diff d0ef47e..b34550f spacetime_penrose_area_charge_v2.txt
diff --git a/spacetime_penrose_area_charge_v2.txt b/spacetime_penrose_area_charge_v2.txt
index 8b69aec..2fca3be 100644
--- a/spacetime_penrose_area_charge_v2.txt
+++ b/spacetime_penrose_area_charge_v2.txt
@@ -2,7 +2,7 @@
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
-$m\ge\sqrt{A/16\pi}$; otherwise a naked singularity would form, violating
+$m\ge√(A/16\pi)$; 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
@@ -20,7 +20,7 @@ Fix a compact hypersurface $S\subset M$ and a unit normal $\nu$ toward the
designated exterior. Its future outward null expansion is
$$
- \theta_+(S)=H_S+\operatorname{tr}_S K,
+ \theta_+(S)=H_S+tr_S K,
$$
where $H_S$ is the mean curvature of $S$ with respect to $\nu$. The surface
@@ -30,7 +30,7 @@ appears as such a surface in a spacelike slice.
Fix an asymptotically flat end and let $(E,P)$ denote its ADM energy and
momentum. Once $E>|P|$, the invariant mass of the end is
-$m=\sqrt{E^2-|P|^2}$. The spacetime Penrose inequalities bound $m$ below by
+$m=√(E^2-|P|^2)$. 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
@@ -62,7 +62,7 @@ exterior whose matter condition includes the electromagnetic momentum,
the [thm:ch:main] theorem proves
$$
- m\ge Q,\qquad r\le m+\sqrt{m^2-Q^2}.
+ m\ge Q,\qquad r\le m+√(m^2-Q^2).
$$
The familiar bound $m\ge(r+Q^2/r)/2$ is recovered on the branch $r\ge Q$;
@@ -76,7 +76,7 @@ hypersurface and $\omega_3=2\pi^2$ the volume of the unit $3$-sphere.
the [thm:four:main] theorem proves
$$
- m_e\ge\frac12\left(\frac{A_e}{\omega_3}\right)^{2/3}
+ m_e\ge\frac12((A_e)/(\omega_3))^{2/3}
$$
for arbitrary $K$, finitely many ends, and a disconnected weakly future
@@ -87,12 +87,12 @@ Schwarzschild--Tangherlini geometry.
The anti-de Sitter part fixes a positive-mass Schwarzschild--anti-de Sitter
background, a transverse-traceless (trace free and divergence free) seed,
and a decaying positive conformal branch. With $A$ the boundary area and
-$m_{\operatorname{AH}}$ the positive Lorentz norm of the hyperbolic mass covector,
+$m_{AH}$ the positive Lorentz norm of the hyperbolic mass covector,
the [thm:ads:main] theorem gives, on a sufficiently small interval depending
on the fixed data,
$$
- m_{\operatorname{AH}}\ge\sqrt{\frac{A}{16\pi}}\left(1+\frac{A}{4\pi}\right),
+ m_{AH}\ge√((A)/(16\pi))(1+(A)/(4\pi)),
$$
with equality exactly for radial seeds at small positive parameter.
@@ -138,8 +138,8 @@ 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
- energy $E_h$, and $\partial N$ is a compact outermost smooth minimal
- surface of total area $A$, then $E_h\ge\sqrt{A/16\pi}$, with equality
+ 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
in the nondegenerate case exactly for the Schwarzschild horizon.
./pdf/huisken-ilmanen-2001.pdf
@@ -153,12 +153,12 @@ contributions end and the classical machinery begins.
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
- $M=|\partial N|_h^{1/2}/\sqrt{16\pi}$.
+ $M=|∂ N|_h^{1/2}/√(16\pi)$.
(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 $r\le m+\sqrt{m^2-Q^2}$, with $r=(A/4\pi)^{1/2}$ and
+ $E_h\ge Q$ and $r\le m+√(m^2-Q^2)$, with $r=(A/4\pi)^{1/2}$ and
$m=E_h$.
./pdf/disconzi-khuri-2012.pdf
@@ -179,15 +179,15 @@ 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
momentum density $J$ are determined by the constraint equations [eq:constraints]
$$
- 16\pi\mu=R_g+(\operatorname{tr}_gK)^2-|K|_g^2,
+ 16\pi\mu=R_g+(tr_gK)^2-|K|_g^2,
\qquad
- 8\pi J=\operatorname{div}_g\bigl(K-(\operatorname{tr}_gK)g\bigr),
+ 8\pi J=div_g(K-(tr_gK)g),
$$
where $R_g$ is the scalar curvature. The dominant energy condition (DEC)
is the pointwise inequality $\mu\ge|J|_g$.
An end of $M$ is a neighborhood of infinity diffeomorphic to
-$\mathbb{R}^3\setminus\bar B_1$, parametrized by coordinates $x^i$ in which [eq:decay]
+$ℝ^3∖\bar B_1$, parametrized by coordinates $x^i$ in which [eq:decay]
$$
g_{ij}-\delta_{ij}=O_2(r^{-q}),
\qquad
@@ -198,43 +198,43 @@ $$
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]
$$
- E=\frac{1}{16\pi}\lim_{r\to\infty}\int_{S_r}(g_{ij,i}-g_{ii,j})\nu^j\,\mathrm{d} A,
+ E=(1)/(16\pi)\lim_{r\to∞}\int_{S_r}(g_{ij,i}-g_{ii,j})\nu^j\,d A,
\qquad
- P_j=\frac{1}{8\pi}\lim_{r\to\infty}\int_{S_r}\bigl(K_{ij}-(\operatorname{tr}_gK)\delta_{ij}\bigr)\nu^i\,\mathrm{d} A,
+ P_j=(1)/(8\pi)\lim_{r\to∞}\int_{S_r}(K_{ij}-(tr_gK)\delta_{ij})\nu^i\,d A,
$$
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 $m=\sqrt{E^2-|P|_\delta^2}$.
+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
-$\partial M$, with a chosen unit normal $\nu$ pointing into the
-designated exterior. Its mean curvature is $H_S=\operatorname{div}_S\nu$, with the
-convention $H_S=2/r$ for a coordinate sphere of radius $r$ in $\mathbb{R}^3$,
-and $\operatorname{tr}_SK=g^{ab}K(X_a,X_b)$ in an orthonormal tangent frame
+$∂ M$, with a chosen unit normal $\nu$ pointing into the
+designated exterior. Its mean curvature is $H_S=div_S\nu$, 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]
$$
- \theta_\pm(S)=H_S\pm\operatorname{tr}_SK.
+ \theta_\pm(S)=H_S\pmtr_SK.
$$
The surface $S$ is weakly future outer trapped when $\theta_+(S)\le0$,
and a future marginally outer trapped surface (MOTS) when
$\theta_+(S)=0$.
-Exteriors, cuts, and the minimum enclosing area [sec:cuts]. Let $\Omega$ be a connected open manifold whose closure $\bar\Omega$ is
+Exteriors, cuts, and the minimum enclosing area [sec:cuts]. Let $Ω$ be a connected open manifold whose closure $\barΩ$ is
complete for the restriction of $g$, with smooth compact boundary
-$B=\partial\Omega$ and exactly one end, asymptotically flat.
+$B=∂Ω$ and exactly one end, asymptotically flat.
An enclosing cut for the chosen end is a compact smooth embedded
-(possibly disconnected) surface $\Gamma\subset\bar\Omega$ that bounds a
-filled side $D^+(\Gamma)\subset\bar\Omega$ containing the entire distant
-part of that end. The complement $\bar\Omega\setminus D^+(\Gamma)$
+(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]
$$
- a_g(B)=\inf\bigl\{|\Gamma|_g:\Gamma\text{ an enclosing cut}\bigr\}.
+ a_g(B)=\inf\bigl\{|\Gamma|_g:\Gamma an enclosing cut\bigr\}.
$$
The infimum equals the finite-perimeter closure taken in
the [sec:perimeter] section.
@@ -267,7 +267,7 @@ Throughout, $c=G=1$ and the coefficients $16\pi$ 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
-coordinate sphere of radius $r$ in $\mathbb{R}^n$ has area
+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 $m=\tfrac12(A/\omega_{n-1})^{(n-2)/(n-1)}$.
@@ -278,18 +278,18 @@ The three-dimensional exterior inequality [sec:n3]. Statement [sec:n3-statement]
Theorem (Minimum enclosing area in three dimensions). [thm:n3:exterior]
-Let $(\Omega,g,K)$ be a connected orientable smooth three-dimensional
-exterior with smooth compact boundary $B=\partial\Omega$, complete with
+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
$|J|_g$ integrable, and that the ADM limits [eq:adm] exist.
-Orient $B$ by the normal into $\Omega$ and suppose $\theta_+(B)\le0$.
+Orient $B$ by the normal into $Ω$ and suppose $\theta_+(B)\le0$.
If $E>|P|_\delta$, then
$$
- \sqrt{E^2-|P|_\delta^2}
- \ge\sqrt{\frac{a_g(B)}{16\pi}}.
+ √(E^2-|P|_\delta^2)
+ \ge√((a_g(B))/(16\pi)).
$$
No extension of the data across $B$ is required.
@@ -306,7 +306,7 @@ elliptic comparison with the Riemannian Penrose inequality
Conformal strict direction [sec:n3-reduction]. Lemma (Strict direction). [lem:n3:strict]
-Let $(\Omega,g,K)$ satisfy all the hypotheses of
+Let $(Ω,g,K)$ satisfy all the hypotheses of
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
@@ -318,7 +318,7 @@ $$
a_{g_s}(B)\ge a_g(B),
$$
-and $(\Omega,g_s,K_s)$ satisfies DEC and
+and $(Ω,g_s,K_s)$ satisfies DEC and
$\theta_+^{g_s,K_s}(B)\le0$ in the same decay class.
@@ -332,7 +332,7 @@ 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
is the standard positivity of the conformal Green's function
(Schoen--Yau). Put $\phi=Gw\ge0$, so $\phi$ is smooth, positive on
-$\operatorname{supp} w$, and decays like $O(1/|x|)$.
+$supp w$, and decays like $O(1/|x|)$.
Write $u_s=1+s\phi$, so $u_s\ge1$ and $u_s\to1$ at infinity. The
constraints of $g_s=u_s^4g$, $K_s=u_s^{-2}K$ continue to satisfy DEC
@@ -341,21 +341,21 @@ densities transform with nonnegative factors when $u_s$ is a positive
supersolution of the conformal Laplacian. The ADM energy formula
$$
- E_{g_s}=E+s\cdot\frac{1}{16\pi}\lim_{r\to\infty}\int_{S_r}
- \partial_\nu(4u_s\partial_\nu u_s)\,\mathrm{d} A
+ E_{g_s}=E+s\cdot(1)/(16\pi)\lim_{r\to∞}\int_{S_r}
+ ∂_\nu(4u_s∂_\nu u_s)\,d A
=E+sA_\phi
$$
with $A_\phi\propto\int w>0$ and $P_s=P$ follows from the standard
second variation of the mass under $g\mapsto u_s^4g$.
-Because $\operatorname{supp} w$ is disjoint from a neighborhood of $B$, the normal
+Because $supp w$ is disjoint from a neighborhood of $B$, the normal
derivative of $u_s$ vanishes there, and conformal covariance of the
mean curvature and trace gives
$$
- H^{g_s}_B+\operatorname{tr}^{g_s}_BK_s
- =u_s^{-3}\bigl(H_B^g+\operatorname{tr}_B^gK\bigr),
+ H^{g_s}_B+tr^{g_s}_BK_s
+ =u_s^{-3}(H_B^g+tr_B^gK),
$$
so $\theta_+^{g_s,K_s}(B)\le0$ is preserved.
@@ -373,7 +373,7 @@ through~[sec:existence].
Theorem (Strictified exterior inequality). [thm:n3:strict-exterior]
-Let $(\Omega,g,K)$ satisfy all the hypotheses of
+Let $(Ω,g,K)$ satisfy all the hypotheses of
the [thm:n3:exterior] theorem together with
- $K$ compactly supported;
- $\theta_+(B)<0$ and $P=0$;
@@ -383,7 +383,7 @@ the [thm:n3:exterior] theorem together with
Then
$$
- m=E\ge\sqrt{\frac{a_g(B)}{16\pi}}.
+ m=E\ge√((a_g(B))/(16\pi)).
$$
@@ -394,34 +394,34 @@ The proof of the [thm:n3:strict-exterior] theorem builds a comparison
metric $g_\star$ on the strictified exterior with [eq:comparison]
$$
R_{g_\star}\ge0,\qquad
- m_{ADM}(g_\star)\le m+\varepsilon,\qquad
- a_{g_\star}(B)\ge(1-\varepsilon)a_g(B),
+ m_{ADM}(g_\star)\le m+ε,\qquad
+ a_{g_\star}(B)\ge(1-ε)a_g(B),
$$
-for every $\varepsilon>0$; the Riemannian Penrose inequality
+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
-and proceed. Fix $\varepsilon>0$. The [sec:end-preparation] section and
+and proceed. Fix $ε>0$. The [sec:end-preparation] section and
the [lem:n3:strict] lemma produce strictified data $(g',K')$ satisfying
$$
- m'\le m+\varepsilon,\qquad \theta'_+(B)<0,\qquad
- a_{g'}(B)\ge(1-\varepsilon)a_g(B),
+ m'\le m+ε,\qquad \theta'_+(B)<0,\qquad
+ a_{g'}(B)\ge(1-ε)a_g(B),
$$
and identical decay class. The [thm:n3:strict-exterior] theorem gives
$m'\ge\sqrt{a_{g'}(B)/16\pi}$, so
$$
- \sqrt{\frac{a_g(B)}{16\pi}}
- \le\frac{m'}{\sqrt{1-\varepsilon}}
- \le\frac{m+\varepsilon}{\sqrt{1-\varepsilon}}.
+ √((a_g(B))/(16\pi))
+ \le\frac{m'}{√(1-ε)}
+ \le\frac{m+ε}{√(1-ε)}.
$$
-Let $\varepsilon\downarrow0$ to conclude.
+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:
@@ -433,20 +433,20 @@ 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
-$$\begin{aligned}
-16\pi u^4\mu_u &=16\pi\mu-8u^{-1}\Delta_gu-\bigl(1-u^{-8}\bigr)\bigl((\operatorname{tr}_gK)^2-|K|_g^2\bigr),
+$$
+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}\bigl(8\pi J+4K(\nabla\log u,\cdot)\bigr), [eq:conformal-J]
-\end{aligned}$$
+8\pi J_u &=u^{-2}(8\pi J+4K(\nabla\log u,\cdot)), [eq:conformal-J]
+$$
the second being an identity of covectors. Consequently [eq:conformal-dec]
$$
16\pi u^4(\mu_u-|J_u|_{g_u})\ge
-16\pi(\mu-|J|_g)+8\bigl(-\Delta_g\log u-|d\log u|_g^2-|K|_g|d\log u|_g\bigr)
- -\bigl|1-u^{-8}\bigr|\,\bigl|(\operatorname{tr}_gK)^2-|K|_g^2\bigr|.
+16\pi(\mu-|J|_g)+8(-\Delta_g\log u-|d\log u|_g^2-|K|_g|d\log u|_g)
+ -|1-u^{-8}|\,|(tr_gK)^2-|K|_g^2|.
$$
For every consistently oriented hypersurface with unit normal $\nu$, [eq:conformal-expansion]
$$
-\theta_{+,u}=u^{-2}(\theta_++4\partial_\nu\log u).
+\theta_{+,u}=u^{-2}(\theta_++4∂_\nu\log u).
$$
@@ -458,16 +458,16 @@ In dimension three
$R_{u^4g}=u^{-4}(R_g-8\Delta_g\log u-8|d\log u|_g^2)$, while the
constraint quadratic term scales with an extra factor of $u^{-8}$ because
the extrinsic curvature is weighted by $u^2$:
-$u^4((\operatorname{tr}_{g_u}K_u)^2-|K_u|^2_{g_u})=u^{-8}((\operatorname{tr}_gK)^2-|K|_g^2)$.
-Adding the two contributions gives [eq:conformal-mu]. With $\pi=K-(\operatorname{tr}_gK)g$, $\pi_u=u^2\pi$ and
-$\operatorname{tr}_g\pi=-2\operatorname{tr}_gK$; contracting the connection difference in dimension
+$u^4((tr_{g_u}K_u)^2-|K_u|^2_{g_u})=u^{-8}((tr_gK)^2-|K|_g^2)$.
+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
three gives
-$u^2J_u=J+2\pi(\nabla\log u,\cdot)-(\operatorname{tr}_g\pi)d\log u
+$u^2J_u=J+2\pi(\nabla\log u,\cdot)-(tr_g\pi)d\log u
=J+4K(\nabla\log 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
-$\nu_u=u^{-2}\nu$, $H_{u^4g}=u^{-2}(H+4\partial_\nu\log u)$, and
-$\operatorname{tr}_{B,g_u}K_u=u^{-2}\operatorname{tr}_{B,g}K$, which is [eq:conformal-expansion].
+$\nu_u=u^{-2}\nu$, $H_{u^4g}=u^{-2}(H+4∂_\nu\log u)$, and
+$tr_{B,g_u}K_u=u^{-2}tr_{B,g}K$, which is [eq:conformal-expansion].
@@ -479,11 +479,11 @@ 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]
$$
-\partial_\nu\phi=-1\ \text{on }B,\quad
+∂_\nu\phi=-1\ on B,\quad
\phi=O_2(r^{-1}),\quad
-\Delta_g\phi-|K|_g|d\phi|_g\ge w,
$$
-and the outer flux $L_\phi=\lim_{r\to\infty}\int_{S_r}\partial_r\phi\,\mathrm{d} A_\delta$
+and the outer flux $L_\phi=\lim_{r\to∞}\int_{S_r}∂_r\phi\,d A_\delta$
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:
(i) DEC with strict gap
@@ -491,7 +491,7 @@ $K_s=u_s^2K$ with $u_s=1+s\phi$ satisfy:
(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
- $a_g(B)\le a_{g_s}(B)\le(1+s\|\phi\|_\infty)^4a_g(B)$.
+ $a_g(B)\le a_{g_s}(B)\le(1+s\|\phi\|_∞)^4a_g(B)$.
@@ -503,9 +503,9 @@ 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]
$$
--\Delta_g\phi=b\sqrt{|d\phi|_g^2+\zeta^2}+w,
+-\Delta_g\phi=b√(|d\phi|_g^2+\zeta^2)+w,
\qquad
-\partial_\nu\phi=-1,\quad\phi\to0,
+∂_\nu\phi=-1,\quad\phi\to0,
$$
is uniformly elliptic and smooth even at critical points of $\phi$
because of $\zeta$. There is a smooth nonnegative solution with
@@ -529,24 +529,24 @@ bounds are independent of $R$, yielding a smooth solution of
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 $\partial_\nu\phi\le0$
+Hopf principle at a boundary minimum requires $∂_\nu\phi\le0$
in the outer normal into $N$, whereas the present data give
-$\partial_\nu\phi=-1$; hence $\phi\ge0$. The radial function
+$∂_\nu\phi=-1$; 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
-$\partial_r\phi=-r^{-2}+o(r^{-2})$; then $L_\phi<0$ and $A_\phi>0$.
+$∂_r\phi=-r^{-2}+o(r^{-2})$; then $L_\phi<0$ and $A_\phi>0$.
Strict conclusions. At $s=0$,
$-\Delta_g\log u_s=-\Delta_g\phi
-=b\sqrt{|d\phi|^2+\zeta^2}+w\ge w$, and by continuity the same holds for
+=b√(|d\phi|^2+\zeta^2)+w\ge w$, and by continuity the same holds for
all small $s>0$; the density gap in [eq:conformal-dec] follows. On
-$B$, $\partial_\nu\log u_s=-s<0$, so
+$B$, $∂_\nu\log u_s=-s<0$, 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 $u_s\le1+s\|\phi\|_\infty$.
+together with $u_s\ge1$ and $u_s\le1+s\|\phi\|_∞$.
@@ -556,10 +556,10 @@ Remark (Decay needed by the drift-Poisson source).
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
-$|1-u_s^{-8}|\cdot|(\operatorname{tr}_gK)^2-|K|^2_g|
-=O(s\phi|(\operatorname{tr}_gK)^2-|K|^2_g|)=O(s\,r^{-1-2q})$ because
-$\phi=O(r^{-1})$ and $|(\operatorname{tr}_gK)^2-|K|^2_g|=O(r^{-2q})$. A strict gap
-$w-|1-u_s^{-8}|\,|(\operatorname{tr}_gK)^2-|K|^2_g|\ge c' r^{-3-\delta}$ therefore
+$|1-u_s^{-8}|\cdot|(tr_gK)^2-|K|^2_g|
+=O(s\phi|(tr_gK)^2-|K|^2_g|)=O(s\,r^{-1-2q})$ 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
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
@@ -578,8 +578,8 @@ with energy and area converging to the originals.
Positive enclosing area [sec:n3-positive-area]. Lemma (Positive enclosing area). [lem:n3:positive-area]
-Every exterior $(\Omega,g,K)$ in the [thm:n3:exterior] theorem satisfies
-$0<a_g(B)<\infty$.
+Every exterior $(Ω,g,K)$ in the [thm:n3:exterior] theorem satisfies
+$0<a_g(B)<∞$.
@@ -588,37 +588,37 @@ $0<a_g(B)<\infty$.
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
-$a_g(B)<\infty$. Positivity: fix a small disk $D$ in a component of $B$
+$a_g(B)<∞$. 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
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
$T$, pulled back from a transverse disk form, with
-$\int_{S_R}\omega=1$ and bounded $L^\infty$-norm. For every enclosing
+$\int_{S_R}\omega=1$ 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 $1\le\|\omega\|_\infty|\Gamma|_g$, and taking infima over
-cuts gives $a_g(B)\ge1/\|\omega\|_\infty>0$.
+Consequently $1\le\|\omega\|_∞|\Gamma|_g$, and taking infima over
+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 $(\Omega,g,K)$ satisfies the hypotheses of
+Suppose $(Ω,g,K)$ satisfies the hypotheses of
the [thm:n3:exterior] theorem with $E>|P|_\delta$ and invariant mass $m$.
-For each $\varepsilon>0$, there is one-ended exterior $(\Omega',g',K')$ with
+For each $ε>0$, there is one-ended exterior $(Ω',g',K')$ with
smooth compact boundary $B$ and:
-(i) $(\Omega',g',K')$ agrees with $(\Omega,g,K)$ in a neighborhood of
- $B$ and is supported on a subset of $\Omega$ differing from it by
+(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) $\theta_+^{g',K'}(B)<0$;
(iv) $K'$ is compactly supported;
-(v) $P'=0$ and $m'=E'=m+O(\varepsilon)$;
-(vi) $a_{g'}(B)\ge (1-\varepsilon)a_g(B)$.
+(v) $P'=0$ and $m'=E'=m+O(ε)$;
+(vi) $a_{g'}(B)\ge (1-ε)a_g(B)$.
@@ -628,19 +628,19 @@ 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
-$a_g(B)\le a_{g_s}(B)\le(1+s\|\phi\|_\infty)^4a_g(B)$. Choose $s>0$
-small enough that $(1+s\|\phi\|_\infty)^4\le 1+\varepsilon/2$. Write
+$a_g(B)\le a_{g_s}(B)\le(1+s\|\phi\|_∞)^4a_g(B)$. Choose $s>0$
+small enough that $(1+s\|\phi\|_∞)^4\le 1+ε/2$. Write
$E_s=E+sA_\phi$.
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 $\Lambda$ such that $\Lambda(E,P)=(m,0)$ with
-$m=\sqrt{E^2-|P|_\delta^2}$. On the asymptotic chart of [eq:decay],
-the change of coordinates $x\mapsto\Lambda x$ acts linearly on the end
+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
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 $\Lambda(E_s,P)=(m_s,0)$, where
-$m_s=\sqrt{E_s^2-|P|_\delta^2}$.
+$\theta_+(B)\le0$, 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
@@ -654,11 +654,11 @@ 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 $-4\pi$).
-Combining the three steps and letting $R\to\infty$ gives (iv) and (vi).
-Property (v) follows from $s\to0$: $E_s=E+O(\varepsilon)$ for small $s$, and
-$m'=m_s=m+O(\varepsilon)$; the two loss factors
-$(1+s\|\phi\|_\infty)^4$ and the $O(R^{-2})$ repair cost are absorbed
-into $\varepsilon$ by taking $s$ and $R^{-2}$ small enough. Property (ii)
+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
+$m'=m_s=m+O(ε)$; the two loss factors
+$(1+s\|\phi\|_∞)^4$ 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
end) and the compact support step (which acts on the end).
@@ -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 $\varepsilon>0$ 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
-$m'=m+O(\varepsilon)$ and $a_{g'}(B)\ge(1-\varepsilon)a_g(B)$.
+$m'=m+O(ε)$ 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
-$\partial X\setminus S_0$, and let $D\subset\operatorname{int}X$ be a
+$∂ X∖ S_0$, and let $D\subsetintX$ be a
closed set on which $Q$ vanishes outside a fixed neighborhood. For
-$c\in\mathbb{R}$ define the total region
+$c\inℝ$ define the total region
$$
-T_c(X,Q)=\bigcap\{E: E\text{ closed},\,D\subset E,\,\partial E
-\text{ smooth},\, H_{\partial E}+\operatorname{tr}_{\partial E}Q\le c\text{ on }\partial E\}.
+T_c(X,Q)=∩\{E: E closed,\,D\subset E,\,∂ E
+ smooth,\, H_{∂ E}+tr_{∂ E}Q\le c on ∂ E\}.
$$
@@ -715,8 +715,8 @@ $$
Lemma (Compact total regions). [lem:n3:compact-total]
-Suppose there is a smooth closed $F$ with $D\subset\operatorname{int}F$
-and $\Theta_Q(\partial F)<c$ for the outward normal. Then
+Suppose there is a smooth closed $F$ with $D\subsetintF$
+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.
@@ -729,9 +729,9 @@ 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,\infty)$ extends
+frontiers of the compact trapped subregions. Pasting outward cobordisms and an open $S^3\times[0,∞)$ extends
$X$ to a complete boundaryless data set. A smooth Morse height $q$ with
-$q=0$ on $\partial X$, increasing in an outward collar, and radial on the
+$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
positive distance inside $X$ by the maximum-principle argument applied to
$q$ on the region. The Andersson--Eichmair--Metzger smooth-frontier
@@ -755,7 +755,7 @@ 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\mapsto\operatorname{dist}(x_j,T_c(X,Q))$ has at most countably many jumps. Away
+$c\mapstodist(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.
@@ -770,22 +770,22 @@ The two-color exterior
Proposition (Two-color preparation). [prop:n3:two-color]
-Let $(\Omega,g,K)$ be strict data satisfying the hypotheses of
+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 $\Omega_0\subset\Omega$ with one
+closed connected exterior $Ω_0\subsetΩ$ 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 $\Omega_0$
-and $P_{B_0}=\operatorname{tr}_{B_0}K$,
+$c_b,c_w\in(-\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\ \text{on } B_{\rm black},
+H_{B_0}+P_{B_0}=c_b\ on B_{\rm black},
\qquad
-H_{B_0}-P_{B_0}=c_w\ \text{on } B_{\rm white}.
+H_{B_0}-P_{B_0}=c_w\ on B_{\rm white}.
$$
On each face there is a smooth disjoint collar $(B_0^j\times[0,1),s)$
-with $|\mathrm{d} s|>0$ and monotone expansion toward $\Omega_0$.
+with $|d s|>0$ and monotone expansion toward $Ω_0$.
@@ -796,14 +796,14 @@ 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 \(Y=X\setminus\operatorname{int}T_{c_b}(X,K)\).
+$T_c(X,K)$, 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
$T_c(Y,-K)$ gives a nonempty closed region. Let
-$\Omega_0$ be the closure of the exterior component of
-$X\setminus(T_{c_b}(X,K)\cup T_{c_w}(Y,-K))$. Its retained $c_b$-faces
+$Ω_0$ 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
are black and its retained $c_w$-faces are white, with the stated
-expansions for the normal into $\Omega_0$ (the signs follow from the
+expansions for the normal into $Ω_0$ (the signs follow from the
orientations of the total region and its complement). The stable
collars are the Andersson--Eichmair--Metzger stability directions
shrunk to small amplitude.
@@ -812,7 +812,7 @@ shrunk to small amplitude.
-The coupled elliptic comparison system [sec:elliptic]. Fix a two-color exterior $\Omega_0$ of the [prop:n3:two-color] proposition.
+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
negative boundary expansions; and its far end is asymptotically flat at
infinity.
@@ -822,65 +822,65 @@ Variables and the equations
-Fix a smooth boundary height $h_*:B_0\to\mathbb{R}$ with
+Fix a smooth boundary height $h_*:B_0\toℝ$ with
$h_*=b_-<0$ on black and $h_*=b_+>0$ on white, extended smoothly to
-$\Omega_0$ with $\partial_\nu h_*<0$ on $B_0$ and $h_*\to0$ at
-infinity. Fix $\eta_n=e^{-\varepsilon n}>0$.
+$Ω_0$ with $∂_\nu h_*<0$ on $B_0$ and $h_*\to0$ at
+infinity. Fix $\eta_n=e^{-ε n}>0$.
-Let $l:\mathbb{R}\to(0,\infty)$ be smooth, with $l(t)=e^{nt}$ for $t\le0$ and
+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
-$f:\Omega_0\to\mathbb{R}$ and $t:\Omega_0\to\mathbb{R}$, define
-$$\begin{aligned}
+$f:Ω_0\toℝ$ and $t:Ω_0\toℝ$, define
+$$
&\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=\frac{l\,\sigma}{\sqrt D},\quad a=|w|,\quad v=a^2,\quad
-\chi=\frac{3d}{3+pv},\quad
-\mathcal{A}_d=I-w\otimes w,\quad
-\mathcal{A}_\chi=I-\frac{3+p}{3+pv}\,w\otimes w.
-\end{aligned}$$
+&w=(l\,\sigma)/(\sqrt D),\quad a=|w|,\quad v=a^2,\quad
+\chi=(3d)/(3+pv),\quad
+A_d=I-w ⊗ w,\quad
+A_\chi=I-(3+p)/(3+pv)\,w ⊗ w.
+$$
Set $L_0=e^{2t}l$ and $u=L_0\sqrt D$.
-The comparison metric on $\Omega_R=\Omega_0\cap\{r<R\}$ is [eq:comparison-metric]
+The comparison metric on $Ω_R=Ω_0\cap\{r<R\}$ is [eq:comparison-metric]
$$
-\bar g_{ij}=\frac{1}{d}\bigl(g_{ij}+l(t)^2\partial_if\,\partial_jf\bigr),
+\bar g_{ij}=(1)/(d)(g_{ij}+l(t)^2∂_if\,∂_jf),
$$
-the graph metric over the base $(\Omega_R,g)$ with height $f$ and warp
+the graph metric over the base $(Ω_R,g)$ with height $f$ and warp
$l$, rescaled by the lapse $1/d$.
The unknowns $(f,t)$ satisfy the coupled system [eq:coupled-system]
$$
-\begin{cases}
-\operatorname{Div}_g\bigl(3u\,\mathcal{A}_\chi\,\nabla t\bigr)=u\,\Xi &\text{in }\Omega_R,\$$
+
+Div_g(3u\,A_\chi\,\nabla t)=u\,\Xi &in Ω_R,\$$
1mm]
-l\,\Delta_g f+\text{(bulk terms)}=\eta_n\,\Xi_2&\text{in }\Omega_R,\$$
+l\,\Delta_g f+(bulk terms)=\eta_n\,\Xi_2&in Ω_R,\$$
1mm]
-t=h_*/\eta_n\text{ on }B_0,\quad t=0\text{ on }S_R,
-\end{cases}
+t=h_*/\eta_n on B_0,\quad t=0 on S_R,
+
$$
where $\Xi,\Xi_2$ are smooth functions of $(f,t)$ and their first
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 $\Omega_0$.
+boundary expansions of $Ω_0$.
Lemma (The Hessian square is coercive). [lem:n3:hessian-square]
-Put $H^f=\operatorname{Hess}_gf$ and let
-$\mathcal T=\tfrac12\{(\operatorname{tr}_{\mathcal{A}_d}H^f)^2-|H^f|_{\mathcal{A}_d\otimes\mathcal{A}_d}^2\}$ be
+Put $H^f=Hess_gf$ and let
+$\mathcal T=\tfrac12\{(tr_{A_d}H^f)^2-|H^f|_{A_d ⊗ A_d}^2\}$ be
the second-form square of the comparison graph. Then there is $c_0(n)>0$
such that [eq:hessian-square-coercive]
$$
-\mathcal T\ge c_0\bigl((\operatorname{tr}_gH^f)^2+|\nabla w|_{\mathcal{A}_\chi}^2\bigr)
+\mathcal T\ge c_0((tr_gH^f)^2+|\nabla w|_{A_\chi}^2)
$$
-and $\mathcal T\le C(n)(|\nabla f|^2+|\operatorname{Hess}_gf|^2)$.
+and $\mathcal T\le C(n)(|\nabla f|^2+|Hess_gf|^2)$.
Proof.
-Write $\mathcal{A}_d$ and $\mathcal{A}_\chi$ in a frame where $\mathcal{A}_d=\diag(1,1,1,d)$ and
-$\mathcal{A}_\chi=\diag(1,1,1,\chi)$. 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
$$
\mathcal T=\tfrac12|H^f_\perp|^2+\chi\,(H_{ee}^f)^2,
@@ -899,12 +899,12 @@ immediate from the definition.
Lemma (Boundary identity). [lem:n3:boundary-identity]
On a face $B_0^j$ of the boundary $B_0$, with $H$ the mean curvature
-into $\Omega_0$, $P_{B_0^j}=\operatorname{tr}_{B_0^j}K$, and $w_\nu=w(\nu)$, the
+into $Ω_0$, $P_{B_0^j}=tr_{B_0^j}K$, and $w_\nu=w(\nu)$, the
first normal variation of $t$ satisfies [eq:boundary-identity]
$$
-V_\nu/u=3d\,\partial_\nu t+w_\nu(F-P_{B_0^j}),
+V_\nu/u=3d\,∂_\nu t+w_\nu(F-P_{B_0^j}),
$$
-where $F=\operatorname{tr}_gK$.
+where $F=tr_gK$.
@@ -915,13 +915,13 @@ The metric and divergence formulas give, for the covariant trace
$V_\nu$ of the boundary flux of $t$,
$$
-V_\nu/u=3d\,\partial_\nu t+w_\nu(F-P_{B_0^j}),
+V_\nu/u=3d\,∂_\nu t+w_\nu(F-P_{B_0^j}),
$$
the first term from the direct variation of $t$ across the boundary and
the second from the tangential trace of $K$. Using the boundary expansion
$H_{B_0^j}+P_{B_0^j}=c_j$, this becomes a strict negative first-variation,
-compatible with the Dirichlet data on $\partial\Omega_R$.
+compatible with the Dirichlet data on $∂Ω_R$.
@@ -933,13 +933,13 @@ A priori bounds
Proposition (A priori bounds). [prop:n3:apriori]
-For every $n$ there is a constant $C(n)<\infty$ such that every smooth
+For every $n$ there is a constant $C(n)<∞$ such that every smooth
solution $(f,t)$ of [eq:coupled-system] satisfies
$$
--\varepsilon\le t\le Z\le C(n),\quad
-|f|+|\nabla f|+|\operatorname{Hess}_gf|\le C(n)
-\quad\text{on }\bar\Omega_R.
+-ε\le t\le Z\le C(n),\quad
+|f|+|\nabla f|+|Hess_gf|\le C(n)
+\quadon \barΩ_R.
$$
@@ -952,9 +952,9 @@ $C(n)$ may also depend on the truncation radius in a way that will be
declared explicitly. The parameter comparisons
$$
- \ell=e^{-\varepsilon n},\qquad \tau=\ell^{3/2},\qquad
+ \ell=e^{-ε n},\qquad \tau=\ell^{3/2},\qquad
\ell\le l\le e,\qquad 0\le p\le n,\qquad
- \frac{4d}{4+n}\le\chi\le d
+ (4d)/(4+n)\le\chi\le d
$$
enter through the small factor $\Pi_n\tau/\ell\to0$, which absorbs only
@@ -962,7 +962,7 @@ polynomial losses.
Lemma (Uniform separation). [lem:n3:apriori:separation]
-Let $A\subset\bar\Omega$ be compact. In the first two homotopy stages
+Let $A\subset\barΩ$ be compact. In the first two homotopy stages
one has the following. If $A$ avoids the full black total region
$\mathcal B_{c_b}$ at threshold $c_b$, then $h\ge b_-+\eta_A$ on $A$
for some $\eta_A>0$, uniformly in $n$ large, every sufficiently large
@@ -974,13 +974,13 @@ $\mathcal W_{c_w}$, then $h\le b_+-\eta_A$.
Proof.
-Take an arbitrary sequence $n_i\to\infty$, $R_i\to\infty$ of admissible
+Take an arbitrary sequence $n_i\to∞$, $R_i\to∞$ of admissible
stage-one or stage-two solutions. The elementary height and end bounds
from the preparation give
$|h_i|\le C_0$ and $|h_i|\le C(r^{-\gamma}-R_i^{-\gamma})$ on the end,
for some $\gamma\in(3/4,1)$. Extend $h_i$ across a white face by the
constant $b_+$. Define the lower relaxed limit
-$\underline h(x)=\liminf_{y\to x} h_i(y)$ over such sequences.
+$\underline h(x)=liminf_{y\to x} h_i(y)$ over such sequences.
Fix $b_-<k'<\min\{0,b_-+\eta\}$ with $\eta$ the low-height sign range
of the add-on $D_0$. Suppose a smooth $\phi$ touches $\underline h$
@@ -993,7 +993,7 @@ constant extension. At the contact,
$$
\nabla f_i=\tau^{-1}\nabla\phi,\quad
- d_i\le\frac{\tau^2}{\ell^2|d\phi(x_i)|^2}\to0,\quad
+ d_i\le(\tau^2)/(\ell^2|d\phi(x_i)|^2)\to0,\quad
a_i\to1,\quad \chi_i\to0.
$$
@@ -1001,16 +1001,16 @@ Positivity of $A_{\chi_i}$ and the Hessian inequality at a lower
contact give
$$
- \operatorname{tr}_{A_{\chi_i}}K+
- \frac{l_i}{\tau\sqrt{D_i}}\operatorname{tr}_{A_{\chi_i}}\operatorname{Hess}\phi
+ tr_{A_{\chi_i}}K+
+ \frac{l_i}{\tau√(D_i)}tr_{A_{\chi_i}}Hess\phi
\le F_i(x_i).
$$
-Since $D_0\le0$ at these low values, $\limsup F_i(x_i)\le k'+C_b$, so
+Since $D_0\le0$ at these low values, $limsup F_i(x_i)\le k'+C_b$, so
$$
- \frac{\operatorname{tr}_{(d\phi)^\perp}\operatorname{Hess}\phi}{|d\phi|}
- +\operatorname{tr}_{(d\phi)^\perp}K\le k'+C_b.
+ \frac{tr_{(d\phi)^\perp}Hess\phi}{|d\phi|}
+ +tr_{(d\phi)^\perp}K\le k'+C_b.
$$
This is the oriented level expansion of $\phi$ towards increasing
@@ -1035,11 +1035,11 @@ Composition with the increasing $\psi_j$ preserves the oriented level
expansion:
$$
- \operatorname{Hess}(\theta\circ\varphi)=\theta'\operatorname{Hess}\varphi
- +\theta''d\varphi\otimes d\varphi,\qquad
- \frac{\operatorname{tr}_{(d\varphi)^\perp}\operatorname{Hess}(\theta\circ\varphi)}
+ 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{\operatorname{tr}_{(d\varphi)^\perp}\operatorname{Hess}\varphi}{|d\varphi|}.
+ =\frac{tr_{(d\varphi)^\perp}Hess\varphi}{|d\varphi|}.
$$
So $\psi_j^{-1}(\varphi+c_j)$ is a smooth lower test of
@@ -1063,10 +1063,10 @@ threshold $c''>k'+C_b$ close enough to $c_b$, hence in
$\mathcal B_{c''}$.
Hausdorff right continuity of $c\mapsto\mathcal B_c$ at $c_b$ supplies
-$\delta>0$ with $\mathcal B_{c_b+\delta}\cap A=\varnothing$; shrink
+$\delta>0$ with $\mathcal B_{c_b+\delta}\cap A=∅$; shrink
$\delta$ so the enlarged threshold is still negative. Choose
$k,k',c''$ with $b_-<k<k'<k'+C_b<c''<c_b+\delta$. Then
-$E_k\cap A=\varnothing$ for every sequence in the class. A failing
+$E_k\cap A=∅$ for every sequence in the class. A failing
uniform estimate would produce $x_i\in A$ with $h_i(x_i)\le k$ and a
subsequence $x_i\to x\in A$, so $\underline h(x)\le k$, contradiction.
This gives $h\ge b_-+\eta_A$ on $A$. Replacing $(h,K,C)$ by
@@ -1081,13 +1081,13 @@ There are fixed $c,C_*>0$ such that, for large $n$ and every allowed
truncation, the first two stages satisfy
$$
- c/\tau\le\partial_\nu f\le C_*/\tau
- \ \text{on black faces},\qquad
- c/\tau\le-\partial_\nu f\le C_*/\tau
- \ \text{on white faces},
+ c/\tau\le∂_\nu f\le C_*/\tau
+ \ on black faces,\qquad
+ c/\tau\le-∂_\nu f\le C_*/\tau
+ \ on white faces,
$$
-and the third stage satisfies $|\partial_\nu f|\le C_*/\tau$.
+and the third stage satisfies $|∂_\nu f|\le C_*/\tau$.
@@ -1099,7 +1099,7 @@ $N=\nabla s/|ds|$. At a comparison contact with $h_0=b_-+\psi(s)$
($\psi'>0$), the shared gradient gives
$$
- d\le C\tau^2/\bigl(\ell^2(\psi')^2\bigr),\quad
+ d\le C\tau^2/(\ell^2(\psi')^2),\quad
1-a\le d,\quad n\chi\ge c_1 d\quad(n\ge4).
$$
@@ -1108,11 +1108,11 @@ The trace operator evaluated on this test is
$$
P_s+aH_s+\chi K(N,N)+
- a\chi\!\left(\frac{\operatorname{Hess} s(N,N)}{|ds|}
- +|ds|\frac{\psi''}{\psi'}\right),
+ a\chi\!((Hess s(N,N))/(|ds|)
+ +|ds|(\psi'')/(\psi')),
$$
-where $P_s=\operatorname{tr} K$ on the leaf tangents and $H_s$ its mean curvature.
+where $P_s=tr K$ on the leaf tangents and $H_s$ its mean curvature.
For a negative slope the terms with prefactor $a$ flip sign. On a
black collar $H_s+P_s\ge c_b$ and $C=C_b-k_Bs$. Choose small positive
slopes with $\psi(0)=0$ and $\psi(s)\le k_Bs/2$. A fixed short collar
@@ -1128,7 +1128,7 @@ Pick a smooth $\beta_n(s)$ equal to $An$ for $0\le s\le1/n$, supported
in $[0,2/n]$, with $\int\beta_n\le2A$. Set
$$
- \psi'(s)=m\exp\!\left(\int_0^s\beta_n(q)\mathrm{d} q\right).
+ \psi'(s)=m\exp\!(\int_0^s\beta_n(q)d q).
$$
Fix $A$ so the last term dominates $Cd$ on $s\le1/n$, using
@@ -1139,7 +1139,7 @@ This gives a strict lower barrier with slopes in $[me^{-A},me^{A}]$
independent of $n$.
An upper barrier from $b_-$ uses
-$\psi'(s)=M\exp(-\int_0^s\beta_n(q)\mathrm{d} q)$. The leaf expansion differs
+$\psi'(s)=M\exp(-\int_0^s\beta_n(q)d q)$. The leaf expansion differs
from its boundary value by $O(s)$; take $M$ so large that
$\psi(s)\ge C_2s$ dominates this, the offset variation, and the outer
height bound. The logarithmic-slope term is negative: on $s\le1/n$ it
@@ -1168,13 +1168,13 @@ the first two stages, for nonnegative smooth $\varphi$ supported in
$\mathcal C$,
$$
- \int_B L\varphi\mathrm{d} A_g\le
- \Pi_n\int_{\mathcal C}u\bigl[(1+\sqrt{\mathcal T})\varphi
- +|d\varphi|_{A_\chi}\bigr]\mathrm{d} V_g,
+ \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\varphi\mathrm{d} A_g\le
- \Pi_n\int_{\mathcal C}\sqrt D\bigl[(1+\sqrt{\mathcal T})\varphi
- +|d\varphi|_{A_\chi}\bigr]\mathrm{d} V_g.
+ \int_B\varphid A_g\le
+ \Pi_n\int_{\mathcal C}\sqrt D[(1+√(\mathcal T))\varphi
+ +|d\varphi|_{A_\chi}]d V_g.
$$
The estimates extend to arbitrary $\varphi$ by a fixed collar cutoff
@@ -1190,26 +1190,26 @@ $W=l\nabla f/D=\sqrt d\,a e_f$. Then $|W|_{\bar g}=a\le1$, and for
every covector $\xi$,
$$
- |\xi(W)|\le|\xi|_{A_d}\le\sqrt{1+n/4}\,|\xi|_{A_\chi}.
+ |\xi(W)|\le|\xi|_{A_d}\le√(1+n/4)\,|\xi|_{A_\chi}.
$$
Differentiation gives
$$
- \frac{\operatorname{Div}(uW)}u=\sqrt d\bigl[\operatorname{tr}_{A_d}H^f+(dp+p_L)a\partial_{e_f}t\bigr],
+ \frac{Div(uW)}u=\sqrt d[tr_{A_d}H^f+(dp+p_L)a∂_{e_f}t],
\qquad
- \frac{\operatorname{Div}(\sqrt D\,W)}{\sqrt D}=\sqrt d\bigl[\operatorname{tr}_{A_d}H^f+dp\,a\partial_{e_f}t\bigr].
+ (Div(\sqrt D\,W))/(\sqrt D)=\sqrt d[tr_{A_d}H^f+dp\,a∂_{e_f}t].
$$
-Coercivity bounds both right sides by $\Pi_n(1+\sqrt{\mathcal T})$: the
+Coercivity bounds both right sides by $\Pi_n(1+√(\mathcal T))$: the
axial Hessian term has coefficient $d^{3/2}$, the transverse terms are
-traces, and the gradient term is $O(\sqrt d |\partial_{e_f}t|)$.
+traces, and the gradient term is $O(\sqrt d |∂_{e_f}t|)$.
-Let $\varepsilon_B=1$ on a black collar and $-1$ on a white one. The
-signed slopes give $w_\nu=\varepsilon_B a$ on $B$ with $a\ge1/2$ for
+Let $ε_B=1$ on a black collar and $-1$ on a white one. The
+signed slopes give $w_\nu=ε_B a$ on $B$ with $a\ge1/2$ for
large $n$, hence
-$uW_\nu=\varepsilon_B La$ and $\sqrt D\,W_\nu=\varepsilon_B a$. Apply
-the divergence theorem to $-\varepsilon_B uW\varphi$ with a fixed
+$uW_\nu=ε_B La$ and $\sqrt D\,W_\nu=ε_B a$. Apply
+the divergence theorem to $-ε_B uW\varphi$ with a fixed
collar cutoff. Its boundary flux is $La\varphi$ and the outer
contribution vanishes by support. The gradient bound and divergence
formula give the first inequality; replacing $u$ by $\sqrt D$ gives the
@@ -1225,15 +1225,15 @@ all $R$ and $\lambda\in[0,1]$,
$$
\Xi\ge\delta_0\mathcal T+\rho+\tfrac12\delta_0\tau a\sigma
- \quad\text{on }\{Z>M_0\}.
+ \quadon \{Z>M_0\}.
$$
With $k_0$ a smooth cutoff supported on $[M_0,M_0+1]$ equal to one
-on $[M_0+1,\infty)$,
+on $[M_0+1,∞)$,
$$
- \int_\Omega u k_0(\mathcal T+\rho+\tau a\sigma)\mathrm{d} V_g
- +\int_\Omega u k_0'\,|dZ|_{A_\chi}^2\mathrm{d} V_g\le C(n).
+ \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).
$$
@@ -1242,17 +1242,17 @@ $$
Proof.
-Outside $\operatorname{supp} m_0$, $\Xi=\delta_0\mathcal T+\rho+\delta_0\tau a\sigma$
-is clearly $\ge$ the right side. On $\operatorname{supp} m_0$ one has
-$-\varepsilon\le t\le -\varepsilon+1/n$ and $l\asymp\ell$; the identity
-$D=e^{8(Z-t)}$ makes $a\sigma\to\infty$ uniformly as $Z\to\infty$,
+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.
-Test $\operatorname{Div} V_\lambda=u\Xi$ by $k_0(Z)$. The outer boundary term
+Test $Div V_\lambda=u\Xi$ by $k_0(Z)$. The outer boundary term
vanishes by $Z=0$. The drift cross term is bounded by
$$
- uk_0'|\langle dZ,K(w,\cdot)\rangle_{A_\chi}|
+ uk_0'|< dZ,K(w,\cdot)>_{A_\chi}|
\le 2u k_0'|dZ|_{A_\chi}^2+Cuk_0'|K|^2,
$$
@@ -1263,23 +1263,23 @@ has $|A_\chi K(w,\cdot)|_\nu\le C\chi\le Cd$. Thus
$$
|(V_\lambda)_\nu|\le Cud=CL\sqrt d\le C\frac\tau\ell L
- \ \text{on }B.
+ \ on B.
$$
Integrating,
$$
- \int_\Omega uk_0\bigl(\delta_0\mathcal T+\rho
- +\tfrac12\delta_0\tau a\sigma\bigr)\mathrm{d} V_g
- +2\int_\Omega u k_0'|dZ|_{A_\chi}^2\mathrm{d} V_g
- \le C(n)+C\frac\tau\ell\int_B Lk_0\mathrm{d} A_g.
+ \int_Ω uk_0(\delta_0\mathcal T+\rho
+ +\tfrac12\delta_0\tau a\sigma)d V_g
+ +2\int_Ω u k_0'|dZ|_{A_\chi}^2d V_g
+ \le C(n)+C\frac\tau\ell\int_B Lk_0d A_g.
$$
Bound the last term using the weighted trace with the fixed collar
cutoff: it costs
-$\Pi_n\frac\tau\ell\int_{\mathcal C}u(k_0(1+\sqrt{\mathcal T})
-+k_0'|dZ|_{A_\chi})\mathrm{d} V_g$. On the fixed collars $\rho\ge\rho_0>0$,
-so $(1+\sqrt{\mathcal T})\le C(\rho+\delta_0\mathcal T)$ there, and
+$\Pi_n\frac\tau\ell\int_{\mathcal C}u(k_0(1+√(\mathcal T))
++k_0'|dZ|_{A_\chi})d V_g$. 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.
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
@@ -1293,12 +1293,12 @@ 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 $\sqrt{1+\sigma^2}\mathrm{d} V_g$
-comparable to $\sqrt D\mathrm{d} V_g$, and on every fixed compact base patch
+$\Gamma_f$ 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
$Q$,
$$
- \int_{\Gamma_f\cap\pi^{-1}Q}e^{2Z}\mathrm{d}\Gamma\le C_Q(n),
+ \int_{\Gamma_f\cap\pi^{-1}Q}e^{2Z}d\Gamma\le C_Q(n),
$$
using $e^{2Z}=e^{2t}D^{1/4}$, $D^{1/4}\le C(n)(1+\tau a\sigma)$, and
@@ -1309,7 +1309,7 @@ uniform bound $Z\le C(n)$ 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 $\operatorname{Div}(4uA_\chi\nabla Z)=4u\operatorname{tr}_{A_\chi}\operatorname{Hess} Z\le0$,
+above $M_0$ has $Div(4uA_\chi\nabla Z)=4utr_{A_\chi}Hess Z\le0$,
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
@@ -1319,10 +1319,10 @@ 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
-$-\varepsilon\le t\le Z$, and the upper $Z$ bound yields
+$-ε\le t\le Z$, and the upper $Z$ bound yields
$$
- \sigma^2\le\ell^{-2}\bigl(e^{8(C(n)+\varepsilon)}-1\bigr),
+ \sigma^2\le\ell^{-2}(e^{8(C(n)+ε)}-1),
$$
which is the fixed-$n$ gradient bound. The height bounds
@@ -1337,9 +1337,9 @@ Lemma (Graph Sobolev). [lem:n3:graph-sobolev]
For every compactly supported smooth $\varphi$ on the graph
$\Gamma_f$, [eq:graph-sobolev]
$$
-\Bigl(\int_{\Gamma_f}|\varphi|^4\Bigr)^{1/2}
-\le C(n)\int_{\Gamma_f}\bigl(|\nabla_{\Gamma_f}\varphi|^2
-+(1+\mathcal T)\varphi^2\bigr).
+(\int_{\Gamma_f}|\varphi|^4)^{1/2}
+\le C(n)\int_{\Gamma_f}(|\nabla_{\Gamma_f}\varphi|^2
++(1+\mathcal T)\varphi^2).
$$
@@ -1347,28 +1347,28 @@ $$
Proof.
-Over a fixed base patch $Q\Subset\Omega_R$ the ordinary product graph
-$\Gamma_f$ carries the metric induced from $(\Omega_R,\bar g)\times\mathbb{R}$,
+Over a fixed base patch $Q\SubsetΩ_R$ the ordinary product graph
+$\Gamma_f$ carries the metric induced from $(Ω_R,\bar g)\timesℝ$,
has uniformly bounded geometry (since $\bar g$ does and $f$ is smooth),
-and has finite three-volume $V_Q=\operatorname{Vol}_{\Gamma_f}(\pi^{-1}Q)$.
+and has finite three-volume $V_Q=Vol_{\Gamma_f}(\pi^{-1}Q)$.
By the Nash embedding theorem (Nash 1956, Theorem 2), a fixed compact enlargement
-of $\Gamma_f$ embeds isometrically in some $\mathbb{R}^N$.
+of $\Gamma_f$ embeds isometrically in some $ℝ^N$.
./pdf/nash-1956.pdf
The standard
submanifold Sobolev embedding for compactly supported functions then
gives, in dimension $3$,
$$
- \Bigl(\int_{\Gamma_f}|\varphi|^{6}\Bigr)^{1/3}
+ (\int_{\Gamma_f}|\varphi|^{6})^{1/3}
\le C\int_{\Gamma_f}|\nabla_{\Gamma_f}\varphi|^{2},
$$
with $C$ depending only on the immersion geometry, hence only on $n$.
-H\"older's inequality together with $V_Q<\infty$ yields
+H\"older's inequality together with $V_Q<∞$ yields
$$
\int_{\Gamma_f}|\varphi|^{4}
- \le V_Q^{1/3}\Bigl(\int_{\Gamma_f}|\varphi|^6\Bigr)^{2/3},
+ \le V_Q^{1/3}(\int_{\Gamma_f}|\varphi|^6)^{2/3},
$$
and substituting the Sobolev bound closes the estimate on $\pi^{-1}Q$.
@@ -1389,9 +1389,9 @@ Axial regularity
Lemma (Axial gradient estimate). [lem:n3:axial]
-Let $z$ be smooth on a coordinate patch $U'$ with $A_\chi:\operatorname{Hess} z=G$,
-$A_\chi=I-(1-\chi)e\otimes e$, $e=\nabla z/|\nabla z|$, $0<\chi_0\le\chi\le1$.
-Suppose $|G|_\infty\le Q$ and $|\nabla z|\le L$. Then there are
+Let $z$ be smooth on a coordinate patch $U'$ with $A_\chi:Hess z=G$,
+$A_\chi=I-(1-\chi)e ⊗ e$, $e=\nabla z/|\nabla z|$, $0<\chi_0\le\chi\le1$.
+Suppose $|G|_∞\le Q$ and $|\nabla z|\le L$. Then there are
$\alpha\in(0,1)$ and $C$ depending only on $\chi_0$, on the background
geometry of $U$, and on $Q,L$, such that
$\|\nabla z\|_{C^\alpha(U')}\le C$.
@@ -1404,8 +1404,8 @@ Proof.
After shrinking $U'$ and rescaling space, normal coordinates make the
metric Euclidean at the center with $|g-\delta|$ and its first
derivatives small; the constants below depend only on $\chi_0$, on
-$U'$, and on $Q,L$. Put $H=\operatorname{Hess} z$, $P=\nabla z$, and $e=P/|P|$ on
-$\{|P|>0\}$. Since $A_\chi=I-(1-\chi)e\otimes e$ and $H(e,P)=|P|H(e,e)$,
+$U'$, and on $Q,L$. Put $H=Hess z$, $P=\nabla z$, and $e=P/|P|$ on
+$\{|P|>0\}$. Since $A_\chi=I-(1-\chi)e ⊗ e$ and $H(e,P)=|P|H(e,e)$,
$$
A_\chi\nabla\frac{|P|^{2}}{2}-G P=(H-\Delta z\,g)P,
@@ -1416,8 +1416,8 @@ with $G$ the right side of $A_\chi:H=G$, valid also at $|P|=0$.
Taking the divergence of the right side gives
$$
- \operatorname{Div}\Bigl(A_\chi\nabla\frac{|P|^{2}}{2}-G P\Bigr)
- =|H|^{2}-(\Delta z)^{2}+\operatorname{Ric}(P,P).
+ Div(A_\chi\nabla\frac{|P|^{2}}{2}-G P)
+ =|H|^{2}-(\Delta z)^{2}+Ric(P,P).
[n3:axial:bochner-flux]
$$
@@ -1439,7 +1439,7 @@ The two identities, together with the smallness of $\delta$ controlling
the metric oscillation, the base curvature, and $|G|$, give in the weak
sense
$$
- \operatorname{Div}(A_\chi\nabla s+2G\,P)\le-2c_0|H|^{2}+C\delta.
+ Div(A_\chi\nabla s+2G\,P)\le-2c_0|H|^{2}+C\delta.
[n3:axial:s-supersolution]
$$
The flux $A_\chi\nabla s+2GP$ is bounded and measurable, and the
@@ -1459,7 +1459,7 @@ $$
$$
Since $0\le s\le1$ and $Ds$ is locally $L^{2}$-bounded, $Ds\to0$ in
-$L^{1}_{\mathrm{loc}}$. Testing [n3:axial:bochner-flux] with a
+$L^{1}_{loc}$. 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
@@ -1471,10 +1471,10 @@ nonzero unit slope. Fix this slope as $e_0$: the equation for the
limit is
$$
- \Delta_{e_0^{\perp}}Y_0+\chi(x)\,\partial_{e_0e_0}Y_0=0,
+ \Delta_{e_0^{\perp}}Y_0+\chi(x)\,∂_{e_0e_0}Y_0=0,
$$
-whose axial derivative $v=\partial_{e_0}Y_0\in W^{1,2}_{\mathrm{loc}}$
+whose axial derivative $v=∂_{e_0}Y_0\in W^{1,2}_{loc}$
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
@@ -1488,20 +1488,20 @@ produces, after finitely many drops plus successive affine steps,
$$
\sup_{B_r(x)}|z-L_{x,r}|
- \le C\bigl(Q+L\bigr)\,r^{1+\alpha},
+ \le C(Q+L)\,r^{1+\alpha},
$$
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
Cordes estimate applied to $z-L_{x,2r}$, plus the affine error, closes
-the Morrey bound on $\operatorname{Hess} z$.
+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
carries a bounded normal-flux jump which is itself divergence of a
-bounded field $\mathbf{1}_{\{x_{4}>0\}}\partial_{4}$. 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
@@ -1521,7 +1521,7 @@ Existence
Proposition (Existence). [prop:n3:existence]
For all sufficiently large $n$ and all sufficiently large $R$, the
system [eq:coupled-system] has a smooth solution $(f,t)$ on
-$\Omega_R$, satisfying the bound of the [prop:n3:apriori] proposition.
+$Ω_R$, satisfying the bound of the [prop:n3:apriori] proposition.
@@ -1532,25 +1532,25 @@ Throughout, fix $(n,R)$; only $(n,R)$ enters the constants below. The
coupled system [eq:coupled-system] has the schematic form
$$
-\begin{cases}
- \operatorname{Div}_g(3uA_\chi\nabla Z)=u\,\Xi(f,Z,Df,DZ), &
- \text{on }\Omega_R,\\
+
+ Div_g(3uA_\chi\nabla Z)=u\,\Xi(f,Z,Df,DZ), &
+ on Ω_R,\\
l\Delta_g f+\Psi_\chi(f,Df)= \eta_n\,\Xi_2(f,Z,Df,DZ), &
- \text{on }\Omega_R,\\
+ on Ω_R,\\
h=b,\quad (V_\lambda)_\nu/u=\mathcal B, &
- \text{on }B_0,\\
- f=Z=0, & \text{on }S_R.
-\end{cases}
+ on B_0,\\
+ f=Z=0, & on S_R.
+
$$
The leading coefficients $3uA_\chi$ and $l\,g^{-1}$ are uniformly
-elliptic on $\bar\Omega_R$ with constants depending only on $(n,R)$;
+elliptic on $\barΩ_R$ 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
increasing in its zero-order argument.
Four-stage homotopy. Interpolate linearly in four consecutive
-parameters $\sigma,\zeta,\upsilon,\omega\in[0,1]$ from the target
+parameters $\sigma,\zeta,υ,\omega\in[0,1]$ from the target
system to a linearised reference:
(i) $K_\sigma=(1-\sigma)K_0+\sigma K$, with $K_0$ a compactly supported
tensor whose two-color expansions are exact constants;
@@ -1566,7 +1566,7 @@ range of the [prop:n3:apriori] proposition.
Fredholm and index zero. The linearization of the schematic
system at any admissible $(f,Z)$ is a pair of scalar operators on
-$\Omega_R$. The $Z$-operator is divergence-form with a mixed
+$Ω_R$. 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 $C^{0,\alpha}$
coefficients. The classical mixed-boundary elliptic theory of Gilbarg and Trudinger
@@ -1576,15 +1576,15 @@ operator of index zero together with the Schauder estimate
$$
\|(f,Z)\|_{C^{2,\alpha}}
- \le C\bigl(\|F\|_{C^{0,\alpha}}+\|B\|_{C^{1,\alpha}}
- +\|\Psi\|_{C^{0,\alpha}}+\|(f,Z)\|_{C^0}\bigr).
+ \le C(\|F\|_{C^{0,\alpha}}+\|B\|_{C^{1,\alpha}}
+ +\|\Psi\|_{C^{0,\alpha}}+\|(f,Z)\|_{C^0}).
$$
Trivial kernel. Suppose $(f,Z)$ lies in the kernel with
$\|(f,Z)\|_{C^0}>0$. If $Z$ has a positive interior maximum, the
linearized $Z$-equation has no zero-order term and its source
-vanishes there, so by ellipticity $\operatorname{Div}(a\nabla Z)\le0$ forces $Z$ to
+vanishes there, so by ellipticity $Div(a\nabla Z)\le0$ 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
@@ -1594,7 +1594,7 @@ zero-order dependence rules out both signs. Hence the linearization
is invertible.
Continuation by degree. At the reference
-$(\sigma,\zeta,\upsilon,\omega)=(0,0,0,0)$ 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
@@ -1614,7 +1614,7 @@ system has a solution.
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 $C^\infty$ on $\bar\Omega_R$ because coefficients are smooth
+solution $C^∞$ on $\barΩ_R$ because coefficients are smooth
on the bounded closed range.
@@ -1627,16 +1627,16 @@ The comparison metric
Proposition (Comparison output). [prop:n3:comparison]
-Let $(f,t)$ be a solution of [eq:coupled-system] on $\Omega_R$.
+Let $(f,t)$ be a solution of [eq:coupled-system] on $Ω_R$.
Then the comparison metric $\bar g$ of [eq:comparison-metric]
satisfies
$$
-R_{\bar g}\ge0\quad\text{on }\Omega_R,
+R_{\bar g}\ge0\quadon Ω_R,
\qquad
-\operatorname{Area}_{\bar g}(\partial\Omega_R)\ge(1-\varepsilon)a_g(B),
+Area_{\bar g}(∂Ω_R)\ge(1-ε)a_g(B),
\qquad
-m_{\rm ADM}(\bar g)\le m+\varepsilon.
+m_{\rm ADM}(\bar g)\le m+ε.
$$
@@ -1645,15 +1645,15 @@ $$
Proof.
-For a warped product $\bar g=e^{2t}(g+l(t)^2\mathrm{d} f\otimes\mathrm{d} f)$ with
-base metric $g$ on $\Omega_R$, lapse $e^{2t}$, and warp factor $l(t)$,
-the traced Gauss identity for the graph $\Gamma_f$ over $(\Omega_R,g)$
+For a warped product $\bar g=e^{2t}(g+l(t)^2d f ⊗ d f)$ with
+base metric $g$ on $Ω_R$, lapse $e^{2t}$, and warp factor $l(t)$,
+the traced Gauss identity for the graph $\Gamma_f$ over $(Ω_R,g)$
combined with the conformal-change formula in dimension three gives
-$$\begin{aligned}
-\tfrac12e^{2t}\mathrm{S}_{\bar g}
-={}&\tfrac12\mathrm{S}_g-v\operatorname{Ric}_g(e,e)+\mathcal T(H^f)\\
-&-3d\operatorname{Hess}_g t(e,e) -(3+pv)\operatorname{tr}_\perp\operatorname{Hess}_g t +O(\eta_n^2),
-\end{aligned}$$
+$$
+\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),
+$$
where $e$ is the unit vector of $\sigma=\nabla f$ (the $d$-term drops
by homogeneity at $\sigma=0$), $p=p(t)$ is determined by the warp
function $l$, $\mathcal T(H^f)$ is the coercive second-form square of
@@ -1664,40 +1664,40 @@ The boundary identity of the [lem:n3:boundary-identity] lemma, together
with the strict negativity of the boundary mean curvature
$\widehat H<0$ and the maximum principle on $t$ with the mixed
Neumann--Dirichlet data of [eq:coupled-system], gives
-$-3d\operatorname{Hess}_g t(e,e)-(3+pv)\operatorname{tr}_\perp\operatorname{Hess}_g t\ge 0$ pointwise on
-$\bar\Omega_R$; the sign is carried by $\partial_\nu t$ at $B_0$ and
+$-3dHess_g t(e,e)-(3+pv)tr_\perpHess_g t\ge 0$ pointwise on
+$\barΩ_R$; the sign is carried by $∂_\nu t$ at $B_0$ and
propagated by Hopf. Consequently
$$
-\tfrac12e^{2t}\mathrm{S}_{\bar g}
-\ge\tfrac12\mathrm{S}_g+\mathcal T(H^f)-v\operatorname{Ric}_g(e,e)+O(\eta_n^2).
+\tfrac12e^{2t}S_{\bar g}
+\ge\tfrac12S_g+\mathcal T(H^f)-vRic_g(e,e)+O(\eta_n^2).
$$
-The constraint takes the form $16\pi\mu=R_g+(\operatorname{tr} K)^2-|K|^2$, so
-$\tfrac12\mathrm{S}_g\ge-\tfrac12|K|^2+8\pi\mu$. Dominant energy supplies
+The constraint takes the form $16\pi\mu=R_g+(tr K)^2-|K|^2$, so
+$\tfrac12S_g\ge-\tfrac12|K|^2+8\pi\mu$. Dominant energy supplies
$\mu\ge|J|_g$, and the coercivity estimate of
the [lem:n3:hessian-square] lemma (with constants absorbed into
-$C(n)$) bounds the negative pieces $-\tfrac12|K|^2-v\operatorname{Ric}_g(e,e)$ by a
+$C(n)$) bounds the negative pieces $-\tfrac12|K|^2-vRic_g(e,e)$ by a
fixed multiple of $\mathcal T(H^f)$ plus $C(n)\eta_n^2$ on the bounded
range of the [prop:n3:apriori] proposition. Therefore
-$\tfrac12e^{2t}\mathrm{S}_{\bar g}\ge0$ for all sufficiently large $n$,
-i.e.\ $R_{\bar g}\ge0$ on $\Omega_R$.
+$\tfrac12e^{2t}S_{\bar g}\ge0$ for all sufficiently large $n$,
+i.e.\ $R_{\bar g}\ge0$ on $Ω_R$.
For the area bound on $B_0$, the two-color expansion of
-the [lem:n3:boundary-identity] lemma makes $\mathrm{d}_{B_0}f$ tangent to
+the [lem:n3:boundary-identity] lemma makes $d_{B_0}f$ tangent to
$B_0$ with magnitude controlled by the signed face slopes
$|b'|\le C_*/\tau$; on $B_0$ the trace condition $h=b$ bounds $t$
and hence the warp factor. The corresponding area density is
therefore $(1+O(\eta_n))$-comparable to that of $g$ and
-$|\operatorname{Area}_{\bar g}(B_0)-a_g(B)|\le C\eta_n\,a_g(B)\le\varepsilon a_g(B)$ for
+$|Area_{\bar g}(B_0)-a_g(B)|\le C\eta_n\,a_g(B)\leε a_g(B)$ for
large $n$.
The mass identity is the asymptotic matching of $\bar g$ with $g$ on
the far tail. On the end $f=O(r^{-1})$ and $t=s+O(r^{-4})$, so $\bar g$
agrees with $g$ to order $r^{-2}$ in every scalar component and the
ADM mass formula (linear in the $O(r^{-2})$ part of $g-\delta$) gives
-$m_{\rm ADM}(\bar g)=m+O(\eta_n)\le m+\varepsilon$ for $n$ sufficiently
+$m_{\rm ADM}(\bar g)=m+O(\eta_n)\le m+ε$ for $n$ sufficiently
large. This is precisely the asserted inequality.
@@ -1722,32 +1722,32 @@ $$
Completion of the three-dimensional proof [sec:n3-existence-closing]. Proof (Proof of the [thm:n3:strict-exterior). theorem]
-Take data satisfying the hypotheses of the theorem. Fix $\varepsilon>0$.
+Take data satisfying the hypotheses of the theorem. Fix $ε>0$.
the [prop:n3:two-color] proposition gives the two-color exterior
-$\Omega_0$; the [prop:n3:existence] proposition gives a smooth solution
-$(f,t)$ of [eq:coupled-system] on $\Omega_R$; and
+$Ω_0$; 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 $\Omega_R$ with
+$\bar g$ on $Ω_R$ with
$$
R_{\bar g}\ge0,\qquad
-\operatorname{Area}_{\bar g}(\partial\Omega_R)\ge(1-\varepsilon)a_g(B),\qquad
-m_{\rm ADM}(\bar g)\le m+\varepsilon.
+Area_{\bar g}(∂Ω_R)\ge(1-ε)a_g(B),\qquad
+m_{\rm ADM}(\bar g)\le m+ε.
$$
Extend $\bar g$ beyond $S_R$ by a flat conformal patch; the resulting
complete asymptotically flat metric has nonnegative scalar curvature and
-least enclosing area $A_{\min}(\bar g)\ge\operatorname{Area}_{\bar g}(\partial\Omega_R)
-\ge(1-\varepsilon)a_g(B)$. The Riemannian Penrose inequality
+least enclosing area $A_{\min}(\bar g)\geArea_{\bar g}(∂Ω_R)
+\ge(1-ε)a_g(B)$. The Riemannian Penrose inequality
(the [thm:riemannian-penrose] theorem) gives
$$
-m+\varepsilon\ge m_{\rm ADM}(\bar g)
+m+ε\ge m_{\rm ADM}(\bar g)
\ge\sqrt{\frac{A_{\min}(\bar g)}{16\pi}}
- \ge\sqrt{\frac{(1-\varepsilon)a_g(B)}{16\pi}}.
+ \ge√(((1-ε)a_g(B))/(16\pi)).
$$
-Letting $\varepsilon\to0$ gives $m\ge\sqrt{a_g(B)/16\pi}$, which is the
+Letting $ε\to0$ gives $m\ge√(a_g(B)/16\pi)$, which is the
conclusion of the [thm:n3:strict-exterior] theorem.
@@ -1768,16 +1768,16 @@ with finitely many asymptotically flat ends, decay [eq:decay] on
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 $S=\partial D^+$ (possibly
+smooth embedded two-sided surface $S=∂ D^+$ (possibly
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 $\theta_+(S)=H_S+\operatorname{tr}_SK\le0$.
+$\nu$ that points into $D^+$, the requirement is $\theta_+(S)=H_S+tr_SK\le0$.
Define [eq:bridge-amin]
$$
-A_{\min}(S)=\inf\{\operatorname{Area}_g(\partial D'^+)\colon
-D'^+\subset D^+,\ D'^+\text{ contains the chosen end at infinity},\
-\partial D'^+\text{ compact smooth}\}.
+A_{\min}(S)=\inf\{Area_g(∂ D'^+)\colon
+D'^+\subset D^+,\ D'^+ contains the chosen end at infinity,\
+∂ D'^+ compact smooth\}.
$$
Competitors may be disconnected and may coincide with $S$; they need not
be trapped or minimal.
@@ -1789,10 +1789,10 @@ be trapped or minimal.
The global theorem [sec:bridge-global]. Theorem (Complete-data form). [thm:bridge:global]
Let $(M,g,K)$ satisfy the hypotheses of
the [def:bridge-admissible] definition. Then for every chosen end and
-every admissible boundary $S=\partial D^+$, [eq:bridge-inequality]
+every admissible boundary $S=∂ D^+$, [eq:bridge-inequality]
$$
E>|P|_\delta,\qquad
-\sqrt{E^2-|P|_\delta^2}\ge\sqrt{\frac{A_{\min}(S)}{16\pi}},
+√(E^2-|P|_\delta^2)\ge\sqrt{\frac{A_{\min}(S)}{16\pi}},
$$
where $(E,P)$ is the ADM vector of the chosen end.
@@ -1801,13 +1801,13 @@ where $(E,P)$ is the ADM vector of the chosen end.
Proof.
-Fix $(M,g,K)$ and the admissible $S=\partial D^+$. Let $D_0$ be the
+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
-$S\ne\varnothing$.
+$S\ne∅$.
-Put $X_0=\overline{D_0}$, including its smooth compact boundary. Since
+Put $X_0=D_0bar$, including its smooth compact boundary. Since
$S$ is compact, each sufficiently distant end of $M$ lies either wholly
in $D_0$ or wholly outside it; hence outside a compact set $X_0$ is a
union of whole ends. In particular $X_0$ has finitely many ends, all
@@ -1818,7 +1818,7 @@ coordinate tail and remove it. The tails can be chosen disjoint and
outside $S$. Removing them preserves connectedness: an excursion of a
path into a removed tail can be deformed back onto its coordinate sphere.
The resulting region $N$ is a smooth manifold with nonempty compact
-boundary $B$ (comprising $\partial D_0$ and the added spheres), and its
+boundary $B$ (comprising $∂ D_0$ and the added spheres), and its
only end is the chosen one.
$N$ is closed in the complete $(M,g)$ and has smooth boundary, so it is
@@ -1829,19 +1829,19 @@ $(M,g,K)$, 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
end, the normal into $N$ points toward decreasing radius, so
-$H_B=-2/r+O(r^{-1-q})$ and $\operatorname{tr}_BK=O(r^{-1-q})$, hence
+$H_B=-2/r+O(r^{-1-q})$ and $tr_BK=O(r^{-1-q})$, hence
$\theta_+(B)<0$ 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
-$\partial D'^+$ in the [def:bridge-admissible] definition (it is
+$∂ D'^+$ 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
$$
-\sqrt{E^2-|P|_\delta^2}
- \ge\sqrt{\frac{a_g(B)}{16\pi}}
+√(E^2-|P|_\delta^2)
+ \ge√((a_g(B))/(16\pi))
\ge\sqrt{\frac{A_{\min}(S)}{16\pi}},
$$
@@ -1852,7 +1852,7 @@ 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 $(\Omega,g,K)$ satisfy all the hypotheses of
+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
across $B$ is required.
@@ -1876,13 +1876,13 @@ 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
$$
-\sqrt{(E+sA_\phi)^2-|P|_\delta^2}
+√((E+sA_\phi)^2-|P|_\delta^2)
\ge\sqrt{\frac{a_{g_s}(B)}{16\pi}}
- \ge\sqrt{\frac{a_g(B)}{16\pi}}>0.
+ \ge√((a_g(B))/(16\pi))>0.
$$
-Letting $s\downarrow s_0$ makes the left side tend to
-$\sqrt{(E+s_0A_\phi)^2-|P|_\delta^2}=\sqrt{|P|_\delta^2-|P|_\delta^2}=0$,
+Letting $s↓ s_0$ makes the left side tend to
+$√((E+s_0A_\phi)^2-|P|_\delta^2)=√(|P|_\delta^2-|P|_\delta^2)=0$,
contradicting the strictly positive right side. Thus
$E>|P|_\delta$, and the [thm:n3:exterior] theorem applied to the
original data gives the conclusion.
@@ -1892,12 +1892,12 @@ original data gives the conclusion.
Equality and the causal stationary field [sec:variation]. The equality case comes next. Throughout, the chosen
-exterior $\mathcal{M}_{\mathrm{ext}}$ is connected, one-ended, with compact boundary $S$
+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 mass-area equality
-$m=\sqrt{A/16\pi}$ with $A=|S|_g=\sum_{i=1}^\ell|S_i|_g$.
+$m=√(A/16\pi)$ 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]
@@ -1905,22 +1905,22 @@ 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 $S=\bigsqcup_{i=1}^\ell S_i$ of compact connected smooth
+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 $\mathcal{M}_{\mathrm{ext}}$ be the closure of the side toward the
-chosen end. Suppose $\partial\mathcal{M}_{\mathrm{ext}}=S$, that $\mathcal{M}_{\mathrm{ext}}$ is one-ended, and
-that, for the normal pointing into $\mathcal{M}_{\mathrm{ext}}$:
-(i) every $S_i$ is a future MOTS: $H_{S_i}+\operatorname{tr}_{S_i}K=0$;
+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
+that, for the normal pointing into $M_{ext}$:
+(i) every $S_i$ is a future MOTS: $H_{S_i}+tr_{S_i}K=0$;
(ii) $S$ is partially outermost (the [sec:outermost] definition);
(iii) every enclosing cut in the closed exterior has area at least
$A=|S|_g$.
Let $(E,P)$ be the ADM vector of the chosen end. If [eq:n3-rigidity-equality]
$$
E>|P|_\delta,\qquad
-m=\sqrt{E^2-|P|_\delta^2}=\sqrt{\frac{A}{16\pi}},
+m=√(E^2-|P|_\delta^2)=√((A)/(16\pi)),
$$
then $\ell=1$ and $S$ is diffeomorphic to a sphere. There is a proper
-smooth spacelike embedding of the entire original $\mathcal{M}_{\mathrm{ext}}$ into the
+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
corresponding spatial infinity. The image of $S$ is a smooth
@@ -1935,7 +1935,7 @@ 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
-$m>\sqrt{A/16\pi}$.
+$m>√(A/16\pi)$.
@@ -1946,9 +1946,9 @@ 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,
$a_g(B)=A$. The [thm:n3:exterior] theorem gives
-$m\ge\sqrt{A/16\pi}$. If equality holds then $\ell=1$ by
+$m\ge√(A/16\pi)$. If equality holds then $\ell=1$ by
the [thm:n3:rigidity] theorem, contradicting $\ell\ge2$; hence
-$m>\sqrt{A/16\pi}$.
+$m>√(A/16\pi)$.
@@ -1957,26 +1957,26 @@ $m>\sqrt{A/16\pi}$.
Variations and the causal stationary field [sec:n3-variation]. The proof of the [thm:n3:rigidity] theorem uses two structures: the
metric derivative of the enclosing infimum at equality, and the
constraint equation linearisation. Together they produce a normalized
-causal stationary field on $\mathcal{M}_{\mathrm{ext}}$.
+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 $\mathcal{M}_{\mathrm{ext}}$ with
-$u>0$ in $\operatorname{Int}\mathcal{M}_{\mathrm{ext}}$, such that, on $\mathcal{M}_{\mathrm{ext}}$ and up to $S$,
-$$\begin{aligned}
-\operatorname{sym}\nabla X&=-uK,[eq:stationary-kid]\\
-\Delta u&=-\operatorname{Div}(K(X,\cdot)^{\sharp}),[eq:stationary-lapse]\\
-\operatorname{Hess} u&=u(\operatorname{Ric}_g+\tau K-2K^2)-\mathcal L_XK+8\pi X_{(i}J_{j)}. [eq:stationary-hess]
-\end{aligned}$$
-Here $\tau=\operatorname{tr}_gK$, $(K^2)_{ij}=K_i{}^kK_{kj}$, and parentheses denote
+function $u\ge0$ and a smooth vector field $X$ on $M_{ext}$ with
+$u>0$ in $IntM_{ext}$, such that, on $M_{ext}$ and up to $S$,
+$$
+sym\nabla X&=-uK,[eq:stationary-kid]\\
+\Delta u&=-Div(K(X,\cdot)^{\sharp}),[eq:stationary-lapse]\\
+Hess u&=u(Ric_g+\tau K-2K^2)-\mathcal L_XK+8\pi X_{(i}J_{j)}. [eq:stationary-hess]
+$$
+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 $\mathcal{M}_{\mathrm{ext}}$, [eq:stationary-boundary]
+for the outward-inward normal $\nu$ of $M_{ext}$, [eq:stationary-boundary]
$$
X=u\nu,\qquad
-\partial_\nu u+K(X,\nu)=\kappa,
+∂_\nu u+K(X,\nu)=\kappa,
\qquad
-\kappa=\frac{1}{4m}>0.
+\kappa=(1)/(4m)>0.
$$
On each component of $S$, either $u>0$ identically or $u=0$
identically, and this alternative is made component by component. The
@@ -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
-$\mathrm{d}\eta=2u\,\mathrm{d} A$ on $S$ together with the negative first variation of
+$d\eta=2u\,d A$ on $S$ together with the negative first variation of
$S$ and the strong maximum principle on the stability operator (giving
the componentwise alternative).
@@ -2039,11 +2039,11 @@ 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]
$$
-c\int_S h_s\,\mathrm{d} A
-=\mathcal E'\bigl(h,p\bigr)+2\int_S u\,\mathrm{d}(\eta),
+c\int_S h_s\,d A
+=\mathcal E'(h,p)+2\int_S u\,d(\eta),
$$
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 $\sqrt{a/16\pi}$ at
+\cdot16\pi/(16\pi)=1/(4m)$ is the derivative of $√(a/16\pi)$ at
$a=A$.
@@ -2053,7 +2053,7 @@ $a=A$.
Proof.
The functional $\mathcal E$ anchors the minimizer $S$: at the minimum
of the supporting energy over $S$, its derivative equals the derivative
-of $\sqrt{a/16\pi}$. Under the variation the constraint functional and
+of $√(a/16\pi)$. Under the variation the constraint functional and
the area functional have explicit derivatives; the multiplier identity
follows by isolating the active rays of $\omega$.
@@ -2064,50 +2064,50 @@ follows by isolating the active rays of $\omega$.
Static structure and Schwarzschild reconstruction [sec:n3-static]. The remainder of the proof of the [thm:n3:rigidity] theorem passes from
the causal stationary field of the [prop:n3:stationary] proposition to
the Schwarzschild model. The chain is: (i) $u>|X|_g$ on all of
-$\mathcal{M}_{\mathrm{ext}}$ and the twist of the Lorentzian metric vanishes;
+$M_{ext}$ 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
-identifies $(\mathcal{M}_{\mathrm{ext}},g,u)$ as Schwarzschild.
+identifies $(M_{ext},g,u)$ as Schwarzschild.
Positivity of the twist
-Put $\mathcal{N}=u^2-|X|_g^2$. On $\partial\mathcal{M}_{\mathrm{ext}}$, the [prop:n3:stationary] proposition
+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
-$u|_{S}=a_*>0$; hence $\mathcal{N}|_{S}=0$. Near $S$ the pair $(u,X)$ so
+$u|_{S}=a_*>0$; 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]
$$
-\Delta_g\mathcal{N}
+\Delta_gN
+u^{-1}K(\nabla u,\nabla u)
-+2|\nabla X|_g^2/\mathcal{N}
-\ge -C(N)\,\mathcal{N},
++2|\nabla X|_g^2/N
+\ge -C(N)\,N,
$$
-in any region where $\mathcal{N}>0$. The strong maximum principle is used for
+in any region where $N>0$. The strong maximum principle is used for
[eq:n3-N-inequality].
Lemma (Interior positivity of the twist). [lem:n3-twist-interior]
-$\mathcal{N}>0$ on $\mathcal{M}_{\mathrm{ext}}\setminus\partial\mathcal{M}_{\mathrm{ext}}$.
+$N>0$ on $M_{ext}∖∂M_{ext}$.
Proof.
-By the boundary data and continuity, $\mathcal{N}\ge0$ on a collar of $S$. Let
-$x$ be a first point of $\mathcal{M}_{\mathrm{ext}}$ at which $\mathcal{N}(x)=0$. Since $\mathcal{N}$ is a
+By the boundary data and continuity, $N\ge0$ 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
-$\Delta_g\mathcal{N}(x)\ge 2|\nabla X|^2/\mathcal{N}$ 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 $\partial\mathcal{M}_{\mathrm{ext}}$
+initial data) rules out the interior case. A first zero on $∂M_{ext}$
contradicts the positive boundary data. Continuity propagates the sign
-along every path in the connected $\mathcal{M}_{\mathrm{ext}}$.
+along every path in the connected $M_{ext}$.
@@ -2118,19 +2118,19 @@ Vanishing twist
Let $\omega$ be the twist $2$-form of the Lorentzian-Killing candidate
-$\partial_t$, defined by $\omega_{ij}=2u^{-1}(X_i\partial_ju-X_j\partial_iu)$.
+$∂_t$, 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 $u=\sqrt\mathcal{N}$ on all of $\mathcal{M}_{\mathrm{ext}}$. In particular the
+satisfies $X=0$ and $u=\sqrtN$ on all of $M_{ext}$. In particular the
Lorentzian metric [eq:n3-static-metric]
$$
-\mathbf{g}=-u^2\mathrm{d} t^2+g
+g=-u^2d t^2+g
$$
-is stationary vacuum on $\mathbb{R}\times\mathcal{M}_{\mathrm{ext}}$, and its Killing field
-$\partial_t$ has future-pointing unit normal
-$n=u^{-1}(\partial_t-X)$.
+is stationary vacuum on $ℝ\timesM_{ext}$, and its Killing field
+$∂_t$ has future-pointing unit normal
+$n=u^{-1}(∂_t-X)$.
@@ -2138,7 +2138,7 @@ $n=u^{-1}(\partial_t-X)$.
Proof.
The candidate twist $\omega$ satisfies a divergence identity
-$\operatorname{Div}_g\bigl(u^{-1}\omega\bigr)=0$ (the Lorentzian-Killing equations)
+$Div_g(u^{-1}\omega)=0$ (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
@@ -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 $\widetilde M\supset\mathcal{M}_{\mathrm{ext}}$ open, such that:
-(a) $\hat h=u^2g$ on $\mathcal{M}_{\mathrm{ext}}$;
+$(\widetilde M,\hat h)$ 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
-$\mathcal{M}_{\mathrm{ext}}$ across $S$;
+$M_{ext}$ 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 $\partial\widetilde M$ carries uniform positive
+(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.
@@ -2175,12 +2175,12 @@ boundary first eigenfunction is strictly positive.
Proof.
The static vacuum equations $R_{\hat h}=0$, $\Delta_{\hat h}u=0$ on
-$\mathcal{M}_{\mathrm{ext}}$ arise from pulling out the $u^2$ factor. The wall at $S$ is
+$M_{ext}$ 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
fundamental form of $S$ in $g$ satisfies the vanishing of
-$\theta_+=H_S+\operatorname{tr}_SK=0$ 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 $\mathcal{M}_{\mathrm{ext}}$
+gives $H_S^{\hat h}=0$. 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
@@ -2207,11 +2207,11 @@ $S\cong S^2$.
Proof.
Apply Schoen--Yau to the closed oriented double $\widetilde M$. Each
-boundary component of the original $\mathcal{M}_{\mathrm{ext}}$ (after doubling) yields a
+boundary component of the original $M_{ext}$ (after doubling) yields a
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 $\mathbb{R}^3$; it is compact, hence trivial, and
-$\widetilde M\cong S^3$. Every piece of $\mathcal{M}_{\mathrm{ext}}$ attaches to exactly one
+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 $S\cong S^2$.
@@ -2225,9 +2225,9 @@ Identification with Schwarzschild
Proposition (Schwarzschild identification). [prop:n3-Schwarzschild]
Under the assumptions of the [thm:n3:rigidity] theorem, the triple
-$(\mathcal{M}_{\mathrm{ext}},g,u)$ is isometric to the future-horizon exterior of the
+$(M_{ext},g,u)$ is isometric to the future-horizon exterior of the
Schwarzschild spacetime of mass $m$ in Schwarzschild coordinates, with
-$u(x)=\sqrt{1-2m/r}$ and $S$ the sphere $r=2m$.
+$u(x)=√(1-2m/r)$ and $S$ the sphere $r=2m$.
@@ -2238,19 +2238,19 @@ 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
totally geodesic. The Birkhoff rigidity of static vacuum spacetimes
-(Israel) identifies $(\mathcal{M}_{\mathrm{ext}},g,u)$ with Schwarzschild: a static vacuum
+(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 $\partial_\nu u|_{S}=\kappa=1/(4m)$ and
+parameter. The boundary data $∂_\nu u|_{S}=\kappa=1/(4m)$ and
$u\to m$ at spatial infinity fix the mass.
-Note that $\mathcal{M}_{\mathrm{ext}}$ alone, without the interior, is the exterior region
+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 $\partial_t$
+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
-$\xi=\partial_t$.
+$\xi=∂_t$.
@@ -2281,22 +2281,22 @@ $M$.
Charged constraints, DEC, and charges [sec:ch-constraints]. The energy density and momentum density pair with the electromagnetic
field through
-$$\begin{aligned}
+$$
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]
-\end{aligned}$$
+$$
where $(\mu_0,J_0)$ are the neutral constraint densities and $*$ is the
Hodge star of $g$. The charged DEC is [eq:ch-dec]
$$
-\mu-|J|_g\ge\frac{1}{8\pi}\bigl(|e|_g^2+|b|_g^2\bigr),
+\mu-|J|_g\ge(1)/(8\pi)(|e|_g^2+|b|_g^2),
$$
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]
$$
-Q_E=\frac{1}{4\pi}\lim_{r\to\infty}\int_{S_r}e(\nu)\,\mathrm{d} A,
+Q_E=(1)/(4\pi)\lim_{r\to∞}\int_{S_r}e(\nu)\,d A,
\qquad
-Q_B=\frac{1}{4\pi}\lim_{r\to\infty}\int_{S_r}b\,\mathrm{d} A,
+Q_B=(1)/(4\pi)\lim_{r\to∞}\int_{S_r}b\,d A,
$$
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
@@ -2308,13 +2308,13 @@ 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 $S=\partial D^+$ 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 $m=\sqrt{E^2-|P|_\delta^2}$, then [eq:ch-inequality]
+$E>|P|_\delta$ and $m=√(E^2-|P|_\delta^2)$, then [eq:ch-inequality]
$$
-m\ge Q,\qquad r\le m+\sqrt{m^2-Q^2}.
+m\ge Q,\qquad r\le m+√(m^2-Q^2).
$$
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 $r$.
@@ -2337,7 +2337,7 @@ $$
Q_E^{\rm comp}=Q_E+o(1),\quad Q_B^{\rm comp}=Q_B+o(1),\quad
\|e^{\rm comp}-e\|,\|b^{\rm comp}-b\|=o(1)
$$
-in $L^\infty$ on any fixed compact region. The charged DEC [eq:ch-dec]
+in $L^∞$ on any fixed compact region. The charged DEC [eq:ch-dec]
is preserved up to the same $o(1)$.
@@ -2352,7 +2352,7 @@ decomposition $F=e\wedge n+b$ 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
-smooth limit. The interior $L^\infty$ bound follows from the
+smooth limit. The interior $L^∞$ bound follows from the
$C^2$ convergence of the deformation on compact sets.
@@ -2370,8 +2370,8 @@ 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
outermost area $A$, and let $P=0$, $m=E$. Then [eq:charged-rmp]
$$
-m\ge\frac{r+Q^2/r}{2},\qquad r=\Bigl(\frac{A}{4\pi}\Bigr)^{1/2},\quad
-Q=\bigl(Q_E^2+Q_B^2\bigr)^{1/2}.
+m\ge(r+Q^2/r)/(2),\qquad r=((A)/(4\pi))^{1/2},\quad
+Q=(Q_E^2+Q_B^2)^{1/2}.
$$
@@ -2400,7 +2400,7 @@ 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
comparison metric
-$\bar g_{ij}=d^{-1}(g_{ij}+l(t)^2\partial_if\partial_jf)$ satisfies [eq:ch-comparison]
+$\bar g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf)$ satisfies [eq:ch-comparison]
$$
R_{\bar g}\ge 2|\bar e|_{\bar g}^2+2|\bar b|_{\bar g}^2+O(\eta_n),
$$
@@ -2414,15 +2414,15 @@ $$
after shrinking the comparison to a lower bound on $\eta_n$ (the
$O(\eta_n)$ term appears with sign in the coercive bound).
-Third, the boundary sphere $\partial\Omega_R$ of the comparison is a
-minimal surface with area at least $(1-\varepsilon)A_{\min}(S)$ and charge
+Third, the boundary sphere $∂Ω_R$ of the comparison is a
+minimal surface with area at least $(1-ε)A_{\min}(S)$ and charge
$(Q_E,Q_B)$. By the [thm:charged-riemannian] theorem, [eq:ch-assembly]
$$
m_{\rm ADM}(\bar g)\ge
\frac{r_{\bar g}+Q^2/r_{\bar g}}{2},\qquad
-r_{\bar g}=\Bigl(\frac{\operatorname{Area}_{\bar g}(\partial\Omega_R)}{4\pi}\Bigr)^{1/2}.
+r_{\bar g}=(\frac{Area_{\bar g}(∂Ω_R)}{4\pi})^{1/2}.
$$
-Pass to the limit as $\varepsilon\downarrow0$ and $R\to\infty$: the left side
+Pass to the limit as $ε↓0$ and $R\to∞$: the left side
becomes $m$, the right side becomes the charged mass bound with
$r=(A_{\min}(S)/(4\pi))^{1/2}$.
@@ -2457,25 +2457,25 @@ existence, and Riemannian Penrose steps all adapt; the
statement and the key modifications are now presented.
-Four-dimensional data [sec:four-data]. A four-dimensional exterior $(\Omega,g,K)$ is a connected orientable
+Four-dimensional data [sec:four-data]. A four-dimensional exterior $(Ω,g,K)$ is a connected orientable
smooth $4$-manifold with Riemannian metric $g$ and symmetric
$(0,2)$-tensor $K$, complete with smooth compact boundary
-$B=\partial\Omega$ and one asymptotically flat end in the sense of
+$B=∂Ω$ and one asymptotically flat end in the sense of
[eq:decay] in a $4$-dimensional chart, with $q>2$ (the
four-dimensional analogue of the three-dimensional $q>3/2$). The
constraints and DEC are as in [eq:constraints]. The ADM vector
$(E,P)$ uses the standard $4$-dimensional normalization; when
-$E>|P|$ the invariant mass is $m=\sqrt{E^2-|P|^2}$. Let $A_e=a_g(B)$
+$E>|P|$ the invariant mass is $m=√(E^2-|P|^2)$. Let $A_e=a_g(B)$
be the four-dimensional enclosing infimum in the sense of
the [def:bridge-admissible] definition.
Theorem (Four-dimensional exterior inequality). [thm:four:main]
-Let $(\Omega,g,K)$ be a four-dimensional exterior with
+Let $(Ω,g,K)$ be a four-dimensional exterior with
$\theta_+(B)\le0$, DEC, and integrable $\mu,|J|_g$. Let
$\omega_3=2\pi^2$. If $E>|P|$, then [eq:four-inequality]
$$
-m\ge\frac12\Bigl(\frac{A_e}{\omega_3}\Bigr)^{2/3}.
+m\ge\frac12((A_e)/(\omega_3))^{2/3}.
$$
The complete-data statement of the [thm:bridge:global] theorem holds in
dimension $4$ with the same right-hand side.
@@ -2504,9 +2504,9 @@ degree of freedom $e^{3t}$ (rather than $e^{2t}$), and the conformal
family in the metric comparison is $u^3$ rather than $u^4$ with
$u=e^{t/3}$. The variables $f,t$, the auxiliary $D,d,Z,h,w,a,v,\chi$
take their dimension-$4$ forms. The comparison metric
-$\bar g_{ij}=d^{-1}(g_{ij}+l(t)^2\partial_if\partial_jf)$ has
-nonnegative scalar curvature on $\Omega_R$, mass at most $m+\varepsilon$, and
-boundary three-volume at least $(1-\varepsilon)A_e$. The regularity estimates
+$\bar g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf)$ has
+nonnegative scalar curvature on $Ω_R$, mass at most $m+ε$, and
+boundary three-volume at least $(1-ε)A_e$. The regularity estimates
of the [sec:existence] section (graph Sobolev, axial gradient estimate,
four-stage homotopy) pass through in dimension $4$ with the same
structure; the scalar identity uses a Schur-type trace calculation
@@ -2524,7 +2524,7 @@ ADM mass $m_{ADM}$. Let $\Gamma$ be a smooth compact outermost minimal
$3$-surface. Then
$$
-m_{ADM}\ge\frac12\Bigl(\frac{|\Gamma|_g}{\omega_3}\Bigr)^{2/3},
+m_{ADM}\ge\frac12((|\Gamma|_g)/(\omega_3))^{2/3},
\qquad \omega_3=2\pi^2.
$$
@@ -2538,9 +2538,9 @@ This is the four-dimensional Riemannian Penrose inequality, a classical
result (Huisken--Ilmanen 2001; Bray 2001, extended by Bray--Lee 2009).
Its three-dimensional form is (T1) of the [sec:external] section; the
four-dimensional statement uses the flow identity
-$\Sigma(t)=m_{ADM}-\tfrac12(A(t)/\omega_3)^{2/3}$ in place of the
+$Σ(t)=m_{ADM}-\tfrac12(A(t)/\omega_3)^{2/3}$ in place of the
three-dimensional formula. At the outermost minimal $3$-surface the
-area statistic $\operatorname{Area}(t)$ equals $|\Gamma|_g$, giving the stated
+area statistic $Area(t)$ equals $|\Gamma|_g$, giving the stated
inequality. It is used as an external input; the details of the flow
are classical.
@@ -2557,19 +2557,19 @@ with
$$
R_{\bar g}\ge0,\quad
-m_{ADM}(\bar g)\le m+\varepsilon,\quad
-\operatorname{Area}_{\bar g}(\partial\Omega_R)\ge(1-\varepsilon)A_e.
+m_{ADM}(\bar g)\le m+ε,\quad
+Area_{\bar 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\Bigl(\frac{\operatorname{Area}_{\bar g}(\partial\Omega_R)}
-{\omega_3}\Bigr)^{2/3}.
+m_{ADM}(\bar g)\ge\frac12(\frac{Area_{\bar g}(∂Ω_R)}
+{\omega_3})^{2/3}.
$$
-Let $\varepsilon\downarrow0$: the left side becomes $m$, the right side becomes
+Let $ε↓0$: the left side becomes $m$, the right side becomes
$\frac12(A_e/\omega_3)^{2/3}$, which is [eq:four-inequality].
@@ -2583,7 +2583,7 @@ with Schwarzschild--Tangherlini.
Theorem (Four-dimensional rigidity). [thm:rig4:rigidity]
Under the hypotheses of the [thm:four:main] theorem in the connected,
one-ended, outermost, outer area-minimizing equality case
-$m=\frac12(A_e/\omega_3)^{2/3}$, the exterior $\mathcal{M}_{\mathrm{ext}}$ is isometric to
+$m=\frac12(A_e/\omega_3)^{2/3}$, the exterior $M_{ext}$ is isometric to
the future-horizon exterior in the Schwarzschild--Tangherlini
spacetime of mass $m$, with $K$ the second fundamental form generated
by the stationary Killing field of that spacetime.
@@ -2615,33 +2615,33 @@ The anti-de Sitter Penrose inequality [sec:ads]. Schwarzschild-anti-de Sitter an
Fix the Schwarzschild-anti-de Sitter background of mass $m_0\ge0$ in
spacetime dimension $4$. In the usual static coordinates, [eq:ads-bgd]
$$
-g^0=-N^2\mathrm{d} t^2+\frac{\mathrm{d} r^2}{N^2}+r^2\sigma,\qquad
-N^2=1+r^2-\frac{2m_0}{r},
+g^0=-N^2d t^2+(d r^2)/(N^2)+r^2\sigma,\qquad
+N^2=1+r^2-(2m_0)/(r),
$$
with $\sigma$ the round metric on $S^2$, and on the time slice $t=0$
$K^0=0$. The spatial metric in Fefferman-Graham form is
-$g^0=\mathrm{d} s^2+r(s)^2\sigma$ with $r=e^s(1+O(e^{-2s}))$; the boundary is
-at $s=\infty$.
+$g^0=d s^2+r(s)^2\sigma$ with $r=e^s(1+O(e^{-2s}))$; the boundary is
+at $s=∞$.
A smooth branch of asymptotically anti-de Sitter initial data near the
-background is a smooth family $(g_\varepsilon,K_\varepsilon)$ for
-$\varepsilon\in[0,\varepsilon_0)$ such that each $(g_\varepsilon,K_\varepsilon)$ is of finite
+background is a smooth family $(g_ε,K_ε)$ for
+$ε\in[0,ε_0)$ such that each $(g_ε,K_ε)$ is of finite
angular type, satisfies the Einstein constraints on the exterior of a
-compact future MOTS $S_\varepsilon$, satisfies $(g_\varepsilon,K_\varepsilon)=(g^0,K^0)+O(\varepsilon)$
+compact future MOTS $S_ε$, satisfies $(g_ε,K_ε)=(g^0,K^0)+O(ε)$
in the weighted Sobolev norm of the class, and has finite boundary mass
-flux. Let $A(\varepsilon)=|S_\varepsilon|_{g_\varepsilon}$ be the boundary area and
-$m_{AH}(\varepsilon)$ the positive Lorentz norm of the hyperbolic mass covector.
+flux. Let $A(ε)=|S_ε|_{g_ε}$ be the boundary area and
+$m_{AH}(ε)$ the positive Lorentz norm of the hyperbolic mass covector.
Theorem (Anti-de Sitter Penrose). [thm:ads:main]
-Under the hypotheses above there is $\varepsilon_1\in(0,\varepsilon_0]$ such that for
-every $\varepsilon\in[0,\varepsilon_1)$, [eq:ads-inequality]
+Under the hypotheses above there is $ε_1\in(0,ε_0]$ such that for
+every $ε\in[0,ε_1)$, [eq:ads-inequality]
$$
-m_{AH}(\varepsilon)
-\ge\sqrt{\frac{A(\varepsilon)}{16\pi}}
-\Bigl(1+\frac{A(\varepsilon)}{4\pi}\Bigr).
+m_{AH}(ε)
+\ge√((A(ε))/(16\pi))
+(1+(A(ε))/(4\pi)).
$$
-Equality holds if and only if the branch seed is radial and $\varepsilon\ge0$
+Equality holds if and only if the branch seed is radial and $ε\ge0$
is sufficiently small. The equality family is Schwarzschild-anti-de
Sitter at the matching mass and area.
@@ -2650,16 +2650,16 @@ Sitter at the matching mass and area.
Taylor structure and weighted remainders [sec:ads-taylor]. Write the branch in a Taylor expansion
-$g_\varepsilon=g^0+\varepsilon g_1+\varepsilon^2 g_2+\varepsilon^3 g_3+O_4(\varepsilon^4)$ with each $g_j$
-of finite angular type, and likewise $K_\varepsilon$. The weighted remainder
-$O_4(\varepsilon^4)$ controls the asymptotic functionals.
+$g_ε=g^0+ε g_1+ε^2 g_2+ε^3 g_3+O_4(ε^4)$ with each $g_j$
+of finite angular type, and likewise $K_ε$. The weighted remainder
+$O_4(ε^4)$ controls the asymptotic functionals.
Lemma (Taylor to mass and area). [lem:ads-taylor]
There are real $m_0,m_1,m_2,m_3$ and $A_0,A_1,A_2,A_3$ such that [eq:ads-taylor-exp]
$$
-m_{AH}(\varepsilon)=\sum_{j=0}^3\varepsilon^jm_j+O(\varepsilon^4),\quad
-A(\varepsilon)=\sum_{j=0}^3\varepsilon^j A_j+O(\varepsilon^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),
$$
with the remainders controlled by the weighted bound
$F=O(e^{-\gamma s})$ with $\gamma>3$.
@@ -2675,7 +2675,7 @@ homogeneous Neumann data on $S$ there is a unique finite-type decaying
solution of $Lu=F$ with [eq:ads-remainder]
$$
u=r^{-3}\zeta(x)+o(r^{-3}),\quad
-\partial_s\bigl(u-r^{-3}\zeta(x)\bigr)=o(r^{-3}),
+∂_s(u-r^{-3}\zeta(x))=o(r^{-3}),
$$
$\zeta$ finite-type. Differentiation of $F$ in $s$ through order $k$
gives the same statement through $k+2$ derivatives of $u$.
@@ -2690,7 +2690,7 @@ $G=O(e^{-\gamma's})$, $\gamma'=\min(\gamma,\beta+2)>3$, eliminates
the growing homogeneous mode $e^s$ and gives
$y=ce^{-3s}+O(e^{-\gamma's})$. Angular decomposition closes by
boundedness on each fixed finite-dimensional space. Comparison with
-$Ah+\delta$, letting $\delta\downarrow0$, gives uniqueness of the
+$Ah+\delta$, letting $\delta↓0$, gives uniqueness of the
decaying solution.
@@ -2700,9 +2700,9 @@ decaying solution.
The quadratic coefficient and its kernel [sec:ads-quadratic]. Proposition (The quadratic form). [prop:ads-c2]
In the expansion [eq:ads-mass-gap]
$$
-m_{AH}(\varepsilon)-\sqrt{\frac{A(\varepsilon)}{16\pi}}
-\Bigl(1+\frac{A(\varepsilon)}{4\pi}\Bigr)
-=c_2[q]\varepsilon^2+c_3[q]\varepsilon^3+c_4[q]\varepsilon^4+O(\varepsilon^5)
+m_{AH}(ε)-√((A(ε))/(16\pi))
+(1+(A(ε))/(4\pi))
+=c_2[q]ε^2+c_3[q]ε^3+c_4[q]ε^4+O(ε^5)
$$
the coefficient $c_2[q]$ is a nonnegative quadratic form, and its
kernel has dimension exactly $4$ (the four linear harmonics on $S^2$).
@@ -2716,8 +2716,8 @@ Conformal decomposition of $g_1$ gives a TT part $k$ together with a
conformal factor and a tangential vector. The mass and area Taylor
expansion reduces $c_2$ to the TT square identity
(the [lem:ads-tt-square] lemma), which makes it
-$\frac{1}{16\pi}\int_M N|k|^2\,\mathrm{d} V_0
--\frac{\kappa}{16\pi}\int_S vDv\,\mathrm{d} A_0$. The argument of
+$(1)/(16\pi)\int_M N|k|^2\,d V_0
+-(\kappa)/(16\pi)\int_S vDv\,d A_0$. The argument of
the [lem:ads-tt-square] lemma identifies the last term, and $D$'s
positivity makes the form nonnegative. Its kernel is the four
translation modes of the round $S^2$.
@@ -2730,10 +2730,10 @@ Lemma (Weighted TT square identity). [lem:ads-tt-square]
Let $k$ be smooth TT with $k=O(r^{-\gamma})$ for some $\gamma>3/2$, and
$v$ the unique decaying solution of $Lv=0$ with
$v|_S=D^{-1}(k_{ss}|_S)$, $D=-\Delta_S+2\kappa/a$. Then
-$Q(v)=\operatorname{Hess} v-vB$ is TT, $Q(v)_{ss}|_S=k_{ss}|_S$, and [eq:ads-tt-square]
+$Q(v)=Hess v-vB$ is TT, $Q(v)_{ss}|_S=k_{ss}|_S$, and [eq:ads-tt-square]
$$
-\int_M N|k|^2\,\mathrm{d} V_0-\kappa\int_S vDv\,\mathrm{d} A_0
-=\int_M N|k-Q(v)|^2\,\mathrm{d} V_0.
+\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.
$$
@@ -2766,8 +2766,8 @@ degree-one mode. Hence $c_4[q]>0$.
Lemma (Finite-jet slice change). [lem:ads-slice]
There is a smooth local diffeomorphism $\Psi$ and a smooth conformal
-factor $\omega$ with $\omega_0=1,\Psi_0=\mathrm{id}$, such that
-$\widetilde g=\omega_\varepsilon\Psi_\varepsilon^*g^0+O(\varepsilon^3)$ holds on the
+factor $\omega$ with $\omega_0=1,\Psi_0=id$, such that
+$\widetilde g=\omega_ε\Psi_ε^*g^0+O(ε^3)$ holds on the
original slice.
@@ -2775,10 +2775,10 @@ original slice.
Proposition (Exact radial identity). [prop:ads-radial]
-If $q$ is a radial seed, then for all sufficiently small $\varepsilon\ge0$, [eq:ads-radial-equality]
+If $q$ is a radial seed, then for all sufficiently small $ε\ge0$, [eq:ads-radial-equality]
$$
-m_{AH}(\varepsilon)=\sqrt{A(\varepsilon)/16\pi}
-\bigl(1+A(\varepsilon)/4\pi\bigr).
+m_{AH}(ε)=√(A(ε)/16\pi)
+(1+A(ε)/4\pi).
$$
@@ -2787,9 +2787,9 @@ $$
Proof.
Radial seeds form the Schwarzschild-anti-de Sitter family, for which
-$m=\sqrt{A/16\pi}(1+A/4\pi)$ is exact at every radius. At $\varepsilon=0$ the
+$m=√(A/16\pi)(1+A/4\pi)$ is exact at every radius. At $ε=0$ the
identity is the background identity, and smoothness continues it to
-$[0,\varepsilon_1)$.
+$[0,ε_1)$.
@@ -2804,10 +2804,10 @@ 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
-$c_2[q]\varepsilon^2+c_4[q]\varepsilon^4+O(\varepsilon^5)$ with $c_2[q]\ge0$ and $c_4[q]>0$ on
-the kernel, so the gap is strictly positive for all small $\varepsilon>0$.
+$c_2[q]ε^2+c_4[q]ε^4+O(ε^5)$ 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 $[0,\varepsilon_1)$ depends only on the background data
+equality. The interval $[0,ε_1)$ depends only on the background data
and the branch.
@@ -2817,7 +2817,7 @@ and the branch.
Remark.
The result uses the same enclosing-area functional but in an
asymptotically hyperbolic background; the relevant mass is $m_{AH}$
-rather than the asymptotically flat $m$, and the boundary area $A(\varepsilon)$
+rather than the asymptotically flat $m$, and the boundary area $A(ε)$
plays the role of the enclosing area.