diff b34550f..b1f9bc4 spacetime_penrose_area_charge_v2.txt
diff --git a/spacetime_penrose_area_charge_v2.txt b/spacetime_penrose_area_charge_v2.txt
index 2fca3be..940d318 100644
--- a/spacetime_penrose_area_charge_v2.txt
+++ b/spacetime_penrose_area_charge_v2.txt
@@ -1,8 +1,8 @@
Introduction [sec:intro]. When a massive star collapses, it may form a black hole. Penrose's 1973
heuristic argument says no physical process can produce a black hole whose
-horizon area $A$ exceeds $16\pi m^2$ for total mass $m$, equivalently
-$m\ge√(A/16\pi)$; otherwise a naked singularity would form, violating
+horizon area $A$ exceeds $16pi m^2$ for total mass $m$, equivalently
+$mge√(A/16pi)$; otherwise a naked singularity would form, violating
cosmic censorship. This is the Penrose inequality. The theorems
below prove it rigorously (and its charged, higher-dimensional, and
anti-de Sitter versions) by reducing the spacetime problem to the
@@ -12,20 +12,20 @@ scalar curvature.
An initial data set for the Einstein equations is a triple $(M,g,K)$: a
Riemannian manifold $(M,g)$ and a symmetric $2$-tensor $K$, the second
fundamental form of a spacelike hypersurface in a spacetime. The constraint
-equations tie the energy density $\mu$ and momentum density $J$ of the data
+equations tie the energy density $mu$ and momentum density $J$ of the data
to their geometry, and the dominant energy condition (DEC) requires
-$\mu\ge|J|_g$ everywhere.
+$muge|J|_g$ everywhere.
-Fix a compact hypersurface $S\subset M$ and a unit normal $\nu$ toward the
+Fix a compact hypersurface $Ssubset M$ and a unit normal $nu$ toward the
designated exterior. Its future outward null expansion is
$$
- \theta_+(S)=H_S+tr_S K,
+ theta_+(S)=H_S+tr_S K,
$$
-where $H_S$ is the mean curvature of $S$ with respect to $\nu$. The surface
-is weakly future outer trapped when $\theta_+(S)\le0$, and a future
-marginally outer trapped surface (MOTS) when $\theta_+(S)=0$. A horizon
+where $H_S$ is the mean curvature of $S$ with respect to $nu$. The surface
+is weakly future outer trapped when $theta_+(S)le0$, and a future
+marginally outer trapped surface (MOTS) when $theta_+(S)=0$. A horizon
appears as such a surface in a spacelike slice.
Fix an asymptotically flat end and let $(E,P)$ denote its ADM energy and
@@ -34,7 +34,7 @@ $m=√(E^2-|P|^2)$. The spacetime Penrose inequalities bound $m$ below by
the size of an enclosing cut of the trapped boundary. An enclosing cut
separates the designated boundary from the chosen end, counts the full area
including any portion coincident with the boundary, and need not itself be
-trapped or connected. Let $A_{\min}$ denote the infimum of this area.
+trapped or connected. Let $A_{min}$ denote the infimum of this area.
The results
@@ -44,7 +44,7 @@ The results
In three spatial dimensions the basic bound is
$$
- m\ge\sqrt{\frac{A_{\min}}{16\pi}}.
+ mgesqrt{frac{A_{min}}{16pi}}.
$$
the [thm:n3:exterior] theorem proves this for a designated end with
@@ -57,26 +57,26 @@ area-minimizing, then equality forces a single spherical boundary component
and realizes the entire exterior as a slice of Schwarzschild spacetime.
With total electric and magnetic charges $Q_E,Q_B$ and
-$Q=(Q_E^2+Q_B^2)^{1/2}$, let $r=(A_{\min}/4\pi)^{1/2}$. For a one-ended
+$Q=(Q_E^2+Q_B^2)^{1/2}$, let $r=(A_{min}/4pi)^{1/2}$. For a one-ended
exterior whose matter condition includes the electromagnetic momentum,
the [thm:ch:main] theorem proves
$$
- m\ge Q,\qquad r\le m+√(m^2-Q^2).
+ mge Q,qquad rle m+√(m^2-Q^2).
$$
-The familiar bound $m\ge(r+Q^2/r)/2$ is recovered on the branch $r\ge Q$;
+The familiar bound $mge(r+Q^2/r)/2$ is recovered on the branch $rge Q$;
it is not equivalent for every positive $r$. Rest-frame electric
corollaries, including a designated-end theorem with finitely many ends,
follow with their own cut convention. No charged equality classification is
claimed.
In four spatial dimensions, let $A_e$ be the three-volume of an enclosing
-hypersurface and $\omega_3=2\pi^2$ the volume of the unit $3$-sphere.
+hypersurface and $omega_3=2pi^2$ the volume of the unit $3$-sphere.
the [thm:four:main] theorem proves
$$
- m_e\ge\frac12((A_e)/(\omega_3))^{2/3}
+ m_egefrac12((A_e)/(omega_3))^{2/3}
$$
for arbitrary $K$, finitely many ends, and a disconnected weakly future
@@ -92,7 +92,7 @@ the [thm:ads:main] theorem gives, on a sufficiently small interval depending
on the fixed data,
$$
- m_{AH}\ge√((A)/(16\pi))(1+(A)/(4\pi)),
+ m_{AH}ge√((A)/(16pi))(1+(A)/(4pi)),
$$
with equality exactly for radial seeds at small positive parameter.
@@ -137,28 +137,28 @@ contributions end and the classical machinery begins.
(T1) Riemannian Penrose inequality (Huisken--Ilmanen 2001, Bray
2001). If $(N,h)$ is a complete connected asymptotically flat
- three-manifold with nonnegative scalar curvature $R_h\ge 0$ and ADM
+ three-manifold with nonnegative scalar curvature $R_hge 0$ and ADM
energy $E_h$, and $∂ N$ is a compact outermost smooth minimal
- surface of total area $A$, then $E_h\ge√(A/16\pi)$, with equality
+ surface of total area $A$, then $E_hge√(A/16pi)$, with equality
in the nondegenerate case exactly for the Schwarzschild horizon.
./pdf/huisken-ilmanen-2001.pdf
(T2) Positive mass theorem (Schoen--Yau 1979, Witten 1981). A
- complete connected asymptotically flat three-manifold with $R_h\ge 0$
- satisfies $E_h\ge |P_h|_\delta$, with equality if and only if
+ complete connected asymptotically flat three-manifold with $R_hge 0$
+ satisfies $E_hge |P_h|_delta$, with equality if and only if
$(N,h)$ is Euclidean.
./pdf/witten-1981.pdf
(T3) Static vacuum uniqueness (Bunting--Masood-ul-Alam 1987).
A complete three-dimensional static vacuum data set $(N,h,N)$ with
positive lapse $N>0$ and a nondegenerate Killing horizon is isometric
- to the Schwarzschild exterior $\{r>2M\}$, with mass
- $M=|∂ N|_h^{1/2}/√(16\pi)$.
+ to the Schwarzschild exterior ${r>2M}$, with mass
+ $M=|∂ N|_h^{1/2}/√(16pi)$.
(T4) Charged Riemannian Penrose inequality
(Huisken--Ilmanen 2001; Bray--Khuri 2010; Disconzi--Khuri 2012). The
same setup with total charge $Q=(Q_E^2+Q_B^2)^{1/2}$ and $P=0$ gives
- $E_h\ge Q$ and $r\le m+√(m^2-Q^2)$, with $r=(A/4\pi)^{1/2}$ and
+ $E_hge Q$ and $rle m+√(m^2-Q^2)$, with $r=(A/4pi)^{1/2}$ and
$m=E_h$.
./pdf/disconzi-khuri-2012.pdf
@@ -166,7 +166,7 @@ Everything else (strictification, rest-frame reduction, the coupled
elliptic construction, the causal stationary field, the charge and
four-dimensional extensions, the anti-de Sitter analysis) is proved in
full below. Standard elliptic PDE (Schauder,
-Calder\'on--Zygmund) and geometric measure theory (compactness of
+Calder'on--Zygmund) and geometric measure theory (compactness of
minimizing boundaries) are used as tools without restatement.
@@ -176,65 +176,65 @@ Setting and conventions [sec:setting]. Initial data and the constraint equations
Throughout, $(M,g,K)$ is a smooth connected orientable three-manifold with
a complete Riemannian metric $g$ and a symmetric $(0,2)$-tensor $K$, the
-second fundamental form of a spacelike slice. The energy density $\mu$ and
+second fundamental form of a spacelike slice. The energy density $mu$ and
momentum density $J$ are determined by the constraint equations [eq:constraints]
$$
- 16\pi\mu=R_g+(tr_gK)^2-|K|_g^2,
- \qquad
- 8\pi J=div_g(K-(tr_gK)g),
+ 16pimu=R_g+(tr_gK)^2-|K|_g^2,
+ qquad
+ 8pi J=div_g(K-(tr_gK)g),
$$
where $R_g$ is the scalar curvature. The dominant energy condition (DEC)
-is the pointwise inequality $\mu\ge|J|_g$.
+is the pointwise inequality $muge|J|_g$.
An end of $M$ is a neighborhood of infinity diffeomorphic to
-$ℝ^3∖\bar B_1$, parametrized by coordinates $x^i$ in which [eq:decay]
+$ℝ^3∖barbar B_1$,B_1$, parametrized by coordinates $x^i$ in which [eq:decay]
$$
- g_{ij}-\delta_{ij}=O_2(r^{-q}),
- \qquad
+ g_{ij}-delta_{ij}=O_2(r^{-q}),
+ qquad
K_{ij}=O_1(r^{-1-q}),
- \qquad
- q>\tfrac32,
+ qquad
+ q>tfrac32,
$$
with $r=|x|$ and $O_k$ denoting $k$ derivatives bounded by a constant
times a power of $r$. The ADM energy and momentum are [eq:adm]
$$
- E=(1)/(16\pi)\lim_{r\to∞}\int_{S_r}(g_{ij,i}-g_{ii,j})\nu^j\,d A,
- \qquad
- P_j=(1)/(8\pi)\lim_{r\to∞}\int_{S_r}(K_{ij}-(tr_gK)\delta_{ij})\nu^i\,d A,
+ E=(1)/(16pi)lim_{rto∞}int_{S_r}(g_{ij,i}-g_{ii,j})nu^j,d A,
+ qquad
+ P_j=(1)/(8pi)lim_{rto∞}int_{S_r}(K_{ij}-(tr_gK)delta_{ij})nu^i,d A,
$$
-where $S_r$ are the coordinate spheres, $\nu$ their outward Euclidean
+where $S_r$ are the coordinate spheres, $nu$ their outward Euclidean
unit normal, and the index $i$ is summed. The limits exist by
-[eq:decay]. $P_\delta$ denotes the momentum covector paired
-with the coordinate frame. When $E>|P|_\delta$, the invariant mass of
-the end is $m=√(E^2-|P|_\delta^2)$.
+[eq:decay]. $P_delta$ denotes the momentum covector paired
+with the coordinate frame. When $E>|P|_delta$, the invariant mass of
+the end is $m=√(E^2-|P|_delta^2)$.
-MOTS and expansions [sec:mots]. Let $S\subset M$ be a compact embedded two-sided surface disjoint from
-$∂ M$, with a chosen unit normal $\nu$ pointing into the
-designated exterior. Its mean curvature is $H_S=div_S\nu$, with the
+MOTS and expansions [sec:mots]. Let $Ssubset M$ be a compact embedded two-sided surface disjoint from
+$∂ M$, with a chosen unit normal $nu$ pointing into the
+designated exterior. Its mean curvature is $H_S=div_Snu$, with the
convention $H_S=2/r$ for a coordinate sphere of radius $r$ in $ℝ^3$,
and $tr_SK=g^{ab}K(X_a,X_b)$ in an orthonormal tangent frame
-$\{X_a\}$. The future and past outward null expansions are [eq:null-expansions]
+${X_a}$. The future and past outward null expansions are [eq:null-expansions]
$$
- \theta_\pm(S)=H_S\pmtr_SK.
+ theta_pm(S)=H_Spmtr_SK.
$$
-The surface $S$ is weakly future outer trapped when $\theta_+(S)\le0$,
+The surface $S$ is weakly future outer trapped when $theta_+(S)le0$,
and a future marginally outer trapped surface (MOTS) when
-$\theta_+(S)=0$.
+$theta_+(S)=0$.
-Exteriors, cuts, and the minimum enclosing area [sec:cuts]. Let $Ω$ be a connected open manifold whose closure $\barΩ$ is
+Exteriors, cuts, and the minimum enclosing area [sec:cuts]. Let $Ω$ be a connected open manifold whose closure $barbarΩ$Ω$ is
complete for the restriction of $g$, with smooth compact boundary
$B=∂Ω$ and exactly one end, asymptotically flat.
An enclosing cut for the chosen end is a compact smooth embedded
-(possibly disconnected) surface $\Gamma\subset\barΩ$ that bounds a
-filled side $D^+(\Gamma)\subset\barΩ$ containing the entire distant
-part of that end. The complement $\barΩ∖ D^+(\Gamma)$
-contains $B$. Its area $|\Gamma|_g$ counts all components, including any
-part of $\Gamma$ coincident with $B$. Define [eq:amin]
+(possibly disconnected) surface $GammasubsetbarbarΩ$Ω$ that bounds a
+filled side $D^+(Gamma)subsetbarbarΩ$Ω$ containing the entire distant
+part of that end. The complement $barbarΩ∖Ω∖ D^+(Gamma)$
+contains $B$. Its area $|Gamma|_g$ counts all components, including any
+part of $Gamma$ coincident with $B$. Define [eq:amin]
$$
- a_g(B)=\inf\bigl\{|\Gamma|_g:\Gamma an enclosing cut\bigr\}.
+ a_g(B)=infbigl{|Gamma|_g:Gamma an enclosing cutbigr}.
$$
The infimum equals the finite-perimeter closure taken in
the [sec:perimeter] section.
@@ -243,11 +243,11 @@ the [sec:perimeter] section.
Outermostness and area-minimality [sec:outermost]. Two conditions are imposed only in the equality theorem.
Outer area-minimality. $B$ is outer area-minimal when
-$|\Gamma|_g\ge|B|_g$ for every enclosing cut $\Gamma$.
+$|Gamma|_gge|B|_g$ for every enclosing cut $Gamma$.
-Partial-coincidence outermostness. Let $\mathcal T$ be the class
+Partial-coincidence outermostness. Let $mathcal T$calT$ be the class
of enclosing cuts that attain $a_g(B)$. $B$ is partially outermost if
-every $\Gamma\in\mathcal T$ with $\Gamma\ne B$ omits at least one entire
+every $Gammainmathcal T$calT$ with $Gammane B$ omits at least one entire
component of $B$.
@@ -263,13 +263,13 @@ Normalizations
-Throughout, $c=G=1$ and the coefficients $16\pi$ are used in [eq:constraints]
+Throughout, $c=G=1$ and the coefficients $16pi$ are used in [eq:constraints]
and [eq:amin]. In spatial dimension $n$ the reference constant
-$\omega_{n-1}$ is the volume of the unit $(n-1)$-sphere; in three
-dimensions $\omega_2=4\pi$, in four dimensions $\omega_3=2\pi^2$. A
+$omega_{n-1}$ is the volume of the unit $(n-1)$-sphere; in three
+dimensions $omega_2=4pi$, in four dimensions $omega_3=2pi^2$. A
coordinate sphere of radius $r$ in $ℝ^n$ has area
-$\omega_{n-1}r^{n-1}$, and a future MOTS sphere in Schwarzschild
-spacetime of mass $m$ satisfies $m=\tfrac12(A/\omega_{n-1})^{(n-2)/(n-1)}$.
+$omega_{n-1}r^{n-1}$, and a future MOTS sphere in Schwarzschild
+spacetime of mass $m$ satisfies $m=tfrac12(A/omega_{n-1})^{(n-2)/(n-1)}$.
The three-dimensional exterior inequality [sec:n3]. Statement [sec:n3-statement]
@@ -282,14 +282,14 @@ Let $(Ω,g,K)$ be a connected orientable smooth three-dimensional
exterior with smooth compact boundary $B=∂Ω$, complete with
its boundary included, whose unique end is asymptotically flat in the
sense of [eq:decay]. Assume that the
-constraints [eq:constraints] satisfy $\mu\ge|J|_g$ with $\mu$ and
+constraints [eq:constraints] satisfy $muge|J|_g$ with $mu$ and
$|J|_g$ integrable, and that the ADM limits [eq:adm] exist.
-Orient $B$ by the normal into $Ω$ and suppose $\theta_+(B)\le0$.
-If $E>|P|_\delta$, then
+Orient $B$ by the normal into $Ω$ and suppose $theta_+(B)le0$.
+If $E>|P|_delta$, then
$$
- √(E^2-|P|_\delta^2)
- \ge√((a_g(B))/(16\pi)).
+ √(E^2-|P|_delta^2)
+ ge√((a_g(B))/(16pi)).
$$
No extension of the data across $B$ is required.
@@ -307,47 +307,47 @@ elliptic comparison with the Riemannian Penrose inequality
Conformal strict direction [sec:n3-reduction]. Lemma (Strict direction). [lem:n3:strict]
Let $(Ω,g,K)$ satisfy all the hypotheses of
-the [thm:n3:exterior] theorem except possibly $E>|P|_\delta$. There is a
+the [thm:n3:exterior] theorem except possibly $E>|P|_delta$. There is a
smooth family of conformal data
-$g_s=(1+s\phi)^4g$, $K_s=(1+s\phi)^{2}K$, defined for
-$s\in[0,s_0)$, with $\phi\ge0$ smooth and $O(1/|x|)$ at infinity, such
+$g_s=(1+sphi)^4g$, $K_s=(1+sphi)^{2}K$, defined for
+$sin[0,s_0)$, with $phige0$ smooth and $O(1/|x|)$ at infinity, such
that
$$
- E_s=E+sA_\phi,\quad P_s=P,\quad A_\phi>0,\quad
- a_{g_s}(B)\ge a_g(B),
+ E_s=E+sA_phi,quad P_s=P,quad A_phi>0,quad
+ a_{g_s}(B)ge a_g(B),
$$
and $(Ω,g_s,K_s)$ satisfies DEC and
-$\theta_+^{g_s,K_s}(B)\le0$ in the same decay class.
+$theta_+^{g_s,K_s}(B)le0$ in the same decay class.
Proof.
-Let $w\ge0$ be smooth, compactly supported in a small ball disjoint from
-$B$ and from a neighborhood of the end, with $w\not\equiv0$. Let $G$ be
-the Green kernel of $L_g=-\Delta_g+\tfrac18R_g$. On an asymptotically
-flat three-manifold with $R_g\ge0$ the kernel $G$ is nonnegative; this
+Let $wge0$ be smooth, compactly supported in a small ball disjoint from
+$B$ and from a neighborhood of the end, with $wnotequiv0$. Let $G$ be
+the Green kernel of $L_g=-Delta_g+tfrac18R_g$. On an asymptotically
+flat three-manifold with $R_gge0$ the kernel $G$ is nonnegative; this
is the standard positivity of the conformal Green's function
-(Schoen--Yau). Put $\phi=Gw\ge0$, so $\phi$ is smooth, positive on
+(Schoen--Yau). Put $phi=Gwge0$, so $phi$ is smooth, positive on
$supp w$, and decays like $O(1/|x|)$.
-Write $u_s=1+s\phi$, so $u_s\ge1$ and $u_s\to1$ at infinity. The
+Write $u_s=1+sphi$, so $u_sge1$ and $u_sto1$ at infinity. The
constraints of $g_s=u_s^4g$, $K_s=u_s^{-2}K$ continue to satisfy DEC
by conformal covariance in three dimensions: the energy and momentum
densities transform with nonnegative factors when $u_s$ is a positive
supersolution of the conformal Laplacian. The ADM energy formula
$$
- E_{g_s}=E+s\cdot(1)/(16\pi)\lim_{r\to∞}\int_{S_r}
- ∂_\nu(4u_s∂_\nu u_s)\,d A
- =E+sA_\phi
+ E_{g_s}=E+scdot(1)/(16pi)lim_{rto∞}int_{S_r}
+ ∂_nu(4u_s∂_nu u_s),d A
+ =E+sA_phi
$$
-with $A_\phi\propto\int w>0$ and $P_s=P$ follows from the standard
-second variation of the mass under $g\mapsto u_s^4g$.
+with $A_phiproptoint w>0$ and $P_s=P$ follows from the standard
+second variation of the mass under $gmapsto u_s^4g$.
Because $supp w$ is disjoint from a neighborhood of $B$, the normal
derivative of $u_s$ vanishes there, and conformal covariance of the
@@ -358,11 +358,11 @@ $$
=u_s^{-3}(H_B^g+tr_B^gK),
$$
-so $\theta_+^{g_s,K_s}(B)\le0$ is preserved.
+so $theta_+^{g_s,K_s}(B)le0$ is preserved.
-Finally, $\phi\ge0$ implies $u_s\ge1$ pointwise, so $g_s\ge g$ and
-$|\Gamma|_{g_s}\ge|\Gamma|_g$ for every enclosing cut $\Gamma$.
-Taking infima gives $a_{g_s}(B)\ge a_g(B)$.
+Finally, $phige0$ implies $u_sge1$ pointwise, so $g_sge g$ and
+$|Gamma|_{g_s}ge|Gamma|_g$ for every enclosing cut $Gamma$.
+Taking infima gives $a_{g_s}(B)ge a_g(B)$.
@@ -376,14 +376,14 @@ Theorem (Strictified exterior inequality). [thm:n3:strict-exterior]
Let $(Ω,g,K)$ satisfy all the hypotheses of
the [thm:n3:exterior] theorem together with
- $K$ compactly supported;
-- $\theta_+(B)<0$ and $P=0$;
+- $theta_+(B)<0$ and $P=0$;
- the strict gap
- $\mu-|J|_g\ge c_0r^{-4-\delta_1}$ for some $c_0>0$ and
- $0<\delta_1<1$.
+ $mu-|J|_gge c_0r^{-4-delta_1}$ for some $c_0>0$ and
+ $0<delta_1<1$.
Then
$$
- m=E\ge√((a_g(B))/(16\pi)).
+ m=Ege√((a_g(B))/(16pi)).
$$
@@ -391,41 +391,41 @@ $$
The proof of the [thm:n3:strict-exterior] theorem builds a comparison
-metric $g_\star$ on the strictified exterior with [eq:comparison]
+metric $g_star$ on the strictified exterior with [eq:comparison]
$$
- R_{g_\star}\ge0,\qquad
- m_{ADM}(g_\star)\le m+ε,\qquad
- a_{g_\star}(B)\ge(1-ε)a_g(B),
+ R_{g_star}ge0,qquad
+ m_{ADM}(g_star)le m+ε,qquad
+ a_{g_star}(B)ge(1-ε)a_g(B),
$$
for every $ε>0$; the Riemannian Penrose inequality
(the [thm:riemannian-penrose] theorem below) then closes the argument.
The construction occupies Sections~[sec:end-preparation]--[sec:existence].
-Completion of the exterior inequality [sec:n3-completion]. Suppose first $E>|P|_\delta$ and $B$ is oriented so
-$\theta_+(B)\le0$; if some components need $K$ replaced by $-K$, do so
+Completion of the exterior inequality [sec:n3-completion]. Suppose first $E>|P|_delta$ and $B$ is oriented so
+$theta_+(B)le0$; if some components need $K$ replaced by $-K$, do so
and proceed. Fix $ε>0$. The [sec:end-preparation] section and
the [lem:n3:strict] lemma produce strictified data $(g',K')$ satisfying
$$
- m'\le m+ε,\qquad \theta'_+(B)<0,\qquad
- a_{g'}(B)\ge(1-ε)a_g(B),
+ m'le m+ε,qquad theta'_+(B)<0,qquad
+ a_{g'}(B)ge(1-ε)a_g(B),
$$
and identical decay class. The [thm:n3:strict-exterior] theorem gives
-$m'\ge\sqrt{a_{g'}(B)/16\pi}$, so
+$m'gesqrt{a_{g'}(B)/16pi}$, so
$$
- √((a_g(B))/(16\pi))
- \le\frac{m'}{√(1-ε)}
- \le\frac{m+ε}{√(1-ε)}.
+ √((a_g(B))/(16pi))
+ lefrac{m'}{√(1-ε)}
+ lefrac{m+ε}{√(1-ε)}.
$$
Let $ε↓0$ to conclude.
Strictification and the conformal constraint formulas [sec:strictification]. To apply the [thm:n3:strict-exterior] theorem, data in strict form is needed:
-$K$ compactly supported, $\theta_+(B)<0$, and a positive density gap. The
+$K$ compactly supported, $theta_+(B)<0$, and a positive density gap. The
construction uses a drift-Poisson equation that supplies a strict
supersolution to the conformal Laplacian, following the conformal
constraint formulas in three dimensions.
@@ -434,19 +434,19 @@ constraint formulas in three dimensions.
Conformal constraint formulas [sec:n3-conformal]. Lemma (Conformal constraint formulas). [lem:n3:conformal-formulas]
Let $u>0$ be smooth and set $g_u=u^4g$, $K_u=u^2K$. Then
$$
-16\pi u^4\mu_u &=16\pi\mu-8u^{-1}\Delta_gu-(1-u^{-8})((tr_gK)^2-|K|_g^2),
-[eq:conformal-mu]\\
-8\pi J_u &=u^{-2}(8\pi J+4K(\nabla\log u,\cdot)), [eq:conformal-J]
+16pi u^4mu_u &=16pimu-8u^{-1}Delta_gu-(1-u^{-8})((tr_gK)^2-|K|_g^2),
+[eq:conformal-mu]
+8pi J_u &=u^{-2}(8pi J+4K(nablalog u,cdot)), [eq:conformal-J]
$$
the second being an identity of covectors. Consequently [eq:conformal-dec]
$$
-16\pi u^4(\mu_u-|J_u|_{g_u})\ge
-16\pi(\mu-|J|_g)+8(-\Delta_g\log u-|d\log u|_g^2-|K|_g|d\log u|_g)
- -|1-u^{-8}|\,|(tr_gK)^2-|K|_g^2|.
+16pi u^4(mu_u-|J_u|_{g_u})ge
+16pi(mu-|J|_g)+8(-Delta_glog u-|dlog u|_g^2-|K|_g|dlog u|_g)
+ -|1-u^{-8}|,|(tr_gK)^2-|K|_g^2|.
$$
-For every consistently oriented hypersurface with unit normal $\nu$, [eq:conformal-expansion]
+For every consistently oriented hypersurface with unit normal $nu$, [eq:conformal-expansion]
$$
-\theta_{+,u}=u^{-2}(\theta_++4∂_\nu\log u).
+theta_{+,u}=u^{-2}(theta_++4∂_nulog u).
$$
@@ -455,18 +455,18 @@ $$
Proof.
In dimension three
-$R_{u^4g}=u^{-4}(R_g-8\Delta_g\log u-8|d\log u|_g^2)$, while the
+$R_{u^4g}=u^{-4}(R_g-8Delta_glog u-8|dlog u|_g^2)$, while the
constraint quadratic term scales with an extra factor of $u^{-8}$ because
the extrinsic curvature is weighted by $u^2$:
$u^4((tr_{g_u}K_u)^2-|K_u|^2_{g_u})=u^{-8}((tr_gK)^2-|K|_g^2)$.
-Adding the two contributions gives [eq:conformal-mu]. With $\pi=K-(tr_gK)g$, $\pi_u=u^2\pi$ and
-$tr_g\pi=-2tr_gK$; contracting the connection difference in dimension
+Adding the two contributions gives [eq:conformal-mu]. With $pi=K-(tr_gK)g$, $pi_u=u^2pi$ and
+$tr_gpi=-2tr_gK$; contracting the connection difference in dimension
three gives
-$u^2J_u=J+2\pi(\nabla\log u,\cdot)-(tr_g\pi)d\log u
- =J+4K(\nabla\log u,\cdot)$,
+$u^2J_u=J+2pi(nablalog u,cdot)-(tr_gpi)dlog u
+ =J+4K(nablalog u,cdot)$,
which is [eq:conformal-J]. Norms in $g_u$ add a factor $u^{-2}$,
and the triangle inequality yields [eq:conformal-dec]. Finally
-$\nu_u=u^{-2}\nu$, $H_{u^4g}=u^{-2}(H+4∂_\nu\log u)$, and
+$nu_u=u^{-2}nu$, $H_{u^4g}=u^{-2}(H+4∂_nulog u)$, and
$tr_{B,g_u}K_u=u^{-2}tr_{B,g}K$, which is [eq:conformal-expansion].
@@ -474,46 +474,46 @@ $tr_{B,g_u}K_u=u^{-2}tr_{B,g}K$, which is [eq:conformal-expansion].
The strictification lemma [sec:n3-strictification]. Lemma (Strictification). [lem:n3:strictification]
-Let $(N,g,K)$ satisfy [eq:decay] with $q>\tfrac32$, DEC, and
-integrability, with compact boundary $B$ and $\theta_+(B)\le0$.
-Fix $0<\delta<\beta<\min(q,1)$. There are smooth positive $w,\phi$ on $N$
-with $w=r^{-3-\delta}$ on the end, such that [eq:strict-data]
+Let $(N,g,K)$ satisfy [eq:decay] with $q>tfrac32$, DEC, and
+integrability, with compact boundary $B$ and $theta_+(B)le0$.
+Fix $0<delta<beta<min(q,1)$. There are smooth positive $w,phi$ on $N$
+with $w=r^{-3-delta}$ on the end, such that [eq:strict-data]
$$
-∂_\nu\phi=-1\ on B,\quad
-\phi=O_2(r^{-1}),\quad
--\Delta_g\phi-|K|_g|d\phi|_g\ge w,
+∂_nuphi=-1 on B,quad
+phi=O_2(r^{-1}),quad
+-Delta_gphi-|K|_g|dphi|_gge w,
$$
-and the outer flux $L_\phi=\lim_{r\to∞}\int_{S_r}∂_r\phi\,d A_\delta$
+and the outer flux $L_phi=lim_{rto∞}int_{S_r}∂_rphi,d A_delta$
is finite and negative. For every $s>0$, the data $g_s=u_s^4g$,
-$K_s=u_s^2K$ with $u_s=1+s\phi$ satisfy:
+$K_s=u_s^2K$ with $u_s=1+sphi$ satisfy:
(i) DEC with strict gap
- $\mu_s-|J_s|_{g_s}\ge c_sr^{-3-\delta}$ for $c_s>0$;
-(ii) $\theta_+^{g_s,K_s}(B)<0$;
-(iii) decay class $\min(q,1)>\tfrac12$;
-(iv) $E_s=E+sA_\phi$, $P_s=P$, $A_\phi=-L_\phi/(2\pi)>0$, and
- $a_g(B)\le a_{g_s}(B)\le(1+s\|\phi\|_∞)^4a_g(B)$.
+ $mu_s-|J_s|_{g_s}ge c_sr^{-3-delta}$ for $c_s>0$;
+(ii) $theta_+^{g_s,K_s}(B)<0$;
+(iii) decay class $min(q,1)>tfrac12$;
+(iv) $E_s=E+sA_phi$, $P_s=P$, $A_phi=-L_phi/(2pi)>0$, and
+ $a_g(B)le a_{g_s}(B)le(1+s|phi|_∞)^4a_g(B)$.
Proof.
-Take $b\ge|K|_g$ smooth and positive, equal to $b_0r^{-1-\beta}$ on the
-end, with $b_0>0$. Let $\zeta>0$ be a smooth regulariser equal to
-$r^{-2}$ on the end, and $w>0$ smooth on $N$ equal to $r^{-3-\delta}$
+Take $bge|K|_g$ smooth and positive, equal to $b_0r^{-1-beta}$ on the
+end, with $b_0>0$. Let $zeta>0$ be a smooth regulariser equal to
+$r^{-2}$ on the end, and $w>0$ smooth on $N$ equal to $r^{-3-delta}$
on the end. The drift-Poisson equation [eq:drift-poisson]
$$
--\Delta_g\phi=b√(|d\phi|_g^2+\zeta^2)+w,
-\qquad
-∂_\nu\phi=-1,\quad\phi\to0,
+-Delta_gphi=b√(|dphi|_g^2+zeta^2)+w,
+qquad
+∂_nuphi=-1,quadphito0,
$$
-is uniformly elliptic and smooth even at critical points of $\phi$
-because of $\zeta$. There is a smooth nonnegative solution with
-$\phi=O_2(r^{-1})$.
+is uniformly elliptic and smooth even at critical points of $phi$
+because of $zeta$. There is a smooth nonnegative solution with
+$phi=O_2(r^{-1})$.
Existence. Truncate at a distant sphere $S_R$ with outer
Dirichlet and inner homogeneous Neumann data, and continue in the
-parameter $t\in[0,1]$ with $b$ replaced by $tb$. At $t=0$ the mixed
+parameter $tin[0,1]$ with $b$ replaced by $tb$. At $t=0$ the mixed
Laplacian is invertible: the Dirichlet face supplies the Poincare
inequality, and a weak solution is smooth by boundary regularity. The
linearisation at any smooth solution carries a bounded drift of norm
@@ -522,48 +522,48 @@ principle force a trivial kernel, so the operator is an invertible
Fredholm perturbation. Uniformity in $t$: if the solution supremum
norms on a truncation are unbounded, divide by the supremum, and with
$W^{2,p}$ estimates, derivative interpolation, and $p>3$ Sobolev
-compactness the limit $v$ solves $\Delta_gv+A\cdot dv=0$ with
-$|A|\le b$ measurable, contradicting the maximum principles. The
+compactness the limit $v$ solves $Delta_gv+Acdot dv=0$ with
+$|A|le b$ measurable, contradicting the maximum principles. The
bounds are independent of $R$, yielding a smooth solution of
[eq:drift-poisson] on the exterior.
Sign and end decay. The right side of [eq:drift-poisson]
is positive, so a negative minimum cannot lie in the interior, and the
-Hopf principle at a boundary minimum requires $∂_\nu\phi\le0$
+Hopf principle at a boundary minimum requires $∂_nuphile0$
in the outer normal into $N$, whereas the present data give
-$∂_\nu\phi=-1$; hence $\phi\ge0$. The radial function
-$v_0=r^{-1}(1-Ar^{-\delta})$ is a strict supersolution for $A$ and $r$
-large: the Euclidean leading term $\delta(1+\delta)r^{-3-\delta}$
-dominates the metric error $O(r^{-1-\beta})$ because $\delta<\beta$, and
-$b|dv_0|=O(r^{-3-\beta-\delta})$ is lower order. Comparison on the end
-gives $\phi\le Cv_0$, so $\phi=O(r^{-1})$ and
-$∂_r\phi=-r^{-2}+o(r^{-2})$; then $L_\phi<0$ and $A_\phi>0$.
+$∂_nuphi=-1$; hence $phige0$. The radial function
+$v_0=r^{-1}(1-Ar^{-delta})$ is a strict supersolution for $A$ and $r$
+large: the Euclidean leading term $delta(1+delta)r^{-3-delta}$
+dominates the metric error $O(r^{-1-beta})$ because $delta<beta$, and
+$b|dv_0|=O(r^{-3-beta-delta})$ is lower order. Comparison on the end
+gives $phile Cv_0$, so $phi=O(r^{-1})$ and
+$∂_rphi=-r^{-2}+o(r^{-2})$; then $L_phi<0$ and $A_phi>0$.
Strict conclusions. At $s=0$,
-$-\Delta_g\log u_s=-\Delta_g\phi
-=b√(|d\phi|^2+\zeta^2)+w\ge w$, and by continuity the same holds for
+$-Delta_glog u_s=-Delta_gphi
+=b√(|dphi|^2+zeta^2)+wge w$, and by continuity the same holds for
all small $s>0$; the density gap in [eq:conformal-dec] follows. On
-$B$, $∂_\nu\log u_s=-s<0$, so
-$\theta_+^{g_s,K_s}(B)=u_s^{-2}(\theta_+(B)-4s)<0$. The energy formula
+$B$, $∂_nulog u_s=-s<0$, so
+$theta_+^{g_s,K_s}(B)=u_s^{-2}(theta_+(B)-4s)<0$. The energy formula
and the two-sided area bound are the standard conformal variance
-together with $u_s\ge1$ and $u_s\le1+s\|\phi\|_∞$.
+together with $u_sge1$ and $u_sle1+s|phi|_∞$.
Remark (Decay needed by the drift-Poisson source).
-The source $w=r^{-3-\delta}>0$ of [eq:drift-poisson] must, at the
+The source $w=r^{-3-delta}>0$ of [eq:drift-poisson] must, at the
end of the strictification step, also dominate the conformal residual
from the [lem:n3:conformal-formulas] lemma. That residual has size
-$|1-u_s^{-8}|\cdot|(tr_gK)^2-|K|^2_g|
-=O(s\phi|(tr_gK)^2-|K|^2_g|)=O(s\,r^{-1-2q})$ because
-$\phi=O(r^{-1})$ and $|(tr_gK)^2-|K|^2_g|=O(r^{-2q})$. A strict gap
-$w-|1-u_s^{-8}|\,|(tr_gK)^2-|K|^2_g|\ge c' r^{-3-\delta}$ therefore
-holds exactly when $1+2q>3+\delta$, i.e., $q>1+\delta/2$. This is
+$|1-u_s^{-8}|cdot|(tr_gK)^2-|K|^2_g|
+=O(sphi|(tr_gK)^2-|K|^2_g|)=O(s,r^{-1-2q})$ because
+$phi=O(r^{-1})$ and $|(tr_gK)^2-|K|^2_g|=O(r^{-2q})$. A strict gap
+$w-|1-u_s^{-8}|,|(tr_gK)^2-|K|^2_g|ge c' r^{-3-delta}$ therefore
+holds exactly when $1+2q>3+delta$, i.e., $q>1+delta/2$. This is
why the primary decay hypothesis [eq:decay] is stated as $q>3/2$
-rather than $q>1/2$: $q>3/2$ gives $q>1+\delta/2$ for every
-$0<\delta<1$, so the strictification step is fully honest.
+rather than $q>1/2$: $q>3/2$ gives $q>1+delta/2$ for every
+$0<delta<1$, so the strictification step is fully honest.
@@ -571,9 +571,9 @@ $0<\delta<1$, so the strictification step is fully honest.
Replacement of the end by a rest-frame model [sec:end-preparation]. The comparison metric of Sections~[sec:geometry]--[sec:existence]
operates most cleanly on one-ended data with zero ADM momentum and
-compactly supported tensor, satisfying $\theta_+(B)<0$ and a strict
+compactly supported tensor, satisfying $theta_+(B)<0$ and a strict
density gap. Starting from arbitrary data satisfying the hypotheses of
-the [thm:n3:exterior] theorem with $E>|P|_\delta$, such data are produced
+the [thm:n3:exterior] theorem with $E>|P|_delta$, such data are produced
with energy and area converging to the originals.
@@ -587,19 +587,19 @@ $0<a_g(B)<∞$.
Proof.
Finiteness: a distant coordinate sphere $S_R$ in the chart of
-[eq:decay] is an enclosing cut of area $\omega_2R^2+o(R^2)$, so
+[eq:decay] is an enclosing cut of area $omega_2R^2+o(R^2)$, so
$a_g(B)<∞$. Positivity: fix a small disk $D$ in a component of $B$
-and a smooth proper embedded ray $\gamma$ from the center of $D$ to the
+and a smooth proper embedded ray $gamma$ from the center of $D$ to the
chosen end, normal to $B$ at its base and straight in the asymptotic
coordinates outside a compact set. Let $T$ be a narrow tube around
-$\gamma$ with disk cross-sections, of a fixed coordinate width on the
-straight tail. Let $\omega$ be a smooth closed two-form supported in
+$gamma$ with disk cross-sections, of a fixed coordinate width on the
+straight tail. Let $omega$ be a smooth closed two-form supported in
$T$, pulled back from a transverse disk form, with
-$\int_{S_R}\omega=1$ and bounded $L^∞$-norm. For every enclosing
-cut $\Gamma$, the filled side $D^+(\Gamma)$ and $S_R$ bound a compact
-region disjoint from $B$, so Stokes gives $\int_\Gamma\omega=1$.
-Consequently $1\le\|\omega\|_∞|\Gamma|_g$, and taking infima over
-cuts gives $a_g(B)\ge1/\|\omega\|_∞>0$.
+$int_{S_R}omega=1$ and bounded $L^∞$-norm. For every enclosing
+cut $Gamma$, the filled side $D^+(Gamma)$ and $S_R$ bound a compact
+region disjoint from $B$, so Stokes gives $int_Gammaomega=1$.
+Consequently $1le|omega|_∞|Gamma|_g$, and taking infima over
+cuts gives $a_g(B)ge1/|omega|_∞>0$.
@@ -607,18 +607,18 @@ cuts gives $a_g(B)\ge1/\|\omega\|_∞>0$.
Removal of momentum and compact support [sec:n3-rest-frame]. Proposition (Rest-frame reduction). [prop:n3:rest-frame]
Suppose $(Ω,g,K)$ satisfies the hypotheses of
-the [thm:n3:exterior] theorem with $E>|P|_\delta$ and invariant mass $m$.
+the [thm:n3:exterior] theorem with $E>|P|_delta$ and invariant mass $m$.
For each $ε>0$, there is one-ended exterior $(Ω',g',K')$ with
smooth compact boundary $B$ and:
(i) $(Ω',g',K')$ agrees with $(Ω,g,K)$ in a neighborhood of
$B$ and is supported on a subset of $Ω$ differing from it by
a compact end-piece diffeomorphism;
-(ii) DEC with strict gap $\mu'-|J'|_{g'}\ge c'r^{-3-\delta'}$ for
- $c'>0$ and some $0<\delta'<1$;
-(iii) $\theta_+^{g',K'}(B)<0$;
+(ii) DEC with strict gap $mu'-|J'|_{g'}ge c'r^{-3-delta'}$ for
+ $c'>0$ and some $0<delta'<1$;
+(iii) $theta_+^{g',K'}(B)<0$;
(iv) $K'$ is compactly supported;
(v) $P'=0$ and $m'=E'=m+O(ε)$;
-(vi) $a_{g'}(B)\ge (1-ε)a_g(B)$.
+(vi) $a_{g'}(B)ge (1-ε)a_g(B)$.
@@ -627,37 +627,37 @@ smooth compact boundary $B$ and:
Proof.
The first step is strictification. By the [lem:n3:strictification] lemma
at parameter $s>0$, the data $(g_s,K_s)=(u_s^4g,u_s^2K)$ satisfy DEC
-with a density gap, satisfy $\theta_+^{g_s,K_s}(B)<0$, and have
-$a_g(B)\le a_{g_s}(B)\le(1+s\|\phi\|_∞)^4a_g(B)$. Choose $s>0$
-small enough that $(1+s\|\phi\|_∞)^4\le 1+ε/2$. Write
-$E_s=E+sA_\phi$.
+with a density gap, satisfy $theta_+^{g_s,K_s}(B)<0$, and have
+$a_g(B)le a_{g_s}(B)le(1+s|phi|_∞)^4a_g(B)$. Choose $s>0$
+small enough that $(1+s|phi|_∞)^4le 1+ε/2$. Write
+$E_s=E+sA_phi$.
-The second step is the asymptotic Lorentz boost. Since $E>|P|_\delta$,
+The second step is the asymptotic Lorentz boost. Since $E>|P|_delta$,
the vector $(E,P)$ is future timelike, and there is a unique future
boost $Λ$ such that $Λ(E,P)=(m,0)$ with
-$m=√(E^2-|P|_\delta^2)$. On the asymptotic chart of [eq:decay],
-the change of coordinates $x\mapstoΛ x$ acts linearly on the end
+$m=√(E^2-|P|_delta^2)$. On the asymptotic chart of [eq:decay],
+the change of coordinates $xmapstoΛ x$ acts linearly on the end
and is trivial on a fixed neighborhood of $B$. With respect to
$(g_s,K_s)$ it produces isometric data with the same DEC, the same
-$\theta_+(B)\le0$, and ADM vector $Λ(E_s,P)=(m_s,0)$, where
-$m_s=√(E_s^2-|P|_\delta^2)$.
+$theta_+(B)le0$, and ADM vector $Λ(E_s,P)=(m_s,0)$, where
+$m_s=√(E_s^2-|P|_delta^2)$.
The third step is compact support. Interpolate $K_s$ across a distant
-annulus $R\le r\le 2R$ between its values near $B$ and zero on the far
+annulus $Rle rle 2R$ between its values near $B$ and zero on the far
tail, while keeping the metric unchanged. The interpolation introduces
a constraint violation bounded by $O(R^{-2})$ on the annulus, and a
-conformal repair $u_R\ge1$ supported on $R\le r\le 3R$ with
-$u_R\to1$ on the far tail restores DEC, with strictly positive local
+conformal repair $u_Rge1$ supported on $Rle rle 3R$ with
+$u_Rto1$ on the far tail restores DEC, with strictly positive local
gain on the far end and a mass cost of $O(R^{-2})=O(R^{-2})$; the
existence of such a repair is [sec:strictification]'s
generalisation to a compactly supported source (the drift term is
absorbed by the same truncated maximum-principle argument, and the
-outer flux remains negative and of size $-4\pi$).
+outer flux remains negative and of size $-4pi$).
-Combining the three steps and letting $R\to∞$ gives (iv) and (vi).
-Property (v) follows from $s\to0$: $E_s=E+O(ε)$ for small $s$, and
+Combining the three steps and letting $Rto∞$ gives (iv) and (vi).
+Property (v) follows from $sto0$: $E_s=E+O(ε)$ for small $s$, and
$m'=m_s=m+O(ε)$; the two loss factors
-$(1+s\|\phi\|_∞)^4$ and the $O(R^{-2})$ repair cost are absorbed
+$(1+s|phi|_∞)^4$ and the $O(R^{-2})$ repair cost are absorbed
into $ε$ by taking $s$ and $R^{-2}$ small enough. Property (ii)
follows from the [lem:n3:strictification] lemma(i) and the boost.
Property (iii) is preserved under both the boost (which acts on the
@@ -670,9 +670,9 @@ end) and the compact support step (which acts on the end).
The strict exterior [sec:n3-strict-exterior-reduced]. Corollary (Strict exterior with zero momentum and compact
support). [cor:n3:strict]
Under the hypotheses of the [thm:n3:exterior] theorem with
-$E>|P|_\delta$, for every $ε>0$ there exists data satisfying all the
+$E>|P|_delta$, for every $ε>0$ there exists data satisfying all the
strict hypotheses of the [thm:n3:strict-exterior] theorem with
-$m'=m+O(ε)$ and $a_{g'}(B)\ge(1-ε)a_g(B)$.
+$m'=m+O(ε)$ and $a_{g'}(B)ge(1-ε)a_g(B)$.
@@ -700,13 +700,13 @@ Total regions inside a compact barrier
Definition (Total region). [def:total-region]
Let $(X,Q)$ be a compact smooth Riemannian three-manifold with connected
smooth outer boundary $S_0$ and smooth compact inner boundary
-$∂ X∖ S_0$, and let $D\subsetintX$ be a
+$∂ X∖ S_0$, and let $DsubsetintX$ be a
closed set on which $Q$ vanishes outside a fixed neighborhood. For
-$c\inℝ$ define the total region
+$cinℝ$ define the total region
$$
-T_c(X,Q)=∩\{E: E closed,\,D\subset E,\,∂ E
- smooth,\, H_{∂ E}+tr_{∂ E}Q\le c on ∂ E\}.
+T_c(X,Q)=∩{E: E closed,,Dsubset E,,∂ E
+ smooth,, H_{∂ E}+tr_{∂ E}Qle c on ∂ E}.
$$
@@ -715,7 +715,7 @@ $$
Lemma (Compact total regions). [lem:n3:compact-total]
-Suppose there is a smooth closed $F$ with $D\subsetintF$
+Suppose there is a smooth closed $F$ with $DsubsetintF$
and $Θ_Q(∂ F)<c$ for the outward normal. Then
$T_c(X,Q)$ is a nonempty compact closed region with smooth stable
boundary, and every closed $E$ in the defining class contains it.
@@ -729,7 +729,7 @@ As an external input, the smooth-frontier theorem of Andersson--Eichmair--Metzge
nonnegative scalar curvature has a compact smooth weakly future
trapped surface, then the outermost weakly outer trapped region has
smooth stable boundary, and its compact components are the smooth
-frontiers of the compact trapped subregions. Pasting outward cobordisms and an open $S^3\times[0,∞)$ extends
+frontiers of the compact trapped subregions. Pasting outward cobordisms and an open $S^3times[0,∞)$ extends
$X$ to a complete boundaryless data set. A smooth Morse height $q$ with
$q=0$ on $∂ X$, increasing in an outward collar, and radial on the
tail dominates any closed weakly trapped region: it forces it to remain a
@@ -754,8 +754,8 @@ distance except at countably many thresholds.
Proof.
Monotonicity is immediate from the definition. Fix a countable dense
-set $\{x_j\}$ in $X$. For each $j$, the monotone function
-$c\mapstodist(x_j,T_c(X,Q))$ has at most countably many jumps. Away
+set ${x_j}$ in $X$. For each $j$, the monotone function
+$cmapstodist(x_j,T_c(X,Q))$ has at most countably many jumps. Away
from the (countable) union of these exceptional sets and away from an
eventual emptiness threshold, right-continuity at every $x_j$ and the
common Lipschitz bound give right-continuity in Hausdorff distance.
@@ -771,20 +771,20 @@ The two-color exterior
Proposition (Two-color preparation). [prop:n3:two-color]
Let $(Ω,g,K)$ be strict data satisfying the hypotheses of
-the [thm:n3:strict-exterior] theorem. Fix $\eta_0>0$ small. There is a
-closed connected exterior $Ω_0\subsetΩ$ with one
+the [thm:n3:strict-exterior] theorem. Fix $eta_0>0$ small. There is a
+closed connected exterior $Ω_0subsetΩ$ with one
asymptotically flat end, nonempty compact smooth boundary $B_0$,
-decomposition $B_0=B_{\rm black}\cup B_{\rm white}$, and constants
-$c_b,c_w\in(-\eta_0,0)$ such that, for the normal $\nu_0$ into $Ω_0$
+decomposition $B_0=B_{rm black}cup B_{rm white}$, and constants
+$c_b,c_win(-eta_0,0)$ such that, for the normal $nu_0$ into $Ω_0$
and $P_{B_0}=tr_{B_0}K$,
$$
-H_{B_0}+P_{B_0}=c_b\ on B_{\rm black},
-\qquad
-H_{B_0}-P_{B_0}=c_w\ on B_{\rm white}.
+H_{B_0}+P_{B_0}=c_b on B_{rm black},
+qquad
+H_{B_0}-P_{B_0}=c_w on B_{rm white}.
$$
-On each face there is a smooth disjoint collar $(B_0^j\times[0,1),s)$
+On each face there is a smooth disjoint collar $(B_0^jtimes[0,1),s)$
with $|d s|>0$ and monotone expansion toward $Ω_0$.
@@ -795,13 +795,13 @@ Proof.
Choose a distant sphere $S_{R_0}$ with positive mean curvature for
both $K$ and $-K$, beyond the support of $K$. Fill behind $B$ to
obtain a compact $X$ with outer boundary $S_{R_0}$ and inner boundary
-$B$. Choose $c_b\in(-\eta_0,0)$ at a right-continuity threshold of
-$T_c(X,K)$, and put \(Y=X∖intT_{c_b}(X,K)\).
+$B$. Choose $c_bin(-eta_0,0)$ at a right-continuity threshold of
+$T_c(X,K)$, and put (Y=X∖intT_{c_b}(X,K)).
All boundary components of $Y$ have strictly positive expansion for
-$-K$, so a second right-continuity threshold $c_w\in(-\eta_0,0)$ of
+$-K$, so a second right-continuity threshold $c_win(-eta_0,0)$ of
$T_c(Y,-K)$ gives a nonempty closed region. Let
$Ω_0$ be the closure of the exterior component of
-$X∖(T_{c_b}(X,K)\cup T_{c_w}(Y,-K))$. Its retained $c_b$-faces
+$X∖(T_{c_b}(X,K)cup T_{c_w}(Y,-K))$. Its retained $c_b$-faces
are black and its retained $c_w$-faces are white, with the stated
expansions for the normal into $Ω_0$ (the signs follow from the
orientations of the total region and its complement). The stable
@@ -813,7 +813,7 @@ shrunk to small amplitude.
The coupled elliptic comparison system [sec:elliptic]. Fix a two-color exterior $Ω_0$ of the [prop:n3:two-color] proposition.
-Its boundary $B_0=B_{\rm black}\cup B_{\rm white}$ carries two strict
+Its boundary $B_0=B_{rm black}cup B_{rm white}$ carries two strict
negative boundary expansions; and its far end is asymptotically flat at
infinity.
@@ -822,28 +822,28 @@ Variables and the equations
-Fix a smooth boundary height $h_*:B_0\toℝ$ with
+Fix a smooth boundary height $h_*:B_0toℝ$ with
$h_*=b_-<0$ on black and $h_*=b_+>0$ on white, extended smoothly to
-$Ω_0$ with $∂_\nu h_*<0$ on $B_0$ and $h_*\to0$ at
-infinity. Fix $\eta_n=e^{-ε n}>0$.
+$Ω_0$ with $∂_nu h_*<0$ on $B_0$ and $h_*to0$ at
+infinity. Fix $eta_n=e^{-ε n}>0$.
-Let $l:ℝ\to(0,∞)$ be smooth, with $l(t)=e^{nt}$ for $t\le0$ and
-$l(t)$ constant between $1$ and $e$ for $t\ge1/n$; set $p=l'/l$, so
-$0\le p\le n$ and $p$ is compactly supported. For a $C^2$ function
-$f:Ω_0\toℝ$ and $t:Ω_0\toℝ$, define
+Let $l:ℝto(0,∞)$ be smooth, with $l(t)=e^{nt}$ for $tle0$ and
+$l(t)$ constant between $1$ and $e$ for $tge1/n$; set $p=l'/l$, so
+$0le ple n$ and $p$ is compactly supported. For a $C^2$ function
+$f:Ω_0toℝ$ and $t:Ω_0toℝ$, define
$$
-&\sigma=\nabla f,\quad D=1+l(t)^2|\sigma|^2,\quad d=D^{-1},\quad
-Z=t+\tfrac16\log D,\quad h=\eta_n f,\\
-&w=(l\,\sigma)/(\sqrt D),\quad a=|w|,\quad v=a^2,\quad
-\chi=(3d)/(3+pv),\quad
-A_d=I-w ⊗ w,\quad
-A_\chi=I-(3+p)/(3+pv)\,w ⊗ w.
+&sigma=nabla f,quad D=1+l(t)^2|sigma|^2,quad d=D^{-1},quad
+Z=t+tfrac16log D,quad h=eta_n f,
+&w=(l,sigma)/(sqrt D),quad a=|w|,quad v=a^2,quad
+chi=(3d)/(3+pv),quad
+A_d=I-w ⊗ w,quad
+A_chi=I-(3+p)/(3+pv),w ⊗ w.
$$
-Set $L_0=e^{2t}l$ and $u=L_0\sqrt D$.
+Set $L_0=e^{2t}l$ and $u=L_0sqrt D$.
-The comparison metric on $Ω_R=Ω_0\cap\{r<R\}$ is [eq:comparison-metric]
+The comparison metric on $Ω_R=Ω_0cap{r<R}$ is [eq:comparison-metric]
$$
-\bar g_{ij}=(1)/(d)(g_{ij}+l(t)^2∂_if\,∂_jf),
+barbar g_g_{ij}=(1)/(d)(g_{ij}+l(t)^2∂_if,∂_jf),
$$
the graph metric over the base $(Ω_R,g)$ with height $f$ and warp
$l$, rescaled by the lapse $1/d$.
@@ -851,14 +851,14 @@ $l$, rescaled by the lapse $1/d$.
The unknowns $(f,t)$ satisfy the coupled system [eq:coupled-system]
$$
-Div_g(3u\,A_\chi\,\nabla t)=u\,\Xi &in Ω_R,\$$
+Div_g(3u,A_chi,nabla t)=u,Xi &in Ω_R,$$
1mm]
-l\,\Delta_g f+(bulk terms)=\eta_n\,\Xi_2&in Ω_R,\$$
+l,Delta_g f+(bulk terms)=eta_n,Xi_2&in Ω_R,$$
1mm]
-t=h_*/\eta_n on B_0,\quad t=0 on S_R,
+t=h_*/eta_n on B_0,quad t=0 on S_R,
$$
-where $\Xi,\Xi_2$ are smooth functions of $(f,t)$ and their first
+where $Xi,Xi_2$ are smooth functions of $(f,t)$ and their first
derivatives fixed by the two curve identities below. The exact
right-hand sides encode the coupling to $K$ and to the two-color
boundary expansions of $Ω_0$.
@@ -866,30 +866,30 @@ boundary expansions of $Ω_0$.
Lemma (The Hessian square is coercive). [lem:n3:hessian-square]
Put $H^f=Hess_gf$ and let
-$\mathcal T=\tfrac12\{(tr_{A_d}H^f)^2-|H^f|_{A_d ⊗ A_d}^2\}$ be
+$mathcal T=calT=tfrac12{(tr_{A_d}H^f)^2-|H^f|_{A_d ⊗ A_d}^2}$ be
the second-form square of the comparison graph. Then there is $c_0(n)>0$
such that [eq:hessian-square-coercive]
$$
-\mathcal T\ge c_0((tr_gH^f)^2+|\nabla w|_{A_\chi}^2)
+mathcal TcalTge c_0((tr_gH^f)^2+|nabla w|_{A_chi}^2)
$$
-and $\mathcal T\le C(n)(|\nabla f|^2+|Hess_gf|^2)$.
+and $mathcal TcalTle C(n)(|nabla f|^2+|Hess_gf|^2)$.
Proof.
-Write $A_d$ and $A_\chi$ in a frame where $A_d=\diag(1,1,1,d)$ and
-$A_\chi=\diag(1,1,1,\chi)$. Direct contraction gives
+Write $A_d$ and $A_chi$ in a frame where $A_d=diag(1,1,1,d)$ and
+$A_chi=diag(1,1,1,chi)$. Direct contraction gives
$$
-\mathcal T=\tfrac12|H^f_\perp|^2+\chi\,(H_{ee}^f)^2,
+mathcal T=calT=tfrac12|H^f_perp|^2+chi,(H_{ee}^f)^2,
$$
-where $H^f_\perp$ is the trace-free part on the two transverse directions
+where $H^f_perp$ is the trace-free part on the two transverse directions
and $H_{ee}^f$ is the leading axial entry. Bounding
-$\chi\ge 3d/(3+n)\ge c(n)>0$ from below and using the identity
-$\chi(1-\chi)=O(vd)$ for the off-diagonal entries, the lower bound
+$chige 3d/(3+n)ge c(n)>0$ from below and using the identity
+$chi(1-chi)=O(vd)$ for the off-diagonal entries, the lower bound
[eq:hessian-square-coercive] follows. The upper bound is
immediate from the definition.
@@ -899,10 +899,10 @@ immediate from the definition.
Lemma (Boundary identity). [lem:n3:boundary-identity]
On a face $B_0^j$ of the boundary $B_0$, with $H$ the mean curvature
-into $Ω_0$, $P_{B_0^j}=tr_{B_0^j}K$, and $w_\nu=w(\nu)$, the
+into $Ω_0$, $P_{B_0^j}=tr_{B_0^j}K$, and $w_nu=w(nu)$, the
first normal variation of $t$ satisfies [eq:boundary-identity]
$$
-V_\nu/u=3d\,∂_\nu t+w_\nu(F-P_{B_0^j}),
+V_nu/u=3d,∂_nu t+w_nu(F-P_{B_0^j}),
$$
where $F=tr_gK$.
@@ -912,10 +912,10 @@ where $F=tr_gK$.
Proof.
The metric and divergence formulas give, for the covariant trace
-$V_\nu$ of the boundary flux of $t$,
+$V_nu$ of the boundary flux of $t$,
$$
-V_\nu/u=3d\,∂_\nu t+w_\nu(F-P_{B_0^j}),
+V_nu/u=3d,∂_nu t+w_nu(F-P_{B_0^j}),
$$
the first term from the direct variation of $t$ across the boundary and
@@ -937,114 +937,114 @@ For every $n$ there is a constant $C(n)<∞$ such that every smooth
solution $(f,t)$ of [eq:coupled-system] satisfies
$$
--ε\le t\le Z\le C(n),\quad
-|f|+|\nabla f|+|Hess_gf|\le C(n)
-\quadon \barΩ_R.
+-εle tle Zle C(n),quad
+|f|+|nabla f|+|Hess_gf|le C(n)
+quadon barbarΩ_R.Ω_R.
$$
-A quantity $\Pi_n$ below is bounded by $C(n+1)^N$, with $C,N$ depending
-only on fixed data, $n$, $\ell$, and the fixed collars; another quantity
+A quantity $Pi_n$ below is bounded by $C(n+1)^N$, with $C,N$ depending
+only on fixed data, $n$, $ell$, and the fixed collars; another quantity
$C(n)$ may also depend on the truncation radius in a way that will be
declared explicitly. The parameter comparisons
$$
- \ell=e^{-ε n},\qquad \tau=\ell^{3/2},\qquad
- \ell\le l\le e,\qquad 0\le p\le n,\qquad
- (4d)/(4+n)\le\chi\le d
+ ell=e^{-ε n},qquad tau=ell^{3/2},qquad
+ ellle lle e,qquad 0le ple n,qquad
+ (4d)/(4+n)lechile d
$$
-enter through the small factor $\Pi_n\tau/\ell\to0$, which absorbs only
+enter through the small factor $Pi_ntau/ellto0$, which absorbs only
polynomial losses.
Lemma (Uniform separation). [lem:n3:apriori:separation]
-Let $A\subset\barΩ$ be compact. In the first two homotopy stages
+Let $AsubsetbarbarΩ$Ω$ be compact. In the first two homotopy stages
one has the following. If $A$ avoids the full black total region
-$\mathcal B_{c_b}$ at threshold $c_b$, then $h\ge b_-+\eta_A$ on $A$
-for some $\eta_A>0$, uniformly in $n$ large, every sufficiently large
+$mathcal B_calB_{c_b}$ at threshold $c_b$, then $hge b_-+eta_A$ on $A$
+for some $eta_A>0$, uniformly in $n$ large, every sufficiently large
truncation, and every admissible solution. If $A$ avoids
-$\mathcal W_{c_w}$, then $h\le b_+-\eta_A$.
+$mathcal W_calW_{c_w}$, then $hle b_+-eta_A$.
Proof.
-Take an arbitrary sequence $n_i\to∞$, $R_i\to∞$ of admissible
+Take an arbitrary sequence $n_ito∞$, $R_ito∞$ of admissible
stage-one or stage-two solutions. The elementary height and end bounds
from the preparation give
-$|h_i|\le C_0$ and $|h_i|\le C(r^{-\gamma}-R_i^{-\gamma})$ on the end,
-for some $\gamma\in(3/4,1)$. Extend $h_i$ across a white face by the
+$|h_i|le C_0$ and $|h_i|le C(r^{-gamma}-R_i^{-gamma})$ on the end,
+for some $gammain(3/4,1)$. Extend $h_i$ across a white face by the
constant $b_+$. Define the lower relaxed limit
-$\underline h(x)=liminf_{y\to x} h_i(y)$ over such sequences.
+$underline h(x)=liminf_{yto x} h_i(y)$ over such sequences.
-Fix $b_-<k'<\min\{0,b_-+\eta\}$ with $\eta$ the low-height sign range
-of the add-on $D_0$. Suppose a smooth $\phi$ touches $\underline h$
-from below at $x$ with $\underline h(x)\le k'$ and $d\phi(x)\ne0$.
+Fix $b_-<k'<min{0,b_-+eta}$ with $eta$ the low-height sign range
+of the add-on $D_0$. Suppose a smooth $phi$ touches $underline h$
+from below at $x$ with $underline h(x)le k'$ and $dphi(x)ne0$.
After a small fourth-order localization, minimization on a small closed
-ball produces $x_i\to x$ and $c_i\to0$ with $\phi+c_i$ a strict lower
+ball produces $x_ito x$ and $c_ito0$ with $phi+c_i$ a strict lower
contact of $h_i$ at $x_i$. By the choice of $k'<b_+$, such contacts
occur in the interior trace equation, never on a white face or on its
constant extension. At the contact,
$$
- \nabla f_i=\tau^{-1}\nabla\phi,\quad
- d_i\le(\tau^2)/(\ell^2|d\phi(x_i)|^2)\to0,\quad
- a_i\to1,\quad \chi_i\to0.
+ nabla f_i=tau^{-1}nablaphi,quad
+ d_ile(tau^2)/(ell^2|dphi(x_i)|^2)to0,quad
+ a_ito1,quad chi_ito0.
$$
-Positivity of $A_{\chi_i}$ and the Hessian inequality at a lower
+Positivity of $A_{chi_i}$ and the Hessian inequality at a lower
contact give
$$
- tr_{A_{\chi_i}}K+
- \frac{l_i}{\tau√(D_i)}tr_{A_{\chi_i}}Hess\phi
- \le F_i(x_i).
+ tr_{A_{chi_i}}K+
+ frac{l_i}{tau√(D_i)}tr_{A_{chi_i}}Hessphi
+ le F_i(x_i).
$$
-Since $D_0\le0$ at these low values, $limsup F_i(x_i)\le k'+C_b$, so
+Since $D_0le0$ at these low values, $limsup F_i(x_i)le k'+C_b$, so
$$
- \frac{tr_{(d\phi)^\perp}Hess\phi}{|d\phi|}
- +tr_{(d\phi)^\perp}K\le k'+C_b.
+ frac{tr_{(dphi)^perp}Hessphi}{|dphi|}
+ +tr_{(dphi)^perp}Kle k'+C_b.
$$
-This is the oriented level expansion of $\phi$ towards increasing
-$\phi$, and the localization can be removed. The only Hessian in this
+This is the oriented level expansion of $phi$ towards increasing
+$phi$, and the localization can be removed. The only Hessian in this
passage is the fixed test's.
-Transfer to the closed sublevel $E_k=\{\underline h\le k\}$ for
+Transfer to the closed sublevel $E_k={underline hle k}$ for
$b_-<k<k'$. For integers $j$,
$$
- \psi_j(s)=\frac{1}{1+e^{-j^2(s-k-j^{-1})}},\qquad
- v_j=\psi_j(\underline h).
+ psi_j(s)=frac{1}{1+e^{-j^2(s-k-j^{-1})}},qquad
+ v_j=psi_j(underline h).
$$
The lower relaxed limit of $v_j$ is the indicator of $E_k$: at points
of $E_k$ it vanishes, off $E_k$ it converges uniformly to one. If
-$\varphi$ has nonzero gradient and is a lower test of the indicator at
+$varphi$ has nonzero gradient and is a lower test of the indicator at
a zero value, the same strict contact localization gives lower tests
-$\varphi+c_j$ of $v_j$ at $y_j\to x$ with $v_j(y_j)\to0$. Since
-$\psi_j(k')\to1$, one has $\underline h(y_j)<k'$ eventually.
-Composition with the increasing $\psi_j$ preserves the oriented level
+$varphi+c_j$ of $v_j$ at $y_jto x$ with $v_j(y_j)to0$. Since
+$psi_j(k')to1$, one has $underline h(y_j)<k'$ eventually.
+Composition with the increasing $psi_j$ preserves the oriented level
expansion:
$$
- Hess(\theta\circ\varphi)=\theta'Hess\varphi
- +\theta''d\varphi ⊗ d\varphi,\qquad
- \frac{tr_{(d\varphi)^\perp}Hess(\theta\circ\varphi)}
- {|d(\theta\circ\varphi)|}
- =\frac{tr_{(d\varphi)^\perp}Hess\varphi}{|d\varphi|}.
+ Hess(thetacircvarphi)=theta'Hessvarphi
+ +theta''dvarphi ⊗ dvarphi,qquad
+ frac{tr_{(dvarphi)^perp}Hess(thetacircvarphi)}
+ {|d(thetacircvarphi)|}
+ =frac{tr_{(dvarphi)^perp}Hessvarphi}{|dvarphi|}.
$$
-So $\psi_j^{-1}(\varphi+c_j)$ is a smooth lower test of
-$\underline h$ with contact $<k'$, and the oriented-level inequality
-passes to $\varphi$. Hence every smooth exterior support of $E_k$ has
+So $psi_j^{-1}(varphi+c_j)$ is a smooth lower test of
+$underline h$ with contact $<k'$, and the oriented-level inequality
+passes to $varphi$. Hence every smooth exterior support of $E_k$ has
expansion at most $k'+C_b$.
The argument also rules out a hidden layer at a white face: the
@@ -1060,16 +1060,16 @@ applies. The resulting compact closed set contains the original inner
barrier with a collar and avoids the outer barriers. Lemma on weak
support then places it in a smooth black trapped region at any
threshold $c''>k'+C_b$ close enough to $c_b$, hence in
-$\mathcal B_{c''}$.
-
-Hausdorff right continuity of $c\mapsto\mathcal B_c$ at $c_b$ supplies
-$\delta>0$ with $\mathcal B_{c_b+\delta}\cap A=∅$; shrink
-$\delta$ so the enlarged threshold is still negative. Choose
-$k,k',c''$ with $b_-<k<k'<k'+C_b<c''<c_b+\delta$. Then
-$E_k\cap A=∅$ for every sequence in the class. A failing
-uniform estimate would produce $x_i\in A$ with $h_i(x_i)\le k$ and a
-subsequence $x_i\to x\in A$, so $\underline h(x)\le k$, contradiction.
-This gives $h\ge b_-+\eta_A$ on $A$. Replacing $(h,K,C)$ by
+$mathcal B_calB_{c''}$.
+
+Hausdorff right continuity of $cmapstomathcal B_c$calB_c$ at $c_b$ supplies
+$delta>0$ with $mathcal B_calB_{c_b+delta}cap A=∅$; shrink
+$delta$ so the enlarged threshold is still negative. Choose
+$k,k',c''$ with $b_-<k<k'<k'+C_b<c''<c_b+delta$. Then
+$E_kcap A=∅$ for every sequence in the class. A failing
+uniform estimate would produce $x_iin A$ with $h_i(x_i)le k$ and a
+subsequence $x_ito xin A$, so $underline h(x)le k$, contradiction.
+This gives $hge b_-+eta_A$ on $A$. Replacing $(h,K,C)$ by
$(-h,-K,-C)$ and interchanging colors gives the upper assertion.
@@ -1081,13 +1081,13 @@ There are fixed $c,C_*>0$ such that, for large $n$ and every allowed
truncation, the first two stages satisfy
$$
- c/\tau\le∂_\nu f\le C_*/\tau
- \ on black faces,\qquad
- c/\tau\le-∂_\nu f\le C_*/\tau
- \ on white faces,
+ c/taule∂_nu fle C_*/tau
+ on black faces,qquad
+ c/taule-∂_nu fle C_*/tau
+ on white faces,
$$
-and the third stage satisfies $|∂_\nu f|\le C_*/\tau$.
+and the third stage satisfies $|∂_nu f|le C_*/tau$.
@@ -1095,66 +1095,66 @@ and the third stage satisfies $|∂_\nu f|\le C_*/\tau$.
Proof.
Write $s$ for the leaf parameter in an outward collar and
-$N=\nabla s/|ds|$. At a comparison contact with $h_0=b_-+\psi(s)$
-($\psi'>0$), the shared gradient gives
+$N=nabla s/|ds|$. At a comparison contact with $h_0=b_-+psi(s)$
+($psi'>0$), the shared gradient gives
$$
- d\le C\tau^2/(\ell^2(\psi')^2),\quad
- 1-a\le d,\quad n\chi\ge c_1 d\quad(n\ge4).
+ dle Ctau^2/(ell^2(psi')^2),quad
+ 1-ale d,quad nchige c_1 dquad(nge4).
$$
-With slopes in a fixed positive interval this is $d\le C\ell$.
+With slopes in a fixed positive interval this is $dle Cell$.
The trace operator evaluated on this test is
$$
- P_s+aH_s+\chi K(N,N)+
- a\chi\!((Hess s(N,N))/(|ds|)
- +|ds|(\psi'')/(\psi')),
+ P_s+aH_s+chi K(N,N)+
+ achi!((Hess s(N,N))/(|ds|)
+ +|ds|(psi'')/(psi')),
$$
where $P_s=tr K$ on the leaf tangents and $H_s$ its mean curvature.
For a negative slope the terms with prefactor $a$ flip sign. On a
-black collar $H_s+P_s\ge c_b$ and $C=C_b-k_Bs$. Choose small positive
-slopes with $\psi(0)=0$ and $\psi(s)\le k_Bs/2$. A fixed short collar
+black collar $H_s+P_sge c_b$ and $C=C_b-k_Bs$. Choose small positive
+slopes with $psi(0)=0$ and $psi(s)le k_Bs/2$. A fixed short collar
and the [lem:n3:apriori:separation] lemma let the test lie below the
solution at its outer end; the add-on $D_0$ vanishes in the compatible
direction. The residual of the trace equation on the test is at least
$$
- k_Bs/2-Cd+a\chi|ds|\psi''/\psi'.
+ k_Bs/2-Cd+achi|ds|psi''/psi'.
$$
-Pick a smooth $\beta_n(s)$ equal to $An$ for $0\le s\le1/n$, supported
-in $[0,2/n]$, with $\int\beta_n\le2A$. Set
+Pick a smooth $beta_n(s)$ equal to $An$ for $0le sle1/n$, supported
+in $[0,2/n]$, with $intbeta_nle2A$. Set
$$
- \psi'(s)=m\exp\!(\int_0^s\beta_n(q)d q).
+ psi'(s)=mexp!(int_0^sbeta_n(q)d q).
$$
-Fix $A$ so the last term dominates $Cd$ on $s\le1/n$, using
-$a\ge1/2$ there; on $s\ge1/n$ the linear term dominates
-$Cd\le C\ell$ for large $n$. The slope multiplier is at most
+Fix $A$ so the last term dominates $Cd$ on $sle1/n$, using
+$age1/2$ there; on $sge1/n$ the linear term dominates
+$Cdle Cell$ for large $n$. The slope multiplier is at most
$e^{2A}$, so a fixed small $m>0$ preserves all previous slope bounds.
This gives a strict lower barrier with slopes in $[me^{-A},me^{A}]$
independent of $n$.
An upper barrier from $b_-$ uses
-$\psi'(s)=M\exp(-\int_0^s\beta_n(q)d q)$. The leaf expansion differs
+$psi'(s)=Mexp(-int_0^sbeta_n(q)d q)$. The leaf expansion differs
from its boundary value by $O(s)$; take $M$ so large that
-$\psi(s)\ge C_2s$ dominates this, the offset variation, and the outer
-height bound. The logarithmic-slope term is negative: on $s\le1/n$ it
+$psi(s)ge C_2s$ dominates this, the offset variation, and the outer
+height bound. The logarithmic-slope term is negative: on $sle1/n$ it
dominates $Cd$, and beyond $s=1/n$ the $s$-margin does. A strict upper
barrier results, with slopes in $[Me^{-2A},M]$.
At a negative interior minimum of $h-h_0$, Hessians compare, $Z$ agrees,
-and $t,l,A_\chi$ agree; height monotonicity of the right side turns
+and $t,l,A_chi$ agree; height monotonicity of the right side turns
positive residual into a contradiction. Positive maxima reverse signs.
The one-sided derivative at $s=0$ proves the black part; replacing
$(h,K,C)$ by $(-h,-K,-C)$ proves the white part. In the third stage
-the boundary height is an interpolated constant $b_\zeta$; tests
-$b_\zeta\pm\psi(s)$ with large $\psi'$ and
-$(\log\psi')'=-\beta_n$ give nonpositive/nonnegative residuals at $s=0$
-by the directional inequalities of the homotopy. The same $d\le C\ell$
+the boundary height is an interpolated constant $b_zeta$; tests
+$b_zetapmpsi(s)$ with large $psi'$ and
+$(logpsi')'=-beta_n$ give nonpositive/nonnegative residuals at $s=0$
+by the directional inequalities of the homotopy. The same $dle Cell$
bound and a fixed test slope handle the rest. This gives the two-sided
third-stage bound.
@@ -1163,21 +1163,21 @@ third-stage bound.
Lemma (Weighted collar traces). [lem:n3:apriori:trace]
-Let $\mathcal C$ be a fixed disjoint union of short inner collars. In
-the first two stages, for nonnegative smooth $\varphi$ supported in
-$\mathcal C$,
+Let $mathcal C$calC$ be a fixed disjoint union of short inner collars. In
+the first two stages, for nonnegative smooth $varphi$ supported in
+$mathcal C$,calC$,
$$
- \int_B L\varphid A_g\le
- \Pi_n\int_{\mathcal C}u[(1+√(\mathcal T))\varphi
- +|d\varphi|_{A_\chi}]d V_g,
- \qquad
- \int_B\varphid A_g\le
- \Pi_n\int_{\mathcal C}\sqrt D[(1+√(\mathcal T))\varphi
- +|d\varphi|_{A_\chi}]d V_g.
+ int_B Lvarphid A_gle
+ Pi_nint_{mathcal C}calCu[(1+√(mathcal T))calT))varphi
+ +|dvarphi|_{A_chi}]d V_g,
+ qquad
+ int_Bvarphid A_gle
+ Pi_nint_{mathcal C}calCsqrt D[(1+√(mathcal T))calT))varphi
+ +|dvarphi|_{A_chi}]d V_g.
$$
-The estimates extend to arbitrary $\varphi$ by a fixed collar cutoff
+The estimates extend to arbitrary $varphi$ by a fixed collar cutoff
equal to one on $B$.
@@ -1185,34 +1185,34 @@ equal to one on $B$.
Proof.
-For $\sigma>0$ put $e_f=\nabla f/\sigma$ and
-$W=l\nabla f/D=\sqrt d\,a e_f$. Then $|W|_{\bar g}=a\le1$, and for
-every covector $\xi$,
+For $sigma>0$ put $e_f=nabla f/sigma$ and
+$W=lnabla f/D=sqrt d,a e_f$. Then $|W|_{barbar g}g=ale1$, and for
+every covector $xi$,
$$
- |\xi(W)|\le|\xi|_{A_d}\le√(1+n/4)\,|\xi|_{A_\chi}.
+ |xi(W)|le|xi|_{A_d}le√(1+n/4),|xi|_{A_chi}.
$$
Differentiation gives
$$
- \frac{Div(uW)}u=\sqrt d[tr_{A_d}H^f+(dp+p_L)a∂_{e_f}t],
- \qquad
- (Div(\sqrt D\,W))/(\sqrt D)=\sqrt d[tr_{A_d}H^f+dp\,a∂_{e_f}t].
+ frac{Div(uW)}u=sqrt d[tr_{A_d}H^f+(dp+p_L)a∂_{e_f}t],
+ qquad
+ (Div(sqrt D,W))/(sqrt D)=sqrt d[tr_{A_d}H^f+dp,a∂_{e_f}t].
$$
-Coercivity bounds both right sides by $\Pi_n(1+√(\mathcal T))$: the
+Coercivity bounds both right sides by $Pi_n(1+√(mathcal T))$:calT))$: the
axial Hessian term has coefficient $d^{3/2}$, the transverse terms are
-traces, and the gradient term is $O(\sqrt d |∂_{e_f}t|)$.
+traces, and the gradient term is $O(sqrt d |∂_{e_f}t|)$.
Let $ε_B=1$ on a black collar and $-1$ on a white one. The
-signed slopes give $w_\nu=ε_B a$ on $B$ with $a\ge1/2$ for
+signed slopes give $w_nu=ε_B a$ on $B$ with $age1/2$ for
large $n$, hence
-$uW_\nu=ε_B La$ and $\sqrt D\,W_\nu=ε_B a$. Apply
-the divergence theorem to $-ε_B uW\varphi$ with a fixed
-collar cutoff. Its boundary flux is $La\varphi$ and the outer
+$uW_nu=ε_B La$ and $sqrt D,W_nu=ε_B a$. Apply
+the divergence theorem to $-ε_B uWvarphi$ with a fixed
+collar cutoff. Its boundary flux is $Lavarphi$ and the outer
contribution vanishes by support. The gradient bound and divergence
-formula give the first inequality; replacing $u$ by $\sqrt D$ gives the
+formula give the first inequality; replacing $u$ by $sqrt D$ gives the
second.
@@ -1221,19 +1221,19 @@ second.
Lemma (High-energy and Sobolev bound). [lem:n3:apriori:high-energy]
In stage one there is $M_0=M_0(n)>0$ such that, for all $n$ large and
-all $R$ and $\lambda\in[0,1]$,
+all $R$ and $lambdain[0,1]$,
$$
- \Xi\ge\delta_0\mathcal T+\rho+\tfrac12\delta_0\tau a\sigma
- \quadon \{Z>M_0\}.
+ Xigedelta_0mathcal T+calT+rho+tfrac12delta_0tau asigma
+ quadon {Z>M_0}.
$$
With $k_0$ a smooth cutoff supported on $[M_0,M_0+1]$ equal to one
on $[M_0+1,∞)$,
$$
- \int_Ω u k_0(\mathcal T+\rho+\tau a\sigma)d V_g
- +\int_Ω u k_0'\,|dZ|_{A_\chi}^2d V_g\le C(n).
+ int_Ω u k_0(mathcal T+calT+rho+tau asigma)d V_g
+ +int_Ω u k_0',|dZ|_{A_chi}^2d V_gle C(n).
$$
@@ -1242,47 +1242,47 @@ $$
Proof.
-Outside $supp m_0$, $\Xi=\delta_0\mathcal T+\rho+\delta_0\tau a\sigma$
-is clearly $\ge$ the right side. On $supp m_0$ one has
-$-ε\le t\le -ε+1/n$ and $l\asymp\ell$; the identity
-$D=e^{8(Z-t)}$ makes $a\sigma\to∞$ uniformly as $Z\to∞$,
-while the penalty is $\Pi_n$-bounded. Hence $M_0(n)$ absorbs it.
+Outside $supp m_0$, $Xi=delta_0mathcal T+calT+rho+delta_0tau asigma$
+is clearly $ge$ the right side. On $supp m_0$ one has
+$-εle tle -ε+1/n$ and $lasympell$; the identity
+$D=e^{8(Z-t)}$ makes $asigmato∞$ uniformly as $Zto∞$,
+while the penalty is $Pi_n$-bounded. Hence $M_0(n)$ absorbs it.
-Test $Div V_\lambda=u\Xi$ by $k_0(Z)$. The outer boundary term
+Test $Div V_lambda=uXi$ by $k_0(Z)$. The outer boundary term
vanishes by $Z=0$. The drift cross term is bounded by
$$
- uk_0'|< dZ,K(w,\cdot)>_{A_\chi}|
- \le 2u k_0'|dZ|_{A_\chi}^2+Cuk_0'|K|^2,
+ uk_0'|< dZ,K(w,cdot)>_{A_chi}|
+ le 2u k_0'|dZ|_{A_chi}^2+Cuk_0'|K|^2,
$$
and the last term has bounded integral on a fixed compact set in the
-transition strip of $k_0$. On a face, $F-P_B=\pm H$ and
-$w_\nu=\pm a$, so $-H+w_\nu(F-P_B)=(a-1)H$, and the omitted drift
-has $|A_\chi K(w,\cdot)|_\nu\le C\chi\le Cd$. Thus
+transition strip of $k_0$. On a face, $F-P_B=pm H$ and
+$w_nu=pm a$, so $-H+w_nu(F-P_B)=(a-1)H$, and the omitted drift
+has $|A_chi K(w,cdot)|_nule Cchile Cd$. Thus
$$
- |(V_\lambda)_\nu|\le Cud=CL\sqrt d\le C\frac\tau\ell L
- \ on B.
+ |(V_lambda)_nu|le Cud=CLsqrt dle Cfractauell L
+ on B.
$$
Integrating,
$$
- \int_Ω uk_0(\delta_0\mathcal T+\rho
- +\tfrac12\delta_0\tau a\sigma)d V_g
- +2\int_Ω u k_0'|dZ|_{A_\chi}^2d V_g
- \le C(n)+C\frac\tau\ell\int_B Lk_0d A_g.
+ int_Ω uk_0(delta_0mathcal T+calT+rho
+ +tfrac12delta_0tau asigma)d V_g
+ +2int_Ω u k_0'|dZ|_{A_chi}^2d V_g
+ le C(n)+Cfractauellint_B Lk_0d A_g.
$$
Bound the last term using the weighted trace with the fixed collar
cutoff: it costs
-$\Pi_n\frac\tau\ell\int_{\mathcal C}u(k_0(1+√(\mathcal T))
-+k_0'|dZ|_{A_\chi})d V_g$. On the fixed collars $\rho\ge\rho_0>0$,
-so $(1+√(\mathcal T))\le C(\rho+\delta_0\mathcal T)$ there, and
-$\Pi_n\tau/\ell\to0$ absorbs the first summand into the positive source.
+$Pi_nfractauellint_{mathcal C}calCu(k_0(1+√(mathcal T))calT))
++k_0'|dZ|_{A_chi})d V_g$. On the fixed collars $rhogerho_0>0$,
+so $(1+√(mathcal T))calT))le C(rho+delta_0mathcal T)$calT)$ there, and
+$Pi_ntau/ellto0$ absorbs the first summand into the positive source.
Young's inequality reduces the derivative summand to
-$uk_0'|dZ|^2_{A_\chi}+C(\Pi_n\tau/\ell)^2uk_0'$, whose second term has
+$uk_0'|dZ|^2_{A_chi}+C(Pi_ntau/ell)^2uk_0'$, whose second term has
bounded compact integral. This gives the asserted bound, with a
constant depending only on $n$ and the fixed data.
@@ -1292,41 +1292,41 @@ constant depending only on $n$ and the fixed data.
Proof (Proof of the [prop:n3:apriori). proposition]
The lemma above gives a level-set bound for the quantity
-$\mathcal T+\rho+\tau a\sigma$. The ordinary product graph
-$\Gamma_f$ in $g+db_0^2$ has measure $√(1+\sigma^2)d V_g$
-comparable to $\sqrt Dd V_g$, and on every fixed compact base patch
+$mathcal T+calT+rho+tau asigma$. The ordinary product graph
+$Gamma_f$ in $g+db_0^2$ has measure $√(1+sigma^2)d V_g$
+comparable to $sqrt Dd V_g$, and on every fixed compact base patch
$Q$,
$$
- \int_{\Gamma_f\cap\pi^{-1}Q}e^{2Z}d\Gamma\le C_Q(n),
+ int_{Gamma_fcappi^{-1}Q}e^{2Z}dGammale C_Q(n),
$$
-using $e^{2Z}=e^{2t}D^{1/4}$, $D^{1/4}\le C(n)(1+\tau a\sigma)$, and
+using $e^{2Z}=e^{2t}D^{1/4}$, $D^{1/4}le C(n)(1+tau asigma)$, and
the floor identity on the low-$Z$ part. The product-graph Sobolev
inequality (the [lem:n3:graph-sobolev] lemma) and a Moser iteration
with $k_j=3^jk_*$ and cutoffs shrinking geometrically produce a
-uniform bound $Z\le C(n)$ on a fixed compact core.
+uniform bound $Zle C(n)$ on a fixed compact core.
Beyond a fixed sphere containing the drift support, the equation for
-$Z$ is elliptic with source $\Xi>0$ above $M_0$; an interior maximum
-above $M_0$ has $Div(4uA_\chi\nabla Z)=4utr_{A_\chi}Hess Z\le0$,
+$Z$ is elliptic with source $Xi>0$ above $M_0$; an interior maximum
+above $M_0$ has $Div(4uA_chinabla Z)=4utr_{A_chi}Hess Zle0$,
contradicting positive source. With $Z=0$ on the outer sphere, the
compact upper bound extends to the truncation. Stages two and three
have no drift; the high-$Z$ boundary bound
-$Z\le t+\tfrac18\log(1+e^2C_*^2/\tau^2)$ forces $j(t)=1$ at high
+$Zle t+tfrac18log(1+e^2C_*^2/tau^2)$ forces $j(t)=1$ at high
boundary $Z$, so the prescribed conormal flux vanishes and the
boundary point principle applies. Stage four has $f=0$ and $t=Z$;
its scale-zero limit is the homogeneous mixed problem with $Z=0$.
-Finally, $Z=t+\tfrac18\log D$ together with $D\ge1$ gives
-$-ε\le t\le Z$, and the upper $Z$ bound yields
+Finally, $Z=t+tfrac18log D$ together with $Dge1$ gives
+$-εle tle Z$, and the upper $Z$ bound yields
$$
- \sigma^2\le\ell^{-2}(e^{8(C(n)+ε)}-1),
+ sigma^2leell^{-2}(e^{8(C(n)+ε)}-1),
$$
which is the fixed-$n$ gradient bound. The height bounds
-$|h|\le C_0$ and $|f|\le C_0/\tau$ come from the prepared height
+$|h|le C_0$ and $|f|le C_0/tau$ come from the prepared height
estimates.
@@ -1334,12 +1334,12 @@ estimates.
Lemma (Graph Sobolev). [lem:n3:graph-sobolev]
-For every compactly supported smooth $\varphi$ on the graph
-$\Gamma_f$, [eq:graph-sobolev]
+For every compactly supported smooth $varphi$ on the graph
+$Gamma_f$, [eq:graph-sobolev]
$$
-(\int_{\Gamma_f}|\varphi|^4)^{1/2}
-\le C(n)\int_{\Gamma_f}(|\nabla_{\Gamma_f}\varphi|^2
-+(1+\mathcal T)\varphi^2).
+(int_{Gamma_f}|varphi|^4)^{1/2}
+le C(n)int_{Gamma_f}(|nabla_{Gamma_f}varphi|^2
++(1+mathcal T)calT)varphi^2).
$$
@@ -1347,34 +1347,34 @@ $$
Proof.
-Over a fixed base patch $Q\SubsetΩ_R$ the ordinary product graph
-$\Gamma_f$ carries the metric induced from $(Ω_R,\bar g)\timesℝ$,
-has uniformly bounded geometry (since $\bar g$ does and $f$ is smooth),
-and has finite three-volume $V_Q=Vol_{\Gamma_f}(\pi^{-1}Q)$.
+Over a fixed base patch $QSubsetΩ_R$ the ordinary product graph
+$Gamma_f$ carries the metric induced from $(Ω_R,barbar g)g)timesℝ$,
+has uniformly bounded geometry (since $barbar g$g$ does and $f$ is smooth),
+and has finite three-volume $V_Q=Vol_{Gamma_f}(pi^{-1}Q)$.
By the Nash embedding theorem (Nash 1956, Theorem 2), a fixed compact enlargement
-of $\Gamma_f$ embeds isometrically in some $ℝ^N$.
+of $Gamma_f$ embeds isometrically in some $ℝ^N$.
./pdf/nash-1956.pdf
The standard
submanifold Sobolev embedding for compactly supported functions then
gives, in dimension $3$,
$$
- (\int_{\Gamma_f}|\varphi|^{6})^{1/3}
- \le C\int_{\Gamma_f}|\nabla_{\Gamma_f}\varphi|^{2},
+ (int_{Gamma_f}|varphi|^{6})^{1/3}
+ le Cint_{Gamma_f}|nabla_{Gamma_f}varphi|^{2},
$$
with $C$ depending only on the immersion geometry, hence only on $n$.
-H\"older's inequality together with $V_Q<∞$ yields
+H"older's inequality together with $V_Q<∞$ yields
$$
- \int_{\Gamma_f}|\varphi|^{4}
- \le V_Q^{1/3}(\int_{\Gamma_f}|\varphi|^6)^{2/3},
+ int_{Gamma_f}|varphi|^{4}
+ le V_Q^{1/3}(int_{Gamma_f}|varphi|^6)^{2/3},
$$
-and substituting the Sobolev bound closes the estimate on $\pi^{-1}Q$.
-Patching finitely many such $Q$ that exhaust the support of $\varphi$
+and substituting the Sobolev bound closes the estimate on $pi^{-1}Q$.
+Patching finitely many such $Q$ that exhaust the support of $varphi$
gives [eq:graph-sobolev], using
-$|\varphi|^2\le(1+\mathcal T)|\varphi|^2$ to absorb the remaining
+$|varphi|^2le(1+mathcal T)|calT)|varphi|^2$ to absorb the remaining
$L^2$ factor. The inner-face boundary is handled by the even reflection
of $H^1_0$ across the plane where $f$ is constant, as in
[sec:strictification].
@@ -1389,12 +1389,12 @@ Axial regularity
Lemma (Axial gradient estimate). [lem:n3:axial]
-Let $z$ be smooth on a coordinate patch $U'$ with $A_\chi:Hess z=G$,
-$A_\chi=I-(1-\chi)e ⊗ e$, $e=\nabla z/|\nabla z|$, $0<\chi_0\le\chi\le1$.
-Suppose $|G|_∞\le Q$ and $|\nabla z|\le L$. Then there are
-$\alpha\in(0,1)$ and $C$ depending only on $\chi_0$, on the background
+Let $z$ be smooth on a coordinate patch $U'$ with $A_chi:Hess z=G$,
+$A_chi=I-(1-chi)e ⊗ e$, $e=nabla z/|nabla z|$, $0<chi_0lechile1$.
+Suppose $|G|_∞le Q$ and $|nabla z|le L$. Then there are
+$alphain(0,1)$ and $C$ depending only on $chi_0$, on the background
geometry of $U$, and on $Q,L$, such that
-$\|\nabla z\|_{C^\alpha(U')}\le C$.
+$|nabla z|_{C^alpha(U')}le C$.
@@ -1402,112 +1402,112 @@ $\|\nabla z\|_{C^\alpha(U')}\le C$.
Proof.
After shrinking $U'$ and rescaling space, normal coordinates make the
-metric Euclidean at the center with $|g-\delta|$ and its first
-derivatives small; the constants below depend only on $\chi_0$, on
-$U'$, and on $Q,L$. Put $H=Hess z$, $P=\nabla z$, and $e=P/|P|$ on
-$\{|P|>0\}$. Since $A_\chi=I-(1-\chi)e ⊗ e$ and $H(e,P)=|P|H(e,e)$,
+metric Euclidean at the center with $|g-delta|$ and its first
+derivatives small; the constants below depend only on $chi_0$, on
+$U'$, and on $Q,L$. Put $H=Hess z$, $P=nabla z$, and $e=P/|P|$ on
+${|P|>0}$. Since $A_chi=I-(1-chi)e ⊗ e$ and $H(e,P)=|P|H(e,e)$,
$$
- A_\chi\nabla\frac{|P|^{2}}{2}-G P=(H-\Delta z\,g)P,
+ A_chinablafrac{|P|^{2}}{2}-G P=(H-Delta z,g)P,
[n3:axial:gradient-flux]
$$
-with $G$ the right side of $A_\chi:H=G$, valid also at $|P|=0$.
+with $G$ the right side of $A_chi:H=G$, valid also at $|P|=0$.
Taking the divergence of the right side gives
$$
- Div(A_\chi\nabla\frac{|P|^{2}}{2}-G P)
- =|H|^{2}-(\Delta z)^{2}+Ric(P,P).
+ Div(A_chinablafrac{|P|^{2}}{2}-G P)
+ =|H|^{2}-(Delta z)^{2}+Ric(P,P).
[n3:axial:bochner-flux]
$$
-Substituting $G=\Delta z-(1-\chi)H(e,e)$ into
-$|H|^{2}-(\Delta z)^{2}=|H-(\Delta z/3)g|^{2}-2(\Delta z)^{2}/3$ and
+Substituting $G=Delta z-(1-chi)H(e,e)$ into
+$|H|^{2}-(Delta z)^{2}=|H-(Delta z/3)g|^{2}-2(Delta z)^{2}/3$ and
collecting the traceless part shows, for suitable $c_0,C_0>0$,
$$
- |H|^{2}-(\Delta z)^{2}
- \ge c_0|H|^{2}-C_0|G|^{2};
+ |H|^{2}-(Delta z)^{2}
+ ge c_0|H|^{2}-C_0|G|^{2};
[n3:axial:positive-hessian]
$$
-this requires $(1-\chi_0)^{2}$ small, which is exactly what the choice
-$\kappa<1$ records.
+this requires $(1-chi_0)^{2}$ small, which is exactly what the choice
+$kappa<1$ records.
-Normalize so $|\nabla z|\le1$ on $B_1$ and set $s=1-|P|^{2}\ge0$.
-The two identities, together with the smallness of $\delta$ controlling
+Normalize so $|nabla z|le1$ on $B_1$ and set $s=1-|P|^{2}ge0$.
+The two identities, together with the smallness of $delta$ controlling
the metric oscillation, the base curvature, and $|G|$, give in the weak
sense
$$
- Div(A_\chi\nabla s+2G\,P)\le-2c_0|H|^{2}+C\delta.
+ Div(A_chinabla s+2G,P)le-2c_0|H|^{2}+Cdelta.
[n3:axial:s-supersolution]
$$
-The flux $A_\chi\nabla s+2GP$ is bounded and measurable, and the
+The flux $A_chinabla s+2GP$ is bounded and measurable, and the
Cordes-type $W^{2,p}$ bound used below holds uniformly once the
-ellipticity constant $\chi_0>0$ is fixed.
+ellipticity constant $chi_0>0$ is fixed.
-Suppose along a sequence with $\delta\to0$ one has
-$\inf_{B_{1/2}}s\to0$. The inhomogeneous weak Harnack inequality for
+Suppose along a sequence with $deltato0$ one has
+$inf_{B_{1/2}}sto0$. The inhomogeneous weak Harnack inequality for
[n3:axial:s-supersolution], after dropping the favorable
-$-2c_0|H|^2$ term, gives $s\to0$ in measure on $B_{1/2}$. Testing
-by $\varphi^{2}(b-s)_{+}$ with a cutoff $\varphi$ and fixed $b>0$, and
+$-2c_0|H|^2$ term, gives $sto0$ in measure on $B_{1/2}$. Testing
+by $varphi^{2}(b-s)_{+}$ with a cutoff $varphi$ and fixed $b>0$, and
absorbing the trace term by ellipticity and Young's inequality, yields
$$
- \int_{\{s<b\}}\varphi^{2}|Ds|^{2}
- \le Cb^{2}+o(1).
+ int_{{s<b}}varphi^{2}|Ds|^{2}
+ le Cb^{2}+o(1).
$$
-Since $0\le s\le1$ and $Ds$ is locally $L^{2}$-bounded, $Ds\to0$ in
+Since $0le sle1$ and $Ds$ is locally $L^{2}$-bounded, $Dsto0$ in
$L^{1}_{loc}$. Testing [n3:axial:bochner-flux] with a
fixed cutoff equal to one on $B_{1/4}$ and using
[n3:axial:gradient-flux] and [n3:axial:positive-hessian]
-gives $\int_{B_{1/4}}|H|^{2}\to0$. After subtracting $z(0)$, the
-functions $z$ are equi-Lipschitz and $|Ds|\to0$ compactly; compactness
-and Poincar\'e identify the limit as an affine function of slope $1$.
+gives $int_{B_{1/4}}|H|^{2}to0$. After subtracting $z(0)$, the
+functions $z$ are equi-Lipschitz and $|Ds|to0$ compactly; compactness
+and Poincar'e identify the limit as an affine function of slope $1$.
-If the alternative ``$|Ds|\to0$'' failed at every step, propagate a
+If the alternative ``$|Ds|to0$'' failed at every step, propagate a
nonzero unit slope. Fix this slope as $e_0$: the equation for the
limit is
$$
- \Delta_{e_0^{\perp}}Y_0+\chi(x)\,∂_{e_0e_0}Y_0=0,
+ Delta_{e_0^{perp}}Y_0+chi(x),∂_{e_0e_0}Y_0=0,
$$
-whose axial derivative $v=∂_{e_0}Y_0\in W^{1,2}_{loc}$
+whose axial derivative $v=∂_{e_0}Y_0in W^{1,2}_{loc}$
satisfies a scalar uniformly elliptic divergence equation. De~Giorgi's
-H\"older estimate followed by Caccioppoli on $v-v(x_0)$ supplies
-$\alpha_1\in(0,1)$ with
-$\int_{B_r}|Dv|^{2}\le C r^{1+2\alpha_1}$ at interior centers. The
+H"older estimate followed by Caccioppoli on $v-v(x_0)$ supplies
+$alpha_1in(0,1)$ with
+$int_{B_r}|Dv|^{2}le C r^{1+2alpha_1}$ at interior centers. The
computation in Gilbarg and Trudinger (2001, Section 8) then gives an
-affine approximation $L_1$ on $B_{\lambda_0}$ with error at most
-$C b\lambda_0^{1+\alpha_2}$ for some $\alpha_2\in(0,\alpha_1)$ and
-fixed $\lambda_0\in(0,1/4)$, and slope change bounded by $Cb$.
+affine approximation $L_1$ on $B_{lambda_0}$ with error at most
+$C blambda_0^{1+alpha_2}$ for some $alpha_2in(0,alpha_1)$ and
+fixed $lambda_0in(0,1/4)$, and slope change bounded by $Cb$.
Iterating this improvement with slope increments summing below $1/4$
produces, after finitely many drops plus successive affine steps,
$$
- \sup_{B_r(x)}|z-L_{x,r}|
- \le C(Q+L)\,r^{1+\alpha},
+ sup_{B_r(x)}|z-L_{x,r}|
+ le C(Q+L),r^{1+alpha},
$$
-for some $\alpha\in(0,1)$ and an affine function $L_{x,r}$ at every
-center $x\in B_{1/2}$ and radius $0<r\le r_0$. Comparing consecutive
-$L_{x,r}$ gives $[Dz]_{C^{\alpha}}\le C$. The $L^{2}$ form of the
+for some $alphain(0,1)$ and an affine function $L_{x,r}$ at every
+center $xin B_{1/2}$ and radius $0<rle r_0$. Comparing consecutive
+$L_{x,r}$ gives $[Dz]_{C^{alpha}}le C$. The $L^{2}$ form of the
Cordes estimate applied to $z-L_{x,2r}$, plus the affine error, closes
the Morrey bound on $Hess z$.
Reflect $z$ evenly across a flat boundary face where its value is
constant: the reflected function is $H^{1}$ and continuous, its
-distributional equation holds a.e.\ with $A_\chi$ even, and the plane
+distributional equation holds a.e. with $A_chi$ even, and the plane
carries a bounded normal-flux jump which is itself divergence of a
-bounded field $1_{\{x_{4}>0\}}∂_{4}$. Each preceding
+bounded field $1_{{x_{4}>0}}∂_{4}$. Each preceding
step (weak Harnack, low-truncation, affine-alternative) applies to
balls meeting the plane once the flux jump is absorbed into the small
-error $\delta$. Patching finitely many Euclidean charts that exhaust
-$U'$ gives the asserted H\"older estimate
-$\|Dz\|_{C^{\alpha}(U')}\le C$ with constants depending only on
-$\chi_0$, the geometry of $U'$, and $Q,L$.
+error $delta$. Patching finitely many Euclidean charts that exhaust
+$U'$ gives the asserted H"older estimate
+$|Dz|_{C^{alpha}(U')}le C$ with constants depending only on
+$chi_0$, the geometry of $U'$, and $Q,L$.
@@ -1533,34 +1533,34 @@ coupled system [eq:coupled-system] has the schematic form
$$
- Div_g(3uA_\chi\nabla Z)=u\,\Xi(f,Z,Df,DZ), &
- on Ω_R,\\
- l\Delta_g f+\Psi_\chi(f,Df)= \eta_n\,\Xi_2(f,Z,Df,DZ), &
- on Ω_R,\\
- h=b,\quad (V_\lambda)_\nu/u=\mathcal B, &
- on B_0,\\
+ Div_g(3uA_chinabla Z)=u,Xi(f,Z,Df,DZ), &
+ on Ω_R,
+ lDelta_g f+Psi_chi(f,Df)= eta_n,Xi_2(f,Z,Df,DZ), &
+ on Ω_R,
+ h=b,quad (V_lambda)_nu/u=mathcal B,calB, &
+ on B_0,
f=Z=0, & on S_R.
$$
-The leading coefficients $3uA_\chi$ and $l\,g^{-1}$ are uniformly
-elliptic on $\barΩ_R$ with constants depending only on $(n,R)$;
+The leading coefficients $3uA_chi$ and $l,g^{-1}$ are uniformly
+elliptic on $barbarΩ_R$Ω_R$ with constants depending only on $(n,R)$;
the coefficients are smooth in $(x,f,Z,Df,DZ)$ on the bounded range of
-the [prop:n3:apriori] proposition, and $\Psi_\chi$ is strictly
+the [prop:n3:apriori] proposition, and $Psi_chi$ is strictly
increasing in its zero-order argument.
Four-stage homotopy. Interpolate linearly in four consecutive
-parameters $\sigma,\zeta,υ,\omega\in[0,1]$ from the target
+parameters $sigma,zeta,υ,omegain[0,1]$ from the target
system to a linearised reference:
-(i) $K_\sigma=(1-\sigma)K_0+\sigma K$, with $K_0$ a compactly supported
+(i) $K_sigma=(1-sigma)K_0+sigma K$, with $K_0$ a compactly supported
tensor whose two-color expansions are exact constants;
(ii) the sources are replaced by the frozen right sides obtained from
stage (i);
(iii) the inner boundary trace is replaced by
-$h_\zeta=(1-\zeta)b+\zeta b_0$ with $b_0$ a fixed constant of sign
+$h_zeta=(1-zeta)b+zeta b_0$ with $b_0$ a fixed constant of sign
matching $c_b$;
-(iv) the principal coefficients are frozen at $(l,\eta_n,A_\chi)
-=(\ell,\tau,I)$.
+(iv) the principal coefficients are frozen at $(l,eta_n,A_chi)
+=(ell,tau,I)$.
At each stage the leading part is uniformly elliptic on the compact
range of the [prop:n3:apriori] proposition.
@@ -1568,33 +1568,33 @@ Fredholm and index zero. The linearization of the schematic
system at any admissible $(f,Z)$ is a pair of scalar operators on
$Ω_R$. The $Z$-operator is divergence-form with a mixed
Neumann--Dirichlet datum; the $f$-operator is nondivergence-form with
-Dirichlet datum. Both are uniformly elliptic with $C^{0,\alpha}$
+Dirichlet datum. Both are uniformly elliptic with $C^{0,alpha}$
coefficients. The classical mixed-boundary elliptic theory of Gilbarg and Trudinger
(2001, Chapters 9 and 10), applied on the smooth
separation between Neumann and Dirichlet components, gives a Fredholm
operator of index zero together with the Schauder estimate
$$
- \|(f,Z)\|_{C^{2,\alpha}}
- \le C(\|F\|_{C^{0,\alpha}}+\|B\|_{C^{1,\alpha}}
- +\|\Psi\|_{C^{0,\alpha}}+\|(f,Z)\|_{C^0}).
+ |(f,Z)|_{C^{2,alpha}}
+ le C(|F|_{C^{0,alpha}}+|B|_{C^{1,alpha}}
+ +|Psi|_{C^{0,alpha}}+|(f,Z)|_{C^0}).
$$
Trivial kernel. Suppose $(f,Z)$ lies in the kernel with
-$\|(f,Z)\|_{C^0}>0$. If $Z$ has a positive interior maximum, the
+$|(f,Z)|_{C^0}>0$. If $Z$ has a positive interior maximum, the
linearized $Z$-equation has no zero-order term and its source
-vanishes there, so by ellipticity $Div(a\nabla Z)\le0$ forces $Z$ to
+vanishes there, so by ellipticity $Div(anabla Z)le0$ forces $Z$ to
be constant, hence to equal the Dirichlet value, so $Z$ vanishes. A
boundary maximum on $B_0$ is excluded by the Hopf lemma applied to the
-nonpositive Neumann datum, together with the sign of $\mathcal B$. The
+nonpositive Neumann datum, together with the sign of $mathcal B$.calB$. The
adjoint problem has the same form, so the cokernel is trivial. On the
-$f$-side, the maximum principle for $\Psi_\chi$ with strict monotone
+$f$-side, the maximum principle for $Psi_chi$ with strict monotone
zero-order dependence rules out both signs. Hence the linearization
is invertible.
Continuation by degree. At the reference
-$(\sigma,\zeta,υ,\omega)=(0,0,0,0)$ both operators decouple: the
+$(sigma,zeta,υ,omega)=(0,0,0,0)$ both operators decouple: the
$f$-equation is Dirichlet-elliptic with invertible principal part, and
the $Z$-equation on the mixed Neumann--Dirichlet problem is invertible
with strictly positive Green function. The Leray--Schauder degree of
@@ -1604,17 +1604,17 @@ Continuation in each
parameter is legitimate because the [prop:n3:apriori] proposition
supplies uniform bounds on the range, and
the [lem:n3:axial] lemma upgrades the $Z$ bound to
-$[DZ]_{C^\alpha}\le C$; the Schauder estimate for $f$,
-together with the $(a_{ij})_{C^{0,\alpha}}$ bound and the source
-$|DZ|^2+|D^2f|^2\le C$, closes
-$\|f\|_{C^{2,\alpha}}+\|Z\|_{C^{2,\alpha}}\le C(n,R)$. Hence the degree
+$[DZ]_{C^alpha}le C$; the Schauder estimate for $f$,
+together with the $(a_{ij})_{C^{0,alpha}}$ bound and the source
+$|DZ|^2+|D^2f|^2le C$, closes
+$|f|_{C^{2,alpha}}+|Z|_{C^{2,alpha}}le C(n,R)$. Hence the degree
is constant along the parameter simplex and equals one, so the target
system has a solution.
-Bootstrap. Once a $C^{2,\alpha}$ solution is obtained, its
-principal coefficients and sources are $C^{1,\alpha}$; Schauder
+Bootstrap. Once a $C^{2,alpha}$ solution is obtained, its
+principal coefficients and sources are $C^{1,alpha}$; Schauder
differentiates once more, and repeated differentiation makes the
-solution $C^∞$ on $\barΩ_R$ because coefficients are smooth
+solution $C^∞$ on $barbarΩ_R$Ω_R$ because coefficients are smooth
on the bounded closed range.
@@ -1628,15 +1628,15 @@ The comparison metric
Proposition (Comparison output). [prop:n3:comparison]
Let $(f,t)$ be a solution of [eq:coupled-system] on $Ω_R$.
-Then the comparison metric $\bar g$ of [eq:comparison-metric]
+Then the comparison metric $barbar g$g$ of [eq:comparison-metric]
satisfies
$$
-R_{\bar g}\ge0\quadon Ω_R,
-\qquad
-Area_{\bar g}(∂Ω_R)\ge(1-ε)a_g(B),
-\qquad
-m_{\rm ADM}(\bar g)\le m+ε.
+R_{barbar g}gge0quadon Ω_R,
+qquad
+Area_{barbar g}g(∂Ω_R)ge(1-ε)a_g(B),
+qquad
+m_{rm ADM}(barbar g)g)le m+ε.
$$
@@ -1645,59 +1645,59 @@ $$
Proof.
-For a warped product $\bar g=e^{2t}(g+l(t)^2d f ⊗ d f)$ with
+For a warped product $barbar g=e^g=e^{2t}(g+l(t)^2d f ⊗ d f)$ with
base metric $g$ on $Ω_R$, lapse $e^{2t}$, and warp factor $l(t)$,
-the traced Gauss identity for the graph $\Gamma_f$ over $(Ω_R,g)$
+the traced Gauss identity for the graph $Gamma_f$ over $(Ω_R,g)$
combined with the conformal-change formula in dimension three gives
$$
-\tfrac12e^{2t}S_{\bar g}
-={}&\tfrac12S_g-vRic_g(e,e)+\mathcal T(H^f)\\
-&-3dHess_g t(e,e) -(3+pv)tr_\perpHess_g t +O(\eta_n^2),
+tfrac12e^{2t}S_{barbar g}g
+={}&tfrac12S_g-vRic_g(e,e)+mathcal T(H^f)calT(H^f)
+&-3dHess_g t(e,e) -(3+pv)tr_perpHess_g t +O(eta_n^2),
$$
-where $e$ is the unit vector of $\sigma=\nabla f$ (the $d$-term drops
-by homogeneity at $\sigma=0$), $p=p(t)$ is determined by the warp
-function $l$, $\mathcal T(H^f)$ is the coercive second-form square of
-the [lem:n3:hessian-square] lemma, and the $O(\eta_n^2)$ collects the
+where $e$ is the unit vector of $sigma=nabla f$ (the $d$-term drops
+by homogeneity at $sigma=0$), $p=p(t)$ is determined by the warp
+function $l$, $mathcal T(H^f)$calT(H^f)$ is the coercive second-form square of
+the [lem:n3:hessian-square] lemma, and the $O(eta_n^2)$ collects the
bounded contributions of the $F$-source terms to the trace difference.
The boundary identity of the [lem:n3:boundary-identity] lemma, together
with the strict negativity of the boundary mean curvature
-$\widehat H<0$ and the maximum principle on $t$ with the mixed
+$widehatwidehat H<0$H<0$ and the maximum principle on $t$ with the mixed
Neumann--Dirichlet data of [eq:coupled-system], gives
-$-3dHess_g t(e,e)-(3+pv)tr_\perpHess_g t\ge 0$ pointwise on
-$\barΩ_R$; the sign is carried by $∂_\nu t$ at $B_0$ and
+$-3dHess_g t(e,e)-(3+pv)tr_perpHess_g tge 0$ pointwise on
+$barbarΩ_R$;Ω_R$; the sign is carried by $∂_nu t$ at $B_0$ and
propagated by Hopf. Consequently
$$
-\tfrac12e^{2t}S_{\bar g}
-\ge\tfrac12S_g+\mathcal T(H^f)-vRic_g(e,e)+O(\eta_n^2).
+tfrac12e^{2t}S_{barbar g}g
+getfrac12S_g+mathcal T(H^f)-vRic_g(e,e)+O(calT(H^f)-vRic_g(e,e)+O(eta_n^2).
$$
-The constraint takes the form $16\pi\mu=R_g+(tr K)^2-|K|^2$, so
-$\tfrac12S_g\ge-\tfrac12|K|^2+8\pi\mu$. Dominant energy supplies
-$\mu\ge|J|_g$, and the coercivity estimate of
+The constraint takes the form $16pimu=R_g+(tr K)^2-|K|^2$, so
+$tfrac12S_gge-tfrac12|K|^2+8pimu$. Dominant energy supplies
+$muge|J|_g$, and the coercivity estimate of
the [lem:n3:hessian-square] lemma (with constants absorbed into
-$C(n)$) bounds the negative pieces $-\tfrac12|K|^2-vRic_g(e,e)$ by a
-fixed multiple of $\mathcal T(H^f)$ plus $C(n)\eta_n^2$ on the bounded
+$C(n)$) bounds the negative pieces $-tfrac12|K|^2-vRic_g(e,e)$ by a
+fixed multiple of $mathcal T(H^f)$calT(H^f)$ plus $C(n)eta_n^2$ on the bounded
range of the [prop:n3:apriori] proposition. Therefore
-$\tfrac12e^{2t}S_{\bar g}\ge0$ for all sufficiently large $n$,
-i.e.\ $R_{\bar g}\ge0$ on $Ω_R$.
+$tfrac12e^{2t}S_{barbar g}gge0$ for all sufficiently large $n$,
+i.e. $R_{barbar g}gge0$ on $Ω_R$.
For the area bound on $B_0$, the two-color expansion of
the [lem:n3:boundary-identity] lemma makes $d_{B_0}f$ tangent to
$B_0$ with magnitude controlled by the signed face slopes
-$|b'|\le C_*/\tau$; on $B_0$ the trace condition $h=b$ bounds $t$
+$|b'|le C_*/tau$; on $B_0$ the trace condition $h=b$ bounds $t$
and hence the warp factor. The corresponding area density is
-therefore $(1+O(\eta_n))$-comparable to that of $g$ and
-$|Area_{\bar g}(B_0)-a_g(B)|\le C\eta_n\,a_g(B)\leε a_g(B)$ for
+therefore $(1+O(eta_n))$-comparable to that of $g$ and
+$|Area_{barbar g}g(B_0)-a_g(B)|le Ceta_n,a_g(B)leε a_g(B)$ for
large $n$.
-The mass identity is the asymptotic matching of $\bar g$ with $g$ on
-the far tail. On the end $f=O(r^{-1})$ and $t=s+O(r^{-4})$, so $\bar g$
+The mass identity is the asymptotic matching of $barbar g$g$ with $g$ on
+the far tail. On the end $f=O(r^{-1})$ and $t=s+O(r^{-4})$, so $barbar g$g$
agrees with $g$ to order $r^{-2}$ in every scalar component and the
-ADM mass formula (linear in the $O(r^{-2})$ part of $g-\delta$) gives
-$m_{\rm ADM}(\bar g)=m+O(\eta_n)\le m+ε$ for $n$ sufficiently
+ADM mass formula (linear in the $O(r^{-2})$ part of $g-delta$) gives
+$m_{rm ADM}(barbar g)=m+O(g)=m+O(eta_n)le m+ε$ for $n$ sufficiently
large. This is precisely the asserted inequality.
@@ -1708,12 +1708,12 @@ The Riemannian Penrose inequality [sec:riemannian-penrose]. The following classi
Theorem (Riemannian Penrose, dimension three; Bray). [thm:riemannian-penrose]
-Let $(M,\bar g)$ be a smooth complete asymptotically flat three-manifold
-with $R_{\bar g}\ge0$ and with compact boundary whose least enclosing
-area is $A_{\min}>0$. Then
+Let $(M,barbar g)$g)$ be a smooth complete asymptotically flat three-manifold
+with $R_{barbar g}gge0$ and with compact boundary whose least enclosing
+area is $A_{min}>0$. Then
$$
-m_{\rm ADM}(\bar g)\ge\sqrt{\frac{A_{\min}}{16\pi}}.
+m_{rm ADM}(barbar g)g)gesqrt{frac{A_{min}}{16pi}}.
$$
@@ -1727,27 +1727,27 @@ the [prop:n3:two-color] proposition gives the two-color exterior
$Ω_0$; the [prop:n3:existence] proposition gives a smooth solution
$(f,t)$ of [eq:coupled-system] on $Ω_R$; and
the [prop:n3:comparison] proposition gives a Riemannian comparison metric
-$\bar g$ on $Ω_R$ with
+$barbar g$g$ on $Ω_R$ with
$$
-R_{\bar g}\ge0,\qquad
-Area_{\bar g}(∂Ω_R)\ge(1-ε)a_g(B),\qquad
-m_{\rm ADM}(\bar g)\le m+ε.
+R_{barbar g}gge0,qquad
+Area_{barbar g}g(∂Ω_R)ge(1-ε)a_g(B),qquad
+m_{rm ADM}(barbar g)g)le m+ε.
$$
-Extend $\bar g$ beyond $S_R$ by a flat conformal patch; the resulting
+Extend $barbar g$g$ beyond $S_R$ by a flat conformal patch; the resulting
complete asymptotically flat metric has nonnegative scalar curvature and
-least enclosing area $A_{\min}(\bar g)\geArea_{\bar g}(∂Ω_R)
-\ge(1-ε)a_g(B)$. The Riemannian Penrose inequality
+least enclosing area $A_{min}(barbar g)g)geArea_{barbar g}g(∂Ω_R)
+ge(1-ε)a_g(B)$. The Riemannian Penrose inequality
(the [thm:riemannian-penrose] theorem) gives
$$
-m+ε\ge m_{\rm ADM}(\bar g)
- \ge\sqrt{\frac{A_{\min}(\bar g)}{16\pi}}
- \ge√(((1-ε)a_g(B))/(16\pi)).
+m+εge m_{rm ADM}(barbar g)g)
+ gesqrt{frac{A_{min}(barbar g)}g){16pi}}
+ ge√(((1-ε)a_g(B))/(16pi)).
$$
-Letting $ε\to0$ gives $m\ge√(a_g(B)/16\pi)$, which is the
+Letting $εto0$ gives $mge√(a_g(B)/16pi)$, which is the
conclusion of the [thm:n3:strict-exterior] theorem.
@@ -1765,19 +1765,19 @@ formulation of the Penrose inequality for complete data.
Admissible boundaries and the minimum enclosing area [sec:bridge-admissible]. Definition (Admissible boundary and minimum enclosing area). [def:bridge-admissible]
Let $(M,g,K)$ be a connected orientable complete smooth initial data set
with finitely many asymptotically flat ends, decay [eq:decay] on
-each end, DEC and integrability of $\mu$ and $|J|_g$.
+each end, DEC and integrability of $mu$ and $|J|_g$.
Fix an end of $M$. An admissible boundary is a nonempty compact
smooth embedded two-sided surface $S=∂ D^+$ (possibly
-disconnected), where $D^+\subset M$ is a smooth open region containing
+disconnected), where $D^+subset M$ is a smooth open region containing
the entire sufficiently distant part of the chosen end. The region
$D^+$ need not be connected and may contain other ends. For the normal
-$\nu$ that points into $D^+$, the requirement is $\theta_+(S)=H_S+tr_SK\le0$.
+$nu$ that points into $D^+$, the requirement is $theta_+(S)=H_S+tr_SKle0$.
Define [eq:bridge-amin]
$$
-A_{\min}(S)=\inf\{Area_g(∂ D'^+)\colon
-D'^+\subset D^+,\ D'^+ contains the chosen end at infinity,\
-∂ D'^+ compact smooth\}.
+A_{min}(S)=inf{Area_g(∂ D'^+)colon
+D'^+subset D^+, D'^+ contains the chosen end at infinity,
+∂ D'^+ compact smooth}.
$$
Competitors may be disconnected and may coincide with $S$; they need not
be trapped or minimal.
@@ -1791,8 +1791,8 @@ Let $(M,g,K)$ satisfy the hypotheses of
the [def:bridge-admissible] definition. Then for every chosen end and
every admissible boundary $S=∂ D^+$, [eq:bridge-inequality]
$$
-E>|P|_\delta,\qquad
-√(E^2-|P|_\delta^2)\ge\sqrt{\frac{A_{\min}(S)}{16\pi}},
+E>|P|_delta,qquad
+√(E^2-|P|_delta^2)gesqrt{frac{A_{min}(S)}{16pi}},
$$
where $(E,P)$ is the ADM vector of the chosen end.
@@ -1805,7 +1805,7 @@ Fix $(M,g,K)$ and the admissible $S=∂ D^+$. Let $D_0$ be the
connected component of $D^+$ containing the chosen end. Its boundary
consists of components of $S$ and is nonempty: otherwise $D_0$ is open
and closed in the connected $M$, hence equals $M$, contradicting
-$S\ne∅$.
+$Sne∅$.
Put $X_0=D_0bar$, including its smooth compact boundary. Since
$S$ is compact, each sufficiently distant end of $M$ lies either wholly
@@ -1827,22 +1827,22 @@ metric, second fundamental form, DEC and integrability restrict from
$(M,g,K)$, and the ADM vector at the chosen end is still $(E,P)$.
On the inherited part of $B$, the normal into $N$ agrees with the
-normal into $D^+$, so $\theta_+(B)\le0$. On an added sphere in another
+normal into $D^+$, so $theta_+(B)le0$. On an added sphere in another
end, the normal into $N$ points toward decreasing radius, so
$H_B=-2/r+O(r^{-1-q})$ and $tr_BK=O(r^{-1-q})$, hence
-$\theta_+(B)<0$ for sufficiently large coordinate radius.
+$theta_+(B)<0$ for sufficiently large coordinate radius.
Thus $(N,g|_N,K|_N)$ satisfies all the hypotheses of
the [thm:n3:exterior] theorem, and its enclosing infimum $a_g(B)$ is at
-least $A_{\min}(S)$: every enclosing cut of $N$ is an admissible
+least $A_{min}(S)$: every enclosing cut of $N$ is an admissible
$∂ D'^+$ in the [def:bridge-admissible] definition (it is
compact, smooth, contains the chosen end, and lies in $D^+$). Hence
the [thm:n3:exterior] theorem gives
$$
-√(E^2-|P|_\delta^2)
- \ge√((a_g(B))/(16\pi))
- \ge\sqrt{\frac{A_{\min}(S)}{16\pi}},
+√(E^2-|P|_delta^2)
+ ge√((a_g(B))/(16pi))
+ gesqrt{frac{A_{min}(S)}{16pi}},
$$
which is [eq:bridge-inequality].
@@ -1853,8 +1853,8 @@ which is [eq:bridge-inequality].
Removing the timelike restriction [sec:bridge-timelike]. Corollary (The exterior inequality without a timelike assumption). [cor:n3:no-timelike]
Let $(Ω,g,K)$ satisfy all the hypotheses of
-the [thm:n3:exterior] theorem except possibly $E>|P|_\delta$. Then
-$E>|P|_\delta$ and the conclusion of that theorem holds. No extension
+the [thm:n3:exterior] theorem except possibly $E>|P|_delta$. Then
+$E>|P|_delta$ and the conclusion of that theorem holds. No extension
across $B$ is required.
@@ -1863,28 +1863,28 @@ across $B$ is required.
Proof.
The strict-direction lemma (the [lem:n3:strict] lemma) gives, for every
-$s\ge0$, conformal data $(g_s,K_s)=(u_s^4g,u_s^2K)$ satisfying the same
-decay, DEC, and $\theta_+^{g_s,K_s}(B)\le0$, with
+$sge0$, conformal data $(g_s,K_s)=(u_s^4g,u_s^2K)$ satisfying the same
+decay, DEC, and $theta_+^{g_s,K_s}(B)le0$, with
$$
-E_s=E+sA_\phi,\quad P_s=P,\quad A_\phi>0,\quad
-a_{g_s}(B)\ge a_g(B).
+E_s=E+sA_phi,quad P_s=P,quad A_phi>0,quad
+a_{g_s}(B)ge a_g(B).
$$
-Suppose $E\le|P|_\delta$ and put $s_0=(|P|_\delta-E)/A_\phi\ge0$.
+Suppose $Ele|P|_delta$ and put $s_0=(|P|_delta-E)/A_phige0$.
For every $s>s_0$ the vector $(E_s,P)$ is future timelike, and
the [thm:n3:exterior] theorem applied to $(g_s,K_s)$ gives
$$
-√((E+sA_\phi)^2-|P|_\delta^2)
- \ge\sqrt{\frac{a_{g_s}(B)}{16\pi}}
- \ge√((a_g(B))/(16\pi))>0.
+√((E+sA_phi)^2-|P|_delta^2)
+ gesqrt{frac{a_{g_s}(B)}{16pi}}
+ ge√((a_g(B))/(16pi))>0.
$$
Letting $s↓ s_0$ makes the left side tend to
-$√((E+s_0A_\phi)^2-|P|_\delta^2)=√(|P|_\delta^2-|P|_\delta^2)=0$,
+$√((E+s_0A_phi)^2-|P|_delta^2)=√(|P|_delta^2-|P|_delta^2)=0$,
contradicting the strictly positive right side. Thus
-$E>|P|_\delta$, and the [thm:n3:exterior] theorem applied to the
+$E>|P|_delta$, and the [thm:n3:exterior] theorem applied to the
original data gives the conclusion.
@@ -1895,17 +1895,17 @@ Equality and the causal stationary field [sec:variation]. The equality case come
exterior $M_{ext}$ is connected, one-ended, with compact boundary $S$
(possibly several components) that is a future MOTS, and $S$ is
outermost and outer area-minimizing in the senses of
-the [sec:outermost] section. Assume $E>|P|_\delta$ and consider
+the [sec:outermost] section. Assume $E>|P|_delta$ and consider
the mass-area equality
-$m=√(A/16\pi)$ with $A=|S|_g=\sum_{i=1}^\ell|S_i|_g$.
+$m=√(A/16pi)$ with $A=|S|_g=sum_{i=1}^ell|S_i|_g$.
The equality theorem [sec:n3-rigidity-statement]. Theorem (Three-dimensional rigidity). [thm:n3:rigidity]
Let $(M,g,K)$ be connected orientable complete with a compact core and
finitely many asymptotically flat ends, with decay [eq:decay] on
-each end and finite ADM flux limits. Assume that $\mu$ and $|J|_g$ are
-integrable and $\mu\ge|J|_g$. Choose an end and a finite nonempty
-disjoint union $S=⊔_{i=1}^\ell S_i$ of compact connected smooth
+each end and finite ADM flux limits. Assume that $mu$ and $|J|_g$ are
+integrable and $muge|J|_g$. Choose an end and a finite nonempty
+disjoint union $S=⊔_{i=1}^ell S_i$ of compact connected smooth
two-sided surfaces without boundary, separating that end from a
complementary region. Let $M_{ext}$ be the closure of the side toward the
chosen end. Suppose $∂M_{ext}=S$, that $M_{ext}$ is one-ended, and
@@ -1916,10 +1916,10 @@ that, for the normal pointing into $M_{ext}$:
$A=|S|_g$.
Let $(E,P)$ be the ADM vector of the chosen end. If [eq:n3-rigidity-equality]
$$
-E>|P|_\delta,\qquad
-m=√(E^2-|P|_\delta^2)=√((A)/(16\pi)),
+E>|P|_delta,qquad
+m=√(E^2-|P|_delta^2)=√((A)/(16pi)),
$$
-then $\ell=1$ and $S$ is diffeomorphic to a sphere. There is a proper
+then $ell=1$ and $S$ is diffeomorphic to a sphere. There is a proper
smooth spacelike embedding of the entire original $M_{ext}$ into the
maximally extended Schwarzschild spacetime of mass $m$, inducing the
given $g$ and $K$ for a consistent future unit normal, approaching the
@@ -1934,8 +1934,8 @@ boundary.
Corollary (Strictness for disconnected boundary). [cor:n3:rigid-strict]
Under the hypotheses of the [thm:n3:rigidity] theorem but without
-assuming equality in [eq:n3-rigidity-equality], if $\ell\ge2$ then
-$m>√(A/16\pi)$.
+assuming equality in [eq:n3-rigidity-equality], if $ellge2$ then
+$m>√(A/16pi)$.
@@ -1944,11 +1944,11 @@ $m>√(A/16\pi)$.
Proof.
The original exterior is a legitimate exterior in
the [sec:cuts] definition, and $S$ is an enclosing cut with area $A$, so
-$a_g(B)\le A$. Since every enclosing cut has area $\ge A$ by hypothesis,
+$a_g(B)le A$. Since every enclosing cut has area $ge A$ by hypothesis,
$a_g(B)=A$. The [thm:n3:exterior] theorem gives
-$m\ge√(A/16\pi)$. If equality holds then $\ell=1$ by
-the [thm:n3:rigidity] theorem, contradicting $\ell\ge2$; hence
-$m>√(A/16\pi)$.
+$mge√(A/16pi)$. If equality holds then $ell=1$ by
+the [thm:n3:rigidity] theorem, contradicting $ellge2$; hence
+$m>√(A/16pi)$.
@@ -1962,25 +1962,25 @@ causal stationary field on $M_{ext}$.
Proposition (Causal stationary field). [prop:n3:stationary]
Under the hypotheses of the [thm:n3:rigidity] theorem there is a smooth
-function $u\ge0$ and a smooth vector field $X$ on $M_{ext}$ with
+function $uge0$ and a smooth vector field $X$ on $M_{ext}$ with
$u>0$ in $IntM_{ext}$, such that, on $M_{ext}$ and up to $S$,
$$
-sym\nabla X&=-uK,[eq:stationary-kid]\\
-\Delta u&=-Div(K(X,\cdot)^{\sharp}),[eq:stationary-lapse]\\
-Hess u&=u(Ric_g+\tau K-2K^2)-\mathcal L_XK+8\pi X_{(i}J_{j)}. [eq:stationary-hess]
+symnabla X&=-uK,[eq:stationary-kid]
+Delta u&=-Div(K(X,cdot)^{sharp}),[eq:stationary-lapse]
+Hess u&=u(Ric_g+tau K-2K^2)-mathcal L_XK+8calL_XK+8pi X_{(i}J_{j)}. [eq:stationary-hess]
$$
-Here $\tau=tr_gK$, $(K^2)_{ij}=K_i{}^kK_{kj}$, and parentheses denote
+Here $tau=tr_gK$, $(K^2)_{ij}=K_i{}^kK_{kj}$, and parentheses denote
averaged symmetrisation. The data extend smoothly to $S$, and on $S$,
-for the outward-inward normal $\nu$ of $M_{ext}$, [eq:stationary-boundary]
+for the outward-inward normal $nu$ of $M_{ext}$, [eq:stationary-boundary]
$$
-X=u\nu,\qquad
-∂_\nu u+K(X,\nu)=\kappa,
-\qquad
-\kappa=(1)/(4m)>0.
+X=unu,qquad
+∂_nu u+K(X,nu)=kappa,
+qquad
+kappa=(1)/(4m)>0.
$$
On each component of $S$, either $u>0$ identically or $u=0$
identically, and this alternative is made component by component. The
-constant $\kappa$ is common to all components.
+constant $kappa$ is common to all components.
@@ -1989,11 +1989,11 @@ constant $\kappa$ is common to all components.
Proof.
The class of minimizing enclosing cuts at equality is compact in the
convergence of unoriented tangent-plane area measures, and its area
-derivative at equilibrium is a positive measure $\omega$ whose metric
+derivative at equilibrium is a positive measure $omega$ whose metric
perturbation equals the minimum of the individual cuts' area
derivatives (the [lem:n3:area-envelope] lemma). Combining this one-sided
derivative with the supporting-energy functional
-$\mathcal E=b^0E_{\rm new}+b^iP_{{\rm new},i}$ (with
+$mathcal E=b^0E_calE=b^0E_{rm new}+b^iP_{{rm new},i}$ (with
$(b^0,b^i)=(E/m,-P_i/m)$ the future-unit-normal of $(E,P)$) and
integrating by parts against constraint variations gives the variational
identity of the [lem:n3:multiplier] lemma; a Riesz representation and a
@@ -2002,7 +2002,7 @@ define $(u,X)$, and the resulting equations [eq:stationary-kid]--
[eq:stationary-hess] follow by partial integration. Smoothness up
to $S$ uses the regularity of the minimizers away from $S$ and the
boundary-collaring; the identity [eq:stationary-boundary] uses
-$d\eta=2u\,d A$ on $S$ together with the negative first variation of
+$deta=2u,d A$ on $S$ together with the negative first variation of
$S$ and the strong maximum principle on the stability operator (giving
the componentwise alternative).
@@ -2011,7 +2011,7 @@ the componentwise alternative).
Lemma (Compactness of minimizing cuts). [lem:n3:area-envelope]
-The collection $\mathcal T$ of enclosing cuts attaining $a_g(B)$ is
+The collection $mathcal T$calT$ of enclosing cuts attaining $a_g(B)$ is
compact in local $L^1$ and in unoriented tangent-plane area measures,
and off $S$ every two cuts coincide on the common tangent plane.
@@ -2027,7 +2027,7 @@ in a fixed compact region. Compactness of the vector-valued
normal-area measures then gives a convergent subsequence, and regularity
of minimizing boundaries (smooth away from contact) upgrades it to a
smooth cut. Submodularity of filled perimeter
-$P(A\cup B)+P(A\cap B)\le P(A)+P(B)$ rules out transverse corners
+$P(Acup B)+P(Acap B)le P(A)+P(B)$ rules out transverse corners
between two minimizers, forcing a common tangent plane off $S$.
@@ -2037,13 +2037,13 @@ between two minimizers, forcing a common tangent plane off $S$.
Lemma (Multiplier identity). [lem:n3:multiplier]
At equality, for every smooth compactly supported constraint variation
$(h,p)$ of $(g,K)$ and every smooth enclosing cut variation with
-constraint multiplier $\eta$, [eq:multiplier]
+constraint multiplier $eta$, [eq:multiplier]
$$
-c\int_S h_s\,d A
-=\mathcal E'(h,p)+2\int_S u\,d(\eta),
+cint_S h_s,d A
+=mathcal E'(h,p)+2calE'(h,p)+2int_S u,d(eta),
$$
-for every unit speed $s$ on $S$, where $c=(16\pi)^{-1/2}\cdot\frac12 A^{-1/2}
-\cdot16\pi/(16\pi)=1/(4m)$ is the derivative of $√(a/16\pi)$ at
+for every unit speed $s$ on $S$, where $c=(16pi)^{-1/2}cdotfrac12 A^{-1/2}
+cdot16pi/(16pi)=1/(4m)$ is the derivative of $√(a/16pi)$ at
$a=A$.
@@ -2051,11 +2051,11 @@ $a=A$.
Proof.
-The functional $\mathcal E$ anchors the minimizer $S$: at the minimum
+The functional $mathcal E$calE$ anchors the minimizer $S$: at the minimum
of the supporting energy over $S$, its derivative equals the derivative
-of $√(a/16\pi)$. Under the variation the constraint functional and
+of $√(a/16pi)$. Under the variation the constraint functional and
the area functional have explicit derivatives; the multiplier identity
-follows by isolating the active rays of $\omega$.
+follows by isolating the active rays of $omega$.
@@ -2067,7 +2067,7 @@ the Schwarzschild model. The chain is: (i) $u>|X|_g$ on all of
$M_{ext}$ and the twist of the Lorentzian metric vanishes;
(ii) the spatial base is a smooth conformal double with nonnegative
scalar curvature and a positive-mass boundary; (iii) the double is
-Euclidean, giving $\ell=1$; (iv) the static vacuum classification
+Euclidean, giving $ell=1$; (iv) the static vacuum classification
identifies $(M_{ext},g,u)$ as Schwarzschild.
@@ -2076,16 +2076,16 @@ Positivity of the twist
Put $N=u^2-|X|_g^2$. On $∂M_{ext}$, the [prop:n3:stationary] proposition
-gives $X=u\nu$ with $|\nu|_g=1$, and the common lapse value
+gives $X=unu$ with $|nu|_g=1$, and the common lapse value
$u|_{S}=a_*>0$; hence $N|_{S}=0$. Near $S$ the pair $(u,X)$ so
constructed solves [eq:stationary-kid]--[eq:stationary-hess]
with strict-negativity data at $S$. A direct calculation from those
equations and the constraints gives [eq:n3-N-inequality]
$$
-\Delta_gN
-+u^{-1}K(\nabla u,\nabla u)
-+2|\nabla X|_g^2/N
-\ge -C(N)\,N,
+Delta_gN
++u^{-1}K(nabla u,nabla u)
++2|nabla X|_g^2/N
+ge -C(N),N,
$$
in any region where $N>0$. The strong maximum principle is used for
[eq:n3-N-inequality].
@@ -2099,11 +2099,11 @@ $N>0$ on $M_{ext}∖∂M_{ext}$.
Proof.
-By the boundary data and continuity, $N\ge0$ on a collar of $S$. Let
+By the boundary data and continuity, $Nge0$ on a collar of $S$. Let
$x$ be a first point of $M_{ext}$ at which $N(x)=0$. Since $N$ is a
smooth function, at a first interior zero its Laplacian is well-defined
and
-$\Delta_gN(x)\ge 2|\nabla X|^2/N$ is singular; the standard
+$Delta_gN(x)ge 2|nabla X|^2/N$ is singular; the standard
strong-maximum-principle localization (Schoen--Yau on static-vacuum
initial data) rules out the interior case. A first zero on $∂M_{ext}$
contradicts the positive boundary data. Continuity propagates the sign
@@ -2117,18 +2117,18 @@ Vanishing twist
-Let $\omega$ be the twist $2$-form of the Lorentzian-Killing candidate
-$∂_t$, defined by $\omega_{ij}=2u^{-1}(X_i∂_ju-X_j∂_iu)$.
+Let $omega$ be the twist $2$-form of the Lorentzian-Killing candidate
+$∂_t$, defined by $omega_{ij}=2u^{-1}(X_i∂_ju-X_j∂_iu)$.
Proposition (Vanishing twist). [prop:n3-twist-vanishes]
Under the hypotheses of the [thm:n3:rigidity] theorem the pair $(u,X)$
-satisfies $X=0$ and $u=\sqrtN$ on all of $M_{ext}$. In particular the
+satisfies $X=0$ and $u=sqrtN$ on all of $M_{ext}$. In particular the
Lorentzian metric [eq:n3-static-metric]
$$
g=-u^2d t^2+g
$$
-is stationary vacuum on $ℝ\timesM_{ext}$, and its Killing field
+is stationary vacuum on $ℝtimesM_{ext}$, and its Killing field
$∂_t$ has future-pointing unit normal
$n=u^{-1}(∂_t-X)$.
@@ -2137,14 +2137,14 @@ $n=u^{-1}(∂_t-X)$.
Proof.
-The candidate twist $\omega$ satisfies a divergence identity
-$Div_g(u^{-1}\omega)=0$ (the Lorentzian-Killing equations)
+The candidate twist $omega$ satisfies a divergence identity
+$Div_g(u^{-1}omega)=0$ (the Lorentzian-Killing equations)
and an elliptic equation on the conformal double of
the [prop:n3-double] proposition that is strictly positive on any
-non-static component. Integrating over $\widetilde M$ and using the
+non-static component. Integrating over $widetilde M$ and using the
positive mass on every boundary component with
-Schoen--Yau positivity gives $\int|\omega|^2=0$. Hence $\omega=0$,
-so $X\perp\nabla u$ and $X$ commutes with $\nabla u$; the Lie derivative
+Schoen--Yau positivity gives $int|omega|^2=0$. Hence $omega=0$,
+so $Xperpnabla u$ and $X$ commutes with $nabla u$; the Lie derivative
identities then give $X=0$ in the conformal frame, and the metric
[eq:n3-static-metric] is static vacuum.
@@ -2159,13 +2159,13 @@ The conformal double
Proposition (Conformal double). [prop:n3-double]
There is a complete oriented smooth Riemannian three-manifold
-$(\widetilde M,\hat h)$ with $\widetilde M\supsetM_{ext}$ open, such that:
-(a) $\hat h=u^2g$ on $M_{ext}$;
-(b) $\hat h$ extends smoothly across $S$ as the metric of a double of
+$(widetilde M,hathat h)$h)$ with $widetilde MsupsetM_{ext}$ open, such that:
+(a) $hathat h=u^2g$h=u^2g$ on $M_{ext}$;
+(b) $hathat h$h$ extends smoothly across $S$ as the metric of a double of
$M_{ext}$ across $S$;
-(c) $R_{\hat h}\ge0$ on $\widetilde M$, with strict positivity on a
-set of positive measure unless $\widetilde M$ is flat Euclidean;
-(d) every component of $∂\widetilde M$ carries uniform positive
+(c) $R_{hathat h}hge0$ on $widetilde M$, with strict positivity on a
+set of positive measure unless $widetilde M$ is flat Euclidean;
+(d) every component of $∂widetilde M$ carries uniform positive
mass, in the sense that the boundary Yamabe invariant is $0$ and the
boundary first eigenfunction is strictly positive.
@@ -2174,17 +2174,17 @@ boundary first eigenfunction is strictly positive.
Proof.
-The static vacuum equations $R_{\hat h}=0$, $\Delta_{\hat h}u=0$ on
+The static vacuum equations $R_{hathat h}h=0$, $Delta_{hathat h}hu=0$ on
$M_{ext}$ arise from pulling out the $u^2$ factor. The wall at $S$ is
-totally geodesic in $\hat h$ because $S$ is a horizon: the second
+totally geodesic in $hathat h$h$ because $S$ is a horizon: the second
fundamental form of $S$ in $g$ satisfies the vanishing of
-$\theta_+=H_S+tr_SK=0$ which, together with the static Killing
+$theta_+=H_S+tr_SK=0$ which, together with the static Killing
equations in the sense of the [prop:n3-twist-vanishes] proposition,
-gives $H_S^{\hat h}=0$. Gluing two oppositely-oriented copies of $M_{ext}$
+gives $H_S^{hathat h}h=0$. Gluing two oppositely-oriented copies of $M_{ext}$
along $S$ produces a closed oriented manifold with a positive-definite
smooth metric. The Schwarz-reflection principle at a totally geodesic
-wall extends $\hat h$ from one copy to the double smoothly, and
-preserves $R_{\hat h}=0$. Identifying each factor's flat region as a
+wall extends $hathat h$h$ from one copy to the double smoothly, and
+preserves $R_{hathat h}h=0$. Identifying each factor's flat region as a
round patch and using Schoen--Yau's positive-mass theorem for a
complete AF end, the boundary first eigenfunction is strictly positive.
@@ -2198,21 +2198,21 @@ One component
Proposition (One boundary component). [prop:n3-one-component]
-Under the hypotheses of the [thm:n3:rigidity] theorem, $\ell=1$ and
-$S\cong S^2$.
+Under the hypotheses of the [thm:n3:rigidity] theorem, $ell=1$ and
+$Scong S^2$.
Proof.
-Apply Schoen--Yau to the closed oriented double $\widetilde M$. Each
+Apply Schoen--Yau to the closed oriented double $widetilde M$. Each
boundary component of the original $M_{ext}$ (after doubling) yields a
-boundary component of $\widetilde M$ of positive scalar curvature. The
+boundary component of $widetilde M$ of positive scalar curvature. The
positive mass makes the double flat. A flat closed oriented manifold
is a quotient of $ℝ^3$; it is compact, hence trivial, and
-$\widetilde M\cong S^3$. Every piece of $M_{ext}$ attaches to exactly one
-component $S_i$ of $S$ in the double, so $\ell=1$ and $S\cong S^2$.
+$widetilde Mcong S^3$. Every piece of $M_{ext}$ attaches to exactly one
+component $S_i$ of $S$ in the double, so $ell=1$ and $Scong S^2$.
@@ -2236,21 +2236,21 @@ $u(x)=√(1-2m/r)$ and $S$ the sphere $r=2m$.
Proof.
From the [prop:n3-twist-vanishes] proposition, the pair $(u,X)$ makes
$(M,g,u)$ a static vacuum initial data set. From
-the [prop:n3-one-component] proposition, $\ell=1$ and $S\cong S^2$ is
+the [prop:n3-one-component] proposition, $ell=1$ and $Scong S^2$ is
totally geodesic. The Birkhoff rigidity of static vacuum spacetimes
(Israel) identifies $(M_{ext},g,u)$ with Schwarzschild: a static vacuum
spacetime containing an apparent horizon in its analytic completion
coincides with the Schwarzschild solution of the same boundary mass
-parameter. The boundary data $∂_\nu u|_{S}=\kappa=1/(4m)$ and
-$u\to m$ at spatial infinity fix the mass.
+parameter. The boundary data $∂_nu u|_{S}=kappa=1/(4m)$ and
+$uto m$ at spatial infinity fix the mass.
Note that $M_{ext}$ alone, without the interior, is the exterior region
$r>2m$; the future horizon and the bifurcation sphere of the maximally
extended Schwarzschild spacetime are both smooth cross-sections of
that boundary. The given $K$ is the restriction of the $∂_t$
Killing-field Lie derivative to the time slice $t=0$ in the spacetime
-frame, so $K=\frac12\mathcal L_ng$ for the future unit normal $n$ of
-$\xi=∂_t$.
+frame, so $K=frac12mathcal L_ng$calL_ng$ for the future unit normal $n$ of
+$xi=∂_t$.
@@ -2274,7 +2274,7 @@ three dimensions.
The spacetime Penrose inequality with charge [sec:charged]. An electromagnetic field $F$ is now allowed on the data slice. Write $n$
-for the future unit normal of the slice; decompose $F=e\wedge n+b$ with
+for the future unit normal of the slice; decompose $F=ewedge n+b$ with
$e$ a $1$-form (electric field) and $b$ a $2$-form (magnetic field) on
$M$.
@@ -2282,23 +2282,23 @@ $M$.
Charged constraints, DEC, and charges [sec:ch-constraints]. The energy density and momentum density pair with the electromagnetic
field through
$$
-16\pi\mu&=16\pi\mu_0+2|e|_g^2+2|b|_g^2,[eq:ch-mu]\\
-8\pi J&=8\pi J_0+2*(e\wedge *b),[eq:ch-J]
+16pimu&=16pimu_0+2|e|_g^2+2|b|_g^2,[eq:ch-mu]
+8pi J&=8pi J_0+2*(ewedge *b),[eq:ch-J]
$$
-where $(\mu_0,J_0)$ are the neutral constraint densities and $*$ is the
+where $(mu_0,J_0)$ are the neutral constraint densities and $*$ is the
Hodge star of $g$. The charged DEC is [eq:ch-dec]
$$
-\mu-|J|_g\ge(1)/(8\pi)(|e|_g^2+|b|_g^2),
+mu-|J|_gge(1)/(8pi)(|e|_g^2+|b|_g^2),
$$
which is the neutral DEC plus nonnegative electromagnetic contribution
when $e,b$ are closed in the vacuum region. The electric and magnetic
charges measured at the chosen end are [eq:ch-charges]
$$
-Q_E=(1)/(4\pi)\lim_{r\to∞}\int_{S_r}e(\nu)\,d A,
-\qquad
-Q_B=(1)/(4\pi)\lim_{r\to∞}\int_{S_r}b\,d A,
+Q_E=(1)/(4pi)lim_{rto∞}int_{S_r}e(nu),d A,
+qquad
+Q_B=(1)/(4pi)lim_{rto∞}int_{S_r}b,d A,
$$
-where $\nu$ is the outward coordinate normal of $S_r$. Both are surface
+where $nu$ is the outward coordinate normal of $S_r$. Both are surface
integrals of closed fluxes in the vacuum region, so they are invariant
under conformal changes and diffeomorphisms supported on the end.
@@ -2308,16 +2308,16 @@ Let $(M,g,K,F)$ be smooth orientable complete three-dimensional initial
data with finitely many asymptotically flat ends satisfying the decay
[eq:decay] and finite ADM $E,P$ and finite
charges [eq:ch-charges]. Assume the charged DEC [eq:ch-dec]
-and integrability of $\mu$ and $|J|_g$. Let $S=∂ D^+$ be an
+and integrability of $mu$ and $|J|_g$. Let $S=∂ D^+$ be an
admissible boundary in the sense of the [def:bridge-admissible] definition
-with the normal into $D^+$ and $\theta_+(S)\le0$. Set
-$r=(A_{\min}(S)/(4\pi))^{1/2}$ and $Q=(Q_E^2+Q_B^2)^{1/2}$. If
-$E>|P|_\delta$ and $m=√(E^2-|P|_\delta^2)$, then [eq:ch-inequality]
+with the normal into $D^+$ and $theta_+(S)le0$. Set
+$r=(A_{min}(S)/(4pi))^{1/2}$ and $Q=(Q_E^2+Q_B^2)^{1/2}$. If
+$E>|P|_delta$ and $m=√(E^2-|P|_delta^2)$, then [eq:ch-inequality]
$$
-m\ge Q,\qquad r\le m+√(m^2-Q^2).
+mge Q,qquad rle m+√(m^2-Q^2).
$$
-The familiar lower bound $m\ge\frac12(r+Q^2/r)$ is recovered on the
-branch $r\ge m$, and is not equivalent for every positive $r$.
+The familiar lower bound $mgefrac12(r+Q^2/r)$ is recovered on the
+branch $rge m$, and is not equivalent for every positive $r$.
@@ -2334,8 +2334,8 @@ coupled elliptic construction of Sections~[sec:geometry]--
[sec:existence], the field strengths $e$ and $b$ are transported
to the comparison data as closed forms on the same decay class, with [eq:ch-transported]
$$
-Q_E^{\rm comp}=Q_E+o(1),\quad Q_B^{\rm comp}=Q_B+o(1),\quad
-\|e^{\rm comp}-e\|,\|b^{\rm comp}-b\|=o(1)
+Q_E^{rm comp}=Q_E+o(1),quad Q_B^{rm comp}=Q_B+o(1),quad
+|e^{rm comp}-e|,|b^{rm comp}-b|=o(1)
$$
in $L^∞$ on any fixed compact region. The charged DEC [eq:ch-dec]
is preserved up to the same $o(1)$.
@@ -2348,7 +2348,7 @@ Proof.
Each of the three operations is a conformal deformation with a
$C^2$ conformal factor bounded between $1$ and $1+c$ on the
end, or a $C^2$ diffeomorphism supported on the end. The
-decomposition $F=e\wedge n+b$ is conformally invariant on
+decomposition $F=ewedge n+b$ is conformally invariant on
four-dimensional spacetimes when gauged to the data slice's unit
normal; the surface charge [eq:ch-charges] is a topological
integral of closed flux, so $Q_E,Q_B$ are unchanged in the
@@ -2366,11 +2366,11 @@ Ilmanen (2001) and Bray (2001).
Theorem (Charged Riemannian Penrose). [thm:charged-riemannian]
Let $(M,g,e,b)$ be complete smooth asymptotically flat three-dimensional
-data with $R_g\ge 2|e|_g^2+2|b|_g^2$ and finite charges $Q_E,Q_B$.
-Let $\Gamma$ be a smooth minimal surface in the chosen exterior with
+data with $R_gge 2|e|_g^2+2|b|_g^2$ and finite charges $Q_E,Q_B$.
+Let $Gamma$ be a smooth minimal surface in the chosen exterior with
outermost area $A$, and let $P=0$, $m=E$. Then [eq:charged-rmp]
$$
-m\ge(r+Q^2/r)/(2),\qquad r=((A)/(4\pi))^{1/2},\quad
+mge(r+Q^2/r)/(2),qquad r=((A)/(4pi))^{1/2},quad
Q=(Q_E^2+Q_B^2)^{1/2}.
$$
@@ -2381,8 +2381,8 @@ $$
Proof.
This is the classical result (T4) of the [sec:external] section. The
charged inverse mean curvature flow (Bray--Khuri, Disconzi--Khuri)
-preserves the inequality $m(t)\ge\frac12(r(t)+Q^2/r(t))$ along the
-flow, with $r(t)=(A(t)/4\pi)^{1/2}$; at the boundary sphere this
+preserves the inequality $m(t)gefrac12(r(t)+Q^2/r(t))$ along the
+flow, with $r(t)=(A(t)/4pi)^{1/2}$; at the boundary sphere this
becomes the charged Riemannian Penrose inequality of (T4).
@@ -2398,33 +2398,33 @@ gives compactly supported $K$, zero momentum, and energy $m'=m+o(1)$.
Second, the charged elliptic construction: in the coupled elliptic
system of the [sec:elliptic] section add the extra source terms
-$2(|e|^2+|b|^2)$ and $2*(e\wedge *b)$ to the constraint densities. The
+$2(|e|^2+|b|^2)$ and $2*(ewedge *b)$ to the constraint densities. The
comparison metric
-$\bar g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf)$ satisfies [eq:ch-comparison]
+$barbar g_g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf)$ satisfies [eq:ch-comparison]
$$
-R_{\bar g}\ge 2|\bar e|_{\bar g}^2+2|\bar b|_{\bar g}^2+O(\eta_n),
+R_{barbar g}gge 2|barbar e|_e|_{barbar g}g^2+2|barbar b|_b|_{barbar g}g^2+O(eta_n),
$$
-with $|\bar e|_{\bar g}^2, |\bar b|_{\bar g}^2$ the transported field
-densities. The boundedness of $\bar e,\bar b$ and of their covariant
+with $|barbar e|_e|_{barbar g}g^2, |barbar b|_b|_{barbar g}g^2$ the transported field
+densities. The boundedness of $barbar e,e,barbar b$b$ and of their covariant
derivatives on the graph, together with the coercivity of
-the [lem:n3:hessian-square] lemma, absorbs the $O(\eta_n)$ into [eq:ch-comparison-eps]
+the [lem:n3:hessian-square] lemma, absorbs the $O(eta_n)$ into [eq:ch-comparison-eps]
$$
-R_{\bar g}\ge 2|\bar e|_{\bar g}^2+2|\bar b|_{\bar g}^2,
+R_{barbar g}gge 2|barbar e|_e|_{barbar g}g^2+2|barbar b|_b|_{barbar g}g^2,
$$
-after shrinking the comparison to a lower bound on $\eta_n$ (the
-$O(\eta_n)$ term appears with sign in the coercive bound).
+after shrinking the comparison to a lower bound on $eta_n$ (the
+$O(eta_n)$ term appears with sign in the coercive bound).
Third, the boundary sphere $∂Ω_R$ of the comparison is a
-minimal surface with area at least $(1-ε)A_{\min}(S)$ and charge
+minimal surface with area at least $(1-ε)A_{min}(S)$ and charge
$(Q_E,Q_B)$. By the [thm:charged-riemannian] theorem, [eq:ch-assembly]
$$
-m_{\rm ADM}(\bar g)\ge
-\frac{r_{\bar g}+Q^2/r_{\bar g}}{2},\qquad
-r_{\bar g}=(\frac{Area_{\bar g}(∂Ω_R)}{4\pi})^{1/2}.
+m_{rm ADM}(barbar g)g)ge
+frac{r_{barbar g}g+Q^2/r_{barbar g}g}{2},qquad
+r_{barbar g}g=(frac{Area_{barbar g}g(∂Ω_R)}{4pi})^{1/2}.
$$
-Pass to the limit as $ε↓0$ and $R\to∞$: the left side
+Pass to the limit as $ε↓0$ and $Rto∞$: the left side
becomes $m$, the right side becomes the charged mass bound with
-$r=(A_{\min}(S)/(4\pi))^{1/2}$.
+$r=(A_{min}(S)/(4pi))^{1/2}$.
@@ -2472,10 +2472,10 @@ the [def:bridge-admissible] definition.
Theorem (Four-dimensional exterior inequality). [thm:four:main]
Let $(Ω,g,K)$ be a four-dimensional exterior with
-$\theta_+(B)\le0$, DEC, and integrable $\mu,|J|_g$. Let
-$\omega_3=2\pi^2$. If $E>|P|$, then [eq:four-inequality]
+$theta_+(B)le0$, DEC, and integrable $mu,|J|_g$. Let
+$omega_3=2pi^2$. If $E>|P|$, then [eq:four-inequality]
$$
-m\ge\frac12((A_e)/(\omega_3))^{2/3}.
+mgefrac12((A_e)/(omega_3))^{2/3}.
$$
The complete-data statement of the [thm:bridge:global] theorem holds in
dimension $4$ with the same right-hand side.
@@ -2488,13 +2488,13 @@ with the four modifications below.
Strictification and end reduction in dimension four [sec:four-reduction]. The drift-Poisson strictification of the [lem:n3:strictification] lemma
-adapts: in dimension $4$ the conformal family $g\mapsto u^2g$,
-$K\mapsto uK$ preserves the constraints and DEC, and the drift
-source exponent is $w=r^{-4-\delta}$ with $0<\delta<\beta<\min(q,1)$.
-The repaired data satisfy $\theta_+(B)<0$, $\mu-|J|_g\ge c_0r^{-4-\delta_1}$
+adapts: in dimension $4$ the conformal family $gmapsto u^2g$,
+$Kmapsto uK$ preserves the constraints and DEC, and the drift
+source exponent is $w=r^{-4-delta}$ with $0<delta<beta<min(q,1)$.
+The repaired data satisfy $theta_+(B)<0$, $mu-|J|_gge c_0r^{-4-delta_1}$
with $c_0>0$, and $K$ compactly supported. The end-reduction of
the [prop:n3:rest-frame] proposition works in dimension $4$: a distant
-Lorentz boost sends $(E,P)\mapsto(m,0)$, and a conformal support
+Lorentz boost sends $(E,P)mapsto(m,0)$, and a conformal support
truncation on the far end adjusts $m$ and $A_e$ by $O(R^{-3})$.
@@ -2502,16 +2502,16 @@ The four-dimensional elliptic construction [sec:four-elliptic]. The coupled syst
$4$. The scalar identity changes: the lapse appears with the extra
degree of freedom $e^{3t}$ (rather than $e^{2t}$), and the conformal
family in the metric comparison is $u^3$ rather than $u^4$ with
-$u=e^{t/3}$. The variables $f,t$, the auxiliary $D,d,Z,h,w,a,v,\chi$
+$u=e^{t/3}$. The variables $f,t$, the auxiliary $D,d,Z,h,w,a,v,chi$
take their dimension-$4$ forms. The comparison metric
-$\bar g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf)$ has
+$barbar g_g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf)$ has
nonnegative scalar curvature on $Ω_R$, mass at most $m+ε$, and
boundary three-volume at least $(1-ε)A_e$. The regularity estimates
of the [sec:existence] section (graph Sobolev, axial gradient estimate,
four-stage homotopy) pass through in dimension $4$ with the same
structure; the scalar identity uses a Schur-type trace calculation
-$3\Delta_{\bar g}Z=\mathcal T$-type decomposition that gives
-$R_{\bar g}\ge0$ from the DEC.
+$3Delta_{barbar g}gZ=mathcal T$-typecalT$-type decomposition that gives
+$R_{barbar g}gge0$ from the DEC.
The four-dimensional Riemannian Penrose inequality [sec:four-rmp]. The classical four-dimensional Riemannian Penrose inequality is used here.
@@ -2519,13 +2519,13 @@ The four-dimensional Riemannian Penrose inequality [sec:four-rmp]. The classical
Theorem (Four-dimensional Riemannian Penrose). [thm:four-riemannian]
Let $(M,g)$ be a complete smooth asymptotically flat four-dimensional
-Riemannian manifold with nonnegative scalar curvature $R_g\ge0$ and
-ADM mass $m_{ADM}$. Let $\Gamma$ be a smooth compact outermost minimal
+Riemannian manifold with nonnegative scalar curvature $R_gge0$ and
+ADM mass $m_{ADM}$. Let $Gamma$ be a smooth compact outermost minimal
$3$-surface. Then
$$
-m_{ADM}\ge\frac12((|\Gamma|_g)/(\omega_3))^{2/3},
-\qquad \omega_3=2\pi^2.
+m_{ADM}gefrac12((|Gamma|_g)/(omega_3))^{2/3},
+qquad omega_3=2pi^2.
$$
@@ -2538,9 +2538,9 @@ This is the four-dimensional Riemannian Penrose inequality, a classical
result (Huisken--Ilmanen 2001; Bray 2001, extended by Bray--Lee 2009).
Its three-dimensional form is (T1) of the [sec:external] section; the
four-dimensional statement uses the flow identity
-$Σ(t)=m_{ADM}-\tfrac12(A(t)/\omega_3)^{2/3}$ in place of the
+$Σ(t)=m_{ADM}-tfrac12(A(t)/omega_3)^{2/3}$ in place of the
three-dimensional formula. At the outermost minimal $3$-surface the
-area statistic $Area(t)$ equals $|\Gamma|_g$, giving the stated
+area statistic $Area(t)$ equals $|Gamma|_g$, giving the stated
inequality. It is used as an external input; the details of the flow
are classical.
@@ -2552,25 +2552,25 @@ Assembly [sec:four-assembly]. Proof (Proof of the [thm:four:main). theorem]
Apply strictification (the [lem:n3:strictification] lemma in
dimension $4$), end reduction (the [prop:n3:rest-frame] proposition
in dimension $4$), and the coupled elliptic construction
-(the [sec:four-elliptic] section) to obtain comparison metric $\bar g$
+(the [sec:four-elliptic] section) to obtain comparison metric $barbar g$g$
with
$$
-R_{\bar g}\ge0,\quad
-m_{ADM}(\bar g)\le m+ε,\quad
-Area_{\bar g}(∂Ω_R)\ge(1-ε)A_e.
+R_{barbar g}gge0,quad
+m_{ADM}(barbar g)g)le m+ε,quad
+Area_{barbar g}g(∂Ω_R)ge(1-ε)A_e.
$$
Apply the [thm:four-riemannian] theorem to the outermost boundary
sphere:
$$
-m_{ADM}(\bar g)\ge\frac12(\frac{Area_{\bar g}(∂Ω_R)}
-{\omega_3})^{2/3}.
+m_{ADM}(barbar g)g)gefrac12(frac{Area_{barbar g}g(∂Ω_R)}
+{omega_3})^{2/3}.
$$
Let $ε↓0$: the left side becomes $m$, the right side becomes
-$\frac12(A_e/\omega_3)^{2/3}$, which is [eq:four-inequality].
+$frac12(A_e/omega_3)^{2/3}$, which is [eq:four-inequality].
@@ -2583,7 +2583,7 @@ with Schwarzschild--Tangherlini.
Theorem (Four-dimensional rigidity). [thm:rig4:rigidity]
Under the hypotheses of the [thm:four:main] theorem in the connected,
one-ended, outermost, outer area-minimizing equality case
-$m=\frac12(A_e/\omega_3)^{2/3}$, the exterior $M_{ext}$ is isometric to
+$m=frac12(A_e/omega_3)^{2/3}$, the exterior $M_{ext}$ is isometric to
the future-horizon exterior in the Schwarzschild--Tangherlini
spacetime of mass $m$, with $K$ the second fundamental form generated
by the stationary Killing field of that spacetime.
@@ -2599,7 +2599,7 @@ following substitutions: the causal stationary field
twist-vanishing (the [prop:n3-twist-vanishes] proposition) still
uses the positive-mass bootstrap on the conformal double; the
conformal double (the [prop:n3-double] proposition) exists; and
-Schoen--Yau positivity gives $\ell=1$ and $S\cong S^3$. The
+Schoen--Yau positivity gives $ell=1$ and $Scong S^3$. The
static-vacuum classification in spacetime dimension $5$
(Bunting--Masood-ul-Alam) then identifies the exterior with
Schwarzschild--Tangherlini of the same mass.
@@ -2612,20 +2612,20 @@ The anti-de Sitter Penrose inequality [sec:ads]. Schwarzschild-anti-de Sitter an
-Fix the Schwarzschild-anti-de Sitter background of mass $m_0\ge0$ in
+Fix the Schwarzschild-anti-de Sitter background of mass $m_0ge0$ in
spacetime dimension $4$. In the usual static coordinates, [eq:ads-bgd]
$$
-g^0=-N^2d t^2+(d r^2)/(N^2)+r^2\sigma,\qquad
+g^0=-N^2d t^2+(d r^2)/(N^2)+r^2sigma,qquad
N^2=1+r^2-(2m_0)/(r),
$$
-with $\sigma$ the round metric on $S^2$, and on the time slice $t=0$
+with $sigma$ the round metric on $S^2$, and on the time slice $t=0$
$K^0=0$. The spatial metric in Fefferman-Graham form is
-$g^0=d s^2+r(s)^2\sigma$ with $r=e^s(1+O(e^{-2s}))$; the boundary is
+$g^0=d s^2+r(s)^2sigma$ with $r=e^s(1+O(e^{-2s}))$; the boundary is
at $s=∞$.
A smooth branch of asymptotically anti-de Sitter initial data near the
background is a smooth family $(g_ε,K_ε)$ for
-$ε\in[0,ε_0)$ such that each $(g_ε,K_ε)$ is of finite
+$εin[0,ε_0)$ such that each $(g_ε,K_ε)$ is of finite
angular type, satisfies the Einstein constraints on the exterior of a
compact future MOTS $S_ε$, satisfies $(g_ε,K_ε)=(g^0,K^0)+O(ε)$
in the weighted Sobolev norm of the class, and has finite boundary mass
@@ -2634,14 +2634,14 @@ $m_{AH}(ε)$ the positive Lorentz norm of the hyperbolic mass covector.
Theorem (Anti-de Sitter Penrose). [thm:ads:main]
-Under the hypotheses above there is $ε_1\in(0,ε_0]$ such that for
-every $ε\in[0,ε_1)$, [eq:ads-inequality]
+Under the hypotheses above there is $ε_1in(0,ε_0]$ such that for
+every $εin[0,ε_1)$, [eq:ads-inequality]
$$
m_{AH}(ε)
-\ge√((A(ε))/(16\pi))
-(1+(A(ε))/(4\pi)).
+ge√((A(ε))/(16pi))
+(1+(A(ε))/(4pi)).
$$
-Equality holds if and only if the branch seed is radial and $ε\ge0$
+Equality holds if and only if the branch seed is radial and $εge0$
is sufficiently small. The equality family is Schwarzschild-anti-de
Sitter at the matching mass and area.
@@ -2658,26 +2658,26 @@ $O_4(ε^4)$ controls the asymptotic functionals.
Lemma (Taylor to mass and area). [lem:ads-taylor]
There are real $m_0,m_1,m_2,m_3$ and $A_0,A_1,A_2,A_3$ such that [eq:ads-taylor-exp]
$$
-m_{AH}(ε)=\sum_{j=0}^3ε^jm_j+O(ε^4),\quad
-A(ε)=\sum_{j=0}^3ε^j A_j+O(ε^4),
+m_{AH}(ε)=sum_{j=0}^3ε^jm_j+O(ε^4),quad
+A(ε)=sum_{j=0}^3ε^j A_j+O(ε^4),
$$
with the remainders controlled by the weighted bound
-$F=O(e^{-\gamma s})$ with $\gamma>3$.
+$F=O(e^{-gamma s})$ with $gamma>3$.
Lemma (Inverse-asymptotic). [lem:ads-inverse]
-Let $L=-\Delta_{g^0}+3$. If $F$ is smooth finite-type with
-$F=O(e^{-\gamma s})$ for some $\gamma>3$, then for either Dirichlet or
+Let $L=-Delta_{g^0}+3$. If $F$ is smooth finite-type with
+$F=O(e^{-gamma s})$ for some $gamma>3$, then for either Dirichlet or
homogeneous Neumann data on $S$ there is a unique finite-type decaying
solution of $Lu=F$ with [eq:ads-remainder]
$$
-u=r^{-3}\zeta(x)+o(r^{-3}),\quad
-∂_s(u-r^{-3}\zeta(x))=o(r^{-3}),
+u=r^{-3}zeta(x)+o(r^{-3}),quad
+∂_s(u-r^{-3}zeta(x))=o(r^{-3}),
$$
-$\zeta$ finite-type. Differentiation of $F$ in $s$ through order $k$
+$zeta$ finite-type. Differentiation of $F$ in $s$ through order $k$
gives the same statement through $k+2$ derivatives of $u$.
@@ -2686,11 +2686,11 @@ gives the same statement through $k+2$ derivatives of $u$.
Proof.
Variation of constants for $y''+2y'-3y=G$ with
-$G=O(e^{-\gamma's})$, $\gamma'=\min(\gamma,\beta+2)>3$, eliminates
+$G=O(e^{-gamma's})$, $gamma'=min(gamma,beta+2)>3$, eliminates
the growing homogeneous mode $e^s$ and gives
-$y=ce^{-3s}+O(e^{-\gamma's})$. Angular decomposition closes by
+$y=ce^{-3s}+O(e^{-gamma's})$. Angular decomposition closes by
boundedness on each fixed finite-dimensional space. Comparison with
-$Ah+\delta$, letting $\delta↓0$, gives uniqueness of the
+$Ah+delta$, letting $delta↓0$, gives uniqueness of the
decaying solution.
@@ -2700,8 +2700,8 @@ decaying solution.
The quadratic coefficient and its kernel [sec:ads-quadratic]. Proposition (The quadratic form). [prop:ads-c2]
In the expansion [eq:ads-mass-gap]
$$
-m_{AH}(ε)-√((A(ε))/(16\pi))
-(1+(A(ε))/(4\pi))
+m_{AH}(ε)-√((A(ε))/(16pi))
+(1+(A(ε))/(4pi))
=c_2[q]ε^2+c_3[q]ε^3+c_4[q]ε^4+O(ε^5)
$$
the coefficient $c_2[q]$ is a nonnegative quadratic form, and its
@@ -2716,8 +2716,8 @@ Conformal decomposition of $g_1$ gives a TT part $k$ together with a
conformal factor and a tangential vector. The mass and area Taylor
expansion reduces $c_2$ to the TT square identity
(the [lem:ads-tt-square] lemma), which makes it
-$(1)/(16\pi)\int_M N|k|^2\,d V_0
--(\kappa)/(16\pi)\int_S vDv\,d A_0$. The argument of
+$(1)/(16pi)int_M N|k|^2,d V_0
+-(kappa)/(16pi)int_S vDv,d A_0$. The argument of
the [lem:ads-tt-square] lemma identifies the last term, and $D$'s
positivity makes the form nonnegative. Its kernel is the four
translation modes of the round $S^2$.
@@ -2727,13 +2727,13 @@ translation modes of the round $S^2$.
Lemma (Weighted TT square identity). [lem:ads-tt-square]
-Let $k$ be smooth TT with $k=O(r^{-\gamma})$ for some $\gamma>3/2$, and
+Let $k$ be smooth TT with $k=O(r^{-gamma})$ for some $gamma>3/2$, and
$v$ the unique decaying solution of $Lv=0$ with
-$v|_S=D^{-1}(k_{ss}|_S)$, $D=-\Delta_S+2\kappa/a$. Then
+$v|_S=D^{-1}(k_{ss}|_S)$, $D=-Delta_S+2kappa/a$. Then
$Q(v)=Hess v-vB$ is TT, $Q(v)_{ss}|_S=k_{ss}|_S$, and [eq:ads-tt-square]
$$
-\int_M N|k|^2\,d V_0-\kappa\int_S vDv\,d A_0
-=\int_M N|k-Q(v)|^2\,d V_0.
+int_M N|k|^2,d V_0-kappaint_S vDv,d A_0
+=int_M N|k-Q(v)|^2,d V_0.
$$
@@ -2755,7 +2755,7 @@ A change of spacetime slice at $2$-jet order in the Taylor expansions
coefficient of the nonradial directions, preserves the boundary area
through the required order, and reduces $c_4$ to a second TT square
built from the transverse coefficient near the original boundary. This
-gives $c_4[q]\ge0$. Equality would force the changed metric to be a
+gives $c_4[q]ge0$. Equality would force the changed metric to be a
round warped-product $2$-jet; conformality of the original branch then
gives a differential identity incompatible with a nonzero decaying
degree-one mode. Hence $c_4[q]>0$.
@@ -2765,9 +2765,9 @@ degree-one mode. Hence $c_4[q]>0$.
Lemma (Finite-jet slice change). [lem:ads-slice]
-There is a smooth local diffeomorphism $\Psi$ and a smooth conformal
-factor $\omega$ with $\omega_0=1,\Psi_0=id$, such that
-$\widetilde g=\omega_ε\Psi_ε^*g^0+O(ε^3)$ holds on the
+There is a smooth local diffeomorphism $Psi$ and a smooth conformal
+factor $omega$ with $omega_0=1,Psi_0=id$, such that
+$widetilde g=omega_εPsi_ε^*g^0+O(ε^3)$ holds on the
original slice.
@@ -2775,10 +2775,10 @@ original slice.
Proposition (Exact radial identity). [prop:ads-radial]
-If $q$ is a radial seed, then for all sufficiently small $ε\ge0$, [eq:ads-radial-equality]
+If $q$ is a radial seed, then for all sufficiently small $εge0$, [eq:ads-radial-equality]
$$
-m_{AH}(ε)=√(A(ε)/16\pi)
-(1+A(ε)/4\pi).
+m_{AH}(ε)=√(A(ε)/16pi)
+(1+A(ε)/4pi).
$$
@@ -2787,7 +2787,7 @@ $$
Proof.
Radial seeds form the Schwarzschild-anti-de Sitter family, for which
-$m=√(A/16\pi)(1+A/4\pi)$ is exact at every radius. At $ε=0$ the
+$m=√(A/16pi)(1+A/4pi)$ is exact at every radius. At $ε=0$ the
identity is the background identity, and smoothness continues it to
$[0,ε_1)$.
@@ -2804,7 +2804,7 @@ Proof.
Combine the [lem:ads-taylor] lemma and
Propositions~[prop:ads-c2], [prop:ads-c4], [prop:ads-radial].
On nonradial seeds, the gap in [eq:ads-mass-gap] is
-$c_2[q]ε^2+c_4[q]ε^4+O(ε^5)$ with $c_2[q]\ge0$ and $c_4[q]>0$ on
+$c_2[q]ε^2+c_4[q]ε^4+O(ε^5)$ with $c_2[q]ge0$ and $c_4[q]>0$ on
the kernel, so the gap is strictly positive for all small $ε>0$.
On radial seeds the exact identity [eq:ads-radial-equality] gives
equality. The interval $[0,ε_1)$ depends only on the background data