Section~\ref{sec:charts}.
All maps and jets in this Section fix zero. A coefficient bound
$\exp(o(s_k))$ means an upper bound $\exp(\eta_k s_k)$ with
$\eta_k\to0$; its rate of convergence may depend on parameters already
fixed. No assertion of uniformity as those parameters vary is intended.
For polynomial automorphisms put
$G_{k,l}=G_{l-1}\circ\cdots\circ G_k$ and $G_{k,k}=\Id$.
\begin{proposition}[Polynomial models]\label{prop:polynomial-models}
Fix an integer $d\ge2$ and a macro width $0<h<1$. After starting at a
sufficiently late macro endpoint, there is a grid $s_k\to\infty$ with
macro endpoints $S_j=e^{hj}$ and the following properties. Each macro
interval is either one bad step or is divided into regular steps of
length comparable to $\sqrt{S_j}$. Thus, eventually,
\begin{equation}
c_h\sqrt{s_k}\le\Delta s_k:=s_{k+1}-s_k
\le(e^h-1)s_k.
\label{eq:dyn:grid-size}
\end{equation}
For $F_k=F_{s_k,s_{k+1}}$ there are polynomial local coordinate changes
$H_k$ and polynomial automorphisms $G_k$ of $\C^n$ satisfying
\begin{equation}
J_d(H_{k+1}\circ F_k)=J_d(G_k\circ H_k).
\label{eq:dyn:jet-conjugacy}
\end{equation}
The degrees of $H_k,G_k,G_k^{-1}$ are bounded independently of $k$.
The coefficients of $H_k$ and its inverse jet are $\exp(o(s_k))$;
$H_k'(0)$ and its inverse have polynomial bounds in $s_k$.
For a constant $C_d$ depending only on the fixed jet order and dimension,
\begin{align}
\norm{G_k}_{\mathrm{coeff}}+\norm{G_k^{-1}}_{\mathrm{coeff}}
&\le \exp\bigl(C_d\Delta s_k+o(s_k)\bigr),
\label{eq:dyn:coefficient-bound}\\
\norm{G_k'(0)^{-1}}
&\le s_k^{C}\exp(\Delta s_k/2).
\label{eq:dyn:center-inverse}
\end{align}
In particular, the coefficient $C_d$ of $\Delta s_k$ is independent of
$h$. Finally, for some finite $D$,
\begin{equation}
\deg G_{0,k}\le D s_k
\quad\text{on an unbounded subsequence of macro endpoints.}
\label{eq:dyn:forward-degree}
\end{equation}
\end{proposition}
First we normalize the Jacobian jets coherently along the grid.
On regular steps we then construct polynomial models whose nonlinear
terms and intervening linear changes preserve bounds for the degree of
each component. On bad steps, realization of the full normalized jet
costs a fixed degree factor. The last part of the proof charges these
factors to growth of $q$ and obtains \eqref{eq:dyn:forward-degree}.
The distinction between the constant $C_d$ in
\eqref{eq:dyn:coefficient-bound} and constants hidden in $o(s_k)$ will
be used in Section~\ref{sec:uniformization}.
\subsection{Coherent volume jets}
On the uniform transition ball, the germs $F_k$ are nonsingular and have
uniformly bounded derivatives of each fixed order on a smaller ball.
Their center derivatives are contractions, and chart normalization gives
\begin{equation}
\abs{\det F_k'(0)}=e^{-\Delta s_k/2},\qquad
\norm{F_k'(0)^{-1}}\le e^{\Delta s_k/2}.
\label{eq:dyn:transition-linear}
\end{equation}
The second inequality follows because every singular value is at most
one and their product is $e^{-\Delta s_k/2}$.
Choose the holomorphic logarithm vanishing at zero of
\[
P_k(z)=\log\frac{\det F_k'(z)}{\det F_k'(0)}.
\]
The upper bound on its real part is $C+\Delta s_k/2$. The coefficient
estimate for a holomorphic function with bounded real part, applied on
concentric balls and with $P_k(0)=0$, therefore gives
$\norm{J_{d-1}P_k}\le C_d(1+\Delta s_k)$.
The volume correction uses only the spacing bounds
\eqref{eq:dyn:grid-size}, so it can be constructed before the macro
intervals are classified. On any such grid the jet series
\begin{equation}
\ell_k=\sum_{l\ge k}J_{d-1}
\bigl(P_l\circ F_{s_k,s_l}\bigr)
\label{eq:dyn:volume-series}
\end{equation}
converges. Indeed Proposition~\ref{prop:shrinking-charts} bounds each
coefficient of the summand by
\[
C_d(1+\Delta s_l)
\min\{1,\exp(C_d s_k-\sigma s_l)\}.
\]
The absence of a constant term in $P_l$ is essential here. For
$s_l\le C'_d s_k$ the step lengths telescope and the number of steps is
polynomial in $s_k$. Beyond that range the displayed exponential is
summable, since the grid eventually has spacing at least one and
$\Delta s_l\le(e^h-1)s_l$. Thus $\ell_k$ has polynomial coefficient
bounds in $s_k$. The composition identity for transitions yields
\begin{equation}
\ell_k=J_{d-1}\bigl(P_k+\ell_{k+1}\circ F_k\bigr).
\label{eq:dyn:volume-cocycle}
\end{equation}
Let $E_k=J_{d-1}\exp(\ell_k)$ and define a polynomial change
\[
A_k(z)=\left(z_1,\ldots,z_{n-1},
\int_0^{z_n}E_k(z_1,\ldots,z_{n-1},\zeta)\,d\zeta\right).
\]
It is tangent to the identity, has degree at most $d$, and satisfies
$J_{d-1}\log\det A_k'=\ell_k$. Its coefficients and inverse jet have
polynomial bounds in $s_k$. In the new coordinates the normalized
logarithmic Jacobian jet of $A_{k+1}F_kA_k^{-1}$ is
\[
J_{d-1}\bigl(\ell_{k+1}\circ F_k+P_k-\ell_k\bigr)\circ A_k^{-1}=0.
\]
This records the sign of the volume correction explicitly. We shall
call a jet with Jacobian determinant equal, through degree $d-1$, to
its center determinant a \emph{constant-volume jet}.
\begin{lemma}[Realization of constant-volume jets]\label{lem:volume-jets}
For fixed $n,d$, every invertible constant-volume $d$-jet fixing zero
is the $d$-jet of a polynomial automorphism of $\C^n$. Its degree and
inverse degree are bounded by a constant $D_d$. Its coefficients and
inverse coefficients are bounded polynomially in the coefficients of
the given jet and its inverse linear part. These bounds do not depend
on the grid or on $h$.
\end{lemma}
\begin{proof}
Normalize the derivative to the identity. If a volume-preserving
polynomial automorphism agrees with the required jet through degree
$m-1$, the first discrepancy is a homogeneous vector polynomial $X_m$
of degree $m$ with $\operatorname{div}X_m=0$: this is the degree $m-1$
part of the Jacobian determinant condition.
The space of such fields is spanned by
\begin{equation}
X(z)=l(z)^m v,\qquad l\in(\C^n)^*,\quad v\in\C^n,\quad l(v)=0.
\label{eq:dyn:shear-fields}
\end{equation}
This is the divergence-free shear lemma of Anders\'en
\cite{AndersenVolume1990}; see also
\cite[proof of Theorem~3.1]{ForstnericLarusson}.
We include the finite-dimensional proof to track the coefficient bounds.
Pair a linear functional on
homogeneous vector polynomials with $l^m v$. The result has the form
$A(l)(v)$, where $A(l)$ is a covector-valued homogeneous polynomial of
degree $m$. If the functional annihilates all fields in
\eqref{eq:dyn:shear-fields}, then $A(l)$ is proportional to $l$.
The polynomial identities $l_iA_j(l)=l_jA_i(l)$ show that
$A(l)=B(l)l$ for a homogeneous polynomial $B$ of degree $m-1$.
Such functionals are exactly the image of the transpose of divergence,
since $\operatorname{div}(l^m v)=m l(v)l^{m-1}$. Taking annihilators
proves the spanning assertion. In dimension one the divergence kernel
in these degrees is zero, so this argument also covers $n=1$.
Select a finite spanning list of the fields
\eqref{eq:dyn:shear-fields}, once for each $2\le m\le d$.
The time-$c$ map of such a field is the polynomial shear
$z\mapsto z+c l(z)^m v$, whose inverse has the minus sign; both have
Jacobian determinant one. A fixed linear decomposition of $X_m$
therefore gives a finite composition of shears correcting degree $m$
without changing lower degrees. Induction through $m=d$, followed by
the original linear map, proves realization. There are a fixed number
of shears, and each operation on the truncated jets is polynomial in
the preceding coefficients and the inverse linear part. This proves
all the coefficient and degree bounds, including those for the inverse.
Restoring the original linear map gives the realizing automorphism
the determinant of that linear map; the correcting shears have
determinant one.
\end{proof}
\subsection{Rate order without commuting matrices}
Consider one interval in logarithmic time, of length $T>0$.
Singular-value decompositions in its start and end unitary tangent
coordinates give derivative
\begin{equation}
D=\diag(e^{-\mu_1T/2},\ldots,e^{-\mu_nT/2}),\qquad
0\le\mu_1\le\cdots\le\mu_n,\qquad
\sum_i\mu_i=\frac{\Delta s}{T}.
\label{eq:dyn:diagonal-rates}
\end{equation}
The volume changes $A_k$ do not change these linear parts.
We need an order statement for the rate forms, and not merely their
ordered eigenvalues. Normalize the cometric at the start to $I$ in a
fixed covector basis. In the diagonalized basis its value at the end
is $\diag(e^{\mu_iT})$. On the complex strip $0\le\Re z\le T$, the
holomorphic covector with components $a_i e^{-\mu_i z/2}$ has squared
norm $\abs a^2$ on both boundary lines. Its squared norm is bounded
and subharmonic by Theorem~\ref{thm:joint-positivity}, applied with
$t=t_{\mathrm{start}}e^{\Re z}$; equivalently the complex time variable
there is shifted by $z/2$. The bounded subharmonic maximum principle
on the strip gives the same upper bound inside. Consequently the
cometric is bounded above by the flat exponential interpolation of its
endpoint values. Differentiating this matrix inequality at the two
ends gives
\begin{equation}
(t\Ric)_{\mathrm{start}}\le\diag(\mu_i),\qquad
\diag(\mu_i)\le(t\Ric)_{\mathrm{end}},
\label{eq:dyn:endpoint-rate-order}
\end{equation}
where each matrix is expressed in the corresponding endpoint unitary
covector coordinates. The inequalities have opposite endpoint signs
because the matrix difference vanishes at both ends and is
nonnegative inside.
At a shared endpoint of two steps, let $U_k$ send old tangent
components to new tangent components. It is unitary; a row covector
$v$ in the new coordinates has old row $vU_k$. Combining the two
inequalities in \eqref{eq:dyn:endpoint-rate-order} gives
\begin{equation}
\sum_l\mu_l^{\mathrm{old}}\abs{(vU_k)_l}^2
\le\sum_i\mu_i^{\mathrm{new}}\abs{v_i}^2.
\label{eq:dyn:loewner-jump}
\end{equation}
This proof allows the instantaneous Ricci matrices to fail to commute.
In particular, the sorted rates are nondecreasing from step to step.
\subsection{The macro grid and propagated weights}
The positive weights below will bound ordinary polynomial degrees
component by component. For a polynomial map
$P=(P_1,\ldots,P_n):\C^n\to\C^n$ with $\deg P_l\le w_l$, one has
\[
\deg(P_1^{\beta_1}\cdots P_n^{\beta_n})
\le\sum_l\beta_lw_l.
\]
Thus a nonlinear map preserves these degree bounds if every monomial
$z^\beta$ in component $i$ satisfies
$\sum_l\beta_lw_l\le w_i$. A linear change between two weight systems
preserves the bounds provided every nonzero entry sends an old
coordinate to a new coordinate of at least its weight. We shall
enforce both rules on regular steps. Bad steps may multiply degrees
by a fixed factor; their defining rate growth will pay for that factor.
Put $Q_j=q(S_j)$ and choose an integer $N\ge1$ with
$e^{-hN}\le a_0$. A macro index $j$ is \emph{flagged} when
\begin{equation}
\log Q_{j+1}-\log Q_{j-N}>M,
\label{eq:dyn:flag}
\end{equation}
where the fixed threshold $M$ will be chosen sufficiently large below.
Declare bad the union of the expanded flag intervals
$[j-N,j+1+N)$ in the integer macro index line. Each bad macro is a
single step. Divide every remaining macro interval into equal steps
whose lengths are comparable to $\sqrt{S_j}$, for example by taking
the number of pieces to be the nearest larger integer to
$(S_{j+1}-S_j)/\sqrt{S_j}$. Once $j$ is large this gives
\eqref{eq:dyn:grid-size}.
If macro $j$ is regular, it is not flagged. On each of its steps,
\eqref{eq:dyn:rate-input} and \eqref{eq:dyn:endpoint-rate-order} imply
\begin{equation}
b_0Q_{j-N}\le\mu_i\le Q_{j+1},\qquad
bQ_j\le\mu_i\le b^{-1}Q_j
\label{eq:dyn:regular-rates}
\end{equation}
for a fixed $b>0$ depending on $M,b_0$. These constants may depend on
$h$ after all parameter choices have been made.
For a regular macro set $\theta_j=Q_j/j^2$ and choose bounded increasing
multipliers $\lambda_j$ on the whole macro sequence by
\[
\lambda_{j+1}/\lambda_j=1+K_0/j^2.
\]
Their product is bounded, since $\sum j^{-2}<\infty$; an overall
constant factor is free. Choose $0<\epsilon_0<1/(10n)$.
For each regular macro, form a finite directed graph whose vertices
are the pairs $(k,i)$ consisting of a step in that macro and one of
its coordinates. Give this vertex raw weight
$\lambda_j(\mu_i-\theta_j)$, where $\mu_i$ is its rate in step $k$.
Draw an edge to a vertex in the same or the next step precisely when
the destination rate is at least the source rate minus
$\epsilon_0\theta_j$. Define $w_{k,i}$ as the maximum raw weight
among vertices from which $(k,i)$ is reachable, allowing paths of
length zero. This maximum exists because the graph is finite, even
when same-step edges form cycles. Assign the weight to both ends of
its diagonal step.
This propagation loses at most $n\epsilon_0\theta_j$ in rates,
independently of the number of steps. To verify the assertion, follow
any permitted chain and keep its successive lower records. Once a
sorted index has appeared, its rate at a later occurrence is at least
its earlier rate, by \eqref{eq:dyn:loewner-jump}. A new lower record
can therefore occur only on the first visit to an index; each such
record lowers the previous record by at most
$\epsilon_0\theta_j$. There are at most $n$ indices. It follows that
\begin{equation}
\lambda_j(\mu_i-\theta_j)\le w_{k,i}
\le\lambda_j(\mu_i-0.9\theta_j).
\label{eq:dyn:weights}
\end{equation}
In particular, whenever a permitted passage is possible the weights
do not decrease.
The same nondecrease can be required across adjacent regular macros.
Indeed, write $\mu'$ for a new rate and $\mu$ for an old one, with
$\mu'\ge\mu-\epsilon_0\theta_j$. A lower bound on the new raw weight
minus the upper bound on the old propagated weight is
\[
(\lambda_{j+1}-\lambda_j)\mu'
-\lambda_{j+1}\theta_{j+1}
+(0.9-\epsilon_0)\lambda_j\theta_j.
\]
Since $\mu'\ge bQ_{j+1}$, choosing $K_0$ large in terms of $b$ makes
this expression nonnegative at every sufficiently large index.
Thus the passage rule is valid also at regular macro junctions.
Increase the overall scale of $\lambda$ so that all weights are at
least one after the eventual starting index.
Call a nonlinear monomial $z^\beta$ in component $i$ \emph{allowed}
when $\sum_l\beta_lw_{k,l}\le w_{k,i}$, and \emph{forbidden} otherwise.
For a forbidden monomial of degree $m\ge2$,
\eqref{eq:dyn:weights} gives
\begin{equation}
\sum_l\beta_l\mu_l-\mu_i
>(0.9m-1)\theta_j\ge0.8\theta_j.
\label{eq:dyn:forbidden-gap}
\end{equation}
For an allowed monomial, every variable occurring in it has strictly
smaller weight than its output: the weights are positive and $m\ge2$.
Thus allowed nonlinear maps are triangular after ordering coordinates
by weight.
\subsection{Aligning the linear jumps}
\begin{lemma}[Linear alignment]\label{lem:linear-alignment}
On every run of regular steps there are unipotent matrices $L_k^+$,
with off-diagonal entries only in rows of smaller weight than their
columns, such that, on putting $L_k^-=D_k^{-1}L_k^+D_k$, the diagonal
step derivative remains $D_k$ and every regular jump becomes
\begin{equation}
J_k=L_{k+1}^-U_k(L_k^+)^{-1},\qquad
(J_k)_{il}=0\quad\hbox{if }w_{k+1,i}<w_{k,l}.
\label{eq:dyn:aligned-jump}
\end{equation}
The matrices $L_k^+$ and their inverses have polynomial bounds in the
macro index $j$, while
\begin{equation}
\norm{L_k^- -I}
\le j^C\exp\left(-c\frac{\sqrt{S_j}}{j^2}\right).
\label{eq:dyn:small-start-change}
\end{equation}
The estimates are uniform in finite terminal horizons, and hold on an
infinite regular run by a coefficientwise limit.
\end{lemma}
\begin{proof}
Work backward, taking $L_k^+=I$ at the last step of a finite run or
horizon. Suppose the next small start adjustment is known and set
$B=L_{k+1}^-U_k$. For a real threshold $x$, let $A_x$ be the span of
rows of $B$ whose new weights are at most $x$, and let $E_x$ be the old
coordinate row space with weights at most $x$.
Projection $P_{E_x}:A_x\to E_x$ is injective, and its inverse on its
image has norm $O(j)$. For the proof assume $A_x\ne0$ and let $r_x$
be the largest new rate among its defining rows. Every old rate
outside $E_x$ exceeds $r_x+\epsilon_0\theta_j$: otherwise the passage
rule would make that old weight at most the weight of a selected new
row. For any unit $a\in A_x$, \eqref{eq:dyn:loewner-jump} and the
small next adjustment imply that its old rate Rayleigh quotient is at
most $r_x+o(\theta_j)$. To see the size of this error, express $a$ as
a combination of the selected rows of $L_{k+1}^-U_k$; its coefficient
norm is $1+o(1)$, and the error is bounded by the operator norm of
$L_{k+1}^- -I$ times the largest old or new rate. The regular rate
bounds make that error $o(\theta_j)$, also at a macro junction.
Nonnegativity of all old rates now gives
\[
(r_x+\epsilon_0\theta_j)
(1-\norm{P_{E_x}a}^2)
\le r_x+o(\theta_j).
\]
As $r_x\le C Q_j$, this implies
$\norm{P_{E_x}a}^2\ge c/j^2$. Empty or full coordinate spaces give
the same assertion in its evident form.
We give the graph extension explicitly. List the distinct spaces
$E_x$ increasingly. For each such space retain the largest required
$A_x$ with that $E_x$, so that the required $A_x$ remain nested.
Starting with the full row space, descend through the list. Suppose
the next larger space $V_+$ already projects isomorphically to $E_+$
and contains the current $A=A_x$. Write $T_+:E_+\to V_+$ for its
inverse projection, $E=E_x$, $Y=P_EA$, and $R_A:Y\to A$ for the
inverse of $P_E|_A$. Define
\begin{equation}
T_E(e)=R_A(P_Ye)+T_+((I-P_Y)e),\qquad e\in E.
\label{eq:dyn:graph-extension}
\end{equation}
Here orthogonal projection $P_Y$ is taken in $E$. Both terms lie in
$V_+$, projection to $E$ is $e$, and the range contains $A$.
Moreover $\norm{T_E}\le\norm{R_A}+\norm{T_+}$.
There are at most $n$ nontrivial stages, so the resulting graph lifts
have polynomial bounds in $j$.
For each coordinate row, take its lift at its own weight threshold.
These rows form $L_k^+$. The rows up to every threshold span the
corresponding nested graph: they lie there and their projections form
a triangular basis of its coordinate space. Each lifted row equals
its coordinate row plus terms of strictly higher weight. Hence
$L_k^+=I+N_k$ with $N_k$ nilpotent, and
$(L_k^+)^{-1}=\sum_{r=0}^{n-1}(-N_k)^r$ has polynomial bounds.
The inclusion $A_x(L_k^+)^{-1}\subset E_x$ is exactly the zero pattern
in \eqref{eq:dyn:aligned-jump}.
For every nonzero off-diagonal entry in row $i$, column $l$, the
within-step passage rule gives $\mu_l-\mu_i>\epsilon_0\theta_j$.
Conjugation therefore gives
\[
(D_k^{-1}L_k^+D_k)_{il}
=(L_k^+)_{il}\exp\bigl(- (\mu_l-\mu_i)\Delta\tau_k/2\bigr).
\]
By \eqref{eq:dyn:regular-rates},
$\theta_j\Delta\tau_k\ge c\sqrt{S_j}/j^2$.
This proves \eqref{eq:dyn:small-start-change}; its exponential absorbs
every polynomial bound needed in the preceding backward step.
After a sufficiently late start the induction thus closes uniformly
in the horizon. A diagonal subsequence of finite-horizon matrices
proves the assertion on infinite runs; the zero conditions and all
bounds are closed under that limit.
\end{proof}
\subsection{Homological equations and polynomial automorphisms}
Choose a single primary coordinate system at each grid endpoint.
Use the adjusted start coordinates if a regular step starts there;
otherwise use the adjusted end coordinates of a regular step just
ended, if there is one; use the volume coordinates in all other cases.
These choices are a linear change, of polynomial size together with
its inverse, following $A_k$. For a regular step, before the possible
jump to the next regular start, write the resulting transition as
$\widehat F_k$, with derivative $D_k$. Its actual transition to the
next primary coordinates is $J_k\widehat F_k$, with $J_k=I$ when the
next step is bad. It has a constant-volume $d$-jet and polynomially
bounded coefficients. This follows by formal composition and
inversion of the primary coordinate jets; only their local germs are
used.
We construct tangent-to-identity polynomial jets $h_k$ at regular
starts, taking $h_k=I$ elsewhere. For a regular step seek
$G_k=J_kg_k$, where $g_k$ has linear part $D_k$ and only allowed
nonlinear terms, satisfying
\begin{equation}
g_k\circ h_k
=J_k^{-1}\circ h_{k+1}\circ J_k\circ\widehat F_k
\quad\text{through degree }d.
\label{eq:dyn:homological-identity}
\end{equation}
In homogeneous degree $m$, once lower degrees have been fixed, this
is
\begin{equation}
g_k^{(m)}+D_kh_k^{(m)}
=J_k^{-1}h_{k+1}^{(m)}(J_kD_kz)+B_k^{(m)}(z),
\label{eq:dyn:homological-layer}
\end{equation}
where $B_k^{(m)}$ contains only already known data.
Put only forbidden coefficients into $h_k^{(m)}$ and only allowed
coefficients into $g_k^{(m)}$. If $\Pi_k$ projects to the forbidden
coefficients, the equation for $h_k^{(m)}$ is a backward affine
recursion with linear operator
\begin{equation}
T_kX=\Pi_k\left[D_k^{-1}J_k^{-1}X(J_kD_kz)\right].
\label{eq:dyn:backward-operator}
\end{equation}
Conjugation by $J_k$ has polynomial norm in $j$ at each fixed degree.
After it has been expanded, a coefficient in output $i$ and input
$\beta$ receives the diagonal factor
$\exp[-(\sum_l\beta_l\mu_l-\mu_i)\Delta\tau_k/2]$.
Projection and \eqref{eq:dyn:forbidden-gap} consequently give
\begin{equation}
\norm{T_k}\le j^{C_m}
\exp\left(-c\frac{\sqrt{S_j}}{j^2}\right).
\label{eq:dyn:backward-contraction}
\end{equation}
This estimate holds for every incoming homogeneous polynomial $X$;
no support restriction on $X$ has been imposed before conjugation.
Solve first on a finite regular horizon with terminal correction the
identity, and proceed by degree. Inductively the lower-layer forcing,
including division by $D_k$, has coefficients bounded by
$\exp(C'_m\sqrt{s_k})$: regular steps have
$\Delta s_k\asymp\sqrt{s_k}$, and
$\norm{D_k^{-1}}\le e^{\Delta s_k/2}$.
The envelopes $\exp(C_m\sqrt{s_k})$ have bounded consecutive ratios,
because $\sqrt{s_{k+1}}-\sqrt{s_k}$ is bounded on regular steps.
Thus \eqref{eq:dyn:backward-contraction} is eventually less than
one half after weighting by these envelopes. The backward geometric
series solves \eqref{eq:dyn:homological-layer} with
\begin{equation}
\norm{h_k}+\norm{J_dh_k^{-1}}+\norm{g_k}
+\norm{g_k^{-1}}_{\mathrm{coeff}}
\le\exp(C_d'\sqrt{s_k}),
\label{eq:dyn:regular-coefficients}
\end{equation}
uniformly in the terminal horizon. Here $C_d'$ may depend on all
fixed grid parameters. Taking coefficientwise diagonal limits gives
the same construction on an infinite regular run.
Each $g_k$ is a triangular polynomial automorphism: all nonlinear
inputs have smaller weights than their output. Its inverse is found
by successive substitution in increasing weight order. Its degree
and inverse degree are bounded in terms of $n,d$, and its determinant
is the constant $\det D_k$.
The coordinate jets $h_k$ also preserve volume through degree $d-1$.
This is not a property assumed of the forbidden projection. In a
finite-horizon problem, differentiate the complete jet identity
\eqref{eq:dyn:homological-identity}. The determinant of
$\widehat F_k'$ has the same constant jet as $\det D_k$, and $g_k$ has
that determinant exactly. If $\det h_{k+1}'=1$ through degree $d-1$,
then the determinant identity forces the same statement for $h_k$.
Backward induction from the identity terminal jet proves it, and it
passes to the coefficientwise limits.
At a bad macro step realize, by Lemma~\ref{lem:volume-jets}, the jet of
the transition between the two primary coordinate systems after their
endpoint corrections $h_k,h_{k+1}$. It is a constant-volume jet by
what was just proved. Enlarge $D_d$ to bound the degrees and inverse
degrees of these models and of the regular models.
Let $H_k$ be the degree-$d$ truncation of the total primary change
followed by $h_k$. It has the required center derivative and inverse
jet, and \eqref{eq:dyn:jet-conjugacy} follows.
For precision, on a bad step the realized input jet has coefficient
bound $\exp(o(s_k))$, whereas its inverse linear part is bounded by
$\exp(\Delta s_k/2+o(s_k))$, by
\eqref{eq:dyn:transition-linear}. Lemma~\ref{lem:volume-jets} costs a
fixed polynomial in these two quantities. Its polynomial exponent
depends only on $n,d$. Therefore it gives
\eqref{eq:dyn:coefficient-bound} with a coefficient $C_d$ independent
of $h,N,M$. All endpoint and regular costs
$\exp(C(d,h,N,M)\sqrt{s_k})$ remain in $\exp(o(s_k))$.
Finally the linear part of the jet conjugacy is
$G_k'(0)=H_{k+1}'(0)F_k'(0)H_k'(0)^{-1}$; the polynomial bounds on the
endpoint linear changes give \eqref{eq:dyn:center-inverse}.
\subsection{Charging bad components to rate growth}
\begin{lemma}[Forward degree control]\label{lem:degree-control}
The flag threshold $M$ can be chosen so that regular macro starts
occur arbitrarily late, and the polynomial models constructed above,
initialized at any sufficiently late such start, satisfy
\eqref{eq:dyn:forward-degree}.
\end{lemma}
\begin{proof}
Set
\[
W=2N+1,\qquad D_*=D_d,\qquad
B_* =\max\{0,\log(2/b_0)\},\qquad \alpha=\frac{M}{3W}.
\]
Choose $M$ so large that
\begin{equation}
\alpha\ge\log D_*+B_*/W,\qquad \alpha>h.
\label{eq:dyn:flag-threshold}
\end{equation}
A flag $f$ has expanded interval $I_f=[f-N,f+1+N)$, of length $W$,
and growth interval $[f-N,f+1]$, on which $\log Q$ increases by more
than $M$.
Consider a full finite bad component $[u,v)$. Choose a maximal
disjoint collection of its expanded flag intervals. Their concentric
triples cover the component: any unselected equal-length interval
meets a selected one and is contained in its triple. Thus there are
at least $(v-u)/(3W)$ selected intervals. Their growth intervals are
disjoint up to endpoints and lie in $[u,v-N]$. Monotonicity gives
\begin{equation}
\log\frac{Q_{v-N}}{Q_u}\ge\alpha(v-u).
\label{eq:dyn:full-bad-growth}
\end{equation}
Every full component has length $L=v-u\ge W$, so
\eqref{eq:dyn:flag-threshold} implies
\begin{equation}
D_*^L Q_u\le e^{-B_*L/W}Q_{v-N}
\le(b_0/2)Q_{v-N}.
\label{eq:dyn:bad-reset-cost}
\end{equation}
We now track the ordinary degree of each component of the accumulated
forward polynomial. At a regular start bound these degrees by the
respective weights. Allowed nonlinear terms preserve that bound
under composition, and \eqref{eq:dyn:aligned-jump} preserves it at a
regular jump. If the next macro is bad, retain the last end weights;
at entry to $[u,v)$ their maximum is at most
$\lambda_{u-1}Q_u$. Each bad map multiplies maximum degree by at most
$D_*$. At the next regular macro $v$, its flag test fails, and hence
$Q_v\le e^M Q_{v-N}$. For sufficiently large $v$ every new start
weight is at least
\[
\lambda_v\bigl(b_0Q_{v-N}-Q_v/v^2\bigr)
\ge\lambda_v(b_0/2)Q_{v-N}.
\]
As $\lambda_v\ge\lambda_{u-1}$,
\eqref{eq:dyn:bad-reset-cost} resets all degree bounds exactly to the
new weights. There is no multiplicative loss for a completed bad
component.
Prefixes require a separate estimate. For a bad prefix $[u,j)$,
the expanded flag intervals wholly contained in the prefix cover
$[u,j-W)$, when this interval is nonempty. Indeed an expanded interval
covering a point before $j-W$ ends before $j$, and none crosses the
left endpoint $u$ of the component. The same disjoint selection gives
\begin{equation}
\log(Q_j/Q_u)\ge\alpha(j-u-W)_+,
\qquad D_*^{j-u}Q_u\le D_*^W Q_j.
\label{eq:dyn:bad-prefix-growth}
\end{equation}
This proof also applies to a prefix of an infinite bad component.
Such a component would give
$\log(Q_j/S_j)\ge(\alpha-h)j-O(1)\to\infty$.
It is impossible: choose continuous-clock points $x_r\to\infty$ with
$q(x_r)\le3x_r$, and let $S_{j_r}\le x_r<S_{j_r+1}$.
Then $Q_{j_r}\le3e^h S_{j_r}$, contradicting that growth.
Thus there are regular macro starts arbitrarily far out.
Start at one of them, after every fixed-parameter threshold above,
and label this grid point $k=0$, retaining the original large macro
indices $j$. The accumulated map is initially the identity and its
component degrees are one, so the initial weight bound holds after
the overall scaling already allowed for $\lambda$. Regular
propagation, the exact full-component reset, and the single prefix
loss in \eqref{eq:dyn:bad-prefix-growth} give, at all macro endpoints,
\[
\deg G_{0,k(j)}\le
C\max\{e^M,D_*^W\}Q_j,
\]
where $C$ can be taken to depend on the bounded supremum of the
multipliers. On a regular macro one uses
$Q_{j+1}\le e^M Q_j$; inside a bad component one uses the prefix
estimate. Applying the bound at the endpoints $j_r$ just selected
proves \eqref{eq:dyn:forward-degree}. This uses only a subsequence
with $Q_j=O(S_j)$, never an all-time bound on $q(s)/s$.
\end{proof}
Lemmas~\ref{lem:volume-jets}, \ref{lem:linear-alignment}, and
\ref{lem:degree-control}, together with the homological construction,
complete the proof of Proposition~\ref{prop:polynomial-models}.
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/dynamics.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/ellipsoids.tex ====
\section{Fiber ellipsoids and joint positivity}\label{sec:ellipsoids}
We turn the disc inequality into a scalar comparison by optimizing a
Hermitian form in each cotangent fiber. Throughout this Section, $t$ is
physical time, and $0<t<T$ for a fixed finite $T$. Thus $t$ plays the
role of the center-time parameter in Proposition~\ref{prop:disc-inequality}.
Let $v$ be its lower semicontinuous disc envelope. We use
\begin{equation}\label{ell:envelope-bounds}
N(x,\xi,0)\le v(x,\xi,t)\le N(x,\xi,t),
\qquad v(x,c\xi,t)=|c|^2v(x,\xi,t).
\end{equation}
The matrices below represent Hermitian forms on cotangent columns by
$\xi^*B\xi$. Accordingly, contraction against a horizontal Levi form
$H$ is $\tr(B^{\mathsf T}H)$; in particular,
$\Delta_{h_t}f=\tr(b_t^{\mathsf T}\dd_xf)$.
\subsection{The optimizer and its contacts}
The optimizer/contact-resolution argument is a Hermitian adaptation of
John's optimal-ellipsoid contact method \cite{John1948}. The
lower-semicontinuity details are proved here; the common-base product
support and the ensuing PDE comparison are further arguments below,
not consequences imported from the classical real convex-body theorem.
For each $(x,t)$, let $B(x,t)$ be the feasible matrix maximizing the
determinant:
\begin{equation}\label{ell:optimizer}
\det B(x,t)=\max\bigl\{\det D:
D>0,\ \xi^*D\xi\le v(x,\xi,t)\text{ for every }\xi\bigr\},
\qquad \mathcal L=\log\det B.
\end{equation}
The ellipsoid $\{\xi:\xi^*D\xi<1\}$ of a feasible matrix contains
$\{\xi:v(x,\xi,t)<1\}$. Maximizing $\det D$ minimizes its Euclidean
volume in these fiber coordinates.
\begin{lemma}\label{ell:optimizer-contacts}
The maximum in \eqref{ell:optimizer} exists and is unique.
The function $\mathcal L$ is lower semicontinuous in each holomorphic
coordinate chart, and
\begin{equation}\label{ell:volume-gap}
F_0:=\log\det b_t-\mathcal L
\quad\text{satisfies}\quad 0\le F_0\le\rho.
\end{equation}
At any fixed point, choose linear cotangent coordinates in which $B=I$.
There are finitely many unit columns $\xi_i$ and positive numbers
$\alpha_i$ such that
\begin{equation}\label{ell:contact-resolution}
v(x,\xi_i,t)=1,\qquad
\sum_{i=1}^m\alpha_i\xi_i\xi_i^*=I,
\qquad \sum_{i=1}^m\alpha_i=n.
\end{equation}
The largest weight can be made arbitrarily small by repeating columns.
\end{lemma}
\begin{proof}
The initial cometric $b_0$ is feasible by
\eqref{ell:envelope-bounds}. Every feasible matrix satisfies $D\le b_t$.
The constraints therefore define a compact set if positive semidefinite
matrices are temporarily admitted. A determinant maximizer in that set
has determinant at least $\det b_0>0$, so is positive definite.
Strict concavity of $\log\det$ on the positive cone gives uniqueness.
It also follows that $\det b_0\le\det B\le\det b_t$, which proves
\eqref{ell:volume-gap}, since
$\rho=\log\det b_t-\log\det b_0$.
Fix a feasible positive matrix $D$ at $(x_0,t_0)$. For $0<\varepsilon<1$,
the candidate $(1-\varepsilon)D$ has a strictly positive margin on the
unit fiber sphere. Lower semicontinuity of $v$ and compactness of that
sphere preserve this margin for all nearby $(x,t)$. Consequently
\[
\liminf_{(x,t)\to(x_0,t_0)}\mathcal L(x,t)
\ge\log\det D+n\log(1-\varepsilon).
\]
Take $D=B(x_0,t_0)$ and then $\varepsilon\downarrow0$.
This proves the asserted semicontinuity without asserting continuity of
the optimizing matrix. Under a change of holomorphic base coordinates,
both $\mathcal L$ and $\log\det b_t$ acquire the same time-independent
pluriharmonic summand. Thus $F_0$ is a globally defined upper
semicontinuous function.
Normalize $B(x_0,t_0)=I$, using a constant linear base-coordinate change
and its induced cotangent change. On the unit sphere put
\[
E=\{\xi:v(x_0,\xi,t_0)=1\}.
\]
This is compact, because $v\ge|\xi|^2$ at the chosen base point.
Suppose that a Hermitian matrix $H$ is strictly negative on $E$.
It is uniformly negative in a neighborhood of $E$ on the sphere;
on the complementary compact set, $v-1$ has a positive minimum.
It follows that $I+\varepsilon H$ is feasible for all sufficiently
small $\varepsilon>0$. Optimality then implies $\tr H\le0$.
The cone generated by $\xi\xi^*$, $\xi\in E$, is closed: its generators
have trace one, so bounded trace bounds the total coefficient in any
conic combination. If $I$ were outside this cone, finite-dimensional
separation would give $H$ with $\tr H>0$ and $\xi^*H\xi\le0$ on $E$.
Subtracting a sufficiently small positive multiple of $I$ contradicts
the preceding paragraph. Hence $I$ belongs to the cone. A finite
conic representation gives \eqref{ell:contact-resolution}, after
discarding zero coefficients. Taking its trace gives $\sum_i\alpha_i=n$.
Replacing each pair $(\xi_i,\alpha_i)$ by $r$ copies of
$(\xi_i,\alpha_i/r)$ preserves every identity and divides the largest
weight by $r$.
\end{proof}
\begin{proposition}\label{prop:ellipsoid-comparison}
For the optimizer in \eqref{ell:optimizer}, every smooth lower test
$\lambda$ of $\mathcal L$ at an interior point $(x_0,t_0)$ satisfies
\begin{equation}\label{ell:volume-viscosity}
\partial_t\lambda(x_0,t_0)
\ge e^{-F_0(x_0,t_0)}\Delta_{h_{t_0}}\lambda(x_0,t_0).
\end{equation}
Thus $\mathcal L$ satisfies this inequality in the lower viscosity
sense on $M\times(0,T)$.
\end{proposition}
To prove this scalar inequality, we construct simultaneous contact
jets at a common base point and time and transfer them to lower tests
of $v$. The transferred tests must have positive vertical Levi blocks,
as required by
Proposition~\ref{prop:disc-inequality}. We first construct a support
for the weighted sum of the contact values, then obtain simultaneous
contacts and add their Schur-complement inequalities.
\subsection{A product support with positive vertical Levi form}
Fix a smooth lower test $\lambda$ of $\mathcal L$ at $(x_0,t_0)$,
and choose the normalized contact list in that fiber. Allow the
columns to vary independently, and set
\begin{equation}\label{ell:rectangular-matrix}
M_c=(\sqrt{\alpha_1}\xi_1,\ldots,\sqrt{\alpha_m}\xi_m),
\qquad A=M_cM_c^*.
\end{equation}
Near the original tuple $M_0$, the matrix $M_c$ has rank $n$.
Let $\Pi(M_c)$ be its row plane in $\Gr(n,m)$.
For every nearby $(x,t)$ and every full-rank tuple,
\begin{equation}\label{ell:product-support}
\sum_i\alpha_i v(x,\xi_i,t)
\ge\tr(B(x,t)A)
\ge n+\mathcal L(x,t)+\log\det A.
\end{equation}
Indeed apply $a\ge1+\log a$ to the positive eigenvalues of
$B^{1/2}AB^{1/2}$. Equality in the second inequality holds exactly
when $B^{1/2}AB^{1/2}=I$.
Here is the Levi calculation that will control each separate fiber.
For a full-rank complex $n\times m$ matrix $M$, put $A=MM^*$.
For a matrix variation $Z$, differentiating along $M+zZ$ gives
\begin{align}
\dd\log\det(MM^*)[Z,\bar Z]
&=\tr(A^{-1}ZZ^*)
-\tr(A^{-1}ZM^*A^{-1}MZ^*) \notag\\
&=\tr\bigl(A^{-1}Z(I-M^*A^{-1}M)Z^*\bigr)\ge0.
\label{ell:rectangular-levi}
\end{align}
The middle factor in the final expression is an orthogonal projection.
This form is the pullback by $\Pi$ of the Grassmannian form induced
by the Pl\"ucker embedding. In particular, for a variation $V_i$ in
the $i$th unweighted column alone,
\begin{equation}\label{ell:single-column}
\dd_{\xi_i}\log\det A[V_i,\bar V_i]
=\alpha_i\bigl(1-\alpha_i\xi_i^*A^{-1}\xi_i\bigr)
V_i^*A^{-1}V_i.
\end{equation}
At any tuple with $A=I$ and $|\xi_i|=1$, the form after division by
$\alpha_i$ is therefore $(1-\alpha_i)I$. Repeating the contact
columns leaves their matrix resolution unchanged and makes these
normalized vertical forms as close to $I$ as needed.
Subtracting a small multiple of
$(|x-x_0|^2+|t-t_0|^2)^2$ from $\lambda$ makes its contact strict on a compact
coordinate cylinder without changing its time derivative or spatial
Levi form there. Choose a smooth nonnegative function $\eta$ on
$\Gr(n,m)$, vanishing only at $\Pi(M_0)$, and scale it so that
\begin{equation}\label{ell:grassmann-penalty}
\dd(\eta\circ\Pi)\le\dd\log\det A.
\end{equation}
Such a function exists: the squared Euclidean distance from the
orthogonal projector of a plane to the projector of $\Pi(M_0)$ is
smooth and vanishes at that plane alone, and its Levi form is bounded
above by a constant times the positive Grassmannian metric.
Fix $0<\theta<1$, and use the smooth support
\begin{equation}\label{ell:localized-support}
g_\theta(x,t,\xi_1,\ldots,\xi_m)
=n+\lambda(x,t)+\log\det A-\theta\eta(\Pi).
\end{equation}
The difference between the left side of \eqref{ell:product-support}
and $g_\theta$ is nonnegative. At equality, strictness forces
$(x,t)=(x_0,t_0)$, the trace inequality forces $A=I$, and the penalty
forces $\Pi=\Pi(M_0)$. Thus $M_c=UM_0$ for a unitary $U\in U(n)$.
In particular each unweighted column still has norm one. Equality
also requires each column to remain a contact of $v$.
The equality set is therefore a compact \emph{subset} of this unitary
orbit. It is nonempty, since it contains the original contact tuple.
Choose a bounded open neighborhood $O$ containing this equality set,
with compact closure in the base-time cylinder and in the full-rank
matrix locus. Its boundary avoids the equality set. Lower
semicontinuity gives a positive gap on $\partial O$ between the sum
and its support. All columns on $\overline O$, and their small
neighborhoods, lie in one bounded cotangent-coordinate region.
\subsection{Simultaneous contacts with a common base}
We use Alexandrov's almost-everywhere second-order differentiability
theorem \cite{Aleksandrov1939}, in the form of
\cite[Appendix A, Theorem~A.2]{CIL}. The contact-set and translated-test
arguments below follow the Jensen and inf-convolution framework
of \cite[Appendix A, Lemmas~A.3--A.5]{CIL}. We give the local details
that retain one shared base and time and introduce no negative
error in the vertical Levi form.
Use the real coordinate vector $q=(\Re x,\Im x,t,\Re\xi,\Im\xi)$ and
define, with sources restricted to a fixed slightly larger compact
coordinate region,
\begin{equation}\label{ell:inf-convolution}
v^\kappa(q)=\inf_z\left\{v(z)+\frac{|q-z|^2}{2\kappa}\right\}.
\end{equation}
The infimum is attained. On the region used here,
$v^\kappa\le v$ and any minimizing source satisfies
$|z-q|\le C\sqrt\kappa$, by the local upper bound and nonnegativity
in \eqref{ell:envelope-bounds}. Sources are consequently interior
for small $\kappa$. Moreover,
\begin{equation}\label{ell:epigraph-limit}
q_j\to q,\quad\kappa_j\downarrow0
\quad\Longrightarrow\quad
\liminf_j v^{\kappa_j}(q_j)\ge v(q).
\end{equation}
This follows by applying lower semicontinuity to minimizing sources.
The function $v^\kappa(q)-|q|^2/(2\kappa)$ is concave, being an
infimum of affine functions. Thus $v^\kappa$ is semiconcave and has
a full real second-order expansion almost everywhere.
Let $Q=(x,t,\xi_1,\ldots,\xi_m)$ denote the product coordinates,
understood as real coordinates when differentiating in time, and put
\[
f_\kappa(Q)=\sum_i\alpha_i v^\kappa(x,\xi_i,t)-g_\theta(Q).
\]
This function is semiconcave. The shared variables $x,t$ have not
been duplicated. By \eqref{ell:epigraph-limit}, the positive boundary
gap on $O$ persists for all sufficiently small $\kappa$, whereas
$f_\kappa\le0$ at the original equality tuple. For any sequence
$\delta_\kappa\downarrow0$, all minima of
$f_\kappa(Q)-p\cdot Q$ on $\overline O$, $|p|\le\delta_\kappa$,
are interior for small $\kappa$. Any limit of such minimizing points
lies in the original equality set: the minimum values have upper
limit at most zero, and \eqref{ell:epigraph-limit} gives the reverse
inequality at each limit.
For completeness, these contacts form a set of positive Lebesgue
measure for each fixed small $\kappa$ and $\delta_\kappa>0$.
Let $E_\kappa$ be the set of minimizers for all slopes in that closed
ball. Continuity of $f_\kappa$ and the boundary gap make $E_\kappa$
a compact subset of $O$. At a contact, its supporting lower affine function and
semiconcavity imply differentiability, with $Df_\kappa=p$.
If $x,y\in E_\kappa$ are close and $C_\kappa$ is a local
semiconcavity constant, then
\[
0\le f_\kappa(y)-f_\kappa(x)-Df_\kappa(x)\cdot(y-x)
\le\tfrac12C_\kappa|x-y|^2.
\]
Combining the lower support at $x$ with the upper quadratic bound
based at $y$, evaluated at $y+h$, gives
\[
(Df_\kappa(x)-Df_\kappa(y))\cdot h
\le\tfrac12C_\kappa\bigl(|x-y|^2+|h|^2\bigr).
\]
Choose $h$ of length $|x-y|$ in the direction of the gradient
difference. This proves that the gradient restricted to the contact
set is locally Lipschitz. Its image contains the entire slope ball,
so that contact set cannot have measure zero. Compact interior
localization makes the preceding argument valid in finitely many
coordinate balls. Notice that no strictifying quadratic has been
added to $g_\theta$.
For each $i$, the exceptional set where $v^\kappa$ lacks an Alexandrov
expansion is null. The projection
\[
Q\longmapsto(x,t,\xi_i)
\]
is a surjective real linear map; its inverse image of that exceptional
set is null by Fubini. A finite union remains null. Hence we may
choose a contact $Q_\kappa\in E_\kappa$ at which every factor has a
full real second-order expansion simultaneously. Write
\[
a_i^\kappa=\partial_t v^\kappa(x_\kappa,\xi_i^\kappa,t_\kappa),
\qquad W_i^\kappa=\dd_{x,\xi}v^\kappa
(x_\kappa,\xi_i^\kappa,t_\kappa).
\]
Differentiating the common-base minimum gives
\begin{equation}\label{ell:time-sum}
\sum_i\alpha_i a_i^\kappa
=\partial_t\lambda(x_\kappa,t_\kappa)+(p_\kappa)_t,
\qquad |p_\kappa|\le\delta_\kappa,
\end{equation}
and, for every complex $X,V_1,\ldots,V_m$,
\begin{equation}\label{ell:aggregate-levi}
\sum_i\alpha_i W_i^\kappa[(X,V_i),\overline{(X,V_i)}]
\ge\dd_x\lambda[X,\bar X]
+(1-\theta)\dd\log\det A[V,\bar V].
\end{equation}
Here the forms on the right are evaluated at $Q_\kappa$.
To justify the passage from real to complex Hessians, if $H$ is a
real Hessian and $Z$ is a complex spatial direction, its Levi
quadratic form is
\[
\tfrac14\bigl(H[Z_{\R},Z_{\R}]
+H[JZ_{\R},JZ_{\R}]\bigr).
\]
Apply the real Hessian inequality in both directions, with time
component zero. The linear perturbation has zero Hessian, and
\eqref{ell:grassmann-penalty} gives
\eqref{ell:aggregate-levi}. Thus this is an inequality of complex
Levi forms, rather than an identification of real and complex
Schur complements.
Let $C_i^\kappa$ denote the vertical block of $W_i^\kappa$.
Take $X=0$ and vary only the $i$th column in
\eqref{ell:aggregate-levi}. Equation~\eqref{ell:single-column}
and convergence to the compact equality set give
\begin{equation}\label{ell:vertical-positive}
C_i^\kappa\ge
\bigl((1-\theta)(1-\max_j\alpha_j)-o(1)\bigr)I.
\end{equation}
The $o(1)$ tends to zero as $\kappa\downarrow0$, with the list and
$\theta$ fixed. Replicate the list beforehand so that
$\max_i\alpha_i<1$; these vertical blocks are then uniformly positive.
We finally transfer these jets to the original envelope. Fix
$\kappa$ and the chosen contact, and let $z_i$ be a minimizing source
in \eqref{ell:inf-convolution} for
$q_i=(x_\kappa,t_\kappa,\xi_i^\kappa)$.
Subtract $\varepsilon|h|^2$ from the real second-order expansion of
$v^\kappa(q_i+h)$. For every $\varepsilon>0$ this gives a smooth
quadratic lower test $\phi_{i,\varepsilon}$ in a sufficiently small
neighborhood of $q_i$. Since
\[
v^\kappa(q_i+h)
\le v(z_i+h)+\frac{|q_i-z_i|^2}{2\kappa},
\]
the translated function
\begin{equation}\label{ell:translated-test}
\psi_{i,\varepsilon}(z_i+h)
=\phi_{i,\varepsilon}(q_i+h)
-\frac{|q_i-z_i|^2}{2\kappa}
\end{equation}
is a lower test of $v$ at $z_i$. Translation preserves its time
derivative $a_i^\kappa$ and its Levi form
$W_i^\kappa-\varepsilon I$.
Apply Proposition~\ref{prop:disc-inequality}. Its canonical
cotangent contraction has constant coefficients in these coordinates,
so the possibly different source base points and times cause no
coefficient error. If $C_i^\kappa\ge cI$ and $0<\varepsilon<c/2$,
it supplies $K_{i,\varepsilon}\ge0$ with
\[
K_{i,\varepsilon}\ge\Schur(W_i^\kappa-\varepsilon I),
\qquad
a_i^\kappa\ge
\tr\bigl((C_i^\kappa-\varepsilon I)^{\mathsf T}
K_{i,\varepsilon}\bigr)
\ge(c-\varepsilon)\tr K_{i,\varepsilon}.
\]
In particular $a_i^\kappa\ge0$. At this fixed contact, let
$\varepsilon\downarrow0$ first. The displayed trace bound gives a
convergent subsequence, and continuity of the Schur complement on
positive vertical blocks yields matrices $K_i^\kappa$ satisfying
\begin{equation}\label{ell:transferred-inequality}
K_i^\kappa\ge0,\qquad
K_i^\kappa\ge\Schur(W_i^\kappa),\qquad
a_i^\kappa\ge\tr\bigl((C_i^\kappa)^{\mathsf T}K_i^\kappa\bigr).
\end{equation}
The temporary Hessian decrease has thus disappeared before taking
any limit in $\kappa$. In particular it cannot be magnified by a
later vertical minimization.
\subsection{The scalar volume inequality}
\begin{proof}[Completion of the proof of Proposition~\ref{prop:ellipsoid-comparison}]
Use the preceding construction at the lower contact, with $B=I$.
Set $\mathcal K_\kappa=\sum_i\alpha_iK_i^\kappa$.
We first show
\begin{equation}\label{ell:schur-aggregate}
\mathcal K_\kappa\ge0,
\qquad \mathcal K_\kappa\ge\dd_x\lambda(x_\kappa,t_\kappa).
\end{equation}
For a Hermitian form with blocks
$W=\left(\begin{smallmatrix}H&D\\D^*&C\end{smallmatrix}\right)$,
$C>0$, completing the square gives
\[
W[(X,V),\overline{(X,V)}]
=X^*(H-DC^{-1}D^*)X
+(V+C^{-1}D^*X)^*C(V+C^{-1}D^*X).
\]
Its minimum over the complex vertical vector $V$ is
$\Schur(W)[X,\bar X]$. The extra term on the right of
\eqref{ell:aggregate-levi} is nonnegative by
\eqref{ell:rectangular-levi}. Minimizing the left side independently
over every $V_i$ therefore proves
\[
\sum_i\alpha_i\Schur(W_i^\kappa)
\ge\dd_x\lambda(x_\kappa,t_\kappa).
\]
Equation~\eqref{ell:transferred-inequality} gives
\eqref{ell:schur-aggregate}.
Write $c_{\theta,\alpha}=(1-\theta)(1-\max_i\alpha_i)>0$.
Equations~\eqref{ell:time-sum}, \eqref{ell:vertical-positive}, and
\eqref{ell:transferred-inequality} imply
\begin{equation}\label{ell:aggregate-trace}
\partial_t\lambda(x_\kappa,t_\kappa)+(p_\kappa)_t
\ge\bigl(c_{\theta,\alpha}-o(1)\bigr)\tr\mathcal K_\kappa.
\end{equation}
The positive aggregate matrices are bounded. Let $\kappa\downarrow0$
and pass to a subsequence to obtain a matrix $\mathcal K_{\theta,\alpha}$
such that
\[
\mathcal K_{\theta,\alpha}\ge0,\qquad
\mathcal K_{\theta,\alpha}\ge\dd_x\lambda(x_0,t_0),\qquad
\partial_t\lambda(x_0,t_0)
\ge c_{\theta,\alpha}\tr\mathcal K_{\theta,\alpha}.
\]
Now let $\theta\downarrow0$ and repeat the construction with increasingly
replicated lists, so that $\max_i\alpha_i\downarrow0$.
The aggregate matrices still act on the fixed $n$-dimensional
horizontal space and have uniformly bounded trace. Another
subsequence gives
\begin{equation}\label{ell:limiting-majorant}
\mathcal K\ge0,\qquad \mathcal K\ge\dd_x\lambda(x_0,t_0),
\qquad \partial_t\lambda(x_0,t_0)\ge\tr\mathcal K.
\end{equation}
This also specifies the parameter order: fix the list and $\theta$;
regularize and select simultaneous contacts; remove the lower-test
Hessian decrease at each fixed contact; let $\kappa$ and the slopes
tend to zero; finally let $\theta$ and the maximum weight tend to zero.
No bound uniform in the number of individual contact jets is needed.
At the normalized point $B=I\le b_{t_0}$, and
$F_0=\log\det b_{t_0}$. All eigenvalues of $b_{t_0}$ are at least
one, so each is at most their product. Consequently
\[
0<e^{-F_0}b_{t_0}\le I.
\]
Using the two matrix inequalities in
\eqref{ell:limiting-majorant}, in the indicated order, gives
\[
\partial_t\lambda
\ge\tr\mathcal K
\ge e^{-F_0}\tr(b_{t_0}^{\mathsf T}\mathcal K)
\ge e^{-F_0}\tr(b_{t_0}^{\mathsf T}\dd_x\lambda)
=e^{-F_0}\Delta_{h_{t_0}}\lambda.
\]
The positivity of $\mathcal K$ justifies the middle comparison even
when $\dd_x\lambda$ is indefinite. This proves
\eqref{ell:volume-viscosity}.
\end{proof}
\subsection{Equality of the envelope and the norm}
\begin{theorem}\label{thm:joint-positivity}
For the flow in Theorem~\ref{thm:localized-flow}, the function
\[
(x,\xi,w)\longmapsto N(x,\xi,e^{w+\bar w})
\]
is plurisubharmonic on $T^{*(1,0)}M\times\C$.
For every finite $T$, its disc envelope satisfies $v=N$ on
$T^{*(1,0)}M\times(0,T)$.
\end{theorem}
\begin{proof}
In any holomorphic base coordinates let $\ell=\log\det b_t$.
The flow equation and the Ricci-form identity give
\[
\partial_t\ell=R_{h_t},\qquad
\dd_x\ell=\Ric(h_t),\qquad
\Delta_{h_t}\ell=R_{h_t}.
\]
If $\varphi$ is an upper test of $F_0=\ell-\mathcal L$, then
$\lambda=\ell-\varphi$ is a lower test of $\mathcal L$.
Proposition~\ref{prop:ellipsoid-comparison} therefore says
\begin{equation}\label{ell:scalar-gap-equation}
\partial_t\varphi
\le e^{-F_0}\Delta_{h_t}\varphi
+(1-e^{-F_0})R_{h_t}
\end{equation}
at contact. The coefficient $F_0$ there equals the test value
$\varphi$. By \eqref{ell:volume-gap}, $F_0$ is upper semicontinuous,
locally bounded, and between zero and $\rho$. Since $\rho$ is smooth
and vanishes initially, extending $F_0$ by zero at $t=0$ preserves
upper semicontinuity. Apply Proposition~\ref{prop:scalar-comparison}
to \eqref{ell:scalar-gap-equation} on each interval $[0,T']$ with
$T'<T$. Since $T'$ is arbitrary, $F_0=0$ for every $t<T$.
Its proof does not require completeness of $h_t$ at positive time.
We have $B\le b_t$ and $\det B=\det b_t$. The positive matrix
$b_t^{-1/2}Bb_t^{-1/2}\le I$ has determinant one; every one of its
eigenvalues is therefore one. Hence $B=b_t$, and feasibility and
\eqref{ell:envelope-bounds} imply
\[
N(x,\xi,t)=\xi^*B\xi\le v(x,\xi,t)\le N(x,\xi,t).
\]
This proves $v=N$.
For the raw disc infimum $v^{\rm raw}$ of \eqref{var:envelope},
we have $v\le v^{\rm raw}$, while constant discs give $v^{\rm raw}\le N$.
Thus $v^{\rm raw}=N$ too, so $N$ obeys the submean inequality for
every admissible disc, not merely for regularized centers.
Given a holomorphic disc $(f,W)$ in $Y\times\C$, put
$t_0=e^{W(0)+\overline{W(0)}}$ and $w=W-W(0)$.
Choose $T$ larger than the maximum of $e^{W+\bar W}$ on the closed
disc. The defining disc inequality then gives
\[
N(f(0),t_0)
\le\frac1{2\pi}\int_0^{2\pi}
N\bigl(f(e^{i\vartheta}),e^{W(e^{i\vartheta})+
\overline{W(e^{i\vartheta})}}\bigr)\,d\vartheta.
\]
It suffices to use discs smooth past their boundary, including all
sufficiently small coordinate discs. Since the joint norm is smooth,
these submean inequalities are precisely its plurisubharmonicity.
The horizon $T$ was arbitrary.
\end{proof}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/ellipsoids.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/flow.tex ====
\section{A global flow and scalar comparison by localization}
\label{sec:flow}
We first construct the flow with estimates on fixed initial compact
sets. Completeness of its positive-time slices will be proved in
Proposition~\ref{prop:harnack-completeness}.
\begin{theorem}[Localized flow]\label{thm:localized-flow}
There is a smooth K\"ahler--Ricci flow $h_t$ on $M$, for
$0\le t<\infty$, with $h_0=g$. There are smooth real functions $U,P$
and $\rho$ such that
\begin{equation}\label{flow:identities}
\begin{gathered}
0<h_t\le g,\qquad
h_t=g-t\Ric(g)+\dd U,\qquad U(\cdot,0)=0,\\
P=U_t=\log\frac{\det h_t}{\det g},\qquad
\rho=-P\ge0,\qquad R_{h_t}=-P_t\ge0.
\end{gathered}
\end{equation}
For each finite $T$, there is $C_T$ such that
\begin{equation}\label{flow:rho-average}
\sup_{0\le t\le T}
\frac{1}{\Vol_gB_g(o,L)}\int_{B_g(o,L)}\rho(x,t)\,dV_g(x)
\le C_TL\qquad(L\ge1).
\end{equation}
At every fixed $x$, $\rho(x,t)\le C_x(1+t)$ for all $t\ge0$.
Furthermore, on each finite time interval there is a smooth positive
function $Q$ and a constant $c>0$ with
\begin{equation}\label{flow:proper-supercaloric}
Q\ge c(1+r^2),\qquad
(\partial_t-\Delta_{h_t})Q\ge c\tr_{h_t}g.
\end{equation}
Smoothness at $t=0$ means smoothness with all one-sided time derivatives.
\end{theorem}
\subsection{Complete Hermitian approximations}
Take regular values $L\to\infty$ of the exhaustion $u$ in
Proposition~\ref{prop:initial-data}, and put $\Omega_L=\{u<L\}$.
Choose a fixed smooth nonnegative function $F$ on $[0,1)$, zero on
$[0,1/2]$, which equals $-2\log(1-v)$ near $v=1$. On $\Omega_L$ set
\[
f=F(u/L),\qquad \alpha=e^fg.
\]
The estimates for $u$ imply
\begin{equation}\label{flow:conformal-bounds}
|\partial f|_\alpha\le C/L,
\qquad |\dd f|_\alpha\le C/L^2.
\end{equation}
For example, near the boundary $e^f=(1-u/L)^{-2}$, so the factors
$(1-u/L)^{-1}$ and $(1-u/L)^{-2}$ in the first and second derivatives
are canceled by the conformal norm. On the transition region
$u\asymp L$, the bound $|\dd u|_g\le C/u$ gives the same estimates.
The metric $\alpha$ is complete on $\Omega_L$: approaching the regular
boundary has infinite length for the factor $(1-u/L)^{-1}$, and the
closure of $\Omega_L$ is compact. For each fixed $L$ it has bounded
geometry of every order in holomorphic charts. Indeed choose finitely
many ordinary charts on the compact closure and, near a point of
boundary distance comparable to $1-u/L$, rescale their coordinate
radii by a sufficiently small fixed multiple of $1-u/L$. In the
rescaled charts $\alpha$ and its inverse are uniformly bounded, and
every derivative of its coefficients is bounded. The constants here
are allowed to depend on $L$ and on each derivative order.
Run Chern--Ricci flow on this complete Hermitian background:
\begin{equation}\label{flow:approximation}
K=\alpha-t\Ric(\alpha)+\dd U,\qquad
U_t=P=\log\frac{\det K}{\det\alpha},\qquad U(\cdot,0)=0.
\end{equation}
Until the passage to the local limit below, $U,P$ refer to the
approximating solution.
Bounded-geometry short-time existence is
\cite[Definition~2.1 and Lemma~2.1]{LeeTam}. We establish estimates
on every closed bounded-geometry interval of this solution; the
resulting continuation intervals increase to infinity as $L$ increases.
Write $\mathscr L_K=\partial_t-\Delta_K$.
Differentiation of \eqref{flow:approximation} gives
\begin{equation}\label{flow:scalar-evolutions}
\mathscr L_KP=-\tr_K\Ric(\alpha),\qquad
\mathscr L_KP_t=-|\Ric(K)|_K^2,
\qquad P_t(\cdot,0)\le C/L^2.
\end{equation}
The last inequality follows from
$\Ric(\alpha)=\Ric(g)-n\dd f$, initial Ricci positivity, and
\eqref{flow:conformal-bounds}. In particular,
\begin{equation}\label{flow:upper-potentials}
P_t\le C/L^2,
\qquad P\le Ct/L^2,
\qquad U\le Ct^2/(2L^2).
\end{equation}
These are bounded maximum-principle estimates on an already existing
bounded-geometry interval.
Fix a horizon $D>0$. Suppose that $\phi\le0$ is smooth and
\begin{equation}\label{flow:phi-condition}
\alpha-D\Ric(\alpha)+D\dd\phi\ge c\alpha
\quad(c>0).
\end{equation}
Define
\begin{equation}\label{flow:Q-definition}
Q=(D-t)P+U-D\phi+nt.
\end{equation}
Using
$\mathscr L_KU=P-n+\tr_K(\alpha-t\Ric(\alpha))$ gives the exact
identity
\begin{equation}\label{flow:Q-comparison}
\mathscr L_KQ
=\tr_K\{\alpha-D\Ric(\alpha)+D\dd\phi\}
\ge c\tr_K\alpha,
\qquad Q\ge0.
\end{equation}
The initial value is $-D\phi\ge0$. For bounded $\phi$, the maximum
principle proves the last assertion directly; adding a vanishing
positive multiple of the proper function $f$ is an equivalent
localization. The same proof applies to functions with downward
logarithmic poles, since $Q\to+\infty$ at those poles.
For a fixed $L$, choose finitely many canonical sections with no common
zero on $\overline{\Omega_L}$. A constant shift of
$\log\sum_i|S_i|_g^2$ is bounded, nonpositive, and has complex Hessian
at least $\Ric(g)$. It satisfies
\[
\alpha-D\Ric(\alpha)+D\dd\phi
\ge \alpha+nD\dd f\ge(1-CD/L^2)\alpha.
\]
Thus for fixed $D$ and sufficiently large $L$ we may take $c=1/2$.
Equations~\eqref{flow:upper-potentials} and
\eqref{flow:Q-comparison} give both bounds for $P$ on $0\le t\le D/2$:
indeed $(D-t)P\ge D\phi-U-nt$. Integrating also gives both bounds
for $U$. These bounds may depend on $L$.
\subsection{The largest metric eigenvalue}
We next bound the largest eigenvalue of $K$ relative to $\alpha$.
The bound will tend to one as $L\to\infty$ wherever $Q$ stays bounded
independently of $L$. The local potential estimates below will supply
this control on each fixed compact set, yielding $h_t\le g$ in the
limit. At a point use $g$-normal holomorphic
coordinates diagonalizing $K$, write $K_{i\bar i}=\lambda_i$, and
suppose that $\lambda_1$ is largest. Put $q=K_{1\bar1}$ and
\[
W=\frac{K_{1\bar1}}{e^fg_{1\bar1}}.
\]
The Rayleigh quotient in this fixed coordinate direction is a smooth
lower support for the largest relative eigenvalue and agrees with it
at the point. It therefore suffices for a maximum-principle test of
that eigenvalue. Wherever $W\ge1$, direct differentiation gives
\begin{equation}\label{flow:eigenvalue-raw}
\mathscr L_K\log W
\le -\frac1q\sum_{i,j}
\frac{|\partial_1K_{i\bar j}|^2}{\lambda_i\lambda_j}
+|\partial\log q|_K^2
+CL^{-2}\tr_K\alpha.
\end{equation}
Here are the derivative exchanges in this formula. Since $K-\alpha$
is closed,
\[
\partial_1\partial_{\bar1}K_{i\bar i}
-\partial_i\partial_{\bar i}K_{1\bar1}
=\partial_1\partial_{\bar1}\alpha_{i\bar i}
-\partial_i\partial_{\bar i}\alpha_{1\bar1}.
\]
In the chosen coordinates the right side is
$(e^f)_{1\bar1}-(e^f)_{i\bar i}$; the second derivatives of $g$
cancel by its K\"ahler symmetries. Dividing their contraction by
$q\ge e^f$ bounds the error by $CL^{-2}\tr_K\alpha$.
Differentiating the denominator $g_{1\bar1}$ contributes
$-\sum_i\lambda_i^{-1}R_g(i,\bar i,1,\bar1)\le0$.
The remaining conformal derivatives have the same controlled size.
These facts give \eqref{flow:eigenvalue-raw}.
For $i\ne1$, the first-derivative exchange is
\[
\partial_1K_{i\bar1}=\partial_iq-e^ff_i.
\]
The terms with $j=1$ in the negative square in
\eqref{flow:eigenvalue-raw} therefore dominate
\[
\left|\partial\log q-W^{-1}\sum_{i\ne1}f_i\,dz_i\right|_K^2.
\]
Expanding this square and using
$\partial\log q=\partial\log W+\partial f$ at the point yields
\begin{equation}\label{flow:eigenvalue-gradient}
\mathscr L_K\log W
\le CL^{-2}\tr_K\alpha
+2W^{-1}|\partial f|_K|\partial\log W|_K.
\end{equation}
Set $a=L^{-1}$ and
\[
J(v)=av+\log(1+av)\qquad(v\ge0).
\]
At a positive maximum of $\log W-J(Q)$, one has
\[
W\ge e^{aQ}(1+aQ),\qquad
\partial\log W=J'(Q)\partial Q,
\quad J'\ge a,\quad J'\le2a,\quad
-J''=\frac{a^2}{(1+aQ)^2}.
\]
The chain rule, \eqref{flow:Q-comparison}, and
\eqref{flow:eigenvalue-gradient} give
\[
\mathscr L_K(\log W-J(Q))
\le (CL^{-2}-ca)\tr_K\alpha
+2W^{-1}J'|\partial f|_K|\partial Q|_K
+J''|\partial Q|_K^2.
\]
Young's inequality absorbs the middle term into half of
$-J''|\partial Q|_K^2$: its remaining coefficient is bounded by
\[
C\frac{W^{-2}(J')^2}{-J''}|\partial f|_K^2
\le C|\partial f|_K^2
\le CL^{-2}\tr_K\alpha.
\]
For sufficiently large $L$, this contradicts the maximum principle.
To attain the maximum on $\Omega_L$, test first with the additional
penalty $-\delta f$. Its Hessian term is controlled by
$C\delta L^{-2}\tr_K\alpha$, and its gradient terms have the same
bound after Young's inequality. Send $\delta\downarrow0$ afterwards.
At $t=0$, $W=1$ and $Q\ge0$. We conclude
\begin{equation}\label{flow:eigenvalue-bound}
K\le e^{J(Q)}\alpha.
\end{equation}
With the bounded choice of $\phi$, $Q$ is bounded above by
\eqref{flow:upper-potentials}. Thus \eqref{flow:eigenvalue-bound}
gives an upper metric bound. The lower bound for $P$, hence for
$\det K/\det\alpha$, then bounds the least eigenvalue below.
We spell out the continuation implication in the uniform holomorphic
charts. Let $\iota$ be the real symmetric representation of a Hermitian
matrix and write
\[
F(B)=\tfrac12\log\det_{\R}B,\qquad
S(x,t)=\iota(\alpha-t\Ric(\alpha)),\qquad
T(D^2U)=\iota(\dd U).
\]
Then $U_t-F(S+T(D^2U))=-\log\det_{\C}\alpha$.
Two-sided metric equivalence puts the transformed Hessian $S+T(D^2U)$
in a fixed compact convex positive matrix range. The function $F$
is smooth, concave and uniformly elliptic near this range. The map
$T$ is linear, preserves nonnegativity, and has norm comparable to
the original norm on nonnegative real symmetric matrices. The smooth
background coefficients have uniform local H\"older bounds.
These are the hypotheses of the transformed-Hessian parabolic
estimate \cite[Theorem~5.1]{ChuRegularity}. Its bound for the spatial
second derivatives and first time derivative depends on $\|U\|_\infty$
and these data, without a prior bound for the full real Hessian.
Schauder estimates then give higher derivatives and continuation on
each fixed approximating background, as in
\cite[proof of Theorem~4.1]{LeeTam}.
For initial estimates, extend $U$ by zero to negative times and use
$S(x,t_+)$, where $t_+=\max(t,0)$. Each fixed approximating solution
is already smooth one-sided at zero by short-time existence.
Since $U(\cdot,0)=U_t(\cdot,0)=0$, its spatial second derivatives
and first time derivative extend continuously by zero. Thus the
extension is $C^2$ in the parabolic sense required by Theorem~5.1
of \cite{ChuRegularity} and solves the extended equation also at zero.
The time-Lipschitz background satisfies that theorem's H\"older
hypothesis, so the same estimate applies across the initial surface.
Spatial differentiation and the linear equation for $P$, with smooth
one-sided coefficients and initial data, give successive initial
Schauder estimates. On each fixed compact these estimates are
uniform in $L$ once the local metric bounds below hold and $\alpha=g$
there. Thus the approximating flows exist on $[0,D/2]$ whenever
$L$ is sufficiently large for the chosen $D$.
\subsection{The local limit and its potential estimates}
The bounds needed to pass to a limit use other choices of $\phi$.
For any canonical section $S$ from
Proposition~\ref{prop:initial-data}, take
\begin{equation}\label{flow:singular-weight}
\phi=\log|S|_g^2-nf-\epsilon u^2-A_\epsilon,
\qquad 0<\epsilon\le c_0/D,
\qquad A_\epsilon=C_S'(1+\epsilon^{-1}).
\end{equation}
The growth bound for $S$ makes $\phi\le0$ after choosing $C_S'$.
Since $|\dd u^2|_g\le C$, the canonical-current inequality gives
\[
\alpha-D\Ric(\alpha)+D\dd\phi
\ge\alpha-CD\epsilon g\ge\tfrac12\alpha
\]
when $c_0$ is small. The comparison is made away from the zeros of
$S$, where $\phi$ is smooth; the downward poles localize it away
from those zeros. Finite sums of squared norms may be used as well.
Fix a compact set on which the chosen section does not vanish.
For all sufficiently large $L$ it lies in $\{u<L/2\}$, where $f=0$.
Equation~\eqref{flow:upper-potentials} bounds $Q$ above there by a
constant independent of $L$. Equation~\eqref{flow:Q-comparison}
bounds $P$ below there, and \eqref{flow:eigenvalue-bound} gives a local
upper metric bound with $e^{J(Q)}\to1$. The determinant lower bound
then gives a positive local lower metric bound. Finitely many such
nonvanishing sections cover an arbitrary compact set. Local
parabolic estimates on slightly larger compact sets give all
derivatives, including the one-sided initial estimates, uniformly
in $L$. A diagonal subsequence over compacts and increasing horizons
converges smoothly to $U,P,h$ with $h_0=g$. The approximations have
$\alpha=g$ on each fixed compact for large $L$, so
\eqref{flow:approximation} passes to the K\"ahler equation there.
The limit of \eqref{flow:eigenvalue-bound} is $h_t\le g$;
the limits of \eqref{flow:upper-potentials} give $P\le0$ and $P_t\le0$.
This proves \eqref{flow:identities}.
The $Q$ comparison also passes to the limit. For a weight
$\phi=\log|S|_g^2-\epsilon u^2-A_\epsilon$ and $t\le D/2$, it gives,
using $U\le0$,
\begin{equation}\label{flow:rho-pointwise}
\rho(x,t)\le
\frac{-D\phi(x)+nt}{D-t}
\le2\bigl(-\log|S(x)|_g^2+\epsilon u(x)^2+A_\epsilon+n\bigr).
\end{equation}
Take $D=2(T+1)$ and, on a ball of large radius $L$, take
$\epsilon$ comparable to $L^{-1}$ and at most $c_0/D$.
Proposition~\ref{prop:initial-data} then bounds the average of the
right side by $C_TL$. Bounded radii are absorbed into the constant.
This proves \eqref{flow:rho-average}. At a fixed $x$, choose $S(x)\ne0$,
put $D=2(t+1)$, and take $\epsilon=c_0/D$. The same inequality gives
$\rho(x,t)\le C_x(1+t)$.
Finally replace $\log|S|_g^2$ by $\Xi$. For every sufficiently small
$\epsilon>0$, the smooth function
$\phi_\epsilon=\Xi-\epsilon u^2-A_\epsilon$ is nonpositive after
a finite constant shift and gives a smooth nonnegative $Q_\epsilon$
with the differential inequality in \eqref{flow:Q-comparison}, now
with $\alpha=g$. The difference
\[
Q_\epsilon-Q_{\epsilon/2}
=\frac{D\epsilon}{2}u^2+D(A_\epsilon-A_{\epsilon/2})
\]
shows that $Q_\epsilon\ge c u^2-C$ because
$Q_{\epsilon/2}\ge0$. Adding a constant gives a positive $Q$
satisfying \eqref{flow:proper-supercaloric}. Theorem~\ref{thm:localized-flow}
is proved.
\subsection{A cutoff adapted to the evolving metric}
The comparison arguments below require a positive integrable weight
$\chi$ with $\dd\chi\le C\chi h_t$. A polynomial weight $u^{-k}$
gives only a Hessian bound by $Cu^{-k-2}g$, and $h_t$ need not be
uniformly comparable to $g$ at infinity. We therefore construct a
function $H$ whose complex Hessian approaches $g-h_t$ at infinity.
The factor $\exp(-AH/u^2)$, for a suitable $A>0$, will contribute
the term $-Au^{-2}(g-h_t)$ to the logarithmic Hessian, absorbing
the errors measured in the initial metric.
Fix $T<\infty$ and work, if necessary, on a slightly larger finite
time interval. Put
\[
\eta_t=g-h_t,
\qquad a_t=\tr_g\eta_t.
\]
The forms $\eta_t$ are nonnegative, closed, and bounded by $g$;
their traces satisfy $0\le a_t\le n$.
The potential identity gives
\[
a_t=tR_g-\Delta_gU.
\]
Integrate against a cutoff from Section~\ref{sec:preliminaries} equal
to one on $B(o,L)$, and use
$-U=\int_0^t\rho(\cdot,s)\,ds$,
\eqref{prelim:averages}, and \eqref{flow:rho-average}. Moving the
Laplacian onto the cutoff gives
\begin{equation}\label{flow:eta-average}
\frac{1}{\Vol B(o,L)}\int_{B(o,L)}a_t\,dV_g
\le C_Tt/L\qquad(L\ge1,\ 0\le t\le T).
\end{equation}
For large dyadic $L$, define
\[
H_{L,t}=-\int_0^{L^2}H_ba_t\,db.
\]
There are $\delta,\beta>0$ such that, on $B(o,C_0L)$,
\begin{align}
|H_{L,t}|+L|\partial H_{L,t}|_g
&\le C_TL^{2-\delta}t^\beta,
\label{flow:heat-correction-size}\\
\dd H_{L,t}&=\eta_t+O(t/L)g,
\label{flow:heat-correction-hessian}\\
\partial_tH_{L,t}&\ge H_{L,t}/t-C_TL
\qquad(t>0).
\label{flow:heat-correction-time}
\end{align}
We give the heat estimates and the time identity separately.
Gaussian upper bounds and relative volume comparison, combined with
\eqref{flow:eta-average}, give, for some fixed $p>2$,
\begin{equation}\label{flow:heat-integrand}
H_ba_t(x)\le
C_T\min\left\{1,\frac{t}{L}\left(\frac{L}{\sqrt b}\right)^p\right\},
\qquad x\in B(o,C_0L),\quad0<b\le L^2.
\end{equation}
For clarity, when $\sqrt b\ll L$ the ball $B(x,\sqrt b)$ may be
small relative to $B(o,L)$. Relative volume comparison loses at
most a fixed power of $L/\sqrt b$; outside $B(o,CL)$, Gaussian
annuli absorb both that power and the polynomial volume growth.
The estimate $a_t\le n$ supplies the first term in the minimum.
For the gradient, use the heat-kernel estimate
\[
|\nabla_xH_b(x,y)|
\le \frac{C}{\sqrt b}\,
\frac{\exp(-c\,d_g(x,y)^2/b)}
{\sqrt{\Vol B(x,\sqrt b)\Vol B(y,\sqrt b)}}.
\]
It follows from \cite[Theorem~1.3, equation~(1.6)]{SoupletZhang}
and the Gaussian upper bound in \cite[Corollary~3.1]{LiYau},
absorbing the factor $1+d_g(x,y)^2/b$ into the Gaussian.
Integrating its absolute value against $a_t\ge0$ and using the same
relative-volume and annular estimates as above gives
$C_Tb^{-1/2}(t/L)(L/\sqrt b)^p$; increase the fixed $p>2$ if needed.
The separate bound $a_t\le n$ gives $Cb^{-1/2}$.
Thus the gradient obeys the minimum in
\eqref{flow:heat-integrand}, multiplied by $C/\sqrt b$.
Splitting both integrals at
$b_*=L^2(t/L)^{2/p}$ shows that their bounds are respectively
\[
C_TL^{2-2/p}t^{2/p},\qquad
C_TL^{1-1/p}t^{1/p}.
\]
After increasing the finite-time constant, we can take
$\delta=\beta=1/p$ in \eqref{flow:heat-correction-size}.
Lemma~\ref{lem:hodge-heat} gives
\[
\dd H_{L,t}=\eta_t-\mathcal H_{L^2}\eta_t.
\]
The last form is nonnegative, and its trace is bounded by $C_Tt/L$
by \eqref{flow:heat-integrand}. This proves
\eqref{flow:heat-correction-hessian}.
The trace identity also gives
\begin{equation}\label{flow:H-potential-identity}
H_{L,t}=-t\int_0^{L^2}H_bR_g\,db+H_{L^2}U-U.
\end{equation}
To justify it, insert initial-metric cutoffs in
$\int_0^{L^2}H_b\Delta_gU\,db=H_{L^2}U-U$.
Both $U$ and $\Delta_gU=tR_g-a_t$ have polynomially bounded absolute
ball integrals, by \eqref{flow:rho-average},
\eqref{prelim:averages}, and $0\le a_t\le n$.
Gaussian bounds for the scalar heat kernel and its gradient, together
with the absolute Laplacian bounds for the cutoffs, make the shell
errors tend to zero. This proves the identity distributionally and
then smoothly locally. Differentiating it in $t$ is justified by
the same ball-integral bounds on $P=-\rho$; alternatively integrate
the identity over a time interval and then differentiate locally.
Subtracting $H_{L,t}/t$ gives
\[
\partial_tH_{L,t}-H_{L,t}/t
=H_{L^2}(P-U/t)-(P-U/t).
\]
Since $P_t\le0$, $P-U/t\le0$ and $0\le U/t-P\le\rho$.
Its heat average is at most $C_TL$, by
\eqref{flow:rho-average} and Gaussian annular summation. The
unaveraged term has the favorable sign. This proves
\eqref{flow:heat-correction-time}.
Choose a nonnegative smooth dyadic partition of unity $\zeta_L(u)$,
with scales $L\asymp u$, and put
$H=\sum_L\zeta_L(u)H_{L,t}$, using finitely many harmless initial
scales on a fixed compact. The partition derivatives have sizes
$O(u^{-1})$ and $O(u^{-2})$ in gradient and complex Hessian.
Equations~\eqref{flow:heat-correction-size}--\eqref{flow:heat-correction-time}
give
\begin{equation}\label{flow:glued-correction}
\begin{gathered}
|H|\le C_Tu^{2-\delta}t^\beta,
\qquad |\partial H|_g\le C_Tu^{1-\delta}t^\beta,\\
\dd H=\eta_t+O(u^{-\delta}t^\beta+t/u)g,
\qquad H_t\ge H/t-C_Tu.
\end{gathered}
\end{equation}
These functions are smooth in space and continuously differentiable
in time, one-sided at $t=0$, which is all that the integration below
uses. In fact \eqref{flow:H-potential-identity} gives their first
time derivatives in terms of $H_{L^2}P$ and local smooth functions.
The uniform absolute ball-integral bounds for $P$ give continuity of
that heat average, including at $t=0$, by local convergence and
uniform Gaussian tail estimates. Spatial regularity follows from
the heat equation and \eqref{flow:heat-correction-hessian}.
\begin{lemma}[Integrable comparison cutoff]\label{flow:cutoff-lemma}
For every sufficiently large fixed integer $k$, there is a positive
function $\chi(x,t)$ on $M\times[0,T]$, smooth in space and continuously
differentiable in time one-sided at the initial surface, such that
\begin{equation}\label{flow:cutoff-bounds}
\begin{gathered}
C_T^{-1}u^{-k}\le\chi\le C_Tu^{-k},\qquad
|\partial\chi|_g\le C_T\chi/u,\\
\dd\chi\le C_T\chi h_t,
\qquad \partial_t\chi\le C_T(1+t^{\beta-1})\chi
\quad(t>0).
\end{gathered}
\end{equation}
\end{lemma}
\begin{proof}
Take
\[
\chi=u^{-k}\exp(-AH/u^2).
\]
For fixed $A$, \eqref{flow:glued-correction} gives the two-sided
size estimate and the gradient estimate. Expanding
$\chi^{-1}\dd\chi=\dd\log\chi+
\partial\log\chi\otimes\bar\partial\log\chi$ gives at infinity
\[
\chi^{-1}\dd\chi
\le u^{-2}\{-A(g-h_t)+C_kg+o(1)g\},
\]
uniformly on the fixed time interval. The terms containing
$H/u^2$ or its gradient are $o(u^{-2})g$ after $A$ is fixed.
Choose $A>C_k+1$ and then take $u$ sufficiently large. The negative
multiple of $g$ absorbs the remaining error and leaves
$\chi^{-1}\dd\chi\le Au^{-2}h_t$. On the remaining compact,
smoothness and positivity of $h_t$ give the asserted bound. Finally,
$\chi_t/\chi=-AH_t/u^2$ and \eqref{flow:glued-correction} give
\[
\chi_t/\chi\le A|H|/(tu^2)+C_T/u
\le C_T(1+t^{\beta-1}).
\]
The exponent $\beta>0$ makes this time coefficient integrable at zero.
\end{proof}
\subsection{Minimality and nonlinear scalar comparison}
\begin{proposition}[Scalar comparisons]\label{prop:scalar-comparison}
Fix a finite $T>0$. The following conclusions hold for the flow of
Theorem~\ref{thm:localized-flow}.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The function $\rho$ is the minimal nonnegative solution with
zero initial values of
\begin{equation}\label{flow:rho-source}
(\partial_t-\Delta_{h_t})\rho
=\tr_{h_t}\Ric(g).
\end{equation}
There are nonnegative smooth supercaloric functions $\rho_j$ on
$M\times[0,T]$ such that
\begin{equation}\label{flow:rho-tails}
0\le\rho_j\le\rho,\qquad
\rho_j\ge\rho-C_j,\qquad
\rho_j\longrightarrow0
\text{ locally uniformly on }M\times[0,T].
\end{equation}
Each fixed $\rho_j$ has all spatial and one-sided time
derivatives smooth at $t=0$ and bounded on compact spacetime
sets.
\item Suppose $v$ is upper semicontinuous, bounded on compact
spacetime sets, $0\le v\le\rho$, and satisfies, in the upper
viscosity sense on $0<t<T$,
\begin{equation}\label{flow:nonlinear-comparison}
\partial_tv\le e^{-v}\Delta_{h_t}v+(1-e^{-v})R_{h_t}.
\end{equation}
Then $v=0$ on $M\times[0,T)$.
\end{enumerate}
Neither assertion requires completeness of $h_t$.
\end{proposition}
Before proving the proposition, we isolate the smooth majorant used
in part \textup{(ii)}. Its construction uses compact-domain parabolic
theory and will be given immediately after the comparison proof.
\begin{lemma}[Smooth majorant]\label{lem:scalar-majorant}
Fix $T>0$ and let $v$ be upper semicontinuous, bounded on compact
spacetime sets, and satisfy $0\le v\le\rho$ on $M\times[0,T]$.
Suppose that $v$ satisfies \eqref{flow:nonlinear-comparison} in the
upper viscosity sense for $0<t<T$. There is a function $w$, smooth
on $M\times(0,T)$ and continuous on $M\times[0,T)$, such that
\[
v\le w\le\rho,\qquad
w_t=e^{-w}\Delta_{h_t}w+(1-e^{-w})R_{h_t},\qquad
w(\cdot,0)=0.
\]
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:scalar-comparison}]
The Ricci identity
$\Ric(h_t)=\Ric(g)+\dd\rho$, traced with $h_t$, proves
\eqref{flow:rho-source}, since $\rho_t=R_{h_t}$.
Its source is nonnegative.
Solve the source equation with zero initial and lateral boundary
data on a smooth relatively compact exhaustion, first as a weak
energy solution \cite[Chapter~III, Theorem~4.1]{LSU}, and then use
local parabolic regularity. A source need not vanish at the
initial lateral corner, so full corner compatibility is not
asserted. This bounded-domain construction and its energy
estimates pass between a finite set of coordinate charts; it
requires no completeness of the evolving metric.
Domain comparison makes these solutions increase, and
comparison with $\rho$ bounds them above. Their limit $\rho_{\min}$
is the minimal nonnegative potential. Local estimates give a smooth
solution for $t>0$ and continuous zero initial values, since it is
bounded by the smooth function $\rho=O_K(t)$ on each compact.
The difference $z=\rho-\rho_{\min}$ satisfies
\[
0\le z\le\rho,\qquad z_t=\Delta_{h_t}z,
\qquad z(\cdot,0)=0.
\]
Use $\chi$ from Lemma~\ref{flow:cutoff-lemma}, with $k>2n+3$,
and a cutoff $p_N=p(u/N)$ which is one on $u\le N$ and zero on
$u\ge2N$. Set
\[
I_N(t)=\int_M z\chi p_N\,dV_{h_t}.
\]
Since $\partial_tdV_{h_t}=-R_{h_t}dV_{h_t}$ and $R_{h_t}\ge0$,
integration by parts on the compact support of $p_N$ gives
\begin{equation}\label{flow:linear-mass}
I_N'(t)\le C_T(1+t^{\beta-1})I_N(t)+E_N(t),
\qquad |E_N(t)|\le C_TN^{2n-1-k}.
\end{equation}
Here the error contains the derivatives falling on $p_N$. Its
complex Hessian, including the gradient cross terms with $\chi$,
has $g$-norm at most $C_TN^{-k-2}$ on $N\le u\le2N$.
The pointwise cofactor identity and $0<h_t\le g$ imply, for any
real $(1,1)$-form $B$,
\begin{equation}\label{flow:cofactor}
|\tr_{h_t}B|\,dV_{h_t}
\le C_n|B|_g\,dV_g.
\end{equation}
Indeed diagonalize $h_t$ relative to $g$; the coefficient multiplying
each diagonal entry of $B$ is the product of the other $n-1$
eigenvalues, each at most one. This also covers $n=1$.
Equation~\eqref{flow:rho-average} bounds the integral of $z$ on the
shell by $C_TN^{2n+1}$, proving the error bound.
The mass $I=\int z\chi\,dV_h$ is finite. Its tails tend to zero
uniformly in $t$ by $z\le\rho$, $dV_h\le dV_g$, and the same
dyadic shell estimate. Its initial limit is zero by local
convergence and this tail control. Integrating
\eqref{flow:linear-mass} from a positive initial time, sending
$N\to\infty$, and then sending that time to zero gives
$I(t)\le\int_0^tC_T(1+s^{\beta-1})I(s)\,ds$.
Gronwall's inequality gives $I=0$, and positivity of $\chi$ gives
$z=0$. Thus $\rho=\rho_{\min}$.
For the tails, choose smooth spatial cutoffs $0\le\theta_j\le1$
increasing to one, each compactly supported and eventually equal
to one near each fixed compact. Let $\rho_j$ be the minimal
zero-data potential of
\[
F_j=(1-\theta_j)\tr_{h_t}\Ric(g).
\]
It is nonnegative and supercaloric. Linearity on each exhaustion
domain and monotone limits give the decomposition of $\rho$ into
this exterior-source potential and the minimal potential of
$\theta_j\tr_h\Ric(g)$. The latter is at most
$T\sup_{M\times[0,T]}\theta_j\tr_h\Ric(g)<\infty$ by the scalar
maximum principle on the domains. This proves
$\rho_j\ge\rho-C_j$. Monotone convergence of the nonnegative
Dirichlet Green integrals, first in the domains and then for the
source cutoffs, shows that $\rho_j\downarrow0$ pointwise.
We record the initial regularity needed later. On every fixed
compact neighborhood the coefficients of $\Delta_{h_t}$ and the
source $F_j$ are smooth and uniformly parabolic through $t=0$.
The domination $0\le\rho_j\le\rho=O_K(t)$ supplies continuous
zero initial values.
For clarity, this initial regularity is local in space. Extend
the smooth coefficients and source from a coordinate neighborhood
to a smooth uniformly parabolic Cauchy problem, and solve it with
zero initial data \cite[Chapter~IV, Theorem~5.1]{LSU}.
Subtract this local particular solution from $\rho_j$ on a smaller
neighborhood. The difference is $O(t)$ there and solves the
homogeneous equation for $t>0$. A spatially localized energy
estimate from time $\delta>0$, followed by $\delta\downarrow0$,
puts the difference in the local weak energy class through zero;
the squared $L^2$ norm of its initial value on the fixed compact
is $O(\delta^2)$ and tends to zero. Extend that difference by zero to
negative times, and extend the coefficients smoothly. Its zero
initial trace removes a distributional boundary term, so it
solves the homogeneous equation across the initial surface.
Interior regularity \cite[Chapter~III, Theorem~12.1]{LSU} makes it
smooth there. Adding back the particular solution proves the
claimed smooth one-sided jets. This argument extends the
homogeneous difference, not the forced potential, by zero.
In particular,
\[
\partial_t\rho_j(x,0)=F_j(x,0),
\]
and higher initial derivatives are determined recursively from
$(\rho_j)_t=\Delta_{h_t}\rho_j+F_j$; they are smooth functions of
$x$. No regularity at a distant Dirichlet corner is used. Dini's
theorem on compact spacetime sets now gives the local uniform
convergence in \eqref{flow:rho-tails}. If an auxiliary smooth
extension across $t=0$ is desired on a compact, it can be chosen
to match these jets. Extension by zero to negative time is not
asserted.
For \textup{(ii)}, let $w$ be the smooth majorant from
Lemma~\ref{lem:scalar-majorant}. Thus $v\le w\le\rho$, and $w$
solves equality in \eqref{flow:nonlinear-comparison}.
Multiplying its equation by $e^w$ gives the useful
volume-form identity
\begin{equation}\label{flow:nonlinear-volume}
\partial_t\bigl[(e^w-1)dV_{h_t}\bigr]
=(\Delta_{h_t}w)dV_{h_t}.
\end{equation}
For example, the cancellation uses
$\partial_tdV_h=-R_hdV_h$ and
$e^w(1-e^{-w})R_h=(e^w-1)R_h$.
Integrate \eqref{flow:nonlinear-volume} against $\chi p_N$ and
move the Laplacian onto that compactly supported function. Since
$w\le e^w-1$ and $\Delta_h\chi\le C_T\chi$, the interior term is
bounded by a constant times
\[
M_N(t)=\int_M(e^w-1)\chi p_N\,dV_h.
\]
The time derivative of $\chi$ is bounded by
$C_T(1+t^{\beta-1})\chi$. The exterior error is again
$O(N^{2n-1-k})$, using $w\le\rho$ and
\eqref{flow:cofactor}. Moreover
\[
0\le(e^w-1)dV_h\le e^\rho dV_h=dV_g.
\]
Thus the full weighted mass is finite, its tails are uniformly
small, and its initial limit is zero by dominated convergence on
compacts and $\chi\asymp u^{-k}$. The same limiting Gronwall
argument as above forces it to vanish. Hence $w=0$, and $0\le v\le w$
proves \textup{(ii)}.
\end{proof}
\subsection{Constructing the smooth majorant}
\begin{proof}[Proof of Lemma~\ref{lem:scalar-majorant}]
We construct the majorant on relatively compact domains and then
pass to a limit.
On a smooth relatively compact domain
$\Omega$ and for a fixed time less than $T$, consider equality in
\eqref{flow:nonlinear-comparison}. Zero and $\rho$ are respectively
a subsolution and a supersolution. To check the latter, use
\[
R_{h_t}-\Delta_{h_t}\rho=\tr_{h_t}\Ric(g)\ge0.
\]
The function $\rho+2\alpha$ is also a supersolution for every
constant $\alpha>0$.
Take constant initial data $\alpha$. On $\partial\Omega$ choose
a smooth $\zeta:[0,\infty)\to[0,1]$, equal to one near zero and
supported in $[0,1]$, and put
\[
\varphi(x,t)=\rho(x,t)+\alpha
-e^{-\alpha}R_g(x)t\zeta(t/\delta).
\]
For $\delta\sup_{\partial\Omega}R_g\le\alpha/2$ this lies between
$\rho+\alpha/2$ and $\rho+\alpha$.
It has $\varphi(\cdot,0)=\alpha$ and
$\varphi_t(\cdot,0)=(1-e^{-\alpha})R_g$, which are exactly the
value and first equation compatibility conditions.
We give the local existence and continuation argument on this
fixed compact domain. We use the compatible linear Dirichlet
Schauder theorem \cite[Chapter~IV, Theorem~5.2]{LSU} and its
uniform small-time inverse estimate
\cite[Chapter~IV, Theorem~5.4]{LSU}.
These Euclidean-domain statements have the same compact-manifold
form by the following finite-chart construction for a smooth
uniformly parabolic linear operator $L$. Choose a partition of
unity $\chi_i$ and cutoffs $\psi_i=1$ near $\supp\chi_i$ in
finitely many interior and boundary charts. Local domains are
smoothly capped outside these supports, and $\psi_i$ vanishes near
every artificial boundary. If $E_i$ is the local zero-data inverse,
then $Sf=\sum_i\psi_i E_i(\chi_i f)$ satisfies
$LSf=f+\mathcal Rf$ for the global linear operator $L$.
The remainder consists of first- and zeroth-order commutators
$[L,\psi_i]E_i(\chi_i f)$. On the source space
$f(\cdot,0)=0$, the local solutions have zero initial trace.
Parabolic interpolation and the uniform inverse bounds give
$\|\mathcal R\|_{C^{\beta,\beta/2}\to C^{\beta,\beta/2}}
\le C\epsilon^\theta$ on a time interval of length $\epsilon$,
with $\theta>0$; for $0<\beta<1$ one can take
$\theta=(1-\beta)/2$. The remainder preserves this source space.
Thus $S(I+\mathcal R)^{-1}$ is the required global inverse for
small $\epsilon$. All constants here may depend on $\Omega$.
Extend the boundary data to a smooth function $p$ with
\[
p(x,0)=\alpha,\qquad
p_t(x,0)=c(x):=(1-e^{-\alpha})R_g(x)
\quad\hbox{throughout }\Omega.
\]
For example, in a smooth boundary collar with projection $\pi$
and cutoff $\vartheta=1$ near the boundary, take
\[
p(x,t)=\alpha+t c(x)
+\vartheta(x)\{\varphi(\pi(x),t)-\alpha-tc(\pi(x))\},
\]
and use $\alpha+tc(x)$ outside the collar. The bracket and its
first time derivative vanish at zero.
Set $L=\partial_t-e^{-p}\Delta_h$ and seek
$v_{\Omega,\alpha}=p+z$, with zero
initial and lateral data for $z$. Its equation is
\[
Lz=\mathcal N(z):=
(e^{-p-z}-e^{-p})\Delta_h(p+z)
+(1-e^{-p-z})R_h-Lp.
\]
For every such $z\in C^{2+\beta,1+\beta/2}$,
$\mathcal N(z)(x,0)=0$ throughout $\Omega$.
The linear inverse is therefore applied to compatible sources.
If $z(\cdot,0)=0$, then
$\|z\|_\infty\le\epsilon\|z_t\|_\infty$; spatial interpolation
and the time H\"older quotient give
\[
\|z\|_{C^{\beta,\beta/2}}
\le C\epsilon^{1-\beta/2}
\|z\|_{C^{2+\beta,1+\beta/2}}.
\]
On a fixed ball in the latter space, the product and composition
estimates consequently give
\[
\|\mathcal N(z)-\mathcal N(w)\|_{C^{\beta,\beta/2}}
\le C\epsilon^{1-\beta/2}
\|z-w\|_{C^{2+\beta,1+\beta/2}}.
\]
The coefficient of every highest-derivative difference has this
small factor. Also $\mathcal N(0)$ is smooth and vanishes at
$t=0$, so its $C^{\beta,\beta/2}$ norm tends to zero with $\epsilon$.
The linear inverse and the contraction theorem give a local
Dirichlet solution. Classical comparison with the barriers gives
$0\le v_{\Omega,\alpha}\le\rho+2\alpha$.
For continuation, write
$\Delta_h=a^{ij}\partial_i\partial_j+b^j\partial_j$ in fixed
real coordinates and set $q=e^{v_{\Omega,\alpha}}$. Directly,
\[
q_t-\partial_i\left(\frac{a^{ij}}q\,\partial_jq\right)
=\frac{b^j-\partial_i a^{ij}}q\,\partial_jq+(q-1)R_h.
\]
The barriers bound $q$ above and away from zero. The leading
matrix is uniformly elliptic, and the displayed drift and forcing
are bounded independently of $\nabla q$ on the fixed cylinder.
Boundary as well as interior H\"older estimates therefore apply
\cite[Chapter~III, Theorem~10.1]{LSU}; the exterior-measure
condition there holds for the smooth compact boundary.
Hence $v_{\Omega,\alpha}$ is H\"older up to the parabolic boundary.
Its original equation now has H\"older coefficients and source,
so \cite[Chapter~IV, Theorem~5.2]{LSU} gives a controlled
$C^{2+\gamma,1+\gamma/2}$ norm on the closed slab for some
$0<\gamma<1$. In particular its second spatial derivatives are
bounded there. Only this finite corner regularity is asserted
from the first compatibility condition.
On slabs separated from the initial surface, differentiated
boundary Schauder estimates give all higher bounds, since the
coefficients and lateral data are smooth. At a hypothetical
positive maximal time the solution thus has smooth limiting
initial data satisfying the boundary equation jets. The same
local construction, with those data and their first equation jet
in place of $\alpha$ and $c$, restarts it. This proves existence
up to the prescribed finite horizon. No constants near the
remote boundaries need be uniform as $\Omega$ exhausts $M$.
Viscosity comparison with this smooth solution gives
$v\le v_{\Omega,\alpha}$. To see the only nonlinear point in that
comparison, at a positive interior maximum of
$e^{-Ct}(v-v_{\Omega,\alpha})$ the test has the same spatial Hessian
as $v_{\Omega,\alpha}$. The difference of the right sides is bounded
by a constant times $v-v_{\Omega,\alpha}$, because the derivative
in the value variable is
$e^{-v}(R_h-\Delta_hv_{\Omega,\alpha})$, bounded on the compact slab.
Choose $C$ larger than this bound and use a strict increasing time
penalty. The initial and lateral gaps are positive, so no boundary
maximum interferes. This proves the comparison directly, consistently
with the usual viscosity framework \cite{CIL}.
Exhaust $M$, let $\alpha\downarrow0$, and use local uniformly
parabolic estimates to extract a smooth interior limit $w$ with
\[
v\le w\le\rho,\qquad
w_t=e^{-w}\Delta_hw+(1-e^{-w})R_h,
\qquad w(\cdot,0)=0.
\]
The limit at the initial surface follows from the barriers on each
compact; no uniform estimates near the remote lateral boundaries
are required.
\end{proof}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/flow.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/introduction.tex ====
\section{Introduction}\label{sec:introduction}
In his 1982 problem survey, Yau asked whether a complete noncompact
K\"ahler manifold with positive holomorphic bisectional curvature must
be biholomorphic to complex Euclidean space \cite[\S9(b), p.~45]{YauProblems}.
We prove the conjecture under its pointwise positivity hypothesis.
\begin{theorem}\label{thm:uniformization}
Let $M$ be a connected noncompact complex manifold of complex dimension
$n\geq1$. If $M$ admits a smooth complete K\"ahler metric with strictly
positive holomorphic bisectional curvature, then $M$ is biholomorphic
to $\C^n$.
\end{theorem}
Strict positivity in Theorem~\ref{thm:uniformization} is pointwise.
In particular, the theorem assumes neither a uniform positive lower
bound nor a finite upper bound for the curvature. No volume-growth,
noncollapsing, Ricci-pinching, or topological hypothesis is added.
Its conclusion identifies the entire complex manifold with $\C^n$;
it does not assert that the given metric is Euclidean.
In particular, $M$ is Stein and contractible.
It resolves Yau's uniformization conjecture for strictly positive
holomorphic bisectional curvature in every complex dimension.
\subsection{Antecedents and the remaining analytic difficulties}
The one-dimensional case belongs to the classical relation between
curvature and conformal type. A complete noncompact Riemann surface
of positive Gaussian curvature is conformally the plane, by
Cohn-Vossen's topology theorem and the parabolicity theorem of
Blanc--Fiala--Huber \cite[Satz~8]{CohnVossen}\cite[Theorem~15]{Huber}.
In the compact setting, the Frankel conjecture was resolved by
Mori and by Siu--Yau: positive holomorphic bisectional curvature
characterizes complex projective space \cite{MoriAmple,SiuYauFrankel}.
Yau's noncompact question asks for an analogous characterization
of complex Euclidean space while allowing unrestricted geometry
at infinity \cite[\S9(b), p.~45]{YauProblems}.
One route to this problem studies holomorphic functions and
algebraic embeddings. Greene--Wu showed that positive real sectional
curvature yields a smooth strictly convex exhaustion and hence
Steinness \cite[Theorem~10]{GreeneWuConvex1976}.
Mok obtained an affine-algebraic embedding under positive bisectional
curvature, maximal volume growth, and quadratic scalar-curvature
decay \cite{MokEmbedding1984}.
These results connect curvature to global function theory while
retaining hypotheses or conclusions different from unrestricted
uniformization. Chen--Zhu proved intrinsic growth and average-curvature
estimates under pointwise positive bisectional curvature
\cite{ChenZhu}. In particular, their linear average-scalar-curvature
bound requires no upper curvature bound, but its constant may
depend on the center. They also pursued algebraic compactification
under bounded positive sectional curvature and a finite top Ricci
integral \cite{ChenZhuPositive2009}.
Ni--Tam developed heat deformation of plurisubharmonic functions
under nonnegative bisectional curvature, without a global upper
curvature bound \cite{NiTamStructure}. Their smoothing theorem
keeps an explicit growth condition on the initial function.
A second route uses geometric flow to construct shrinking holomorphic
charts and then assembles those charts by complex dynamics.
Cao's differential Harnack estimates for K\"ahler--Ricci flow
\cite{CaoHarnack} are an analytic antecedent of this approach.
Chau--Tam established uniformization under bounded nonnegative
bisectional curvature and maximal volume growth
\cite{ChauTamComplex2006}. Their later work developed locally
biholomorphic charts, covering and pseudoconvex-domain conclusions,
and soliton-limit methods under additional curvature-decay or
pinching conditions \cite{ChauTamStein,ChauTamSimply}.
The distinction between a covering, a map onto a domain, and a
biholomorphism onto all of $\C^n$ is essential here.
Liu proved uniformization under nonnegative bisectional curvature and
maximal volume growth, without an upper curvature bound, using a
Gromov--Hausdorff approach and retracting holomorphic vector fields
\cite{LiuUniformization}. Lee--Tam subsequently obtained the same
conclusion under these hypotheses by smoothing the metric and applying
the bounded-curvature theorem \cite{LeeTam}. Both results retain
maximal volume growth.
Recent work has removed assumptions at infinity in important special
cases.\footnote{The cited versions also record AI assistance.
Datar, Pingali, and Seshadri credit ChatGPT pro~5.2 with the normalization
observation used in Lemma~5.13, as well as feedback and discussions
\cite[Section~1.1 and Acknowledgments]{DatarPingaliSeshadri}.
Their later paper credits ChatGPT~5.5 Pro with the auxiliary function
and induction for the B\'ezout estimates, and ChatGPT~5.0 in thinking
mode with the heat-comparison estimate recalled as Lemma~2.2
\cite[Sections~1--3]{DatarPingaliSeshadriTop}.
Wu reports ChatGPT-6 assistance in developing the proofs and the
author's verification of the arguments
\cite[Acknowledgments]{WuConditional}.}
Datar, Pingali, and Seshadri
\cite[Theorem~1]{DatarPingaliSeshadri} prove that every complete noncompact
K\"ahler surface with positive real sectional curvature is biholomorphic
to $\mathbb C^2$. Their method uses Lipschitz plurisubharmonic weights of
finite Monge--Amp\`ere mass and weighted holomorphic functions.
For positive bisectional curvature, their Theorem~2 gives the same
conclusion under strong Steinness, contractibility, and simple
connectivity at infinity. Here strong Steinness means the existence of
a smooth plurisubharmonic exhaustion with bounded gradient.
Wu \cite[Corollaries~1.5--1.6]{WuConditional} removes contractibility
from this criterion: strong Steinness and simple connectivity at infinity
suffice. Wu also obtains the conclusion under positive bisectional curvature
when the latter topological condition holds and the real sectional
curvatures are nonnegative outside a compact set. These are surface results; positive real sectional
curvature is a stronger hypothesis than positive holomorphic bisectional
curvature.
The finite-mass approach also leads to higher-dimensional results.
Datar, Pingali, and Seshadri \cite[Theorem~1.1]{DatarPingaliSeshadriTop}
prove finiteness of the integral of the top power of the Ricci form on
complete noncompact K\"ahler manifolds with positive real sectional
curvature. Their theorem also applies under positive bisectional
curvature when a smooth strictly plurisubharmonic exhaustion with
uniformly bounded gradient is available; with positive bounded sectional
curvature, they obtain quasiprojectivity
\cite[Theorem~1.2]{DatarPingaliSeshadriTop}.
Dat \cite{DatHessian} develops an $m$-Hessian capacity approach to
finite-Monge--Amp\`ere weights in complex dimensions at least three;
its proposed uniformization route retains proper-map and multiplicity
assumptions.
A complementary direction relates holomorphic growth to the asymptotic
geometry of the K\"ahler--Ricci flow. Shi
\cite[Theorem~1.6]{ShiMinimalDegrees} proves an identity between the
minimal growth degree of holomorphic volume forms and the refined
minimal degrees of holomorphic functions for complete noncompact K\"ahler
manifolds with nonnegative bisectional curvature and nonsplitting
universal cover. Under maximal volume growth, the flow results
and coordinates in \cite[Theorem~1.7 and Corollary~1.12]{ShiMinimalDegrees}
connect these holomorphic degrees with Lyapunov exponents. This gives a
quantitative connection between the function-theoretic and flow
approaches to uniformization in that growth regime.
Two issues must be addressed in the unrestricted problem.
First, the initial metric can have unbounded curvature and arbitrarily
collapsed regions. A complete bounded-curvature flow theorem or a
global tensor maximum principle cannot therefore be used without
first establishing its hypotheses. Second, shrinking charts alone
can identify the manifold with a proper domain in $\C^n$.
To obtain surjectivity, one needs quantitative control of their
coordinate changes even when contraction rates vary substantially.
\subsection{Structure of the proof}
Write $g$ for the initial metric and fix $o\in M$.
We construct a global K\"ahler--Ricci flow $h_t$ by complete Hermitian
approximations on exhausting domains. Estimates for the metric and
its logarithmic volume loss
$\rho(x,t)=\log(\det g(x)/\det h_t(x))$ pass to the limit
without global curvature control. A spatial weight adapted to the nonnegative form $g-h_t$
then proves scalar comparison before completeness of $h_t$ is known.
This order is important: completeness is a later consequence of
the Harnack inequality.
The curvature argument takes place on the holomorphic cotangent
bundle. We minimize the boundary squared dual norm over holomorphic
discs, with compact boundary walls and a vanishing area penalty.
The canonical symplectic bivector relates the complex Hessian of
the norm to its time derivative. A quantitative deformation lemma
gives a viscosity inequality for the disc infimum; optimizing a
Hermitian ellipsoid in each fiber converts that inequality into
the scalar comparison already proved. The infimum therefore equals
the evolving squared dual norm. This gives joint positivity in space and
logarithmic time, hence the matrix and scalar Harnack inequalities.
The proof uses three time parameters, each for a different estimate:
\begin{center}
\begin{tabular}{lll}
\toprule
Clock & Definition & Role \\
\midrule
Physical time & $t$ & K\"ahler--Ricci flow \\
Logarithmic time & $\tau=\log t$ & Joint positivity and Harnack estimates \\
Volume clock & $s=\rho(o,t)$ & Chart contraction and polynomial degrees \\
\bottomrule
\end{tabular}
\end{center}
The volume clock becomes strictly increasing by the Ricci positivity
proved in Section~\ref{sec:charts}. Sections~\ref{sec:discs}--\ref{sec:variation}
use $s$ temporarily for the physical time at a disc center; from
Section~\ref{sec:charts} onward it always denotes the volume clock.
The volume clock measures contraction directly. For the transition
$F_{s,u}$ from a centered unitary chart at clock $s$ to one at $u\ge s$,
\[
|\det F_{s,u}'(0)|=e^{-(u-s)/2}.
\]
Writing $R(x,t)$ for the scalar curvature of $h_t$, the speed of the
volume clock in logarithmic time is $q(s)=ds/d\tau=tR(o,t)$.
A static logarithmic-norm estimate bounds the greatest singular length
of an $h_t$-unitary local chart, measured in the initial metric $g$,
by a fixed power of its least singular length. Together with the
determinant loss, this controls contraction in every direction.
Selected time intervals with Ricci pinching then give a contraction
argument for loops. Simple connectivity allows the local inverse
charts to continue coherently over exhausting compact sets, and a
plurisubharmonic Liouville argument gives one contraction exponent
valid on every fixed compact set.
Finally, we normalize the volume jets of the transition maps and
construct polynomial automorphisms with constant Jacobian. The
nonlinear shears used to realize normalized jets have determinant
one; the full models retain their contracting linear determinant.
Intervals of nonuniform contraction are allowed: their degree
cost is charged to growth of $q$ in the volume clock.
Inverse iteration of these models makes the corrected inverse charts
converge to an injective holomorphic map $M\to\C^n$.
A subsequential bound for the degree of the forward compositions,
together with smallness on compact subsets of the image and an
inverse-ball inclusion, rules out a finite maximal radius for centered
balls in the image.
This proves that the limiting coordinate map is onto~all~of~$\C^n$.
These mechanisms are proved in the order in which they are used.
Sections~\ref{sec:preliminaries}--\ref{sec:flow} establish the initial
analytic data and scalar comparison. Sections~\ref{sec:discs}--\ref{sec:ellipsoids}
prove joint positivity by discs and ellipsoids.
Section~\ref{sec:charts} obtains completeness, simple connectivity,
and quantitative shrinking charts.
Sections~\ref{sec:dynamics}--\ref{sec:uniformization} construct the
biholomorphism. In particular, the disc-deformation estimate and
the forward-degree estimate are stated with the uniformities
needed at their later applications.
\subsection{Conventions}
We use the real curvature convention in which
$\Rm(a,b,b,a)$ is the sectional-curvature numerator of the plane
spanned by $a,b$. The hypothesis is
\[
\Rm(u,v,v,u)+\Rm(u,Jv,Jv,u)>0
\]
for every pair of real unit tangent vectors $u,v$ at every point.
By the K\"ahler symmetries, this is equivalent to Griffiths strict
positivity of $T^{1,0}M$ in complex notation. In complex dimension
one the displayed sum equals Gaussian curvature.
We identify a Hermitian tensor $(a_{i\bar j})$ with its real
$(1,1)$-form, suppressing the factor $\sqrt{-1}$ when no confusion
can result. Thus $\dd f$ denotes the complex Hessian of $f$,
$\Delta_h f=\tr_h\dd f$, and
\[
\partial_t h_t=-\Ric(h_t),\qquad
\Ric(h)=-\dd\log\det h.
\]
All Laplacian, curvature, and volume conventions in the proof use
this normalization. The dual of a Hermitian metric includes the
transpose dictated by the holomorphic cotangent pairing; matrix
contractions below retain it explicitly when necessary.
The initial complete metric is $g$, a fixed base point is $o$,
and $r(x)=d_g(o,x)$. Constants may change between estimates.
Subscripts indicate dependencies when those dependencies matter.
For Hermitian forms, $A=O(b)h$ means $-Cbh\leq A\leq Cbh$.
The abbreviation psh means plurisubharmonic. A positive constant
extracted from curvature on a specified compact set is never
understood to be uniform over all of $M$.
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/introduction.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/preliminaries.tex ====
\section{Functions, sections, and smoothing on the initial manifold}
\label{sec:preliminaries}
Throughout this Section, distances, balls, norms, and volume are those of
the complete initial metric $g$. Fix $o\in M$ and put $r(x)=\dist_g(o,x)$.
Constants may depend on $g$ and $o$. In particular, the estimates below
are not estimates uniform over all choices of center.
We prepare the functions and estimates used in the flow and disc
arguments. Holomorphic interpolation supplies global functions and
tangent fields for controlling analytic discs, and canonical sections
with growth bounds for the flow's potential barriers. Heat smoothing
produces a smooth exhaustion with controlled complex Hessian; combined
with the canonical sections, its cutoffs yield averaged curvature
estimates.
\begin{proposition}[Initial analytic data]\label{prop:initial-data}
The following objects and estimates are available.
\begin{enumerate}[label=\textup{(\roman*)}]
\item There is a smooth strictly plurisubharmonic function $\psi$ with
$\psi\le C(1+r)$.
\item Holomorphic functions and holomorphic tangent fields interpolate
arbitrary finite jets at a finite set of points. There are finitely
many holomorphic functions defining an injective immersion
$F:M\longrightarrow\C^N$, and finitely many global holomorphic
tangent fields spanning $T^{1,0}M$ at every point.
\item At every point $x$ there is a global holomorphic canonical section
$S$ with $S(x)\ne0$ and
$\log|S|_g^2\le C_S(1+r)$. There is also a smooth function $\Xi$
such that
\begin{equation}\label{prelim:xi}
\dd\Xi\ge\Ric(g),\qquad \Xi\le C(1+r^{3/2}).
\end{equation}
\item There is a smooth exhaustion $u\ge1$ such that
\begin{equation}\label{prelim:exhaustion}
C^{-1}(1+r)\le u\le C(1+r),\qquad
|\partial u|_g\le C,\qquad |\dd u|_g\le C/u.
\end{equation}
\item If $S(o)\ne0$ is chosen as in \textup{(iii)}, then, for $L\ge1$,
\begin{equation}\label{prelim:averages}
\frac{1}{\Vol B(o,L)}\int_{B(o,L)}
\bigl|\log|S|_g^2\bigr|\,dV_g\le CL,
\qquad
\frac{1}{\Vol B(o,L)}\int_{B(o,L)}R_g\,dV_g\le C/L.
\end{equation}
\end{enumerate}
Neither $F$ nor $\psi$ is asserted to be proper. The exhaustion in
\textup{(iv)} has a two-sided complex Hessian bound.
\end{proposition}
\subsection{The weight and holomorphic interpolation}
Let $\gamma$ be a unit-speed ray starting at $o$ and let
\[
b(x)=\lim_{j\to\infty}\{d_g(o,\gamma(j))-d_g(x,\gamma(j))\}.
\]
The triangle inequality gives monotonicity of the expressions in braces,
their boundedness on compact sets, and $|b(x)-b(y)|\le d_g(x,y)$.
We recall the complex Hessian comparison that makes $b$ strictly
plurisubharmonic; this is the Busemann input in the argument of
Chen--Zhu \cite[Section~2]{ChenZhu}.
Take $x$ in a fixed compact set and a minimizing segment of length $l$
from $x$ to $\gamma(j)$. For an initial vector $X$, use its parallel
translate multiplied by $1-\sigma/l$ as a comparison variation field,
and use the same field construction for $JX$. The second variation of
length is computed on their perpendicular components. Its curvature
contribution is the negative of
\[
\int_0^l(1-\sigma/l)^2
\{\Rm(\dot\gamma,X_\parallel,X_\parallel,\dot\gamma)
+\Rm(\dot\gamma,JX_\parallel,JX_\parallel,\dot\gamma)\}
\,d\sigma.
\]
Removing radial components does not change this expression. It is a
positive multiple of the bisectional curvature before the minus sign.
On a fixed short initial subsegment, positivity is bounded below for
all unit initial vectors and all unit initial directions based in the
chosen compact. The derivative term in the index form is $O(l^{-1})$.
Consequently the sum of the two second derivatives of the distance is
at most $-c_K|X|^2+O(l^{-1})|X|^2$, with $c_K>0$.
This comparison does not require a smooth distance function. Vary the
initial endpoint along a holomorphic germ and use the comparison paths
to obtain smooth upper supports for distance. The sum of the endpoint
acceleration terms vanishes for the two real directions of that germ.
Their negatives give lower supports for the signed distance, with the
same strict Levi lower bound. An upper test for the signed distance
also lies above its lower support and therefore has this Levi lower
bound. Local uniform passage to $b$ gives the same viscosity inequality.
Equivalently, testing on complex lines gives the subharmonic
inequality. Thus $b$ is strictly plurisubharmonic, with a positive Levi
lower bound on each compact. Richberg approximation, with pointwise
error less than one, gives a smooth strictly plurisubharmonic $\psi$
with $\psi\le1+r$ after adding a constant
\cite{Richberg}\cite[Section~4, Corollary~2]{GreeneWu}.
Here are the bundle choices for the interpolation step. Write
$K=\bigwedge^nT^{*(1,0)}M$. A function, a canonical section, and a
tangent field can be regarded respectively as a holomorphic top form
with coefficients in
\begin{equation}\label{prelim:coefficient-bundles}
K^{-1},\qquad \boldone,\qquad T^{1,0}M\otimes K^{-1}.
\end{equation}
The first has nonnegative curvature, the second is flat, and the last
has nonnegative Nakano curvature: it is $E\otimes\det E$ for the
Griffiths-positive bundle $E=T^{1,0}M$, so the
Demailly--Skoda tensor-with-determinant theorem applies
\cite[Theorem~2.6]{Demailly}. These statements refer to the coefficient bundles
in \eqref{prelim:coefficient-bundles}; no Nakano positivity of $E$
itself is being assumed.
For completeness, prescribe jets of order $m$ at a finite set $A$.
Choose disjoint coordinate balls and local holomorphic sections with
those jets, multiply them by cutoffs equal to one on smaller balls,
and call the resulting smooth section $s_0$. Its $\dbar$ is supported
in a compact set disjoint from $A$. Use the weight
\[
k\psi+\sum_{a\in A}c_m\vartheta_a\log|z_a|^2,
\qquad c_m>n+m,
\]
where each $\vartheta_a$ is one near $a$ and supported in the chosen
coordinate ball. The negative Hessian terms of the cutoff logarithms
are supported in a fixed compact. A sufficiently large $k$ makes the
total weight strictly plurisubharmonic and dominates those terms.
The inverse-curvature norm of $\dbar s_0$ is integrable, since its
support avoids the poles. The complete K\"ahler $L^2$ estimate for
bundle-valued top forms gives $v$ with
$\dbar v=\dbar s_0$ and finite weighted norm
\cite{Hormander}\cite[Theorem~5.1]{Demailly}. The singular-weight statement follows by
regularizing the logarithms, using uniform estimates, and taking a
weak limit. Near a point of $A$, $v$ is holomorphic; integrability
against $|z_a|^{-2c_m}$ forces its Taylor coefficients through order
$m$ to vanish. Hence $s_0-v$ has the prescribed jets. This also proves
point separation by including two points in $A$.
For a canonical section, the same construction gives
\[
\int_M |S|_g^2 e^{-k\psi}\,dV_g<\infty.
\]
The cutoff pole weights can be dropped in this estimate at the price
of a constant. The function $|S|_g^2$ is subharmonic: away from its
zeros, $\dd\log|S|_g^2=\Ric(g)$, and its squared norm has the same
nonnegative distributional Laplacian across the zeros. The local
mean-value inequality on a radius-one ball
\cite[Theorem~2.1]{LiSchoen}, together with the upper bound for
$\psi$, yields
\[
|S(x)|_g^2\le
\frac{C_S}{\Vol B(x,1)}\exp\bigl(C_S(1+r(x))\bigr).
\]
Nonnegative Ricci curvature gives
$\Vol B(x,1)\ge c(1+r(x))^{-2n}$ by Bishop--Gromov relative volume
comparison with a fixed ball at $o$ \cite[\S2.1]{GromovBetti}.
Its polynomial loss is absorbed by the exponential.
This proves the growth assertion in Proposition~\ref{prop:initial-data};
compare the canonical-section construction in
\cite[Section~3]{ChenZhu}.
We explain why finite tuples suffice, using the dimension-count
argument underlying parametric transversality.
The compact-open spaces of holomorphic functions and sections are
Fr\'echet and hence Baire spaces. Fix a compact family of points,
or of distinct pairs of points. Surjectivity of finite-jet evaluation,
openness of matrix rank, and a finite cover give a finite-dimensional
space of perturbations whose evaluation derivative is onto on a
neighborhood of that compact family. This derivative is independent
of the perturbation parameter, since the evaluations are affine
linear. Thus the universal evaluation is a submersion there.
For each smooth stratum of the forbidden locus, its inverse image
has real codimension equal to that of the stratum. If this codimension
exceeds the real dimension of the point or pair space, the inverse
image has dimension less than the perturbation space. Its projection
to that space has measure zero: cover it by countably many compact
coordinate patches; on each patch the projection is Lipschitz, and
covering a $d$-dimensional cube by $O(\delta^{-d})$ cubes shows that
its image in $\R^p$, $p>d$, has volume at most $C\delta^{p-d}$.
Consequently arbitrarily small perturbations avoid the forbidden
strata on the prescribed compact family.
For $N$ functions, the rank-$r$ stratum of $N\times n$ derivative
matrices has complex codimension $(N-r)(n-r)\ge N-n+1$ when $r<n$.
For $N\ge2n+1$ this exceeds $n$.
The equal-values condition on distinct pairs has complex codimension
$N>2n$. Avoidance on a compact is open in the compact-open topology,
using Cauchy estimates for derivatives. A countable exhaustion of
$M$ and of $M\times M\setminus\operatorname{diag}$, followed by Baire
intersection, therefore gives one finite injective immersion.
For $m$ tangent sections the least deficient-rank codimension is
$m-n+1>n$ when $m\ge2n$. The same zeroth-jet argument gives a finite
spanning family. Neither conclusion asserts properness.
Choose countably many canonical sections $S_i$ with no common zero.
Write $a_i=|S_i|_g^2$. The bound
$a_i\le\exp(C_i(1+r))$ implies
\[
a_i\le e^{B_i}\exp(1+r^{3/2})
\]
for a finite $B_i$, because a linear function of $r$ is bounded above
by $r^{3/2}$ plus a constant. Choose $c_i>0$ so small that
$c_ie^{B_i}\le2^{-i}$ and that every derivative of $c_i a_i$ of order
at most $i$ has norm at most $2^{-i}$ on the first $i$ exhaustion
compacts. The sum $\sum c_i a_i$ converges smoothly locally, is
positive, and is bounded above by $\exp(1+r^{3/2})$. In a local
canonical frame it is $\det(g)^{-1}$ times a sum of squares of
holomorphic functions. Thus
$\Xi=\log\sum c_i|S_i|_g^2$ satisfies \eqref{prelim:xi}.
\subsection{Hodge heat without a curvature upper bound}
To smooth the distance truncations used below, we need to preserve an
upper bound for their complex Hessians. Scalar heat estimates will
then control the trace and give the missing lower Hessian bound.
We also record how heat evolution of a bounded closed $(1,1)$-form
produces a scalar potential for its change; this identity will be used
again when constructing the flow comparison cutoff.
Let $H_a$ denote scalar heat for $\Delta_g$. Normalize the Hodge heat
operator $\mathcal H_a$ on real $(1,1)$-forms so that it agrees with
$H_a$ after taking the trace. The normalization is also chosen so
that its generator on closed real $(1,1)$-forms is
$A\mapsto\dd\tr_g A$.
The scalar heat kernel has total mass one. Indeed, fix its pole $y$ and
let $\chi_R$ be a Lipschitz distance cutoff equal to one on $B_g(y,R)$,
supported in $B_g(y,2R)$, and satisfying $|\nabla\chi_R|\leq C/R$.
The Gaussian estimate of Li--Yau \cite{LiYau} and the kernel gradient estimate of
Souplet--Zhang \cite[Theorem~1.3]{SoupletZhang}, together with
Bishop--Gromov comparison \cite[\S2.1]{GromovBetti}, give
\[
\int_{R\leq d_g(x,y)\leq2R}|\nabla_xH_s(x,y)|\,dV_g(x)
\leq C s^{-1/2}(1+R/\sqrt{s})^N e^{-cR^2/s}
\]
for a fixed dimensional exponent $N$. Integrating the heat equation
against $\chi_R$ from $\varepsilon$ to $t\leq T$ and then letting
$\varepsilon\downarrow0$ therefore yields
\[
\left|\int_M\chi_R(x)H_t(x,y)\,dV_g(x)-1\right|
\leq \frac{C}{R}\int_0^T
s^{-1/2}(1+R/\sqrt{s})^N e^{-cR^2/s}\,ds.
\]
The right side tends to zero as $R\to\infty$. Positivity and the
sub-Markov property of the minimal kernel allow passage to the total
mass, proving $H_t1=1$. The cutoffs here require only completeness and
the first derivative of distance; they precede the smooth exhaustion
constructed below.
\begin{lemma}[Heat order and commutation]\label{lem:hodge-heat}
For bounded real $(1,1)$-forms, Hodge heat is well defined by cutoff
limits of the complete $L^2$ semigroup. It preserves order and obeys
\[
-Cg\le A\le Cg\quad\Longrightarrow\quad
-Cg\le\mathcal H_aA\le Cg,
\qquad \tr_g\mathcal H_aA=H_a\tr_g A.
\]
If $A$ is also smooth and closed, then
\begin{equation}\label{prelim:hodge-identity}
\mathcal H_aA
=A+\dd\int_0^aH_b\tr_g A\,db.
\end{equation}
If a Lipschitz function $f$ is constant off a compact set and
$\dd f\le Cg$ as currents, then
\begin{equation}\label{prelim:hessian-preservation}
\dd H_af\le Cg\qquad(a>0).
\end{equation}
\end{lemma}
\begin{proof}
First take smooth compactly supported data. On a relatively compact
smooth domain use the componentwise zero-Dirichlet realization of
the Hodge heat equation. Its Weitzenb\"ock formula is a connection
Laplacian plus a zeroth-order curvature reaction. At a null vector
$X$ of a nonnegative Hermitian form $A$, diagonalize $A$ in a unitary
frame. The reaction on $(X,\bar X)$ is a positive fixed multiple of
\[
\sum_j R_g(X,\bar X,e_j,\bar e_j)A_{j\bar j}\ge0;
\]
the Ricci-times-$A$ terms vanish there. The tensor maximum principle
therefore preserves the nonnegative cone. Applying the same argument
to $Cg-A$, with nonnegative boundary data, proves the upper bound by
$Cg$. The trace satisfies the scalar heat equation, including the
zero boundary condition. This is the null-eigenvector mechanism in
\cite[Section~2]{NiTamStructure}.
We specify the complete realization and the exhaustion step.
On compactly supported smooth forms put $D=d+d^*$.
Completeness gives essential self-adjointness of $D^2$ and hence of
$D$: a deficiency vector for $D$ would lie in the kernel of
$(D^2)^*+1$, which is zero
\cite[Corollary~2.10 and Remark~2.12]{BravermanMilatovicShubin}.
The closed quadratic form of the resulting Hodge Laplacian is
\[
q(A)=\|dA\|_2^2+\|d^*A\|_2^2,
\qquad
V=\overline{C^\infty_c}^{\,(\|\cdot\|_2^2+q)^{1/2}}.
\]
For a nested smooth exhaustion $\Omega_j$, let $V_j$ be the same
closure of forms supported in $\Omega_j$, extended by zero.
On a fixed domain, local ellipticity identifies this with the
full zero boundary-value section domain. It is not an absolute
or relative Hodge boundary condition. The spaces $V_j$ are nested
and their union is dense in $V$.
For $\lambda>0$, the domain resolvent applied to an $L^2$ form,
and extended by zero, is the orthogonal projection of the complete
resolvent onto $V_j$ for the inner product $q+\lambda\langle\cdot,\cdot\rangle$:
both satisfy the same variational identity against $V_j$.
The projections therefore converge in that norm. Strong resolvent
convergence and the common lower bound zero imply strong heat
convergence \cite[Section~6.6, Theorem~6.31 and Corollary~6.33]{Teschl2009}.
Here $\Delta_j$ denotes the nonnegative domain Hodge Laplacian.
Zero extension of its heat means $e^{-a\Delta_j}P_j$, where $P_j$
is restriction followed by zero extension; it does not mean
extending its generator by zero.
The cone and trace assertions pass to this limit and then to bounded
data by cutoff and local regularity. More explicitly, the complete
Hodge semigroup has the off-diagonal estimate
\begin{equation}\label{prelim:off-diagonal}
\|\boldone_E\mathcal H_a\boldone_F\|_{2\to2}
\le C\exp\bigl(-c\dist(E,F)^2/a\bigr).
\end{equation}
To see that this estimate needs no curvature upper bound, use the
self-adjoint first-order operator $d+d^*$. Its principal symbol has
norm equal to the covector norm, so the wave evolution has finite
propagation speed \cite[Theorem~1.1 and Section~4]{McIntoshMorris}.
The Gaussian Fourier representation of
$\exp(-a(d+d^*)^2)$ then gives \eqref{prelim:off-diagonal}, with the
inessential change of constants required by our normalization
\cite[Theorem~3.4]{CoulhonSikora}.
Since $\Vol B(o,R)\le CR^{2n}$, dyadic annular decomposition shows
that the cutoff evolutions of a bounded form are Cauchy in $L^2$ on
each compact, uniformly for bounded positive times. Interior
parabolic estimates, whose coefficients are bounded on the compact
under consideration, give smooth convergence away from $a=0$.
Closedness is proved using the complete semigroup, not the
Dirichlet operators on domains. Let $\zeta_R$ be a compactly supported
Lipschitz cutoff equal to one on $B(o,R)$, supported on $B(o,2R)$,
and satisfying $|d\zeta_R|\le C/R$. The complete Hodge semigroup
commutes with $d$ on the $L^2$ domains, by the self-adjoint de Rham
complex.
The fixed time normalization does not affect the spectral identity
$D e^{-aD^2}=e^{-aD^2}D$, and projection to degree one higher gives
the $d$ identity. The data $\zeta_RA$ used here belong to the full
$D$ graph domain by compactly supported $H^1$ approximation.
For bounded closed $A$,
\[
d\mathcal H_a(\zeta_RA)
=\mathcal H_a^{(3)}(d\zeta_R\wedge A).
\]
The data on the right have $L^2$ norm at most $CR^{n-1}$ and support
outside $B(o,R)$. Estimate~\eqref{prelim:off-diagonal} in degree
three shows that they tend to zero on each fixed compact, uniformly
for $0<a\le a_0$. Distributional passage to the limit proves
$d\mathcal H_aA=0$. Local regularity also gives the initial value
$A$ on each compact. K\"ahler commutation on closed forms now gives
\[
\partial_a\mathcal H_aA=\dd\tr_g\mathcal H_aA
=\dd H_a\tr_g A.
\]
Integration proves \eqref{prelim:hodge-identity}, first in
distributions and then as an identity of smooth forms for $a>0$.
The passage from closed form heat to a scalar potential by integrating
its trace is also the mechanism in
\cite[Lemma~2.1 and proof of Theorem~2.1]{NiTamHodge2013}.
Finally put $f_0=f-c$, where $c$ is the constant value at infinity.
This is a compactly supported Lipschitz function. Complete $L^2$
K\"ahler commutation, smooth approximation, and duality give, for a
compactly supported nonnegative Hermitian test form $B$,
\[
\langle\dd H_af_0,B\rangle
=\langle\dd f_0,\mathcal H_aB\rangle.
\]
The metric pairing here identifies the cone of nonnegative Hermitian
forms with its dual; equivalently one pairs the current with the
Hodge star of $B$. Choose $0\le\zeta\le1$ compactly supported and
equal to one near $\supp f_0$. Since $\dd f_0$ is supported there,
\[
\langle\dd f_0,\mathcal H_aB\rangle
=\langle\dd f_0,\zeta\mathcal H_aB\rangle
\le C\int\zeta\tr_g\mathcal H_aB\,dV_g
\le C\int\tr_gB\,dV_g.
\]
Positivity and scalar trace commutation justify the last two steps.
Finally $H_a1=1$ under nonnegative Ricci curvature, so
$\dd H_af=\dd H_af_0$. This proves
\eqref{prelim:hessian-preservation}.
\end{proof}
\subsection{Dyadic smoothing and averaged curvature}
For $L=2^j$, $j\ge0$, choose a fixed smooth nondecreasing scalar
truncation of $r^2/L^2$ which is zero on $r\le L$ and one on
$r\ge2L$, and denote it by $f_L$. It is Lipschitz, with constant
$C/L$. The paired index-form comparison with linear parallel fields
gives $\dd r^2\le Cg$ away from the cut locus and in the upper-support
sense at the cut locus. The derivative-square term introduced by the
truncation is bounded on its annulus. Hence
\[
\dd f_L\le CL^{-2}g
\]
as currents. The word ``smooth'' here describes the scalar
truncation; $f_L$ itself need not be smooth at the cut locus.
Let $w_L=H_{\epsilon L^2}f_L$, with a fixed sufficiently small
$\epsilon>0$. Gaussian heat bounds and the Lipschitz estimate give
\[
\|w_L-f_L\|_\infty
\le (C/L)\sup_x\int d_g(x,y)H_{\epsilon L^2}(x,y)\,dV_g(y)
\le C\sqrt\epsilon.
\]
Nonnegative Ricci curvature is sufficient for these scalar heat
estimates. Applying the Li--Yau gradient inequality to the positive
heat solutions $1+H_af_L$ and $2-H_af_L$ gives, at
$a=\epsilon L^2$,
\[
|\Delta_g w_L|\le C_\epsilon L^{-2},\qquad
|\partial w_L|_g\le C_\epsilon L^{-1}.
\]
Indeed the first translate bounds $\partial_aH_af_L$ from below,
the second bounds it from above, and substituting both time bounds
in the same inequalities bounds the gradient; see
\cite[Theorem~1.3(i)]{LiYau}.
Lemma~\ref{lem:hodge-heat} gives the upper bound for each complex
Hessian eigenvalue, and the lower trace bound then gives the lower
bound for each eigenvalue. Thus $|\dd w_L|_g\le C_\epsilon L^{-2}$.
Choose a fixed smooth nondecreasing step $\theta$ which is zero near
zero and one near one, with the constant regions wide enough to
contain the above heat error. Then $v_L=\theta(w_L)$ is identically
zero on $r\le L$ and one on $r\ge2L$, and
\[
|\partial v_L|_g\le C/L,\qquad |\dd v_L|_g\le C/L^2.
\]
The locally finite sum
\[
u=1+\sum_{j\ge0}2^jv_{2^j}
\]
is comparable to $1+r$. At any point, only a bounded number of
summands have nonzero derivatives; their scales are comparable to
$1+r$. This proves \eqref{prelim:exhaustion}. In particular, scalar
cutoffs $p(u/L)$ have gradient $O(L^{-1})$ and absolute complex
Hessian $O(L^{-2})$, with support and transition region in fixed
multiples of $B(o,L)$.
Put $w=\log|S|_g^2$ for a canonical section with $S(o)\ne0$.
The Poincar\'e--Lelong formula gives
\begin{equation}\label{prelim:canonical-current}
\dd w\ge\Ric(g)
\end{equation}
as currents. The truncations $w_m=\max(w,-m)$ are subharmonic.
The heat submean inequality and the linear upper growth give
\[
w_m(o)\le H_{L^2}w_m(o),\qquad
H_{L^2}(w_m)^+(o)\le CL.
\]
One can justify the submean inequality first on exhaustion domains
and then by Gaussian decay; the positive part has at most linear
growth. For $m$ large, $w_m(o)=w(o)$, so
$H_{L^2}(w_m)^-(o)\le CL+|w(o)|$. The heat-kernel lower bound on
$B(o,L)$ is $c/\Vol B(o,L)$. Monotone convergence as $m\to\infty$
therefore proves the first estimate in \eqref{prelim:averages}.
Take a nonnegative cutoff equal to one on $B(o,L)$ and supported in
$B(o,CL)$ with absolute Laplacian at most $C/L^2$, as just constructed.
Integrating the trace of \eqref{prelim:canonical-current} against it
and moving $\Delta_g$ onto the cutoff gives
\[
\int_{B(o,L)}R_g\,dV_g
\le \frac{C}{L^2}\int_{B(o,CL)}|w|\,dV_g
\le \frac{C}{L}\Vol B(o,L).
\]
The last step uses volume doubling. This proves the second estimate
and completes Proposition~\ref{prop:initial-data}. The average
scalar-curvature conclusion is the estimate of
\cite[Theorem~2]{ChenZhu}; the details above record the particular
cutoffs and absolute logarithmic estimate needed below.
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/preliminaries.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/uniformization.tex ====
\section{Convergence and surjectivity of the coordinates}
\label{sec:uniformization}
We now pass from the polynomial models to a global map. The local
errors must remain small under backward polynomial iteration, and the
subsequential forward degree bound must then rule out a proper image.
We give both steps with their domains and parameter choices.
\subsection{Parameter order and local error estimates}
Decrease the common chart exponent $\sigma$ of
Proposition~\ref{prop:shrinking-charts}, if necessary, so that
$0<\sigma<1$. Choose
\begin{equation}
0<\gamma<\gamma'<\min\{\sigma/2,1\},
\qquad 0<10\nu<\min\{c\gamma,\sigma,\gamma,1\},
\label{eq:unif:decay-parameters}
\end{equation}
with $\gamma$ in the range of its small-ball transition estimate;
decrease its constant $c$ to at most one. Fix the jet order so large
that
\begin{equation}
(d+1)\nu>30.
\label{eq:unif:jet-order}
\end{equation}
Only after this choice fix a sufficiently small $h>0$, as specified
below. Then choose $N,M,b,K_0,\lambda$ as in
Section~\ref{sec:dynamics}, and finally choose a sufficiently late
regular macro start. The estimates in
Proposition~\ref{prop:polynomial-models} hold for this choice.
On a fixed smaller raw transition ball, the jet conjugacy and Cauchy
estimates give
\begin{equation}
\abs{H_{k+1}F_k(z)-G_kH_k(z)}
\le \exp\bigl(A_d\Delta s_k+o(s_k)\bigr)\abs z^{d+1}.
\label{eq:unif:remainder}
\end{equation}
Here $A_d$ is independent of $h$. Indeed the degrees of the
polynomials in this expression are fixed, the $F_k$ are uniformly
bounded on a larger raw ball, and the coefficient estimates of
Proposition~\ref{prop:polynomial-models} bound the two compositions
there. They agree through degree $d$, so the Cauchy remainder has
the displayed factor. This argument uses $H_{k+1}F_k-G_kH_k$
directly; it requires no uniform analytic domain for $H_k^{-1}$.
We shall use two sequences of raw chart coordinates:
\begin{enumerate}[label=(\roman*)]
\item $z_k=f_{s_k}(x)$, uniformly for $x$ in any fixed compact
neighborhood in the exhaustion and all sufficiently large $k$;
\item $z_k=F_{s_a,s_k}(z)$, uniformly for
$\abs z\le e^{-\gamma s_a}$, for any sufficiently large $a$ and
all $k\ge a$.
\end{enumerate}
In the first case the threshold may depend on the compact; in the
second it is uniform in $a$. Continuation and composition of the
transition germs give $z_{k+1}=F_k(z_k)$ in both cases.
The chart estimates and \eqref{eq:unif:decay-parameters} imply,
eventually,
\begin{equation}
\abs{z_k}\le e^{-5\nu s_k},\qquad
\zeta_k:=H_k(z_k),\quad
\abs{\zeta_k}\le e^{-2\nu s_k}.
\label{eq:unif:small-orbits}
\end{equation}
In regime (ii), the case $k=a$ uses $F_{s_a,s_a}=\Id$ and
$\gamma>10\nu$.
Choose $h$ so small that, with $\delta=e^h-1$,
\begin{equation}
\delta<\tfrac1{10},\qquad
A_d\delta<1,\qquad C_d\delta<\nu/4,
\label{eq:unif:macro-choice}
\end{equation}
enlarging $A_d,C_d$ first to cover the finitely many fixed-degree
coefficient estimates used here. By
\eqref{eq:unif:jet-order}--\eqref{eq:unif:small-orbits}, after the
late start the defect satisfies
\begin{equation}
r_k:=\zeta_{k+1}-G_k(\zeta_k),\qquad
\abs{r_k}\le e^{-10s_k}.
\label{eq:unif:orbit-defect}
\end{equation}
All constants dependent on $h,N,M$ multiply $o(s_k)$ in this argument
and are absorbed only by that late start. In particular,
\eqref{eq:unif:macro-choice} does not require choosing $h$ using a
constant which is itself determined by $M$.
The inverse maps have the local Lipschitz bound
\begin{equation}
\Lip\left(G_k^{-1}\bigm|_{B(e^{-\nu s_{k+1}})}\right)
\le e^{\Delta s_k}
\label{eq:unif:inverse-tube}
\end{equation}
at all sufficiently large indices. To check it, the derivative of
$G_k^{-1}$ on this ball is bounded by the sum of
\[
s_k^C e^{\Delta s_k/2}
\quad\hbox{and}\quad
\exp\bigl(C_d\Delta s_k+o(s_k)-\nu s_{k+1}\bigr).
\]
The first term is at most $e^{3\Delta s_k/4}$ eventually, since
$\Delta s_k\ge c_h\sqrt{s_k}$. The second tends to zero by
\eqref{eq:unif:macro-choice}; their sum is at most
$e^{\Delta s_k}$ after increasing the starting index. Integrating
the derivative along segments in this convex ball gives
\eqref{eq:unif:inverse-tube}.
\subsection{Backward approximation inside valid tubes}
\begin{lemma}[Convergence of corrected coordinates]
\label{lem:unif:backward-limit}
For either raw sequence above and every sufficiently large $k$,
\[
\xi_k=\lim_{l\to\infty}G_{k,l}^{-1}\zeta_l
\]
exists locally uniformly in the indicated variables. It is
holomorphic and satisfies
\begin{equation}
\abs{\xi_k-\zeta_k}\le e^{-6s_k},\qquad
\xi_{k+1}=G_k\xi_k.
\label{eq:unif:shadow}
\end{equation}
All applications of \eqref{eq:unif:inverse-tube} take place inside
its stated balls.
\end{lemma}
\begin{proof}
For a terminal index $l$ put $y_l^{(l)}=\zeta_l$ and
$y_i^{(l)}=G_i^{-1}y_{i+1}^{(l)}$ for $i<l$.
Simultaneously establish, by backward induction, that the comparison
segments at step $i$ lie in $B(e^{-\nu s_{i+1}})$ and that
\begin{equation}
\abs{y_k^{(l)}-\zeta_k}
\le \sum_{i=k}^{l-1}
e^{-10s_i}\exp(s_{i+1}-s_k).
\label{eq:unif:finite-error-sum}
\end{equation}
The terminal assertion is immediate. At a backward step compare
$G_i^{-1}y_{i+1}^{(l)}$ with
$G_i^{-1}G_i\zeta_i=\zeta_i$.
Both arguments are close to $\zeta_{i+1}$: the first by the inductive
bound and the second by \eqref{eq:unif:orbit-defect}. Since
$\abs{\zeta_{i+1}}\le e^{-2\nu s_{i+1}}$, they and their joining
segment lie in the required larger tube whenever the error sum has
the bound asserted below. Applying
\eqref{eq:unif:inverse-tube} then gives the next error sum.
Indeed $s_{i+1}\le(1+\delta)s_i$ and the grid eventually has spacing
at least one, so
\[
\sum_{i\ge k}e^{-10s_i+s_{i+1}-s_k}
\le e^{-s_k}\sum_{i\ge k}e^{-(9-\delta)s_i}
\le C e^{-(10-\delta)s_k}\le e^{-6s_k}.
\]
As $\nu<1/10$, this is much smaller than the difference between the
two tube radii. The estimates therefore close the simultaneous
induction without evaluating an inverse outside its proved domain.
For completeness, consecutive terminal approximations are Cauchy
by the same reasoning. At level $l$ their difference is at most
$e^{\Delta s_l}e^{-10s_l}$, and propagation down to $k$ bounds it by
$e^{-10s_l+s_{l+1}-s_k}$. These bounds are summable in $l$.
They are uniform on the compact sets in either regime, so the limits
are holomorphic. The finite identities give
$\xi_{k+1}=G_k\xi_k$, and the infinite error sum proves
\eqref{eq:unif:shadow}.
\end{proof}
\subsection{A coherent injective map on the manifold}
\begin{proposition}[Global coordinates]\label{prop:global-coordinates}
The locally uniform limit on the exhaustion
\begin{equation}
\Psi(x)=\lim_{l\to\infty}G_{0,l}^{-1}H_lf_{s_l}(x)
\label{eq:unif:global-map}
\end{equation}
defines an injective holomorphic map $M\to\C^n$ with $\Psi(o)=0$.
Its image $\Omega$ is open, and $\Psi:M\to\Omega$ is a
biholomorphism. Moreover, on every compact $K\subset\Omega$,
\begin{equation}
\sup_K\abs{G_{0,k}}\le e^{-(\sigma/2)s_k}
\quad\hbox{eventually},
\label{eq:unif:forward-smallness}
\end{equation}
and, at every sufficiently large $k$,
\begin{equation}
G_{0,k}^{-1}\bigl(B(e^{-\gamma's_k})\bigr)\subset\Omega.
\label{eq:unif:inverse-ball-inclusion}
\end{equation}
\end{proposition}
\begin{proof}
On a fixed exhaustion compact choose an index $k$ after which regime
(i) is available. Lemma~\ref{lem:unif:backward-limit} gives $\xi_k$
there, and composing it with the single fixed polynomial
automorphism $G_{0,k}^{-1}$ gives the limit
\eqref{eq:unif:global-map}. On overlaps the inverse charts agree by
continuation, and \eqref{eq:unif:shadow} makes the limits agree.
Thus these maps define a holomorphic $\Psi$ on all of $M$. All maps
fix zero in charts, so $\Psi(o)=0$.
Suppose $\Psi(x)=\Psi(y)$ for two fixed points. For every large $k$,
$G_{0,k}\Psi=\xi_k$ on a compact containing them, and hence
\eqref{eq:unif:shadow} implies
\[
\abs{H_kf_{s_k}(x)-H_kf_{s_k}(y)}\le2e^{-6s_k}.
\]
On the tiny raw-coordinate ball containing these points, write
$H_k(z)=A_kz+E_k(z)$, where $A_k=H_k'(0)$ has conorm at least
$s_k^{-C}$ and $\norm{DE_k}$ is bounded by $\exp(o(s_k))$ times
the radius. Integrating only $DE_k$ along the segment and retaining
the fixed linear term $A_k$ gives
\[
|H_k(z)-H_k(w)|\ge
\bigl(s_k^{-C}-\sup\norm{DE_k}\bigr)|z-w|.
\]
The nonlinear error is eventually less than $s_k^{-C}/2$, so
\begin{equation}
\abs{f_{s_k}(x)-f_{s_k}(y)}\le e^{-5s_k}
\label{eq:unif:raw-separation}
\end{equation}
eventually.
If $x\ne y$, choose a small relatively compact initial metric ball
about $x$ whose closure excludes $y$. Map the straight segment
between the two raw coordinates by $\Phi_{s_k}$, obtaining a path
from $x$ to $y$. The segment stays in the uniform chart ball.
Up to its first exit from the fixed ball about $x$, the compact
metric comparison $e^{-2s_k}g\le h_{t(s_k)}$ from
Proposition~\ref{prop:ricci-windows}, and the uniform chart upper
bound $\Phi_{s_k}^*h_{t(s_k)}\le Cg_{\mathrm{Eucl}}$, bound its
initial-metric length by
$Ce^{s_k}e^{-5s_k}$. This tends to zero and cannot reach the
boundary of that fixed ball. The contradiction proves injectivity.
This argument uses immersed charts only on their indicated sheet;
it requires no global injectivity of $\Phi_{s_k}$.
Nonsingularity follows directly from the same quantitative estimates.
Fix a point and two coordinate neighborhoods $U\Subset V$ whose
closures lie in one compact exhaustion set. The estimates of
regime (i) hold uniformly on $\overline V$ for every sufficiently
large $k$. Differentiating $\Phi_{s_k}f_{s_k}=\Id$, and using
\eqref{chart:metric-decay} and
\eqref{chart:uniform-chart-metric}, gives
\[
|df_{s_k}(v)|\ge c e^{-s_k}|v|_g
\quad\text{on }\overline V.
\]
The coefficient bounds on $H_k$, its polynomially bounded inverse
linear part, and $|f_{s_k}|\le e^{-\sigma s_k}$ therefore imply
\[
|d(H_kf_{s_k})(v)|\ge c s_k^{-C}e^{-s_k}|v|_g.
\]
On the other hand, the shadow bound
$|\xi_k-H_kf_{s_k}|\le e^{-6s_k}$ on $\overline V$ and Cauchy's
estimate on the fixed pair $U\Subset V$ bound its differential on
$U$ by $C e^{-6s_k}|v|_g$. Thus
\[
|d\xi_k(v)|\ge
\bigl(c s_k^{-C}e^{-s_k}-C e^{-6s_k}\bigr)|v|_g>0
\qquad(v\ne0)
\]
for all sufficiently large $k$. Since $\xi_k=G_{0,k}\Psi$ and
$G_{0,k}$ is a polynomial automorphism, $d\Psi$ is nonsingular.
The holomorphic inverse theorem now makes the image $\Omega$ open;
the local inverses agree by the proved injectivity, giving a
holomorphic inverse on $\Omega$. Consequently
$K'=\Psi^{-1}(K)$ is compact whenever $K\subset\Omega$ is compact. On $K'$, the common chart bound
$\abs{f_{s_k}}\le e^{-\sigma s_k}$, the coefficient bound on $H_k$,
and \eqref{eq:unif:shadow} give
\[
\sup_K\abs{G_{0,k}}
\le \exp\bigl(-\sigma s_k+o(s_k)\bigr)+e^{-6s_k},
\]
which proves \eqref{eq:unif:forward-smallness}.
It remains to prove the ball inclusion. Fix a sufficiently large
$k$ and use regime (ii) with $a=k$. For large $l$, the inverse chart
$f_{s_l}$ is defined on the compact image under $\Phi_{s_k}$ of the
closed ball $\overline B(e^{-\gamma s_k})$. The identity
$f_{s_l}\Phi_{s_k}=F_{s_k,s_l}$ holds first as a germ at zero and
then throughout that ball by holomorphic continuation. Therefore
Lemma~\ref{lem:unif:backward-limit} gives
\begin{equation}
\sup_{\abs z\le e^{-\gamma s_k}}
\abs{G_{0,k}\Psi\Phi_{s_k}(z)-H_k(z)}
\le e^{-6s_k}.
\label{eq:unif:chart-shadow}
\end{equation}
Write $A=H_k'(0)$, $r=e^{-\gamma s_k}$, and
\[
T_k=G_{0,k}\Psi\Phi_{s_k},\qquad E_k(z)=T_k(z)-Az.
\]
The map $T_k$ is holomorphic on the fixed chart ball, and fixes zero.
Its shadow estimate and the coefficient bound on $H_k$ give
\[
E_k(0)=0,\qquad \norm{A^{-1}}\le s_k^C,\qquad
\sup_{\abs z\le r}|E_k(z)|
\le \exp(o(s_k))r^2+e^{-6s_k}.
\]
Cauchy's estimate on the smaller ball then gives
\[
\sup_{\abs z\le r/2}\norm{A^{-1}DE_k(z)}
\le C_n s_k^C\bigl(\exp(o(s_k))r+e^{-6s_k}/r\bigr)
\longrightarrow0.
\]
For large $k$ this supremum is at most $1/4$. Moreover,
$\gamma'>\gamma$ implies
$|A^{-1}w|\le r/4$ whenever $|w|\le e^{-\gamma's_k}$,
after increasing the starting index once more. For such a target
$w$, the map
\[
z\longmapsto A^{-1}w-A^{-1}E_k(z)
\]
is $1/4$-Lipschitz on $\overline B(r/2)$ and maps this closed ball
into $B(3r/8)$, because $E_k(0)=0$. Its successive iterates are
Cauchy, stay in the closed ball, and converge to a fixed point $z$.
The fixed-point identity is $T_k(z)=w$. Therefore
$B(e^{-\gamma's_k})\subset G_{0,k}(\Omega)$, which is precisely
\eqref{eq:unif:inverse-ball-inclusion}.
\end{proof}
\subsection{The image is all of affine space}
\label{sec:surjectivity}
We first record the polynomial growth estimate used in the final step.
For a vector polynomial $P$ of degree at most $m$ and $0<R_1<R_2$,
\begin{equation}
\sup_{B(R_2)}\abs P
\le (R_2/R_1)^m\sup_{B(R_1)}\abs P.
\label{eq:unif:polynomial-growth}
\end{equation}
To verify this, restrict each scalar dual pairing of $P$ to a
complex line through zero. The maximum principle in the exterior
of the radius-$R_1$ disc, applied to the polynomial divided by
$z^m$ including its removable value at infinity, gives the scalar
bound. Taking the supremum over unit dual vectors and lines proves
\eqref{eq:unif:polynomial-growth}.
\begin{proof}[Proof of Theorem~\ref{thm:uniformization}]
The preceding analytic and chart constructions, followed by
Propositions~\ref{prop:polynomial-models} and
\ref{prop:global-coordinates}, give a biholomorphism $\Psi:M\to\Omega$
onto an open set containing zero, with the forward smallness and
inverse-ball inclusion proved above. It remains to show that
$\Omega=\C^n$.
Suppose
$R_*=\sup\{R>0:B(R)\subset\Omega\}<\infty$.
It is positive because $0\in\Omega$ and $\Omega$ is open.
By \eqref{eq:dyn:forward-degree} there is an unbounded subsequence
with $\deg G_{0,k}\le Ds_k$ for some finite $D$.
Choose
$0<R_1<R_*<R_2$ so close to $R_*$ that
\begin{equation}
D\log(R_2/R_1)<\sigma/2-\gamma'.
\label{eq:unif:radii-choice}
\end{equation}
The closed smaller ball lies compactly in $\Omega$: it lies inside
some still larger centered ball of radius below $R_*$.
By \eqref{eq:unif:forward-smallness},
\eqref{eq:unif:polynomial-growth}, and
\eqref{eq:unif:radii-choice}, the chosen subsequence satisfies
\[
\sup_{B(R_2)}\abs{G_{0,k}}
\le \exp\left[
-\frac{\sigma}{2}s_k+Ds_k\log(R_2/R_1)\right]
< e^{-\gamma's_k}
\]
for all sufficiently large indices.
Equation~\eqref{eq:unif:inverse-ball-inclusion} then implies
$B(R_2)\subset\Omega$, contradicting the definition of $R_*$.
Therefore $R_*=\infty$, and $\Omega=\C^n$.
The map $\Psi$ in \eqref{eq:unif:global-map} is consequently a
biholomorphism $M\to\C^n$.
\end{proof}
The surjectivity step used the degrees
of the forward compositions $G_{0,k}$ on their available
subsequence; no degree estimate for their inverses, or uniform
instantaneous contraction ratio, is required.
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/uniformization.tex ====
==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/variation.tex ====
\section{A differential inequality for the disc envelope}
\label{sec:variation}
Let $v$ be the lower semicontinuous envelope defined in
\eqref{var:envelope}--\eqref{var:sandwich}, with its fixed horizon $T$.
We continue to denote physical time at a disc center by $s$.
The following inequality is the input to the fiber optimization in
Section~\ref{sec:ellipsoids}.
\begin{proposition}[Differential inequality for the envelope]
\label{prop:disc-inequality}
Suppose that a smooth real function $\varphi$ touches $v$ from below at
$(y_*,s_*)$, where $0<s_*<T$. In holomorphic cotangent coordinates write
\[
W=\dd_Y\varphi(y_*,s_*)=
\begin{pmatrix} A&B\\ B^*&C\end{pmatrix},\qquad C>0.
\]
There is a Hermitian form $K$ on the horizontal coordinate space such that
\begin{equation}
K\ge0,\qquad K\ge A-BC^{-1}B^*,\qquad
\partial_s\varphi(y_*,s_*)\ge\tr(C^{\mathsf T}K).
\label{var:viscosity-inequality}
\end{equation}
The transpose is the tangent--cotangent contraction in these coordinates.
In particular the statement applies to lower tests translated in a fixed
cotangent coordinate chart; its contraction has constant coefficients.
\end{proposition}
We prove the Proposition through localized almost minimizing discs.
The estimate on their deformations will be stated with its constants:
the compact set containing the interiors of the discs is allowed to
depend on the area penalty, whereas the coefficient multiplying that
penalty will depend only on boundary data.
After this deformation estimate, we remove the negative part of the
boundary Hessian and factor the remaining positive form. A backward
clock variation bounds its symplectic cost. The final Schur-complement
limit gives the asserted differential inequality.
\subsection{Localization and the order of approximation}
Subtracting a positive multiple of
$|y-y_*|^4+|s-s_*|^4$ from $\varphi$ makes the contact strict on a compact
coordinate cylinder $\mathcal C\Subset Y\times(0,T)$ without changing
the derivatives in \eqref{var:viscosity-inequality}.
Thus $v-\varphi\ge0$ on $\mathcal C$, with equality only
at $(y_*,s_*)$, and with a positive gap near its boundary. The definition
of lower semicontinuous regularization gives
\[
\inf_{(y,s)\in\mathcal C}\bigl(v^{\rm raw}(y,s)-\varphi(y,s)\bigr)=0.
\]
On an interval slightly larger than $[0,T]$, take $Q$ and $\rho_j$ from
Theorem~\ref{thm:localized-flow} and
Proposition~\ref{prop:scalar-comparison}, and set
\begin{equation}
d_{0,j}(x,t)=j^{-1}Q(x,t)+\rho_j(x,t).
\label{var:base-cost}
\end{equation}
Thus $d_{0,j}\ge0$, $(\partial_t-\Delta_{h_t})d_{0,j}\ge0$, and
$d_{0,j}\to0$ uniformly on compact space--time sets. Choose increasing
$A_j\to\infty$ so that
\begin{equation}
d_{0,j}(x,t)\ge\rho(x,t)+j\quad\hbox{if }u(x)>A_j/3.
\label{var:tail-wall}
\end{equation}
This is possible because $\rho_j\ge\rho-C_j$ and $Q$ has a positive
quadratic lower bound. Choose fixed smooth scalar profiles and set
$\delta_1=\chi(u/A_j)$ and
$\delta_2=\widetilde\chi(N(\cdot,0)/B_j^2)$, with
$0\le\delta_i\le1$, where $\delta_1>0$ exactly when $u<A_j$ and
$\delta_1=1$ when $u\le2A_j/3$, while $\delta_2>0$ exactly when
$N(y,0)<B_j^2$ and $\delta_2=1$ when $N(y,0)\le B_j^2/2$.
Here $B_j\ge j$ will be increased below. Smooth cutoffs flat at their
zero sets are permissible. On the open boundary region put
\begin{equation}
k_i=\delta_i^{-1}-1,\qquad
d(y,t,s)=d_{0,j}(x,t)+(1+s)(k_1(x)+k_2(y)).
\label{var:wall-cost}
\end{equation}
Only the boundary of a competing disc is subject to these spatial
restrictions. Their closed boundary region is a compact set $K_j\subset Y$.
Each fixed $d_{0,j}$ is smooth one-sided at $t=0$, with all derivatives
bounded on compact sets. When needed, use a smooth extension matching
these initial jets; extending it by zero to negative time is unnecessary.
Use the boundary seminorm of Section~\ref{sec:discs},
\[
D(a)^2=\sum_{k\in\mathbb Z}|k|\,\|a_k\|^2,
\]
and add the nonnegative energy
\begin{equation}
\begin{split}
\mathcal E(f,w)={}&\int_\Delta\bigl(x_f^*\omega_g+
f^*G^*\omega_{\rm eucl}+
(\mathfrak g\circ f)^*\omega_{\Gr}
+w^*\omega_{\rm eucl}\bigr)\\
&+D\bigl(\log(\delta_1\delta_2)\circ f\bigr)^2.
\end{split}
\label{var:energy}
\end{equation}
Here $x_f=\pi\circ f$, and $\mathfrak g$ is the Gauss map of the immersion $G$ from
Lemma~\ref{lem:disc-frames}.
Let $\mu_{j,\eps}$ be the infimum, over centers in $\mathcal C$ and discs
with the stated boundary restrictions, of
\begin{equation}
\mathcal F_{j,\eps}(f,w,s)=
\langle N(f,t)+d(f,t,s)\rangle+\eps\mathcal E(f,w)
-\varphi(f(0),s),\qquad t=s e^{w+\bar w}.
\label{var:penalized-functional}
\end{equation}
Infima are sufficient; no minimizing disc is asserted to exist.
Every fixed admissible smooth disc has finite energy. Also every fixed
disc in \eqref{var:envelope} eventually has $\delta_1=\delta_2=1$ on its
boundary. Consequently
\begin{equation}
\mu_{j,0}\longrightarrow0,\qquad
\mu_{j,\eps}\downarrow\mu_{j,0}\quad(\eps\downarrow0).
\label{var:infimum-limits}
\end{equation}
For a minimizing sequence indexed by $\nu$, at fixed $j,\eps>0$,
nonnegativity of the unpenalized excess gives
\begin{equation}
\limsup_{\nu}\eps\mathcal E_\nu
\le\mu_{j,\eps}-\mu_{j,0},\qquad
\lim_{\eps\downarrow0}\limsup_\nu\eps(1+\mathcal E_\nu)=0.
\label{var:vanishing-penalty}
\end{equation}
As $j\to\infty$ after these limits, the centers approach $(y_*,s_*)$ and
\begin{equation}
\langle d\rangle\longrightarrow0,\qquad
\langle N(f,t)\rangle\le C.
\label{var:small-walls}
\end{equation}
Indeed each of $v-\varphi$, the boundary cost $d$, and $\eps\mathcal E$
is nonnegative and bounded by \eqref{var:penalized-functional}.
The strict contact and the boundary gap in $\mathcal C$ imply the
assertion about centers. They therefore have a uniform interior margin
for large $j$, small $\eps$, and late terms of the sequences.
At fixed $j,\eps$ the energies are bounded. Proposition~\ref{prop:disc-compactness}
then confines all images of $f$ to a common compact $K_{j,\eps}'$.
There is no claim that $K_{j,\eps}'$ is independent of $\eps$.
One may require the initial discs to extend holomorphically past the
circle: replacing $(f,w)$ by $(f(r\zeta),w(r\zeta))$, $r\uparrow1$,
preserves their centers and converges in every boundary norm used here.
For each disc its strict boundary inequalities persist for $r$ close
enough to one. Choosing the approximation separately for each term
preserves all infima and minimizing-sequence assertions above.
\subsection{Boundary calculus and holomorphic deformations}
We first give the boundary estimates used to distinguish the two kinds
of constants. The difference-quotient formula for $D$ gives
\begin{equation}
\begin{split}
D(ab)&\le\|a\|_\infty D(b)+\|b\|_\infty D(a),\\
D(\Psi(a))&\le\Lip(\Psi)D(a),\\
|\langle a,\partial_\theta b\rangle|&\le D(a)D(b).
\end{split}
\label{var:boundary-calculus}
\end{equation}
The last inequality follows directly from Fourier series; it is the
pairing between $\dot H^{1/2}$ and $\dot H^{-1/2}$. All three formulas
hold componentwise for finite-dimensional vectors and matrices, with
fixed dimensional constants. Smooth functions of bounded data can
therefore be composed and multiplied with constants determined by their
smooth bounds on the relevant compact sets. The circular Hilbert
transform preserves $D$ on mean-zero functions.
For a disc let $E_f$ be the frame in Lemma~\ref{lem:disc-frames}, and put
\[
X=G\circ f,\qquad U=dG\,E_f,\qquad p=(\delta_1\delta_2)\circ f,
\qquad q=\log p.
\]
In the boundary estimates below, $p,q$ denote these scalar functions.
The symplectic polynomial $p(\mathcal H)$ is distinguished by its matrix
argument. For an integer
$a\ge4$, to be enlarged once below, let $m=p^a$ and let $h$ be the scalar
outer function with boundary modulus $m$ and $h(0)>0$. Thus
\begin{equation}
h(0)=\exp\langle\log m\rangle,
\quad \|h\|_\infty\le1,
\quad D(h)+D(h/m)\le C_a D(q).
\label{var:outer-wall}
\end{equation}
Here $h/m$ denotes only its boundary phase. To verify the last bound,
write that phase as $\exp(i\mathcal H(aq))$; both the exponential on the
imaginary axis and $q\mapsto e^{aq}$ for $q\le0$ are Lipschitz.
The frame and energy estimates give
\begin{equation}
\|U\|_\infty\le1,\qquad
D(X)+D(U)+D(w)+D(q)\le C(1+\sqrt{\mathcal E}).
\label{var:basic-boundary-bounds}
\end{equation}
Constants absorbing fixed area normalizations are harmless. Since
$t\le T$, the exponential restricted to the range of $2\Re w$ gives
$D(t)\le C_TD(w)$, even if some boundary times approach zero.
For $t$ and $s$ held fixed in the spatial Hessian, define
\begin{equation}
W_b=\dd_Y(N+d),\qquad T_b=m^2E_f^*W_bE_f.
\label{var:boundary-symbol}
\end{equation}
For sufficiently large fixed $a$,
\begin{equation}
\|T_b\|_\infty\le C_j,
\qquad D(T_b)^2\le C_j(1+\mathcal E).
\label{var:symbol-bounds}
\end{equation}
For example
$\dd(\delta^{-1})=2\delta^{-3}\partial\delta\otimes\bar\partial\delta
-\delta^{-2}\dd\delta$.
Multiplication by $p^{2a}$ clears all its denominators and leaves smooth
coefficients on the closed boundary region. Every other coefficient
is smooth on $K_j\times[0,T]\times\mathcal C_s$.
Choose $K_j\subset\mathcal W_j\Subset Y$ with compact closure.
The injective immersion $G$ restricted to $\overline{\mathcal W_j}$
has a continuous inverse onto its image by compactness; thus
$G|_{\mathcal W_j}$ is an embedding.
A finite extension and partition construction from $G(\mathcal W_j)$
near $G(K_j)$ expresses these coefficients as bounded smooth functions
of $X$ there.
The same applies to tensor coefficients evaluated on $U$.
Equation~\eqref{var:symbol-bounds} now follows from
\eqref{var:boundary-calculus} and \eqref{var:basic-boundary-bounds}.
These coefficient extensions are used only along the embedded compact
set; no properness of $G$ is asserted.
\begin{lemma}[Deformation estimate, with its uniformities]
\label{lem:disc-deformation}
Use the preceding localization, with $j$ sufficiently large and
$\eps>0$ sufficiently small that the minimizing-sequence centers have
a common interior margin in $\mathcal C$. Fix a bound $L<\infty$
and such a sequence for \eqref{var:penalized-functional}, with energy
bounded by $E_0<\infty$. Let $l_\nu$ be any holomorphic columns, smooth on the
closed disc, such that
\[
\|l_\nu\|_\infty\le L,\qquad D(l_\nu)\le D_0<\infty.
\]
They may depend on the disc. With
$\varphi_{E,c}=E_f(0)^*\dd_Y\varphi(f(0),s)E_f(0)$, one has
\begin{equation}
\limsup_{\nu\to\infty}
\left\{
h(0)^2\varphi_{E,c}(l(0),\overline{l(0)})
-\langle T_b(l,\bar l)\rangle
-C_j(L)\eps\bigl(1+\mathcal E+D(l)^2\bigr)
\right\}\le0.
\label{var:deformation-estimate}
\end{equation}
The constant $C_j(L)$ depends only on $L$, the dimension, the chosen
integer $a$, the fixed time interval and center cylinder, and smooth
coefficient bounds for the metrics, immersion, Gauss map and cutoffs
on a fixed neighborhood of the boundary compact $K_j$.
It is independent of $\eps,E_0,D_0$, of the interior
compact $K_{j,\eps}'$, and of the auxiliary loss introduced in the proof.
In contrast, the existence radius of the deformations, bounds on their
full coefficients, and bounds on Taylor remainders may depend on
$j,L,E_0,D_0,K_{j,\eps}'$, that loss, the interior margin of the
centers, and the fixed~smooth~test~$\varphi$.
\end{lemma}
\begin{proof}
We give the realization, Taylor justification, and actual mixed-derivative
estimate separately.
\emph{Realization of the first field.}
Let $Z_1,\ldots,Z_b$ be the global holomorphic spanning fields on $Y$.
In the immersion $G$, let $A$ be their column matrix along $f$.
If $r=\dim_\C Y$, enumerate all $r$-row, $r$-column minors $a_i$ of $A$.
On $K_{j,\eps}'$ the sum of their squared moduli has a common positive
lower bound.
Their sup norms and Dirichlet norms are bounded at the fixed parameters:
the minors are smooth holomorphic functions of $X$ on this compact,
and $D(X)$ is bounded. Set
\[
\sigma_i=\frac{\overline{a_i}}{\sum_k|a_k|^2},\qquad
b_{ik}=\sigma_i\bar\partial\sigma_k-
\sigma_k\bar\partial\sigma_i.
\]
These $(0,1)$ coefficients have bounded $L^2(\Delta)$ norm. Solve
$\bar\partial Q_{ik}=b_{ik}$, skew-symmetrically, with bounded
$W^{1,2}$ norm. Explicitly, if $b_{ik}=b_{ik,0}\,d\bar\zeta$,
take $Q_{ik}=4\partial_\zeta\Delta_D^{-1}b_{ik,0}$, with the
convention $\Delta=4\partial_\zeta\partial_{\bar\zeta}$.
Here $\Delta_D^{-1}$ is the zero-Dirichlet inverse on the unit disc.
The unit disc has smooth boundary, the principal coefficients are
the identity, and all lower coefficients vanish. The scalar
Dirichlet estimate therefore gives
\[
\|\Delta_D^{-1}b_{ik,0}\|_{W^{2,2}(\Delta)}
\le C\|b_{ik,0}\|_{L^2(\Delta)}
\]
\cite[Theorem~9.15 and Lemma~9.17]{GilbargTrudinger}, proving the
asserted $W^{1,2}$ bound. Linearity preserves skew symmetry.
Smooth input on the closed disc gives smooth output there
\cite[Theorem~8.13]{GilbargTrudinger}.
Here is the precise boundary estimate. For a smooth function $q$
on the closed disc with boundary Fourier coefficients $a_k$, let
$h(re^{i\theta})=\sum_k a_kr^{|k|}e^{ik\theta}$ be its harmonic
extension. Integration by parts gives
\[
\int_\Delta|\nabla q|^2
=\int_\Delta|\nabla h|^2+\int_\Delta|\nabla(q-h)|^2
\ge2\pi\sum_k|k||a_k|^2.
\]
If $q_0(r)$ is the angular average, then for $1/2\le r\le1$
\[
|q_0(1)|^2\le2|q_0(r)|^2
+2(1-r)\int_r^1|q_0'(s)|^2\,ds.
\]
Integrating in $r$ and applying Jensen bounds the constant
coefficient $|a_0|^2$ by $C\|q\|_{W^{1,2}(\Delta)}^2$.
The nonconstant $L^2$ Fourier sum is bounded by its $|k|$-weighted
sum. Consequently
\[
\|q|_{\partial\Delta}\|_{L^2}^2+D(q|_{\partial\Delta})^2
\le C\|q\|_{W^{1,2}(\Delta)}^2.
\]
Applied entrywise to $Q_{ik}$, this bounds its boundary $D$ and
$L^2$ norms. The operator constants are fixed-disc constants;
the input norms can still depend on $K'_{j,\eps}$.
The functions
\[
g_i=\sigma_i+\sum_kQ_{ik}a_k
\]
are holomorphic and $\sum_i g_i a_i=1$: differentiating uses
$\sum_k a_k\sigma_k=1$ and
$\sum_k a_k\bar\partial\sigma_k=0$, while skew symmetry cancels the
quadratic term in the sum.
A $W^{1,2}$ trace need not be bounded. For a large number $P$ let
$\alpha$ be the scalar outer function with positive center value and
boundary modulus
\[
r_P=(1+|Q|^2/P)^{-1}.
\]
All norms and products involving $Q$ in the following boundary estimates
refer to its trace on the circle.
The maps $Q\mapsto\log(1+|Q|^2/P)$ have Lipschitz constant
$C/\sqrt P$, and their $L^2$ norms are at most $C\|Q\|_2/\sqrt P$.
The maps $Q\mapsto r_PQ$ have bounded Lipschitz constant and sup norm
at most $C\sqrt P$. The phase of $\alpha$ is the Hilbert transform of
$\log r_P$. Thus
\[
\|\alpha\|_\infty\le1,\quad
D(\alpha)\le C D(Q)/\sqrt P,\quad
D(\alpha Q)\le C D(Q),\quad
\|\alpha Q\|_\infty\le C\sqrt P.
\]
In the third estimate the $\sqrt P$ from the bounded product multiplies
the $P^{-1/2}$ bound for the phase. Moreover,
\[
\langle1-|\alpha|^2\rangle\le C\|Q\|_2^2/P,
\qquad -\log\alpha(0)=\langle\log(1+|Q|^2/P)\rangle\longrightarrow0.
\]
Given $\eta>0$, choose $P$ uniformly for the sequence so that these
last two quantities are at most $\eta$ and $D(\alpha)\le1$.
The boundary estimates bound the sup norms and $D$ norms of
$\alpha g_i$. These functions are holomorphic, so the maximum principle
also bounds their interior sup norms at the fixed parameters; it is
not applied to the nonholomorphic products $\alpha Q$.
For each minor $a_i$, Cramer's rule supplies a vector $c_i$, holomorphic
in the disc, with $Ac_i=a_iUl$. Namely use the adjugate on its selected
rows and columns and set the other coefficient entries to zero.
The identity follows by expanding the augmented determinants: every
$(r+1)$-minor of $(A,Ul)$ is zero because this matrix still has rank $r$.
This also treats selected minors that vanish identically.
Consequently $c=\sum_i\alpha g_i c_i$ has bounded sup norm and $D$ and
obeys $Ac=\alpha Ul$. Write $c=(c_1,\ldots,c_b)$ and compose the
complex flows of $Z_1,\ldots,Z_b$
with times $z h c_1,\ldots,z h c_b$, starting at $f$.
For a sufficiently small common complex parameter $z$ this gives a
holomorphic family $f_z$, smooth up to the circle, whose first field is
\begin{equation}
V=\left.\partial_z f_z\right|_0=\alpha hE_fl.
\label{var:first-field}
\end{equation}
The existence radius is uniform at the fixed parameters, since the
initial images lie in $K_{j,\eps}'$ and all flow times are bounded.
On the boundary the displacement is $m$ times a uniformly bounded
smooth function. Because $m\le\delta_i^a$, decreasing this radius
makes each new $\delta_i$ comparable to its old value, with constants
between $1/2$ and $3/2$. Thus the spatial boundary restrictions persist.
The variable $w$ and all times $t$ are unchanged, so no uniform margin
below the upper time boundary is needed. The center stays in
$\mathcal C$ by its already established margin.
\emph{Uniform Taylor remainders at the fixed parameters.}
We spell this out because smoothness of individual discs does not by
itself imply the uniform expansion needed for an infimum.
On the controlled interior compact the flow map and all its derivatives
through any fixed finite order are bounded. Differentiating in the
disc variable introduces one derivative of $f$, $h$, or $c$.
Their Dirichlet norms are bounded; for $f$ use local metric equivalence
on $K_{j,\eps}'$ and \eqref{var:energy}. Hence the family and its first
three real parameter derivatives have bounded $W^{1,2}$ norms, with
bounded sup norms for their values. Differentiating the area integrands
through order three yields only bounded coefficients times products of
at most two such first disc derivatives. Cauchy--Schwarz supplies
uniform integrable bounds. The same reasoning applies to the Gauss
map, whose derivatives are bounded on the controlled compact.
At the boundary write $\beta=h/m$. The boundary values of $f_z$ can
be written in ambient coordinates as a smooth function of $X$, $c$,
$\beta$, and $z m$, on a fixed compact range. More precisely, for
either cutoff,
\[
\delta_i(f_z)=\delta_i(f)+m R_i(z,X,c,\beta),\qquad R_i(0,\cdot)=0,
\]
where the real parameter derivatives through order three of $R_i$
have bounded sup and $D$ norms at the fixed parameters, by
\eqref{var:boundary-calculus}. Since
$m/\delta_i=\delta_i^{a-1}\delta_{3-i}^{a}$ is smooth on the closed
boundary region, the function
\[
\log\delta_i(f_z)-\log\delta_i(f)
=\log\bigl(1+(m/\delta_i)R_i\bigr)
\]
and its parameter derivatives through order three have uniformly
bounded $D$ norms. For reciprocal walls, use
\[
\frac1{\delta_i(f_z)}-\frac1{\delta_i(f)}
=-\frac{m}{\delta_i^2}\,
\frac{R_i}{1+(m/\delta_i)R_i}.
\]
The prefactor is smooth and bounded when $a\ge2$. Therefore all
nonzero parameter derivatives of the wall costs through order three
are uniformly bounded, even when the unchanged wall cost is large.
The remaining boundary integrands $N,d_{0,j}$ have bounded derivatives
on $K_j\times[0,T]$. Finally the quadratic seminorm of the logarithmic
wall has three bounded parameter derivatives by its Hilbert space
pairing formula. These observations prove a uniform $O(|z|^3)$
remainder after parameter-circle averaging of the entire functional
minus the test. The constants in this paragraph are allowed to depend
on the interior compact and on $\eta$.
\emph{The actual mixed energy derivative uses boundary constants only.}
Set $\widehat V=dG\,V=\alpha hUl$ on the circle. Equations
\eqref{var:boundary-calculus}--\eqref{var:first-field} imply
\begin{equation}
\|\widehat V\|_\infty\le L,
\qquad D(\widehat V)^2\le
C_j(L)\bigl(1+\mathcal E+D(l)^2\bigr).
\label{var:first-field-bound}
\end{equation}
Here only $\|\alpha\|_\infty\le1$ and $D(\alpha)\le1$ were used;
the corona coefficients have disappeared.
For clarity, consider any one of the closed $(1,1)$ forms $\omega$ in
the three spatial area terms of \eqref{var:energy}. It is a smooth
closed form on $Y$ after pullback. In ambient coordinates near the
boundary write its intrinsic tensor as
$i b_{p\bar q}(X)\,dX^p\wedge d\bar X^q$, extending the coefficients
smoothly from $G(\mathcal W_j)$ near $G(K_j)$ as above. Deformed
boundaries stay intrinsically in $\mathcal W_j$ for a smaller parameter
radius. Closure is used intrinsically
before this extension. Cartan's formula and Stokes give
\begin{equation}
\begin{split}
\left.\partial_z\partial_{\bar z}
\int_\Delta f_z^*\omega\right|_0
=i\int_0^{2\pi}\bigl[&
b_{p\bar q}(X)\widehat V^p\partial_\theta
\overline{\widehat V^q}\\
&+(\partial_{\bar r}b_{p\bar q})(X)
\widehat V^p\overline{\widehat V^r}\,
\partial_\theta\overline{X^q}\bigr]\,d\theta.
\end{split}
\label{var:boundary-area-identity}
\end{equation}
One can also verify the identity using local potentials: the mixed
parameter derivative equals the boundary normal derivative of
$\omega(V,\bar V)$, with the corresponding Stokes normalization.
In \eqref{var:boundary-area-identity}, the holomorphic dependence of
$Gf_z$ eliminates a mixed second parameter field. The identity is
global; it does not require partitioning $\omega$ into nonclosed forms.
The coefficient $b$ has bounded $C^2$ norm determined only by $K_j$.
For $B=\|\widehat V\|_\infty$,
\[
\begin{split}
D(b(X)\widehat V)&\le C_j\bigl(D(\widehat V)+B D(X)\bigr),\\
D((\bar\partial b)(X)\widehat V\overline{\widehat V})
&\le C_j\bigl(BD(\widehat V)+B^2D(X)\bigr).
\end{split}
\]
Pairing each angular derivative by \eqref{var:boundary-calculus}
bounds the absolute value of \eqref{var:boundary-area-identity} by
$C_j(L)(D(X)^2+D(\widehat V)^2)$. This proves the required estimate
for all spatial areas. The $w$ area is unchanged. In particular for
the Euclidean area the identity reduces, up to its fixed normalization,
to $D(\widehat V)^2$.
It remains to check the logarithmic seminorm; its unbounded argument
cannot be handled by an unweighted smooth-composition assertion.
On the boundary put $Z=\alpha\beta Ul$, so that $\widehat V=mZ$.
Then
\[
\|Z\|_\infty\le L,\qquad
D(Z)\le C_j(L)(1+\sqrt{\mathcal E}+D(l)).
\]
Regarding $p=\delta_1\delta_2$ as a smooth ambient function, the tensors
\begin{equation}
\begin{split}
m\partial\log p&=p^{a-1}\partial p,\\
m^2\dd\log p&=p^{2a-1}\dd p
-p^{2a-2}\partial p\otimes\bar\partial p
\end{split}
\label{var:cleared-log-tensors}
\end{equation}
extend smoothly to the closed boundary region.
For $q_z=\log p(f_z)$ their contractions against $Z$ and
$(Z,\bar Z)$ are exactly $\partial_zq_z|_0$ and
$\partial_z\partial_{\bar z}q_z|_0$, respectively. Consequently each
has $D$ bounded by $C_j(L)(1+\sqrt{\mathcal E}+D(l))$.
Write $b=\partial_zq_z|_0$ and $c=\partial_z\partial_{\bar z}q_z|_0$,
and let $B_D$ be the sesquilinear pairing
$B_D(u,v)=\sum_k|k|u_k\overline{v_k}$. Since $q_z$ is real,
differentiating this quadratic seminorm gives exactly
\begin{equation}
\left.\partial_z\partial_{\bar z}D(q_z)^2\right|_0
=2D(b)^2+2\Re B_D(q,c).
\label{var:log-energy-identity}
\end{equation}
Since $D(q)^2\le\mathcal E$, Cauchy--Schwarz proves
\begin{equation}
\left|\left.\partial_z\partial_{\bar z}
\mathcal E(f_z,w)\right|_0\right|
\le C_j(L)(1+\mathcal E+D(l)^2).
\label{var:actual-energy-derivative}
\end{equation}
All constants in \eqref{var:actual-energy-derivative} use boundary
coefficients only, independently of the realization and Taylor constants.
\emph{Testing the infimum and removing the auxiliary loss.}
Every deformed value of \eqref{var:penalized-functional} is at least
$\mu_{j,\eps}$, while the values at $z=0$ approach it. Average over
$|z|=r$, subtract the value at zero, and divide by $r^2$.
First take the minimizing-sequence limit with $r$ fixed, and then
$r\downarrow0$. The uniform remainder just proved yields a
nonnegative lower limit for the mixed derivative of cost minus test.
Its nonpenalty terms are
\[
\langle|\alpha|^2 T_b(l,\bar l)\rangle
-|\alpha(0)|^2h(0)^2\varphi_{E,c}(l(0),\overline{l(0)}).
\]
The center frames and their inverses are bounded independently of
interior area by Lemma~\ref{lem:disc-frames}. Thus the test term is
bounded, as is the boundary symbol by \eqref{var:symbol-bounds}.
Removing $\alpha$ changes these terms by at most $C_{j,L,\varphi}\eta$.
Equation~\eqref{var:actual-energy-derivative} has a coefficient
independent of $\eta$. Send $\eta\downarrow0$ after the sequence and
radius limits. This proves \eqref{var:deformation-estimate} with the
stated uniformities. In particular, none of the constants controlling
the shrinking realization radius multiplies the final $\eps\mathcal E$
error.
\end{proof}
\subsection{Removing the negative part and factoring the boundary form}
The allowance for disc-dependent columns in
Lemma~\ref{lem:disc-deformation} is essential here. Let
$S=(T_b)_-$ be the positive semidefinite negative part of $T_b$.
Matrix clipping is Lipschitz in the Frobenius norm, so
$\|S\|_\infty\le C_j$ and $D(S)^2\le C_j(1+\mathcal E)$.
Let $S_P$ be its Fej\'er mean of order $P$. Positivity of the Fej\'er
kernel preserves its sup bound, and its Fourier multipliers give
\begin{equation}
\|S_P-S\|_2^2\le C_j(1+\mathcal E)/P.
\label{var:fejer-error}
\end{equation}
Each column $l_i(\zeta)=\zeta^{P+1}S_P(\zeta)e_i$ is holomorphic,
vanishes at zero, and has
\[
\|l_i\|_\infty\le C_j,\qquad D(l_i)^2\le C_j(P+1).
\]
The frequency shift has modulus one on the circle, so summing
\eqref{var:deformation-estimate} over these columns tests
$\tr(T_bS_P^2)$. In comparison,
$\tr(T_bS^2)=-\tr(S^3)$.
For fixed $j,\eps$, discard finitely many sequence terms and let
$B_\eps$ bound $1+\mathcal E_\nu$, with
$\eps B_\eps\to0$ as $\eps\downarrow0$, as permitted by
\eqref{var:vanishing-penalty}. Choose an integer $P=P_\eps$, fixed
along this sequence, with
\[
B_\eps/P_\eps\longrightarrow0,
\qquad \eps P_\eps\longrightarrow0;
\]
for example round $\sqrt{B_\eps/\eps}$ upward.
The trace error is at most $C_j\|S_P-S\|_2$ by boundedness of both
matrices. Equations~\eqref{var:deformation-estimate} and
\eqref{var:fejer-error} therefore imply
\begin{equation}
\lim_{\eps\downarrow0}\limsup_\nu
\langle\tr((T_b)_-^3)\rangle=0,
\qquad
\lim_{\eps\downarrow0}\limsup_\nu\|(T_b)_-\|_1=0.
\label{var:negative-part}
\end{equation}
The second conclusion follows by H\"older's inequality in fixed dimension.
This argument uses both sides of the intermediate frequency choice;
a bound merely of the form $\eps\mathcal E=O(1)$ would not suffice.
Fix $\delta>0$. Apply the smooth scalar function
$\lambda\mapsto\delta+(\lambda+\sqrt{\lambda^2+\delta^2})/2$
to $T_b$ and call the resulting positive boundary matrix $L$.
Then
\[
L\ge T_b+\delta I,\qquad L\ge\delta I,
\qquad \|L-(T_b)_+\|\le\tfrac32\delta.
\]
Functional calculus in fixed dimension gives
$D(L)\le C_{j,\delta}D(T_b)$.
Lemma~\ref{lem:boundary-factorization} gives
$L=H^*H$, with $H,H^{-1}$ holomorphic and smooth on the closed disc;
this is the smooth boundary factorization of
\cite[Theorem~2.1]{BerndtssonRosay}, with the quantitative seminorm
estimate established in Section~\ref{sec:discs}.
For every constant unit column $e$,
\[
l=H^{-1}e,\qquad \|l\|_\infty\le\delta^{-1/2},
\qquad D(l)^2\le C_{j,\delta}(1+\mathcal E).
\]
These are permissible disc-dependent tests at fixed $j,\eps,\delta$.
Since $T_b\le L$, their boundary quadratic costs are at most one.
Equation~\eqref{var:vanishing-penalty} now gives
\begin{equation}
h(0)^2\varphi_{E,c}\le(1+o(1))H(0)^*H(0)
\label{var:center-majorant}
\end{equation}
as the sequence limit and then $\eps\downarrow0$ are taken at fixed
$j,\delta$. To see that this is a matrix assertion, if the largest
violation did not tend to zero, choose its unit eigenvector separately
for each disc. Those columns satisfy exactly the same sup and $D$
bounds, contradicting Lemma~\ref{lem:disc-deformation}.
Thus no fixed-test assumption has entered either the Fej\'er test or
the spectral-factor test.
\subsection{The backward clock and the symplectic cost}
Choose cutoffs $0\le\chi_i\le1$ which are one wherever $\delta_i$
differs, or starts to differ, from one. Require
\[
\supp\chi_1\subset\{u>A_j/3\},\qquad
\supp\chi_2\subset\{N(y,0)>c_2B_j^2\}
\]
for a fixed $c_2>0$. Before choosing $B_j$ we can choose $C'_j\ge1$
depending on the base compact, $d_{0,j}$ and the time interval, and a
fixed integer $c_3$, such that
\begin{equation}
\begin{split}
\bigl|p(W_b)-\partial_tN
-\Delta_{h_t}\bigl(d_{0,j}+(1+s)k_1\bigr)\bigr|
\le C'_j(1+B_j)^{c_3}
(\delta_1\delta_2)^{-c_3}\boldone_{\{\chi_2=1\}}.
\end{split}
\label{var:fiber-wall-error}
\end{equation}
Indeed Lemma~\ref{lem:symplectic-evolution} gives the exact equality
without the second wall. The remaining expression is a polynomial
of degree at most two in the Hessian entries. On a compact base,
$N(y,t)$ and $N(y,0)$ are quadratic in the fiber variables, and their
fixed finite derivatives grow polynomially in those variables.
Differentiating the reciprocal cutoff twice introduces at most
three inverse powers of $\delta_2$; the analogous first-wall factors
have at most three inverse powers of $\delta_1$.
Enlarging a fixed integer $c_3$ bounds all these finitely many terms.
This proves \eqref{var:fiber-wall-error}, including that $C'_j$ is
chosen before $B_j$.
Also
\begin{equation}
|\Delta_{h_t}k_1|\le C e^\rho\delta_1^{-3}
\boldone_{\{\chi_1=1\}}.
\label{var:base-wall-error}
\end{equation}
Here the controlled gradient and complex Hessian of $u$ bound
$|\dd k_1|_g$ by $C\delta_1^{-3}$ uniformly in large $A_j$.
Since $0<h_t\le g$, each inverse eigenvalue relative to $g$ is at
most $e^\rho$, proving \eqref{var:base-wall-error}.
Fix integers $a,c$ sufficiently large that $4a\ge c_3+3$ and
$c>c_3+3$, retaining all earlier requirements on $a$. Put
\begin{equation}
\begin{split}
\gamma_0={}&m^4\exp\bigl[-\chi_1(\rho+c\log(j+2))\bigr]\\
&\mathrel{}\cdot\exp\bigl[-\chi_2(
\log C'_j+c\log(j+2)+c\log(1+B_j))\bigr],\\
\gamma={}&\gamma_0/\langle\gamma_0\rangle.
\end{split}
\label{var:clock-weight}
\end{equation}
Choose $B_j$ so large that $\log C'_j/B_j^2\to0$.
The preceding choice of $C'_j$ makes this noncircular.
Since $-\log\delta_i\le k_i$, \eqref{var:small-walls} controls
$\langle-\log m\rangle$. On $\supp\chi_1$,
\eqref{var:tail-wall} controls both $\rho$ and $j$ by $d_{0,j}$.
On $\supp\chi_2$, the pointwise boundary inequality
$N(f,t)\ge N(f,0)$ and \eqref{var:small-walls} give
$\langle\chi_2\rangle\le C/B_j^2$.
It follows, in the stated successive limits, that
\begin{equation}
\langle-\log\gamma_0\rangle\longrightarrow0,
\qquad \langle\gamma_0\rangle\longrightarrow1,
\qquad \langle\log\gamma\rangle\longrightarrow0.
\label{var:clock-loss}
\end{equation}
For example, the second limit also follows from
$e^{\langle\log\gamma_0\rangle}\le\langle\gamma_0\rangle\le1$.
The powers and exponentials in \eqref{var:clock-weight} bound the
weighted errors in \eqref{var:fiber-wall-error} and
\eqref{var:base-wall-error} by a quantity tending to zero. On the
support of each error its corresponding cutoff equals one, so the
cancellation is literal; $m^4$ clears the displayed inverse wall
powers, $e^{-\rho}$ clears $e^\rho$, and
$(C'_j)^{-1}(1+B_j)^{-c}(j+2)^{-c}$ clears the fiber error.
The denominator $\langle\gamma_0\rangle$ tends to one, and $t/s$ is
bounded because centers stay in $\mathcal C$.
Since $(\partial_t-\Delta_{h_t})d_{0,j}\ge0$, we obtain
\begin{equation}
\left\langle\frac ts\gamma(\partial_tN+\partial_t d_{0,j})
+k_1+k_2\right\rangle
\ge\left\langle\frac ts\gamma p(W_b)\right\rangle-o_j(1).
\label{var:clock-comparison}
\end{equation}
Here $o_j(1)$ tends to zero after the earlier limits.
The clock test giving the left side of \eqref{var:clock-comparison}
must run backward. At fixed $j$,
\[
\|\gamma\|_\infty\le C_j,\qquad
D(\gamma)^2\le C_j(1+\mathcal E).
\]
These follow by \eqref{var:boundary-calculus} from the smooth
coefficients in \eqref{var:clock-weight}; the denominator has a
positive lower bound by Jensen and the bounded cost.
For an explicit bound, let $M$ bound the unpenalized boundary cost
along the fixed-$j$ sequences. Then
$\langle k_1+k_2\rangle\le M$,
$\langle-\log p\rangle\le M$, and
$\langle\chi_1\rho\rangle\le M$ by the tail wall.
Hence
\[
\langle-\log\gamma_0\rangle
\le (4a+1)M+2c\log(j+2)+\log C'_j+c\log(1+B_j),
\]
and Jensen gives a positive lower bound for
$\langle\gamma_0\rangle$ depending only on these fixed-$j$ data.
No lower bound on the individual cutoffs or boundary times is used.
Let $\Gamma$ be holomorphic with boundary real part $\gamma-1$ and
$\Gamma(0)=0$. For $s'<s$ set
\[
w'=w+\tfrac12\log(s'/s)\Gamma.
\]
On the circle the new time is $t(s'/s)^\gamma$.
In the interior its exponent is the positive harmonic extension of
$\gamma$. Hence every time decreases, including when the original
disc has arbitrarily little margin below $T$.
The center of $w'$ is still zero.
Comparing the perturbed functional with its infimum, first along the
minimizing sequence and then letting $s'\uparrow s$, gives
\begin{equation}
\partial_s\varphi(f(0),s)
\ge\left\langle\frac ts\gamma(\partial_tN+\partial_t d_{0,j})
+k_1+k_2\right\rangle
-C_j\eps(1+\mathcal E)-o(1).
\label{var:backward-test}
\end{equation}
The sign follows because the quotient denominator $s'-s$ is negative.
For the energy error only the $w$ area changes, and
$D(\Gamma)\le C D(\gamma)$ gives its derivative bound by
Cauchy--Schwarz. Its Taylor expansion is quadratic in $\log(s'/s)$.
For the boundary cost, derivatives of $t(s'/s)^\gamma$ are uniformly
bounded at fixed $j$; $N,d_{0,j}$ are smooth down to $t=0$ on $K_j$.
The possibly unbounded unchanged walls multiply $1+s'$ affinely and
thus have no second time remainder. These facts justify the
sequence-then-difference limit in \eqref{var:backward-test} without a
bound on the imaginary part of $\Gamma$.
Let $\mathcal J_E$ be the bivector components in $E_f$. Its holomorphic
sup bound depends only on $K_j$ by Lemma~\ref{lem:disc-frames}.
Since $p$ is quadratic in the Hermitian form,
$p(W_b)=m^{-4}p(T_b,\mathcal J_E)$.
At fixed $j$, the factor $\gamma m^{-4}$ is bounded. Therefore
\eqref{var:negative-part}, the uniform symbol bounds, and
$L=(T_b)_++O(\delta)$ allow the replacement
\begin{equation}
\left\langle\frac ts\gamma p(W_b)\right\rangle
=\left\langle\frac ts\gamma m^{-4}p(L,\mathcal J_E)\right\rangle
+O_j(\delta)+o(1)
\label{var:positive-replacement}
\end{equation}
after $\eps\downarrow0$.
The vector $(\bigwedge^2H)\mathcal J_E$ is holomorphic and nonzero.
Logarithmic submean for its squared norm, followed by scalar Jensen,
gives
\begin{equation}
\begin{split}
\left\langle\frac ts\gamma m^{-4}p(L,\mathcal J_E)\right\rangle
&\ge \exp\langle\log\gamma\rangle\,
h(0)^{-4}p(H(0)^*H(0),\mathcal J_E(0)).
\end{split}
\label{var:symplectic-submean}
\end{equation}
Indeed $\langle\log(t/s)\rangle=2\Re w(0)=0$ and
$\log h(0)=\langle\log m\rangle$, which give precisely the factors
in \eqref{var:symplectic-submean}. If desired, logarithmic submean
follows by applying submean to $\log(|Z|^2+\eta)$ and then
$\eta\downarrow0$ for the displayed holomorphic vector $Z$.
\subsection{Passing from positive majorants to the required form}
\begin{lemma}[Symplectic cost and the Schur complement]
\label{lem:symplectic-schur}
In a cotangent vector space with canonical bivector
$\mathcal J=\sum_i e_i\wedge f_i$, let $\mathcal H>0$ be Hermitian,
and suppose $\mathcal H\ge W$, where the vertical block $C$ of $W$
is positive. Then
\begin{equation}
S=\Schur(\mathcal H)\ge0,\qquad
S\ge\Schur(W),\qquad p(\mathcal H)\ge\tr(C^{\mathsf T}S).
\label{var:schur-cost}
\end{equation}
\end{lemma}
\begin{proof}
Write $\mathcal H=\left(\begin{smallmatrix}A&B\\B^*&D\end{smallmatrix}\right)$,
$S=A-BD^{-1}B^*$ and $R=D^{-1}B^*$.
The complex-linear change $(x,y)\mapsto(x,y+Rx)$ makes the form
block diagonal with blocks $(S,D)$. It carries the canonical bivector
to
\[
\sum_i e_i\wedge f_i+\sum_{k,i}R_{ki}f_k\wedge f_i.
\]
Its cross and vertical components are orthogonal for the induced
exterior-square form, whence
\[
p(\mathcal H)=\tr(SD^{\mathsf T})+
\left|\sum_{k,i}R_{ki}f_k\wedge f_i\right|_D^2.
\]
The shear need not be symplectic; its additional vertical term is
retained and is nonnegative. Since $D\ge C$ and $S>0$, the last
identity gives $p(\mathcal H)\ge\tr(SC^{\mathsf T})$.
Finally, for each horizontal $x$,
\[
S(x,\bar x)=\inf_y\mathcal H((x,y),\overline{(x,y)})
\ge\inf_y W((x,y),\overline{(x,y)})
=\Schur(W)(x,\bar x).
\]
The last infimum is finite and has the usual Schur expression because
$C>0$. This proves the Lemma.
\end{proof}
\begin{proof}[Completion of the proof of Proposition~\ref{prop:disc-inequality}]
Combine \eqref{var:center-majorant}, \eqref{var:clock-comparison},
\eqref{var:backward-test}, \eqref{var:positive-replacement}, and
\eqref{var:symplectic-submean}. More explicitly, if $r\ge0$ denotes
the error tending to zero in \eqref{var:center-majorant}, the form
with frame matrix
\[
\mathcal H_E=(1+r)h(0)^{-2}H(0)^*H(0)
\]
is a positive majorant of the test Hessian at its center, after an
arbitrarily small enlargement if necessary. Transform it to the
fixed cotangent coordinates. Its symplectic cost is
\[
p(\mathcal H)=(1+r)^2h(0)^{-4}
p(H(0)^*H(0),\mathcal J_E(0)).
\]
For fixed $j,\delta$ first take the minimizing-sequence limits and
$\eps\downarrow0$; then send $\delta\downarrow0$; finally let
$j\to\infty$. Equations~\eqref{var:small-walls} and
\eqref{var:clock-loss} give a diagonal sequence of approaching centers
and positive majorants $\mathcal H$ such that
\[
\limsup p(\mathcal H)\le\partial_s\varphi(y_*,s_*).
\]
No compactness of the complete matrices $\mathcal H$ is needed.
Their test vertical blocks $C_c$ converge to $C>0$, so for late
centers $C_c\ge cI$ with a fixed $c>0$.
Lemma~\ref{lem:symplectic-schur} bounds their positive Schur forms
$S_c$ by $c\tr S_c\le p(\mathcal H)$.
Pass to a convergent subsequence $S_c\to K$. The same Lemma and
continuity of the test Hessian and its Schur complement give
\[
K\ge0,\qquad K\ge\Schur(W),\qquad
\tr(C^{\mathsf T}K)\le\partial_s\varphi(y_*,s_*).
\]
This is \eqref{var:viscosity-inequality} and proves the Proposition.
\end{proof}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/variation.tex ====
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\pdfglyphtounicode{tfm:t5-lmss8/T}{1D5B3}
\pdfglyphtounicode{tfm:texnansi-lmss8/T}{1D5B3}
\pdfglyphtounicode{tfm:ts1-lmss8/T}{1D5B3}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//unicode-map.tex ====