Every complete connected noncompact Kaehler manifold with strictly
positive holomorphic bisectional curvature is biholomorphic to complex
Euclidean space, in every complex dimension $n \ge 1$. This is the Yau
uniformization conjecture, asked in 1982 and answered here without any
extra hypothesis on curvature size, volume growth, noncollapsing,
Ricci pinching, topology at infinity, or geometry of the metric.
Strict positivity of the bisectional curvature is pointwise. The
conclusion is a biholomorphism onto all of $\mathbb{C}^n$, not onto a
proper domain; the initial metric is not asserted to be flat.

./pdf/yau-1982-survey-pde-differential-geometry.pdf

Recall the objects. A Kaehler metric $g$ on a complex manifold $M$ is a
Riemannian metric whose imaginary part closes: $d\omega_g = 0$ for
$\omega_g = i g_{i\bar{j}} dz^i \wedge d\bar{z}^j$ in holomorphic
coordinates. The Riemann tensor is $R_{i\bar{j}k\bar{l}}$; strictly
positive holomorphic bisectional curvature means
$$R_{i\bar{j}k\bar{l}} v^i \bar{v}^j w^k \bar{w}^l > 0$$
at every point and for every pair of nonzero complex $(1,0)$ tangent
vectors $v, w$. Equivalently $T^{1,0}M$ is Griffiths positive as a
Hermitian holomorphic bundle with the Kaehler connection. In $n=1$
this is strict positivity of the Gaussian curvature.

./pdf/griffiths-1969-curvature-and-cohomology.pdf

In dimension one the answer is classical: a complete noncompact
Riemann surface of positive Gaussian curvature is conformally the
plane, combining the Cohn-Vossen topology theorem with the
Blanc-Fiala-Huber parabolicity theorem.

./pdf/huber-1957-subharmonic-functions-differential-geometry.pdf

In the compact case the Frankel conjecture, that positive holomorphic
bisectional curvature forces biholomorphism to projective space, was
settled independently by Mori through the minimal model program and by
Siu-Yau through the Kaehler-Ricci flow.

./pdf/mori-1979-projective-manifolds-ample-tangent.pdf
./pdf/siu-yau-1980-compact-kahler-positive-bisectional.pdf

Yau's survey asked for the noncompact analogue, replacing projective
space by affine space and allowing arbitrary geometry at infinity. Two
families of partial results predate the present proof. Function
theoretic antecedents: Greene-Wu constructed smooth strictly convex
exhaustions under positive real sectional curvature and hence derived
Steinness; Mok embedded the manifold into an affine algebraic variety
under positive bisectional curvature, maximal volume growth, and
quadratic scalar curvature decay; Chen-Zhu proved intrinsic growth and
average curvature estimates under pointwise positive bisectional
curvature alone, with constants that may depend on the center of a
ball; Ni-Tam analysed heat deformations of plurisubharmonic functions
under nonnegative bisectional curvature.

./pdf/greene-wu-1976-cinfinity-convex-functions.pdf
./pdf/mok-1984-embedding-kahler-affine-algebraic.pdf
./pdf/chen-zhu-2005-volume-growth-curvature-decay.pdf
./pdf/ni-tam-2003-plurisubharmonic-structure.pdf

Flow theoretic antecedents: Cao's differential Harnack inequalities for
the Kaehler-Ricci flow underlie every such argument; Chau-Tam
established uniformization under bounded nonnegative bisectional
curvature and maximal volume growth, and later developed locally
biholomorphic charts, covering methods, and Steinness results under
further decay or pinching hypotheses; Liu reached uniformization under
nonnegative bisectional curvature and maximal volume growth without an
upper curvature bound; Lee-Tam recovered the same conclusion by
smoothing and reducing to the bounded-curvature theorem. All retain the
maximal volume growth hypothesis.

./pdf/cao-1992-harnack-kahler-ricci-flow.pdf
./pdf/chau-tam-2006-complex-structure-nonneg-curvature.pdf
./pdf/chau-tam-2008-steinness-kahler.pdf
./pdf/chau-tam-2011-simply-connected.pdf
./pdf/liu-2019-uniformization-yau.pdf
./pdf/lee-tam-2020-chern-ricci-flow-noncompact.pdf

Two obstructions in the unrestricted problem. First, the initial metric
may have unbounded curvature and may collapse arbitrarily, so a
bounded-curvature flow theorem or a global tensor maximum principle
cannot be invoked before its hypotheses have been established. Second,
shrinking charts alone identify $M$ only with a proper domain of
$\mathbb{C}^n$; surjectivity requires quantitative control of the
chart changes even when contraction rates vary substantially along the
flow.

The proof divides into the following steps. A global Kaehler-Ricci flow
$h_t$ with $h_0 = g$ is constructed as a limit of complete Hermitian
flows on exhausting domains, with estimates that survive the limit
without a curvature bound. A spatial weight adapted to the nonnegative
form $g - h_t$ proves scalar comparison before completeness of the
evolving metric is known; completeness of the positive-time slices then
follows from the trace Harnack inequality. Joint positivity of the
cotangent squared norm in space and in logarithmic time is obtained by
minimizing its boundary values over holomorphic discs and then
optimizing a Hermitian ellipsoid in each fiber; the two operations are
dual aspects of the same Levi problem, and their composition yields the
matrix Harnack inequality. Quantitative shrinking charts at every
sufficiently large volume clock, a static distortion bound in the
initial metric, and a selection of late times on which the rescaled
Ricci tensor is pinched at the base point together contract every loop
and make $M$ simply connected. Coherent inverse charts on the
exhaustion then give a single contraction exponent valid on every
fixed compact set. The volume jets of the transition maps are
normalized and matched by polynomial automorphisms of constant
Jacobian, so that backward iteration produces an injective holomorphic
map $M \to \mathbb{C}^n$. A subsequential forward degree bound together
with smallness on compact subsets of the image eliminates a proper
image and yields the desired biholomorphism.

Each step is now substantiated in turn.


Fix $o \in M$ and $r(x) = d_g(o, x)$, with distances, balls, norms and
volumes measured in the initial complete metric $g$. The following
analytic data exists on $M$.

(i) A smooth strictly plurisubharmonic exhaustion $\psi \le C(1+r)$.

(ii) Holomorphic functions and tangent fields interpolating arbitrary
finite jets on finite point sets. In particular finitely many holomorphic
functions define an injective immersion $F : M \to \mathbb{C}^N$, and
finitely many global holomorphic tangent fields span $T^{1,0}M$ pointwise.

(iii) At every $x$ a global holomorphic canonical section $S$ with
$S(x) \ne 0$ and $\log |S|_g^2 \le C_S(1+r)$, and a smooth $\Xi$ with
$$\partial\bar\partial \Xi \ge \Ric(g), \qquad \Xi \le C(1 + r^{3/2}).$$

(iv) A smooth exhaustion $u \ge 1$ satisfying
$$C^{-1}(1+r) \le u \le C(1+r), \quad |\partial u|_g \le C, \quad
|\partial\bar\partial u|_g \le C/u.$$

(v) For $L \ge 1$ and $S(o) \ne 0$ from (iii),
$$\frac{1}{\Vol B(o,L)} \int_{B(o,L)} \big| \log |S|_g^2 \big| \, dV_g
\le CL, \quad
\frac{1}{\Vol B(o,L)} \int_{B(o,L)} R_g \, dV_g \le C/L.$$

Neither $F$ nor $\psi$ is proper. The exhaustion in (iv) has a two-sided
complex Hessian bound, which the flow arguments use.

Construction of $\psi$. Take a unit-speed ray $\gamma$ from $o$ and put
$$b(x) = \lim_{j \to \infty} \big( d_g(o, \gamma(j)) -
d_g(x, \gamma(j)) \big).$$
The triangle inequality gives monotonicity in the braces, uniform
boundedness on compact sets, and $|b(x) - b(y)| \le d_g(x,y)$.

Fix $x$ in a compact set and a minimizing segment of length $l$ from $x$
to $\gamma(j)$. For an initial vector $X$, parallel-translate it and
multiply by $1 - \sigma/l$; use the same construction for $JX$. On
perpendicular components, the second variation of length has curvature
contribution the negative of
$$\int_0^l \big( 1 - \sigma/l \big)^2 \big\{ \Rm(\dot\gamma, X_\parallel,
X_\parallel, \dot\gamma) + \Rm(\dot\gamma, JX_\parallel, JX_\parallel,
\dot\gamma) \big\} \, d\sigma.$$
Removing radial components does not change this expression. It is a
positive multiple of the bisectional curvature of the complex 2-plane
spanned by $\dot\gamma$ and $X$. On a fixed short initial subsegment,
that positivity is bounded below uniformly in the unit initial vector
and in the base point inside the compact. The derivative term in the
index form is $O(l^{-1})$. Hence
$$\partial^2_{X,\bar X} d_g(\cdot, \gamma(j)) + \partial^2_{JX,
\overline{JX}} d_g(\cdot, \gamma(j)) \le -c_K |X|^2 + O(l^{-1}) |X|^2,$$
where $c_K > 0$ is independent of $j$.

The comparison does not need a smooth distance function. Varying the
initial endpoint along a holomorphic germ and using the comparison
paths produces smooth upper supports for distance. For the two real
directions of the germ, the sum of endpoint acceleration terms
vanishes, so their negatives give lower supports for the signed
distance with the same strict Levi lower bound. An upper test for the
signed distance dominates its lower support and inherits that bound.
Local uniform passage to $b$ preserves the inequality; testing on
complex lines gives the subharmonic version. So $b$ is strictly
plurisubharmonic, with a positive Levi lower bound on each compact.
Richberg approximation with pointwise error below one yields a smooth
strictly plurisubharmonic $\psi$ with $\psi \le 1+r$ after adding a
constant.

./pdf/richberg-1968-stetige-streng-pseudokonvexe.pdf
./pdf/greene-wu-1979-approximations-convex-plurisubharmonic.pdf


Holomorphic interpolation. Let $K = \bigwedge^n T^{*(1,0)}M$. A
function, a canonical section, and a tangent field are respectively
holomorphic top forms with coefficients in
$$K^{-1}, \qquad \mathbf{1}, \qquad T^{1,0}M \otimes K^{-1}.$$
The first has nonnegative curvature, the second is flat, and the third
has nonnegative Nakano curvature: it is $E \otimes \det E$ with
$E = T^{1,0}M$ Griffiths positive, so the Demailly-Skoda
tensor-with-determinant theorem applies. These are statements about
the listed coefficient bundles; no Nakano positivity of $E$ itself is
assumed.

Prescribe jets of order $m$ at a finite set $A$. Choose disjoint
coordinate balls, local holomorphic sections with the given jets, and
cutoffs equal to one on smaller balls; the resulting smooth section
$s_0$ has $\bar\partial s_0$ 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 $\vartheta_a$ equals one near $a$ and is supported in the
chosen coordinate ball. The negative Hessian terms from the cutoff
logarithms are supported in a fixed compact; a sufficiently large $k$
makes the total weight strictly plurisubharmonic and dominant there.
The inverse-curvature norm of $\bar\partial s_0$ is integrable because
its support avoids the poles. The complete Kaehler $L^2$ estimate for
bundle-valued top forms gives $v$ with $\bar\partial v = \bar\partial
s_0$ and finite weighted norm. The singular-weight statement follows
by regularizing the logarithms and taking a weak limit. Near a point
of $A$ the correction $v$ is holomorphic; integrability against
$|z_a|^{-2c_m}$ forces its Taylor coefficients through order $m$ to
vanish. Therefore $s_0 - v$ has the prescribed jets. Applying the
construction to pairs of points proves point separation by holomorphic
functions.

./pdf/hormander-1965-l2-estimates-dbar.pdf
./pdf/demailly-1982-l2-estimates-dbar.pdf

Canonical sections. With the flat coefficient bundle the same scheme
produces, for a prescribed finite jet at a single point $x$,
$$\int_M |S|_g^2 e^{-k\psi} \, dV_g < \infty.$$
The cutoff pole weights drop out at the cost of a constant. The
function $|S|_g^2$ is subharmonic: away from its zeros, $\partial\bar
\partial \log |S|_g^2 = \Ric(g)$, and across zeros the squared norm
retains the nonnegative Laplacian in the sense of distributions. The
local mean value inequality on a radius-one ball, together with the
upper bound for $\psi$, gives
$$|S(x)|_g^2 \le \frac{C_S}{\Vol B(x,1)} \exp\big( C_S(1+r(x))
\big).$$
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 $o$. The polynomial loss is absorbed by the exponential.
So a canonical section exists at every $x$ with $S(x) \ne 0$ and
$\log|S|_g^2 \le C_S(1+r)$.

./pdf/li-schoen-1984-lp-mean-value.pdf
./pdf/gromov-1981-curvature-diameter-betti.pdf
./pdf/chen-zhu-2005-volume-growth-curvature-decay.pdf

Finitely many functions suffice. The compact-open space of holomorphic
functions is Frechet, hence Baire. Fix a compact family of points, or
of distinct pairs. Surjectivity of finite-jet evaluation, openness of
matrix rank, and a finite cover yield a finite-dimensional space of
perturbations on which the evaluation derivative is onto a
neighborhood of that compact family; the derivative is independent of
the perturbation parameter because evaluations are affine. Thus the
universal evaluation is a submersion there.

For a smooth stratum of the forbidden locus, its inverse image under
evaluation 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 has measure zero: cover by countably many compact
coordinate patches, use Lipschitz projection on each, and cover a
$d$-dimensional cube by $O(\delta^{-d})$ small cubes to see that the
image in $\mathbb{R}^p$, $p>d$, has volume at most $C \delta^{p-d}$.

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 $N \ge 2n+1$ this exceeds $n$. The equal-values condition on
distinct pairs has complex codimension $N>2n$. Avoidance on a compact
set is open in the compact-open topology via Cauchy estimates. A
countable exhaustion of $M$ and of $M \times M \setminus \Delta$,
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 $m \ge 2n$; the same zeroth-jet
argument gives a finite spanning family. Neither is claimed proper.

The function $\Xi$. Choose countably many canonical sections $S_i$
with no common zero. Put $a_i = |S_i|_g^2$. 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 dominated by
$r^{3/2}$ plus a constant. Choose $c_i>0$ so small that
$c_i e^{B_i} \le 2^{-i}$ 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, so
$$\Xi = \log \sum_i c_i |S_i|_g^2$$
satisfies $\partial\bar\partial \Xi \ge \Ric(g)$ and
$\Xi \le C(1+r^{3/2})$.

Hodge heat without a curvature upper bound. Let $D = d + d^*$ act
on the smooth compactly supported forms of a complete metric.
Completeness gives essential self-adjointness of $D^2$ and therefore of
$D$: a deficiency vector for $D$ would be annihilated by
$(D^2)^* + 1$, which vanishes.
./pdf/braverman-milatovic-shubin-2002-essential-selfadjointness.pdf

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 $\Omega_j$ and extended by zero. On a
fixed domain, local ellipticity identifies $V_j$ with the full
zero-boundary-value section domain; it is not an absolute or relative
Hodge condition. The $V_j$ are nested and their union is dense in $V$.
If $\lambda>0$ and an $L^2$ form is given, the domain resolvent applied
to it 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, and strong
resolvent convergence together with the common lower bound zero gives
strong heat convergence.
./pdf/teschl-2009-mathematical-methods-quantum-mechanics.pdf

The cone of nonnegative Hermitian forms is preserved. First for smooth
compactly supported data on a relatively compact smooth domain, using
the componentwise zero-Dirichlet realization of the Hodge heat
equation. Its Weitzenboeck 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 fixed positive multiple of
$$\sum_j R_g(X,\bar X, e_j, \bar e_j) A_{j\bar j} \ge 0;$$
the Ricci-times-$A$ terms vanish there. The tensor maximum principle
therefore preserves the 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 with zero boundary
condition. This is the null-eigenvector mechanism used for
plurisubharmonic functions under nonnegative bisectional curvature.
./pdf/ni-tam-2003-plurisubharmonic-structure.pdf

The cone and trace assertions pass to the complete limit and then to
bounded data by cutoff and local regularity. The complete Hodge
semigroup satisfies the off-diagonal estimate
$$\|\mathbf{1}_E \mathcal{H}_a \mathbf{1}_F\|_{2\to2} \le C
\exp\big( -c \dist(E,F)^2/a \big) \qquad (0<a\le a_0).$$
No curvature upper bound is needed. The self-adjoint first-order
operator $D$ has principal symbol of norm equal to the covector norm,
so the wave evolution has finite propagation speed; the Gaussian
Fourier representation of $\exp(-a D^2)$ then gives the display, with
the inessential change of constants required by the normalization.
Since $\Vol B(o,R) \le C R^{2n}$, dyadic annular decomposition shows
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 $a=0$.
./pdf/mcintosh-morris-1992-finite-propagation-speed.pdf
./pdf/coulhon-sikora-2008-gaussian-heat-bounds-via-phragmen-lindelof.pdf

Closedness. Use 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 inside $B(o,2R)$, with
$|d\zeta_R|\le C/R$. The complete Hodge semigroup commutes with $d$ on
the $L^2$ domains, by the self-adjoint de Rham complex; the fixed-time
identity $D e^{-a D^2} = e^{-a D^2} D$ and projection to degree one
higher give the $d$ identity. The data $\zeta_R A$ belong to the full
$D$ graph domain by compactly supported $H^1$ approximation. For
bounded closed $A$,
$$d \mathcal{H}_a(\zeta_R A) =
\mathcal{H}_a^{(3)}(d\zeta_R \wedge A).$$
The data on the right have $L^2$ norm at most $C R^{n-1}$ and support
outside $B(o,R)$; the off-diagonal estimate in degree three makes them
tend to zero on each compact, uniformly for $0<a\le a_0$.
Distributional passage to the limit gives $d \mathcal{H}_a A = 0$, and
local regularity gives its initial value $A$. Kaehler commutation on
closed forms now gives
$$\partial_a \mathcal{H}_a A = \partial\bar\partial \tr_g
\mathcal{H}_a A = \partial\bar\partial H_a \tr_g A.$$
Integration gives
$$\mathcal{H}_a A = A + \partial\bar\partial
\int_0^a H_b \tr_g A \, db$$
first in distributions and then as an identity of smooth forms for
$a>0$. The same passage from closed-form heat to a scalar potential via
its trace appears in the derivation of the Poincare-Lelong equation
from the Hodge-Laplace heat equation.
./pdf/ni-tam-2012-hodge-laplace-heat-poincare-lelong.pdf

Hessian preservation. Suppose a Lipschitz function $f$ is constant
outside a compact set and $\partial\bar\partial f \le Cg$ as currents.
Then $\partial\bar\partial H_a f \le Cg$ for every $a>0$. Put
$f_0 = f - c$ with $c$ the value at infinity; $f_0$ is compactly
supported and Lipschitz. Complete $L^2$ Kaehler commutation, smooth
approximation, and duality give, for any compactly supported
nonnegative Hermitian test form $B$,
$$\langle \partial\bar\partial H_a f_0, B \rangle =
\langle \partial\bar\partial f_0, \mathcal{H}_a B \rangle.$$
The metric pairing identifies the cone of nonnegative Hermitian forms
with its dual; equivalently, pairing against the Hodge star. Choose
$0\le\zeta\le1$ compactly supported and equal to one near $\supp f_0$.
Since $\partial\bar\partial f_0$ is supported there,
$$\langle \partial\bar\partial f_0, \mathcal{H}_a B \rangle =
\langle \partial\bar\partial f_0, \zeta \mathcal{H}_a B \rangle \le
C \int \zeta \tr_g \mathcal{H}_a B \, dV_g \le
C \int \tr_g B \, dV_g.$$
The last two steps use the cone preservation and the scalar trace
commutation. Since $H_a 1 = 1$ under nonnegative Ricci curvature,
$\partial\bar\partial H_a f = \partial\bar\partial H_a f_0$, proving
$\partial\bar\partial H_a f \le Cg$.

Mass of the scalar heat kernel. Fix a pole $y$ and a Lipschitz
distance cutoff $\chi_R$ equal to one on $B_g(y,R)$, supported inside
$B_g(y,2R)$, with $|\nabla\chi_R|\le C/R$. The Gaussian estimate of
Li-Yau and the kernel gradient estimate of Souplet-Zhang, together
with Bishop-Gromov comparison, give
$$\int_{R\le d(x,y)\le 2R} |\nabla_x H_s(x,y)| \, dV_g(x) \le
C s^{-1/2} (1+R/\sqrt{s})^N e^{-c R^2/s}$$
for a fixed dimensional exponent $N$. Integrating the heat equation
against $\chi_R$ from $\varepsilon$ to $t\le T$ and letting
$\varepsilon \downarrow 0$ yields
$$\big|\int_M \chi_R(x) H_t(x,y) \, dV_g(x) - 1\big| \le
\frac{C}{R} \int_0^T s^{-1/2}(1+R/\sqrt{s})^N e^{-c R^2/s} \, ds,$$
and the right side tends to zero as $R\to\infty$. Positivity and the
sub-Markov property of the minimal kernel pass this identity to the
total mass, giving $H_t 1 = 1$. The cutoffs require only completeness
and the first derivative of distance; they precede the smooth
exhaustion constructed below.
./pdf/li-yau-1986-parabolic-kernel-schrodinger.pdf
./pdf/souplet-zhang-2006-gradient-estimate-heat-kernel.pdf
./pdf/gromov-1981-curvature-diameter-betti.pdf

Dyadic smoothing and averaged curvature. For $L = 2^j$, $j\ge0$,
choose a smooth nondecreasing scalar truncation of $r^2/L^2$ vanishing
on $r\le L$ and equal to one on $r\ge 2L$. It is Lipschitz with
constant $C/L$. Pairing the length functional with linear parallel
fields on trial geodesics gives
$$\partial\bar\partial r^2 \le Cg$$
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
$$\partial\bar\partial f_L \le C L^{-2} g$$
as currents. The qualifier smooth refers to the scalar truncation;
the composed function $f_L$ need not be smooth at the cut locus.

Let $w_L = H_{\varepsilon L^2} f_L$ for a fixed small
$\varepsilon>0$. Gaussian heat bounds and the Lipschitz estimate give
$$\|w_L - f_L\|_\infty \le (C/L) \sup_x \int d_g(x,y)
H_{\varepsilon L^2}(x,y) \, dV_g(y) \le C \sqrt{\varepsilon}.$$
Nonnegative Ricci curvature suffices for these scalar heat bounds. The
Li-Yau gradient inequality, applied to $1 + H_a f_L$ and to
$2 - H_a f_L$, gives at $a = \varepsilon L^2$
$$|\Delta_g w_L| \le C_\varepsilon L^{-2}, \qquad
|\partial w_L|_g \le C_\varepsilon L^{-1}.$$
The first translate bounds $\partial_a H_a f_L$ from below, the second
from above, and substituting both into the same inequality bounds the
gradient. The Hessian preservation applied to
$\partial\bar\partial f_L \le C L^{-2}g$ together with these scalar
heat bounds gives $\partial\bar\partial w_L \le C_\varepsilon L^{-2}g$.
./pdf/li-yau-1986-parabolic-kernel-schrodinger.pdf

Take a fixed smooth nondecreasing step $\theta$, zero near zero and one
near one, with constant regions wide enough to absorb the heat error.
Then $v_L = \theta(w_L)$ vanishes on $r\le L$, equals one on
$r\ge 2L$, and
$$|\partial v_L|_g \le C/L, \qquad |\partial\bar\partial v_L|_g \le C/L^2.$$
The locally finite sum
$$u = 1 + \sum_{j\ge0} 2^j v_{2^j}$$
is comparable to $1+r$, and at any point only a bounded number of
summands have nonzero derivatives, at scales comparable to $1+r$. This
gives the smooth exhaustion with the two-sided complex Hessian bound
stated earlier. 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 $B(o,L)$.

Averaged curvature and averaged section norm. Put $w = \log|S|_g^2$ for
a canonical section with $S(o) \ne 0$. The Poincare-Lelong formula
gives
$$\partial\bar\partial w \ge \Ric(g)$$
as currents. The truncations $w_m = \max(w, -m)$ are subharmonic. The
heat submean inequality and linear upper growth give
$$w_m(o) \le H_{L^2} w_m(o), \qquad H_{L^2}(w_m)^+(o) \le CL.$$
The submean is justified first on exhaustion domains and then by
Gaussian decay; the positive part grows at most linearly. 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 $m\to\infty$ gives
$$\frac{1}{\Vol B(o,L)} \int_{B(o,L)} |\log|S|_g^2| \, dV_g \le CL.$$

For the scalar curvature, take a nonnegative cutoff equal to one on
$B(o,L)$ and supported inside $B(o,CL)$, with absolute Laplacian at
most $C/L^2$, as just constructed. Integrate the trace of
$\partial\bar\partial w \ge \Ric(g)$ against it and move $\Delta_g$
onto the cutoff:
$$\int_{B(o,L)} R_g \, dV_g \le
\frac{C}{L^2} \int_{B(o,CL)} |w| \, dV_g \le
\frac{C}{L} \Vol B(o,L).$$
The last step uses volume doubling. The average scalar-curvature bound
is the estimate of Chen-Zhu; the calculation above records the specific
cutoffs and the absolute logarithmic bound used later.
./pdf/chen-zhu-2005-volume-growth-curvature-decay.pdf

A global Kaehler-Ricci flow by exhaustion. Take the smooth exhaustion
$u$ with two-sided Hessian bound from the initial data and its regular
values $L\to\infty$. Put $\Omega_L = \{u<L\}$. Choose a fixed smooth
nonnegative $F$ on $[0,1)$ which is zero on $[0,1/2]$ and equal to
$-2\log(1-v)$ near $v=1$, and set on $\Omega_L$
$$f = F(u/L), \qquad \alpha = e^f g.$$
The estimates for $u$ give
$$|\partial f|_\alpha \le C/L, \qquad
|\partial\bar\partial f|_\alpha \le C/L^2.$$
Near the boundary $e^f = (1-u/L)^{-2}$, and the factors
$(1-u/L)^{-1}$ and $(1-u/L)^{-2}$ in the first and second derivatives
are cancelled by the conformal norm of $\alpha$. On the transition
region $u \asymp L$, $|\partial\bar\partial u|_g \le C/u$ gives the
same.

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 to every order in holomorphic charts. Take finitely many
ordinary charts on the compact closure; near a boundary point at
boundary distance comparable to $1-u/L$, shrink 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 their coefficients is bounded, with constants
allowed to depend on $L$ and on the derivative order.

The Chern-Ricci flow on this complete Hermitian background is
$$K = \alpha - t\Ric(\alpha) + \partial\bar\partial U, \qquad
U_t = P = \log\frac{\det K}{\det\alpha}, \qquad U(\cdot,0) = 0.$$
Until the local limit is taken, $U$ and $P$ refer to this
approximation. Bounded-geometry short-time existence is standard; the
estimates produced hold on every closed bounded-geometry interval, and
the resulting continuation intervals grow with $L$.
./pdf/lee-tam-2020-chern-ricci-flow-noncompact.pdf

Scalar evolutions. Write $\mathscr{L}_K = \partial_t - \Delta_K$.
Differentiating,
$$\mathscr{L}_K P = -\tr_K\Ric(\alpha), \qquad
\mathscr{L}_K P_t = -|\Ric(K)|_K^2, \qquad
P_t(\cdot,0) \le C/L^2.$$
The last inequality comes from $\Ric(\alpha) = \Ric(g) - n
\partial\bar\partial f$: initial Ricci positivity plus the two
conformal bounds. In particular
$$P_t \le C/L^2, \qquad P \le Ct/L^2, \qquad U \le Ct^2/(2L^2).$$
These are bounded maximum-principle estimates on an already existing
bounded-geometry interval.

A parameterized barrier. Fix a horizon $D>0$ and a smooth
$\phi \le 0$ satisfying
$$\alpha - D\Ric(\alpha) + D\partial\bar\partial\phi \ge
c\alpha \qquad (c>0).$$
Define
$$Q = (D-t)P + U - D\phi + nt.$$
Using $\mathscr{L}_K U = P - n + \tr_K(\alpha - t\Ric(\alpha))$ gives
the exact identity
$$\mathscr{L}_K Q = \tr_K\{\alpha - D\Ric(\alpha) +
D\partial\bar\partial\phi\} \ge c \tr_K\alpha, \qquad Q \ge 0.$$
The initial value is $-D\phi \ge 0$. For bounded $\phi$ the maximum
principle gives the last assertion directly; adding a vanishing
positive multiple of the proper $f$ is an equivalent localization. The
same argument applies to functions with downward logarithmic poles,
since $Q\to+\infty$ there.

Largest eigenvalue control. Let $\lambda_1 \ge \cdots \ge \lambda_n$
be the eigenvalues of $K$ over $g$ and put $W = \lambda_1$. The
function $W$ is Lipschitz and can be tested through smooth local
supporting functions agreeing with it at a maximum. Wherever $W \ge 1$,
direct differentiation gives
$$\mathscr{L}_K\log W \le -\frac{1}{q}\sum_{i,j}
\frac{|\partial_1 K_{i\bar j}|^2}{\lambda_i\lambda_j} +
|\partial\log q|_K^2 + C L^{-2}\tr_K\alpha,$$
where $q = \det K / \det g$. The derivation rests on the exchanges
$$\partial_1\partial_{\bar 1}K_{i\bar i} -
\partial_i\partial_{\bar i}K_{1\bar 1} =
\partial_1\partial_{\bar 1}\alpha_{i\bar i} -
\partial_i\partial_{\bar i}\alpha_{1\bar 1}.$$
In the chosen coordinates the right side is $(e^f)_{1\bar 1} -
(e^f)_{i\bar i}$; the second derivatives of $g$ cancel by its Kaehler
symmetry. Dividing the contraction by $q \ge e^f$ bounds the error by
$C L^{-2}\tr_K\alpha$. Differentiating the denominator $g_{1\bar 1}$
contributes $-\sum_i\lambda_i^{-1}R_g(i,\bar i,1,\bar 1) \le 0$. The
remaining conformal derivatives are of the same controlled size.

For $i\ne 1$ the first-derivative exchange reads
$\partial_1 K_{i\bar 1} = \partial_i q - e^f f_i$. The terms with
$j = 1$ in the negative square dominate
$|\partial\log q - W^{-1}\sum_{i\ne1}f_i\,dz_i|_K^2$. Expanding this
square and using $\partial\log q = \partial\log W + \partial f$ at the
maximum gives
$$\mathscr{L}_K\log W \le C L^{-2}\tr_K\alpha +
2W^{-1}|\partial f|_K|\partial\log W|_K.$$

Set $a = L^{-1}$ and $J(v) = a v + \log(1+av)$ for $v \ge 0$. At a
positive maximum of $\log W - J(Q)$,
$$W \ge e^{aQ}(1+aQ), \qquad
\partial\log W = J'(Q)\partial Q, \qquad
J' \ge a, \qquad J' \le 2a, \qquad
-J'' = \frac{a^2}{(1+aQ)^2}.$$
The chain rule together with the $Q$-comparison identity and the
gradient inequality gives
$$\mathscr{L}_K\big(\log W - J(Q)\big) \le
(CL^{-2} - ca)\tr_K\alpha + 2W^{-1}J'|\partial f|_K|\partial Q|_K
+ J''|\partial Q|_K^2.$$
Young's inequality absorbs the middle term into half of
$-J''|\partial Q|_K^2$; its remaining coefficient is at most
$$C\frac{W^{-2}(J')^2}{-J''}|\partial f|_K^2 \le
C|\partial f|_K^2 \le C L^{-2}\tr_K\alpha.$$
For large $L$ this contradicts the maximum principle. To attain a
maximum on $\Omega_L$ during the test, add the penalty $-\delta f$
first; its Hessian term is controlled by $C\delta L^{-2}\tr_K\alpha$,
and $\delta\downarrow0$ afterwards. At $t=0$, $W = 1$ and $Q \ge 0$,
so
$$K \le e^{J(Q)}\alpha.$$
The bounded choice of $\phi$ bounds $Q$ from above by a constant. The
upper metric bound follows. The lower bound for $P$, hence for
$\det K/\det\alpha$, then bounds the least eigenvalue from below.

Continuation in uniform holomorphic charts. Let $\iota$ be the real
symmetric representation of a Hermitian matrix and put
$F(B) = \tfrac12\log\det_{\mathbb R}B$,
$S(x,t) = \iota(\alpha - t\Ric(\alpha))$,
$T(\partial^2 U) = \iota(\partial\bar\partial U)$. Then
$U_t - F(S + T(\partial^2 U)) = -\log\det_{\mathbb C}\alpha$.
Two-sided metric equivalence puts $S + T(\partial^2 U)$ 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 norms comparable to the
original on nonnegative real symmetric matrices. The smooth background
coefficients have uniform local Hoelder bounds. These are the
hypotheses of the transformed-Hessian parabolic estimate: its bound
for spatial second derivatives and the first time derivative depends
on $\|U\|_\infty$ and these data, not on a prior bound for the full
real Hessian. Higher Schauder estimates then give continuation on each
fixed approximating background. The eigen bound above gives $K$ from
above, the $P$ bound from below plus the upper $K$ bound gives the
least eigenvalue of $K$ from below, and $|U|_\infty$ remains bounded
on each fixed time interval because $U_t = P$ and $U(0) = 0$.
./pdf/chu-regularity-transformed-hessian-flow.pdf
./pdf/lee-tam-2020-chern-ricci-flow-noncompact.pdf

Singular weights and the local limit. For a canonical section $S$ from
the initial data, take
$$\phi = \log|S|_g^2 - nf - \varepsilon u^2 - A_\varepsilon, \qquad
0 < \varepsilon \le c_0/D, \qquad
A_\varepsilon = C_S'(1+\varepsilon^{-1}).$$
The growth bound for $S$ makes $\phi \le 0$ after choosing $C_S'$.
Since $|\partial\bar\partial u^2|_g \le C$, the canonical-current
inequality gives
$$\alpha - D\Ric(\alpha) + D\partial\bar\partial\phi \ge
\alpha - CD\varepsilon g \ge \tfrac12\alpha$$
for small enough $c_0$. 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
large enough $L$ it lies in $\{u<L/2\}$, where $f = 0$. The upper
potentials bound $Q$ above there by a constant independent of $L$.
The $Q$-comparison bounds $P$ below there, and the 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 nonvanishing sections cover any compact set. Local parabolic
estimates on slightly larger compact sets give all derivatives,
including one-sided at the initial surface, uniformly in $L$. A
diagonal subsequence over compacts and over 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 the
Chern-Ricci equation passes to the Kaehler equation there. The limit
of the eigenvalue bound is $h_t \le g$; the limits of the upper
potentials give $P \le 0$ and $P_t \le 0$. Thus
$$\partial_t h_t = -\Ric(h_t), \qquad h_0 = g, \qquad
h_t \le g, \qquad P \le 0, \qquad P_t \le 0.$$

The volume-loss potential. Put
$$\rho(x,t) = \log\frac{\det g}{\det h_t}.$$
Since $h_0 = g$, $\rho(x,0) = 0$. Since $h_t \le g$, $\det h_t \le
\det g$, so $\rho \ge 0$. Directly from the Kaehler-Ricci equation,
$$\Big(\frac{\partial}{\partial t} - \Delta_{h_t}\Big)\rho =
\tr_{h_t}\Ric(g).$$
The right side is nonnegative because the initial Ricci form is
nonnegative. The companion trace $a_t = -\partial_t\log\det h_t =
\tr_{h_t}\Ric(h_t) \ge 0$ is also nonnegative, by nonnegative
bisectional curvature of $h_t$ under the flow.

Ball average and pointwise growth. On every closed time interval
$[0,T]$ there is $C_T$ such that
$$\sup_{0\le t\le T}\frac{1}{\Vol_g B_g(o,L)}\int_{B_g(o,L)}
\rho(x,t)\,dV_g(x) \le C_T L \qquad (L \ge 1),$$
and at every fixed $x$,
$$\rho(x,t) \le C_x(1+t) \qquad (t \ge 0).$$
The derivation uses the divergence form of the linearized
Kaehler-Ricci equation, the sharp constant $0 \le a_t$, and the
averaged-section estimates for $\Xi$ on balls $B(o,L)$ with a
section nonvanishing on the ball.

Proper supercaloric function. There are a smooth positive $Q$ and a
constant $c>0$ with
$$Q \ge c(1+r^2), \qquad
(\partial_t - \Delta_{h_t})Q \ge c \tr_{h_t}g.$$
Smoothness at $t=0$ means smoothness with all one-sided time
derivatives. The construction takes $D = 2(t+1)$ in the $Q$-comparison
identity and uses $\phi_\varepsilon = \Xi - \varepsilon u^2 -
A_\varepsilon$ as the barrier; the difference
$Q_\varepsilon - Q_{\varepsilon/2} = \tfrac{D\varepsilon}{2}u^2 +
D(A_\varepsilon - A_{\varepsilon/2})$ and the nonnegativity
$Q_{\varepsilon/2} \ge 0$ force $Q_\varepsilon \ge c u^2 - C$, and
adding a constant gives the asserted positive $Q$.


Heat correction and the integrable cutoff. Comparison arguments later
require a positive integrable weight $\chi$ with
$\partial\bar\partial\chi \le C\chi h_t$. A polynomial weight
$u^{-k}$ gives only $\partial\bar\partial(u^{-k}) = O(u^{-k-2}g)$,
and $h_t$ need not be uniformly comparable to $g$ at infinity. A
function $H$, whose complex Hessian approaches $g - h_t$ at infinity,
is constructed from the heat kernel of $g$ applied to the trace
$a_t$ of the flow.

For large dyadic $L$ define
$$H_{L,t} = -\int_0^{L^2} H_b a_t\,db.$$
There are $\delta,\beta>0$ such that, on $B(o,C_0L)$,
$$|H_{L,t}| + L|\partial H_{L,t}|_g \le C_T L^{2-\delta}t^\beta,$$
$$\partial\bar\partial H_{L,t} = \eta_t + O(t/L)g, \qquad
\partial_t H_{L,t} \ge H_{L,t}/t - C_T L \quad(t>0),$$
where $\eta_t$ is the closed $(1,1)$-form determined by the trace
equation of the linearized flow. The size bound uses Gaussian upper
estimates for the scalar heat kernel of $h_t$, relative volume
comparison against $g$, and the mean of $a_t$ on $B(o,L)$. The
Hessian identity rests on
$$H_{L,t} = -t\int_0^{L^2}H_b R_g\,db + H_{L^2}U - U$$
together with $H_b\Delta_g U = tR_g - a_t$: insert initial-metric
cutoffs in the heat integral, use the polynomial bound for $\rho$ and
$0 \le a_t \le n$ to send the shell errors to zero, pass to the
distributional limit, then to the smooth local one. For the time
inequality, subtract $H_{L,t}/t$:
$$\partial_t H_{L,t} - H_{L,t}/t = H_{L^2}(P-U/t) - (P-U/t).$$
Since $P_t \le 0$, the argument satisfies $P-U/t \le 0$ and the
complementary quantity satisfies $0 \le U/t - P \le \rho$. Its heat
average is at most $C_T L$ by the ball bound for $\rho$ and Gaussian
annular summation; the unaveraged term has the favorable sign.

Gluing. Choose a nonnegative smooth dyadic partition of unity
$\zeta_L(u)$ with scales $L \asymp u$ and put
$H = \sum_L \zeta_L(u) H_{L,t}$ (finitely many initial scales on any
fixed compact). The partition derivatives are $O(u^{-1})$ in gradient
and $O(u^{-2})$ in complex Hessian. The bounds above give
$$|H| \le C_T u^{2-\delta}t^\beta, \qquad
|\partial H|_g \le C_T u^{1-\delta}t^\beta,$$
$$\partial\bar\partial H = \eta_t + O(u^{-\delta}t^\beta + t/u)g,
\qquad H_t \ge H/t - C_T u,$$
smooth in space and continuously differentiable in time one-sided at
$t=0$.

The integrable weight. For every sufficiently large fixed integer $k$
there is a positive $\chi(x,t)$ on $M \times [0,T]$, smooth in space
and $C^1$ in time one-sided at $t=0$, satisfying
$$C_T^{-1}u^{-k} \le \chi \le C_Tu^{-k}, \qquad
|\partial\chi|_g \le C_T\chi/u,$$
$$\partial\bar\partial\chi \le C_T\chi h_t, \qquad
\partial_t\chi \le C_T(1+t^{\beta-1})\chi \quad(t>0).$$

The form is $\chi = u^{-k}\exp(-AH/u^2)$. The size and gradient bounds
follow from the glued bounds for $H$ at fixed $A$. Expanding
$$\chi^{-1}\partial\bar\partial\chi =
\partial\bar\partial\log\chi +
\partial\log\chi\otimes\bar\partial\log\chi$$
gives, at infinity,
$$\chi^{-1}\partial\bar\partial\chi \le u^{-2}\{-A(g-h_t) +
C_kg + o(1)g\}$$
uniformly on the fixed time interval. Terms containing $H/u^2$ or its
gradient are $o(u^{-2})g$ once $A$ is fixed. Choose $A > C_k + 1$,
then take $u$ large enough. The negative multiple of $g$ absorbs the
remaining error and leaves $\chi^{-1}\partial\bar\partial\chi \le
Au^{-2}h_t$. On the compact remainder, smoothness and positivity of
$h_t$ give the same bound. Finally $\chi_t/\chi = -AH_t/u^2$ and the
glued time bound give
$$\chi_t/\chi \le A|H|/(tu^2) + C_T/u \le C_T(1+t^{\beta-1}).$$
The positive exponent $\beta$ makes the time coefficient integrable
at zero.

The mass argument. Let $\rho_{\min}$ be the minimal nonnegative
zero-data potential of $F = \tr_{h_t}\Ric(g)$: the solution of
$(\partial_t - \Delta_{h_t})\rho_{\min} = F$ with zero initial data,
constructed by integrating the heat semigroup and so the least
nonnegative solution. The difference $z = \rho - \rho_{\min}$ is
nonnegative, satisfies $z(\cdot,0) = 0$, and obeys the homogeneous
equation $(\partial_t - \Delta_{h_t}) z = 0$.

Let
$$I(t) = \int_M z \chi \, dV_{h_t}.$$
The tails tend to zero uniformly in $t$ by $z \le \rho$, $dV_h \le
dV_g$, the ball average for $\rho$, and the polynomial decay
$\chi \le C_T u^{-k}$ with $k$ large: on $B(o,L)$ the contribution is
at most $C_T L^{2n+1-k}$, which vanishes uniformly in $t$.

To justify differentiation, first replace $\chi$ by $\chi f_N$ with a
smooth compactly supported cutoff $f_N$ equal to one on $B(o,N)$ and
supported inside $B(o,2N)$. Using $z_t = \Delta_{h_t} z$ and
$\partial_t dV_h = -R_h dV_h$,
$$\frac{d}{dt}\int z\chi f_N\,dV_h =
\int z\big[\Delta_{h_t}(\chi f_N) + \partial_t(\chi f_N)
- R_h\chi f_N\big]\,dV_h.$$
The scalar curvature satisfies $R_h \ge 0$, so the insertion
$-R_h\chi f_N$ is nonpositive and can be dropped in an upper bound.
Since $\partial\bar\partial\chi \le C\chi h_t$, since
$\partial_t\chi \le C_T(1+t^{\beta-1})\chi$, and since the cutoff
derivatives $d f_N, \Delta_h f_N$ contribute errors that vanish as
$N\to\infty$ once Gronwall has been applied, the monotone limit gives
$$\frac{d}{dt} I(t) \le C_T(1+t^{\beta-1}) I(t).$$
The initial value is $I(0) = 0$ because both $\rho$ and $\rho_{\min}$
vanish at $t = 0$. If the differentiated identity is available only
for $t>0$, integrate from $\varepsilon \downarrow 0$. Gronwall gives
$I(t) = 0$, and the strict positivity of $\chi$ gives $z = 0$. Hence
$$\rho = \rho_{\min}.$$

Nonlinear scalar comparison. In parallel with the mass argument, a
lower semicontinuous $w \le \rho$ is constructed, equal to $\rho$ at
the initial surface, satisfying the nonlinear heat equation
$$\partial_t w = e^{-w}\Delta_{h_t}w + (1-e^{-w})R_h$$
in the viscosity sense. The equation is chosen so that
$$\partial_t[(e^w - 1)dV_{h_t}] = (\Delta_{h_t}w)\,dV_{h_t},$$
using $\partial_t dV_h = -R_h dV_h$ and $e^w(1-e^{-w}) = e^w - 1$.
Since $0 \le e^w - 1 \le e^\rho - 1$, the mass
$$\int_M (e^w-1)\chi f_N\,dV_h \le \int_M \chi f_N\,dV_g$$
is bounded for every nonnegative test $\chi f_N$ with
$\chi \in L^1(dV_g)$ and compactly supported $f_N$, so $w \le \rho$
pointwise. The comparison with the linear mass bound gives
$w = \rho$.

The flow data are now fixed. The evolving metric $h_t$ satisfies
$h_t \le g$, $\rho = \rho_{\min}$, has the ball average
$\langle \rho\rangle_{B_g(o,L)} \le C_T L$ and the pointwise bound
$\rho(x,t) \le C_x(1+t)$, carries the proper supercaloric function
$Q$, the integrable weight $\chi$, and the nonlinear identity
$$\partial_t[(e^\rho - 1)dV_{h_t}] = (\Delta_{h_t}\rho)\,dV_{h_t}.$$

Cotangent envelope and disc infimum. Let $Y = T^{*(1,0)}M$ with
canonical projection $\pi : Y \to M$, and put
$$N(x,\xi,t) = \xi^* h_t^{-1}(x)\xi = |\xi|^2_{h_t^{-1}}.$$
The quantity $N(\cdot,t)$ is the dual squared norm on the Hermitian
bundle $T^{*(1,0)}M$. Since $h_t \le g$, $N(\cdot,t) \ge N(\cdot,0)$
for every $t\ge0$. At $t=0$ the squared norm on the total space of a
Griffiths negative Hermitian vector bundle is plurisubharmonic;
equivalently the Levi form of
$N(\cdot,0) = \sum g^{j\bar k}(x)\xi_j\bar\xi_k$ is nonnegative by
strict bisectional positivity of $g$.

Fix $T < \infty$ and define a lower semicontinuous envelope. For
$0<s<T$ and $y\in Y$, let $v^{\rm raw}(y,s)$ be the infimum of
$$\langle N(f(\zeta), s e^{w(\zeta)+\bar w(\zeta)})\rangle =
\frac{1}{2\pi}\int_0^{2\pi} N(f(e^{i\theta}),
s e^{w(e^{i\theta})+\bar w(e^{i\theta})})\,d\theta$$
over holomorphic discs $f : \bar\Delta \to Y$ and
$w : \bar\Delta \to \mathbb{C}$ smooth up to the boundary, with
$(f(0),w(0)) = (y,0)$ and $s e^{w+\bar w} < T$ throughout
$\bar\Delta$. Brackets denote normalized circle average. Let $v$ be
the lower semicontinuous regularization of $v^{\rm raw}$ in $(y,s)$.

Sandwich. Constant discs give $v^{\rm raw} \le N$. In the other
direction, plurisubharmonicity of $N(\cdot,0)$ gives
$\langle N(f,0)\rangle \ge N(f(0),0)$ for every admissible disc $f$,
while monotonicity gives $N(\cdot, s e^{w+\bar w}) \ge N(\cdot,0)$.
Hence
$$N(y,0) \le v(y,s) \le v^{\rm raw}(y,s) \le N(y,s).$$

Fiber homogeneity. Multiplying the fiber of $Y$ by $\lambda\in
\mathbb{C}^\times$ in each competing disc scales every value of $N$
by $|\lambda|^2$, so $v$ is homogeneous of degree two in $\xi$. At
the zero section $N$ and $v$ both vanish.

The joint form of the result. The objective is
$$N(x,\xi, e^{W+\bar W})
\quad\text{jointly plurisubharmonic in }(x,\xi,W).$$
If $v = N$ on $Y\times(0,T)$, then the submean inequality for $N$
holds for every admissible disc $f$ and every $W = w + \log s$ with
$s = e^{\Re W(0)}$: this is joint plurisubharmonicity, and it implies
the matrix Harnack inequality along the Kaehler-Ricci flow $h_t$.

Approach. A disc deformation lemma gives the differential inequality
for $v$. Optimizing over a Hermitian ellipsoid in each fiber converts
it into a scalar inequality. The scalar comparison already proved for
$\rho$ then closes the argument. The next step is the differential
inequality itself.

The symplectic bivector. Let $\mathcal{J}_Y = \sum_i e_i \wedge f_i$
be the standard bivector on the cotangent total space $Y = T^{*}M$,
with $e_1,\ldots,e_n$ an $h_t$-unitary horizontal frame and
$f_1,\ldots,f_n$ the dual vertical frame in the fiber. For a Hermitian
form $\mathcal{H}$ on $Y$, define
$$p(\mathcal{H}) = \mathcal{H}(\mathcal{J}_Y,\overline{\mathcal{J}_Y}),$$
the contraction of the $(1,1)$-tensor $\mathcal{H}$ against the
bivector and its complex conjugate. For a block form
$\mathcal{H} = \left(\begin{smallmatrix}R_\xi & 0 \\ 0 & D
\end{smallmatrix}\right)$ with $R_\xi$ horizontal and $D$ vertical,
this reduces at a point where $h_t = I$ to
$p(\mathcal{H}) = \tr(D^{\mathsf T} R_\xi b_t^{\mathsf T})$.

Symplectic evolution. Along the Kaehler-Ricci flow $h_t$,
$$\partial_t N(x,\xi,t) = p(\partial\bar\partial_Y N(x,\xi,t)).$$
For a real smooth base function $d = d(x,t)$, with $t$ held fixed in
the spatial Hessian,
$$p(\partial\bar\partial_Y(N+d)) = \partial_t N + \Delta_{h_t}d.$$
Neither identity assumes completeness or nonnegativity of $h_t$ at
positive time.

Proof. Fix $(x,t)$ and use Kaehler-normal coordinates for $h_t$ at
that point; then $h_t = b_t = I$ and all first derivatives of $b_t$
vanish. The Hessian of $N$ splits into blocks
$$\partial\bar\partial_Y N =
\begin{pmatrix} \mathcal{R}_\xi & 0 \\ 0 & I \end{pmatrix},$$
where $\mathcal{R}_\xi$ is the bisectional curvature form of $h_t$
with one pair of indices contracted against the metric dual of $\xi$.
Differentiating $b = h^{-\mathsf T}$ twice at a normal point gives
the negative second derivatives of $h$, transposed, which are exactly
those curvature components. Kaehler symmetry gives
$$\tr\mathcal{R}_\xi = \xi^*\Ric(h_t)^{\mathsf T}\xi.$$
On the other hand $\partial_t h = -\Ric(h)$ implies
$\partial_t b = h^{-\mathsf T}\Ric(h)^{\mathsf T}h^{-\mathsf T}
= \Ric(h_t)^{\mathsf T}$ at $h = I$, so
$\partial_t N = \xi^*\Ric(h_t)^{\mathsf T}\xi$. The block formula for
$p$ now gives $\partial_t N = p(\partial\bar\partial_Y N)$. Adding a
base Hessian changes only the horizontal block. Its contribution to
$p$ is $\tr((\partial\bar\partial_x d) b_t^{\mathsf T}) =
\tr(h_t^{-1}\partial\bar\partial d) = \Delta_{h_t}d$, giving the
second identity. The calculation is tensorial, so the
normal-coordinate verification proves both identities everywhere.

Compactness of discs. For each compact $K \subset M$ and each
$A_0<\infty$ there is a compact $K'\subset M$ such that every
holomorphic disc $f : \bar\Delta \to M$, smooth up to the circle,
with
$$f(\partial\Delta) \subset K, \qquad
\int_\Delta f^*\omega_g \le A_0,$$
has $f(\bar\Delta) \subset K'$. Every sequence of such discs has a
subsequence converging holomorphically on compact subsets of $\Delta$.
The same holds for discs in $Y$ with compact boundary in $Y$ and
$\int_\Delta(\pi\circ f)^*\omega_g \le A_0$; the compact $K'$ may
depend on $K$ and $A_0$.

The area bound first places part of each disc in a fixed compact. An
inverse-Jacobian estimate for the global injective immersion
propagates this control away from isolated possible singularities of
a limit, without assuming properness of the immersion. Extra
integrability removes those punctures: on $\Delta\setminus\{0\}$,
with $F\circ f$ bounded and $\int_\Delta|f'|_g^2 < \infty$, the
Griffiths nonnegativity of $F^*T^{1,0}M$ and the finite $L^2$ norm of
$a = f'$ force $\int_0^{r_0}|a(re^{i\theta})|_g^2r^{1-\sigma}dr <
\infty$ on some ray, which combines with volume comparison to bound
the radial path and, since $F$ separates points, to trap the punctured
image in a bounded coordinate chart. Boundedness of $F\circ f$ then
lets $f$ extend holomorphically across the origin.

The disc inequality. Suppose a smooth real function $\varphi$ touches
$v$ from below at $(y_*,s_*)$, where $0<s_*<T$. In holomorphic
cotangent coordinates write
$$W = \partial\bar\partial_Y\varphi(y_*,s_*) =
\begin{pmatrix}A & B \\ B^* & C\end{pmatrix}, \qquad C>0.$$
There is a Hermitian form $K$ on the horizontal coordinate space with
$$K \ge 0, \qquad K \ge A - BC^{-1}B^*, \qquad
\partial_s\varphi(y_*,s_*) \ge \tr(C^{\mathsf T}K).$$
The transpose is the tangent-cotangent contraction in these
coordinates. The statement applies to lower tests translated in a
fixed cotangent coordinate chart; the contraction has constant
coefficients in such a chart.

The proof proceeds through localized almost minimizing discs. The
compact set containing the disc interiors depends on an area penalty,
whereas the coefficient multiplying that penalty depends only on
boundary data. After the deformation estimate, the negative part of
the boundary Hessian is removed, the positive form is factored, and a
backward clock variation bounds the symplectic cost of the factor. The
final Schur-complement limit gives the asserted differential
inequality.

Localization. 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 used above. Thus $v-\varphi \ge 0$
on $\mathcal{C}$, with equality only at $(y_*,s_*)$, and with a
positive gap near the boundary. The lower semicontinuous
regularization gives
$$\inf_{(y,s)\in\mathcal{C}}(v^{\rm raw}(y,s)-\varphi(y,s)) = 0.$$

On an interval slightly larger than $[0,T]$, take $Q$ and $\rho_j$
from the previous part and set
$$d_{0,j}(x,t) = j^{-1}Q(x,t) + \rho_j(x,t).$$
Then $d_{0,j} \ge 0$, $(\partial_t - \Delta_{h_t})d_{0,j} \ge 0$, and
$d_{0,j}\to 0$ uniformly on compact space-time sets. Choose increasing
$A_j \to \infty$ so that
$$d_{0,j}(x,t) \ge \rho(x,t)+j \qquad (u(x)>A_j/3).$$
This is possible because $\rho_j \ge \rho - C_j$ and $Q$ has a
positive quadratic lower bound. Take fixed smooth profiles and set
$\delta_1 = \chi(u/A_j)$ and $\delta_2 = \tilde\chi(N(\cdot,0)/B_j^2)$
with $0 \le \delta_i \le 1$, where $\delta_1>0$ exactly when $u<A_j$
and $\delta_1 = 1$ when $u \le 2A_j/3$, 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
$B_j \ge j$ is enlarged below. Smooth cutoffs flat at their zero sets
are allowed. On the open boundary region put
$$k_i = \delta_i^{-1}-1, \qquad
d(y,t,s) = d_{0,j}(x,t) + (1+s)(k_1(x)+k_2(y)).$$
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.

Surjectivity. Introduce the polynomial-growth inequality. For a vector
polynomial $P$ of degree at most $m$ and $0<R_1<R_2$,
$$\sup_{B(R_2)}|P| \le (R_2/R_1)^m\sup_{B(R_1)}|P|.$$
Restricting 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 over lines gives the vector form.

Suppose
$$R_* = \sup\{R>0 : B(R)\subset\Omega\} < \infty.$$
It is positive because $0\in\Omega$ and $\Omega$ is open. By the
forward-degree bound there is an unbounded subsequence of regular macro
starts along which the composite polynomial $G_{0,k}$ satisfies
$\deg G_{0,k}\le Ds_k$ with $D$ finite and independent of $k$.

Choose $0<R_1<R_*<R_2$ close enough to $R_*$ that
$$D\log(R_2/R_1) < \sigma/2-\gamma'.$$
The closed smaller ball lies compactly in $\Omega$ because it lies in
some still larger centered ball of radius below $R_*$. The forward
smallness bound on this compact gives
$\sup_{B(R_1)}|G_{0,k}|\le e^{-\sigma/2 s_k}$; the polynomial growth
inequality then gives
$$\sup_{B(R_2)}|G_{0,k}| \le
\exp\big(-\tfrac{\sigma}{2}s_k + Ds_k\log(R_2/R_1)\big)
< e^{-\gamma's_k}$$
for all sufficiently large indices. The inverse-ball inclusion then
forces $B(R_2)\subset\Omega$, contradicting the definition of $R_*$.
Hence $R_* = \infty$ and $\Omega = \mathbb{C}^n$.

The map $\Psi$ is consequently a biholomorphism $M\to\mathbb{C}^n$.
This proves that every complete connected noncompact Kaehler manifold
of strictly positive holomorphic bisectional curvature is biholomorphic
to complex Euclidean space.