diff b1f9bc4..56bc728 spacetime_penrose_area_charge_v2.txt
diff --git a/spacetime_penrose_area_charge_v2.txt b/spacetime_penrose_area_charge_v2.txt
index 940d318..569adfc 100644
--- a/spacetime_penrose_area_charge_v2.txt
+++ b/spacetime_penrose_area_charge_v2.txt
@@ -1,40 +1,40 @@
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 $16pi m^2$ for total mass $m$, equivalently
-$mge√(A/16pi)$; otherwise a naked singularity would form, violating
+horizon area A exceeds 16pi m^2 for total mass m, equivalently
+mge√(A/16pi); otherwise a naked singularity would form, violating
cosmic censorship. This is the Penrose inequality. The theorems
below prove it rigorously (and its charged, higher-dimensional, and
anti-de Sitter versions) by reducing the spacetime problem to the
classical Riemannian Penrose inequality for a metric of nonnegative
scalar curvature.
-An initial data set for the Einstein equations is a triple $(M,g,K)$: a
-Riemannian manifold $(M,g)$ and a symmetric $2$-tensor $K$, the second
+An initial data set for the Einstein equations is a triple (M,g,K): a
+Riemannian manifold (M,g) and a symmetric 2-tensor K, the second
fundamental form of a spacelike hypersurface in a spacetime. The constraint
-equations tie the energy density $mu$ and momentum density $J$ of the data
+equations tie the energy density mu and momentum density J of the data
to their geometry, and the dominant energy condition (DEC) requires
-$muge|J|_g$ everywhere.
+muge|J|_g everywhere.
-Fix a compact hypersurface $Ssubset M$ and a unit normal $nu$ toward the
+Fix a compact hypersurface Ssubset M and a unit normal nu toward the
designated exterior. Its future outward null expansion is
-$$
+
theta_+(S)=H_S+tr_S K,
-$$
-where $H_S$ is the mean curvature of $S$ with respect to $nu$. The surface
-is weakly future outer trapped when $theta_+(S)le0$, and a future
-marginally outer trapped surface (MOTS) when $theta_+(S)=0$. A horizon
+
+where H_S is the mean curvature of S with respect to nu. The surface
+is weakly future outer trapped when theta_+(S)le0, and a future
+marginally outer trapped surface (MOTS) when theta_+(S)=0. A horizon
appears as such a surface in a spacelike slice.
-Fix an asymptotically flat end and let $(E,P)$ denote its ADM energy and
-momentum. Once $E>|P|$, the invariant mass of the end is
-$m=√(E^2-|P|^2)$. The spacetime Penrose inequalities bound $m$ below by
+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=√(E^2-|P|^2). The spacetime Penrose inequalities bound m below by
the size of an enclosing cut of the trapped boundary. An enclosing cut
separates the designated boundary from the chosen end, counts the full area
including any portion coincident with the boundary, and need not itself be
-trapped or connected. Let $A_{min}$ denote the infimum of this area.
+trapped or connected. Let A_{min} denote the infimum of this area.
The results
@@ -42,13 +42,13 @@ The results
In three spatial dimensions the basic bound is
-$$
+
mgesqrt{frac{A_{min}}{16pi}}.
-$$
+
the [thm:n3:exterior] theorem proves this for a designated end with
-arbitrary $K$ and nonzero ADM momentum; the [thm:bridge:global] theorem
+arbitrary K and nonzero ADM momentum; the [thm:bridge:global] theorem
removes any restriction on the ADM vector and allows a disconnected chosen
exterior region and a disconnected trapped boundary. Equality is treated in
the [thm:n3:rigidity] theorem: if the chosen exterior is connected and
@@ -56,45 +56,45 @@ one-ended and its boundary is a future MOTS that is outermost and outer
area-minimizing, then equality forces a single spherical boundary component
and realizes the entire exterior as a slice of Schwarzschild spacetime.
-With total electric and magnetic charges $Q_E,Q_B$ and
-$Q=(Q_E^2+Q_B^2)^{1/2}$, let $r=(A_{min}/4pi)^{1/2}$. For a one-ended
+With total electric and magnetic charges Q_E,Q_B and
+Q=(Q_E^2+Q_B^2)^{1/2}, let r=(A_{min}/4pi)^{1/2}. For a one-ended
exterior whose matter condition includes the electromagnetic momentum,
the [thm:ch:main] theorem proves
-$$
- mge Q,qquad rle m+√(m^2-Q^2).
-$$
-The familiar bound $mge(r+Q^2/r)/2$ is recovered on the branch $rge Q$;
-it is not equivalent for every positive $r$. Rest-frame electric
+ mge Q, rle m+√(m^2-Q^2).
+
+
+The familiar bound mge(r+Q^2/r)/2 is recovered on the branch rge Q;
+it is not equivalent for every positive r. Rest-frame electric
corollaries, including a designated-end theorem with finitely many ends,
follow with their own cut convention. No charged equality classification is
claimed.
-In four spatial dimensions, let $A_e$ be the three-volume of an enclosing
-hypersurface and $omega_3=2pi^2$ the volume of the unit $3$-sphere.
+In four spatial dimensions, let A_e be the three-volume of an enclosing
+hypersurface and omega_3=2pi^2 the volume of the unit 3-sphere.
the [thm:four:main] theorem proves
-$$
+
m_egefrac12((A_e)/(omega_3))^{2/3}
-$$
-for arbitrary $K$, finitely many ends, and a disconnected weakly future
+
+for arbitrary K, finitely many ends, and a disconnected weakly future
trapped boundary, and the [thm:rig4:rigidity] theorem identifies the
connected, one-ended, outermost, outer area-minimizing equality case as
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
-$m_{AH}$ the positive Lorentz norm of the hyperbolic mass covector,
+and a decaying positive conformal branch. With A the boundary area and
+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,
-$$
+
m_{AH}ge√((A)/(16pi))(1+(A)/(4pi)),
-$$
+
with equality exactly for radial seeds at small positive parameter.
@@ -119,7 +119,7 @@ estimate turns its positive-norm region into a static one, and a complete
conformal double forces Euclidean geometry. In three dimensions the
zero-mass argument uses the spin structure furnished by orientability, and
the boundary identities give every boundary component the full original
-area, forcing one. The last step reconstructs $g$ and $K$ and the smooth
+area, forcing one. The last step reconstructs g and K and the smooth
horizon attachment directly, rather than transferring equality through the
deformations.
@@ -136,30 +136,30 @@ once here, so that a reader can tell where the paper's original
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 $R_hge 0$ and ADM
- energy $E_h$, and $∂ N$ is a compact outermost smooth minimal
- surface of total area $A$, then $E_hge√(A/16pi)$, with equality
+ 2001). If (N,h) is a complete connected asymptotically flat
+ three-manifold with nonnegative scalar curvature R_hge 0 and ADM
+ energy E_h, and ∂ N is a compact outermost smooth minimal
+ surface of total area A, then E_hge√(A/16pi), with equality
in the nondegenerate case exactly for the Schwarzschild horizon.
./pdf/huisken-ilmanen-2001.pdf
(T2) Positive mass theorem (Schoen--Yau 1979, Witten 1981). A
- complete connected asymptotically flat three-manifold with $R_hge 0$
- satisfies $E_hge |P_h|_delta$, with equality if and only if
- $(N,h)$ is Euclidean.
+ complete connected asymptotically flat three-manifold with R_hge 0
+ satisfies E_hge |P_h|_delta, with equality if and only if
+ (N,h) is Euclidean.
./pdf/witten-1981.pdf
(T3) Static vacuum uniqueness (Bunting--Masood-ul-Alam 1987).
- A complete three-dimensional static vacuum data set $(N,h,N)$ with
- positive lapse $N>0$ and a nondegenerate Killing horizon is isometric
- to the Schwarzschild exterior ${r>2M}$, with mass
- $M=|∂ N|_h^{1/2}/√(16pi)$.
+ A complete three-dimensional static vacuum data set (N,h,N) with
+ positive lapse N>0 and a nondegenerate Killing horizon is isometric
+ to the Schwarzschild exterior {r>2M}, with mass
+ M=|∂ N|_h^{1/2}/√(16pi).
(T4) Charged Riemannian Penrose inequality
(Huisken--Ilmanen 2001; Bray--Khuri 2010; Disconzi--Khuri 2012). The
- same setup with total charge $Q=(Q_E^2+Q_B^2)^{1/2}$ and $P=0$ gives
- $E_hge Q$ and $rle m+√(m^2-Q^2)$, with $r=(A/4pi)^{1/2}$ and
- $m=E_h$.
+ same setup with total charge Q=(Q_E^2+Q_B^2)^{1/2} and P=0 gives
+ E_hge Q and rle m+√(m^2-Q^2), with r=(A/4pi)^{1/2} and
+ m=E_h.
./pdf/disconzi-khuri-2012.pdf
Everything else (strictification, rest-frame reduction, the coupled
@@ -174,102 +174,102 @@ Setting and conventions [sec:setting]. Initial data and the constraint equations
-Throughout, $(M,g,K)$ is a smooth connected orientable three-manifold with
-a complete Riemannian metric $g$ and a symmetric $(0,2)$-tensor $K$, the
-second fundamental form of a spacelike slice. The energy density $mu$ and
-momentum density $J$ are determined by the constraint equations [eq:constraints]
-$$
+Throughout, (M,g,K) is a smooth connected orientable three-manifold with
+a complete Riemannian metric g and a symmetric (0,2)-tensor K, the
+second fundamental form of a spacelike slice. The energy density mu and
+momentum density J are determined by the constraint equations [eq:constraints]
+
16pimu=R_g+(tr_gK)^2-|K|_g^2,
- qquad
+
8pi J=div_g(K-(tr_gK)g),
-$$
-where $R_g$ is the scalar curvature. The dominant energy condition (DEC)
-is the pointwise inequality $muge|J|_g$.
-An end of $M$ is a neighborhood of infinity diffeomorphic to
-$ℝ^3∖barbar B_1$,B_1$, parametrized by coordinates $x^i$ in which [eq:decay]
-$$
+where R_g is the scalar curvature. The dominant energy condition (DEC)
+is the pointwise inequality muge|J|_g.
+
+An end of M is a neighborhood of infinity diffeomorphic to
+ℝ^3∖barbar B_1,B_1, parametrized by coordinates x^i in which [eq:decay]
+
g_{ij}-delta_{ij}=O_2(r^{-q}),
- qquad
+
K_{ij}=O_1(r^{-1-q}),
- qquad
+
q>tfrac32,
-$$
-with $r=|x|$ and $O_k$ denoting $k$ derivatives bounded by a constant
-times a power of $r$. The ADM energy and momentum are [eq:adm]
-$$
+
+with r=|x| and O_k denoting k derivatives bounded by a constant
+times a power of r. The ADM energy and momentum are [eq:adm]
+
E=(1)/(16pi)lim_{rto∞}int_{S_r}(g_{ij,i}-g_{ii,j})nu^j,d A,
- qquad
+
P_j=(1)/(8pi)lim_{rto∞}int_{S_r}(K_{ij}-(tr_gK)delta_{ij})nu^i,d A,
-$$
-where $S_r$ are the coordinate spheres, $nu$ their outward Euclidean
-unit normal, and the index $i$ is summed. The limits exist by
-[eq:decay]. $P_delta$ denotes the momentum covector paired
-with the coordinate frame. When $E>|P|_delta$, the invariant mass of
-the end is $m=√(E^2-|P|_delta^2)$.
-
-
-MOTS and expansions [sec:mots]. Let $Ssubset M$ be a compact embedded two-sided surface disjoint from
-$∂ M$, with a chosen unit normal $nu$ pointing into the
-designated exterior. Its mean curvature is $H_S=div_Snu$, with the
-convention $H_S=2/r$ for a coordinate sphere of radius $r$ in $ℝ^3$,
-and $tr_SK=g^{ab}K(X_a,X_b)$ in an orthonormal tangent frame
-${X_a}$. The future and past outward null expansions are [eq:null-expansions]
-$$
+
+where S_r are the coordinate spheres, nu their outward Euclidean
+unit normal, and the index i is summed. The limits exist by
+[eq:decay]. P_delta denotes the momentum covector paired
+with the coordinate frame. When E>|P|_delta, the invariant mass of
+the end is m=√(E^2-|P|_delta^2).
+
+
+MOTS and expansions [sec:mots]. Let Ssubset M be a compact embedded two-sided surface disjoint from
+∂ M, with a chosen unit normal nu pointing into the
+designated exterior. Its mean curvature is H_S=div_Snu, with the
+convention H_S=2/r for a coordinate sphere of radius r in ℝ^3,
+and tr_SK=g^{ab}K(X_a,X_b) in an orthonormal tangent frame
+{X_a}. The future and past outward null expansions are [eq:null-expansions]
+
theta_pm(S)=H_Spmtr_SK.
-$$
-The surface $S$ is weakly future outer trapped when $theta_+(S)le0$,
+
+The surface S is weakly future outer trapped when theta_+(S)le0,
and a future marginally outer trapped surface (MOTS) when
-$theta_+(S)=0$.
+theta_+(S)=0.
-Exteriors, cuts, and the minimum enclosing area [sec:cuts]. Let $Ω$ be a connected open manifold whose closure $barbarΩ$Ω$ is
-complete for the restriction of $g$, with smooth compact boundary
-$B=∂Ω$ and exactly one end, asymptotically flat.
+Exteriors, cuts, and the minimum enclosing area [sec:cuts]. Let Ω be a connected open manifold whose closure barbarΩΩ is
+complete for the restriction of g, with smooth compact boundary
+B=∂Ω and exactly one end, asymptotically flat.
An enclosing cut for the chosen end is a compact smooth embedded
-(possibly disconnected) surface $GammasubsetbarbarΩ$Ω$ that bounds a
-filled side $D^+(Gamma)subsetbarbarΩ$Ω$ containing the entire distant
-part of that end. The complement $barbarΩ∖Ω∖ D^+(Gamma)$
-contains $B$. Its area $|Gamma|_g$ counts all components, including any
-part of $Gamma$ coincident with $B$. Define [eq:amin]
-$$
+(possibly disconnected) surface GammasubsetbarbarΩΩ that bounds a
+filled side D^+(Gamma)subsetbarbarΩΩ containing the entire distant
+part of that end. The complement barbarΩ∖Ω∖ D^+(Gamma)
+contains B. Its area |Gamma|_g counts all components, including any
+part of Gamma coincident with B. Define [eq:amin]
+
a_g(B)=infbigl{|Gamma|_g:Gamma an enclosing cutbigr}.
-$$
+
The infimum equals the finite-perimeter closure taken in
the [sec:perimeter] section.
Outermostness and area-minimality [sec:outermost]. Two conditions are imposed only in the equality theorem.
-Outer area-minimality. $B$ is outer area-minimal when
-$|Gamma|_gge|B|_g$ for every enclosing cut $Gamma$.
+Outer area-minimality. B is outer area-minimal when
+|Gamma|_gge|B|_g for every enclosing cut Gamma.
-Partial-coincidence outermostness. Let $mathcal T$calT$ be the class
-of enclosing cuts that attain $a_g(B)$. $B$ is partially outermost if
-every $Gammainmathcal T$calT$ with $Gammane B$ omits at least one entire
-component of $B$.
+Partial-coincidence outermostness. Let mathcal TcalT be the class
+of enclosing cuts that attain a_g(B). B is partially outermost if
+every Gammainmathcal TcalT with Gammane B omits at least one entire
+component of B.
Regularity and finiteness [sec:fin]. The hypotheses [eq:decay] are the standing
assumption in three dimensions; in four dimensions the analogous
-$O_2$ and $O_1$ decay with the same exponent is assumed. Hypersurface
+O_2 and O_1 decay with the same exponent is assumed. Hypersurface
regularity and boundary expansions are always as stated in the
corresponding theorem. All areas, fluxes, and currents are taken with
-respect to the given $g$.
+respect to the given g.
Normalizations
-Throughout, $c=G=1$ and the coefficients $16pi$ are used in [eq:constraints]
-and [eq:amin]. In spatial dimension $n$ the reference constant
-$omega_{n-1}$ is the volume of the unit $(n-1)$-sphere; in three
-dimensions $omega_2=4pi$, in four dimensions $omega_3=2pi^2$. A
-coordinate sphere of radius $r$ in $ℝ^n$ has area
-$omega_{n-1}r^{n-1}$, and a future MOTS sphere in Schwarzschild
-spacetime of mass $m$ satisfies $m=tfrac12(A/omega_{n-1})^{(n-2)/(n-1)}$.
+Throughout, c=G=1 and the coefficients 16pi are used in [eq:constraints]
+and [eq:amin]. In spatial dimension n the reference constant
+omega_{n-1} is the volume of the unit (n-1)-sphere; in three
+dimensions omega_2=4pi, in four dimensions omega_3=2pi^2. A
+coordinate sphere of radius r in ℝ^n has area
+omega_{n-1}r^{n-1}, and a future MOTS sphere in Schwarzschild
+spacetime of mass m satisfies m=tfrac12(A/omega_{n-1})^{(n-2)/(n-1)}.
The three-dimensional exterior inequality [sec:n3]. Statement [sec:n3-statement]
@@ -278,27 +278,27 @@ The three-dimensional exterior inequality [sec:n3]. Statement [sec:n3-statement]
Theorem (Minimum enclosing area in three dimensions). [thm:n3:exterior]
-Let $(Ω,g,K)$ be a connected orientable smooth three-dimensional
-exterior with smooth compact boundary $B=∂Ω$, 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 $muge|J|_g$ with $mu$ and
-$|J|_g$ integrable, and that the ADM limits [eq:adm] exist.
-Orient $B$ by the normal into $Ω$ and suppose $theta_+(B)le0$.
-If $E>|P|_delta$, then
-$$
+constraints [eq:constraints] satisfy muge|J|_g with mu and
+|J|_g integrable, and that the ADM limits [eq:adm] exist.
+Orient B by the normal into Ω and suppose theta_+(B)le0.
+If E>|P|_delta, then
+
√(E^2-|P|_delta^2)
ge√((a_g(B))/(16pi)).
-$$
-No extension of the data across $B$ is required.
+
+No extension of the data across B is required.
The proof assembles three ingredients: a conformal deformation that
-makes the ADM vector timelike without disturbing $B$
+makes the ADM vector timelike without disturbing B
([sec:n3-reduction]), a preparation of the data to a strict
end-replaced form ([sec:end-preparation]), and the geometric and
elliptic comparison with the Riemannian Penrose inequality
@@ -306,63 +306,63 @@ elliptic comparison with the Riemannian Penrose inequality
Conformal strict direction [sec:n3-reduction]. Lemma (Strict direction). [lem:n3:strict]
-Let $(Ω,g,K)$ satisfy all the hypotheses of
-the [thm:n3:exterior] theorem except possibly $E>|P|_delta$. There is a
+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
-$g_s=(1+sphi)^4g$, $K_s=(1+sphi)^{2}K$, defined for
-$sin[0,s_0)$, with $phige0$ smooth and $O(1/|x|)$ at infinity, such
+g_s=(1+sphi)^4g, K_s=(1+sphi)^{2}K, defined for
+sin[0,s_0), with phige0 smooth and O(1/|x|) at infinity, such
that
-$$
- E_s=E+sA_phi,quad P_s=P,quad A_phi>0,quad
+
+ E_s=E+sA_phi, P_s=P, A_phi>0,
a_{g_s}(B)ge a_g(B),
-$$
-and $(Ω,g_s,K_s)$ satisfies DEC and
-$theta_+^{g_s,K_s}(B)le0$ in the same decay class.
+
+and (Ω,g_s,K_s) satisfies DEC and
+theta_+^{g_s,K_s}(B)le0 in the same decay class.
Proof.
-Let $wge0$ be smooth, compactly supported in a small ball disjoint from
-$B$ and from a neighborhood of the end, with $wnotequiv0$. Let $G$ be
-the Green kernel of $L_g=-Delta_g+tfrac18R_g$. On an asymptotically
-flat three-manifold with $R_gge0$ the kernel $G$ is nonnegative; this
+Let wge0 be smooth, compactly supported in a small ball disjoint from
+B and from a neighborhood of the end, with wnotequiv0. Let G be
+the Green kernel of L_g=-Delta_g+tfrac18R_g. On an asymptotically
+flat three-manifold with R_gge0 the kernel G is nonnegative; this
is the standard positivity of the conformal Green's function
-(Schoen--Yau). Put $phi=Gwge0$, so $phi$ is smooth, positive on
-$supp w$, and decays like $O(1/|x|)$.
+(Schoen--Yau). Put phi=Gwge0, so phi is smooth, positive on
+supp w, and decays like O(1/|x|).
-Write $u_s=1+sphi$, so $u_sge1$ and $u_sto1$ at infinity. The
-constraints of $g_s=u_s^4g$, $K_s=u_s^{-2}K$ continue to satisfy DEC
+Write u_s=1+sphi, so u_sge1 and u_sto1 at infinity. The
+constraints of g_s=u_s^4g, K_s=u_s^{-2}K continue to satisfy DEC
by conformal covariance in three dimensions: the energy and momentum
-densities transform with nonnegative factors when $u_s$ is a positive
+densities transform with nonnegative factors when u_s is a positive
supersolution of the conformal Laplacian. The ADM energy formula
-$$
+
E_{g_s}=E+scdot(1)/(16pi)lim_{rto∞}int_{S_r}
∂_nu(4u_s∂_nu u_s),d A
=E+sA_phi
-$$
-with $A_phiproptoint w>0$ and $P_s=P$ follows from the standard
-second variation of the mass under $gmapsto u_s^4g$.
-Because $supp w$ is disjoint from a neighborhood of $B$, the normal
-derivative of $u_s$ vanishes there, and conformal covariance of the
+with A_phiproptoint w>0 and P_s=P follows from the standard
+second variation of the mass under gmapsto u_s^4g.
+
+Because supp w is disjoint from a neighborhood of B, the normal
+derivative of u_s vanishes there, and conformal covariance of the
mean curvature and trace gives
-$$
+
H^{g_s}_B+tr^{g_s}_BK_s
=u_s^{-3}(H_B^g+tr_B^gK),
-$$
-so $theta_+^{g_s,K_s}(B)le0$ is preserved.
-Finally, $phige0$ implies $u_sge1$ pointwise, so $g_sge g$ and
-$|Gamma|_{g_s}ge|Gamma|_g$ for every enclosing cut $Gamma$.
-Taking infima gives $a_{g_s}(B)ge a_g(B)$.
+so theta_+^{g_s,K_s}(B)le0 is preserved.
+
+Finally, phige0 implies u_sge1 pointwise, so g_sge g and
+|Gamma|_{g_s}ge|Gamma|_g for every enclosing cut Gamma.
+Taking infima gives a_{g_s}(B)ge a_g(B).
@@ -373,81 +373,81 @@ through~[sec:existence].
Theorem (Strictified exterior inequality). [thm:n3:strict-exterior]
-Let $(Ω,g,K)$ 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 $P=0$;
+- K compactly supported;
+- theta_+(B)<0 and P=0;
- the strict gap
- $mu-|J|_gge c_0r^{-4-delta_1}$ for some $c_0>0$ and
- $0<delta_1<1$.
+ mu-|J|_gge c_0r^{-4-delta_1} for some c_0>0 and
+ 0<delta_1<1.
Then
-$$
+
m=Ege√((a_g(B))/(16pi)).
-$$
+
The proof of the [thm:n3:strict-exterior] theorem builds a comparison
-metric $g_star$ on the strictified exterior with [eq:comparison]
-$$
- R_{g_star}ge0,qquad
- m_{ADM}(g_star)le m+ε,qquad
+metric g_star on the strictified exterior with [eq:comparison]
+
+ R_{g_star}ge0,
+ m_{ADM}(g_star)le m+ε,
a_{g_star}(B)ge(1-ε)a_g(B),
-$$
-for every $ε>0$; 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 $ε>0$. The [sec:end-preparation] section and
-the [lem:n3:strict] lemma produce strictified data $(g',K')$ satisfying
-$$
+Completion of the exterior inequality [sec:n3-completion]. Suppose first E>|P|_delta and B is oriented so
+theta_+(B)le0; if some components need K replaced by -K, do so
+and proceed. Fix ε>0. The [sec:end-preparation] section and
+the [lem:n3:strict] lemma produce strictified data (g',K') satisfying
+
- m'le m+ε,qquad theta'_+(B)<0,qquad
+ m'le m+ε, theta'_+(B)<0,
a_{g'}(B)ge(1-ε)a_g(B),
-$$
+
and identical decay class. The [thm:n3:strict-exterior] theorem gives
-$m'gesqrt{a_{g'}(B)/16pi}$, so
-$$
+m'gesqrt{a_{g'}(B)/16pi}, so
+
√((a_g(B))/(16pi))
lefrac{m'}{√(1-ε)}
lefrac{m+ε}{√(1-ε)}.
-$$
-Let $ε↓0$ 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:
-$K$ compactly supported, $theta_+(B)<0$, and a positive density gap. The
+K compactly supported, theta_+(B)<0, and a positive density gap. The
construction uses a drift-Poisson equation that supplies a strict
supersolution to the conformal Laplacian, following the conformal
constraint formulas in three dimensions.
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
-$$
+Let u>0 be smooth and set g_u=u^4g, K_u=u^2K. Then
+
16pi u^4mu_u &=16pimu-8u^{-1}Delta_gu-(1-u^{-8})((tr_gK)^2-|K|_g^2),
[eq:conformal-mu]
8pi J_u &=u^{-2}(8pi J+4K(nablalog u,cdot)), [eq:conformal-J]
-$$
+
the second being an identity of covectors. Consequently [eq:conformal-dec]
-$$
+
16pi u^4(mu_u-|J_u|_{g_u})ge
16pi(mu-|J|_g)+8(-Delta_glog u-|dlog u|_g^2-|K|_g|dlog u|_g)
-|1-u^{-8}|,|(tr_gK)^2-|K|_g^2|.
-$$
-For every consistently oriented hypersurface with unit normal $nu$, [eq:conformal-expansion]
-$$
+
+For every consistently oriented hypersurface with unit normal nu, [eq:conformal-expansion]
+
theta_{+,u}=u^{-2}(theta_++4∂_nulog u).
-$$
+
@@ -455,115 +455,115 @@ $$
Proof.
In dimension three
-$R_{u^4g}=u^{-4}(R_g-8Delta_glog u-8|dlog u|_g^2)$, while the
-constraint quadratic term scales with an extra factor of $u^{-8}$ because
-the extrinsic curvature is weighted by $u^2$:
-$u^4((tr_{g_u}K_u)^2-|K_u|^2_{g_u})=u^{-8}((tr_gK)^2-|K|_g^2)$.
-Adding the two contributions gives [eq:conformal-mu]. With $pi=K-(tr_gK)g$, $pi_u=u^2pi$ and
-$tr_gpi=-2tr_gK$; contracting the connection difference in dimension
+R_{u^4g}=u^{-4}(R_g-8Delta_glog u-8|dlog u|_g^2), while the
+constraint quadratic term scales with an extra factor of u^{-8} because
+the extrinsic curvature is weighted by u^2:
+u^4((tr_{g_u}K_u)^2-|K_u|^2_{g_u})=u^{-8}((tr_gK)^2-|K|_g^2).
+Adding the two contributions gives [eq:conformal-mu]. With pi=K-(tr_gK)g, pi_u=u^2pi and
+tr_gpi=-2tr_gK; contracting the connection difference in dimension
three gives
-$u^2J_u=J+2pi(nablalog u,cdot)-(tr_gpi)dlog u
- =J+4K(nablalog u,cdot)$,
-which is [eq:conformal-J]. Norms in $g_u$ add a factor $u^{-2}$,
+u^2J_u=J+2pi(nablalog u,cdot)-(tr_gpi)dlog u
+ =J+4K(nablalog u,cdot),
+which is [eq:conformal-J]. Norms in g_u add a factor u^{-2},
and the triangle inequality yields [eq:conformal-dec]. Finally
-$nu_u=u^{-2}nu$, $H_{u^4g}=u^{-2}(H+4∂_nulog u)$, and
-$tr_{B,g_u}K_u=u^{-2}tr_{B,g}K$, which is [eq:conformal-expansion].
+nu_u=u^{-2}nu, H_{u^4g}=u^{-2}(H+4∂_nulog u), and
+tr_{B,g_u}K_u=u^{-2}tr_{B,g}K, which is [eq:conformal-expansion].
The strictification lemma [sec:n3-strictification]. Lemma (Strictification). [lem:n3:strictification]
-Let $(N,g,K)$ satisfy [eq:decay] with $q>tfrac32$, DEC, and
-integrability, with compact boundary $B$ and $theta_+(B)le0$.
-Fix $0<delta<beta<min(q,1)$. There are smooth positive $w,phi$ on $N$
-with $w=r^{-3-delta}$ on the end, such that [eq:strict-data]
-$$
-∂_nuphi=-1 on B,quad
-phi=O_2(r^{-1}),quad
+Let (N,g,K) satisfy [eq:decay] with q>tfrac32, DEC, and
+integrability, with compact boundary B and theta_+(B)le0.
+Fix 0<delta<beta<min(q,1). There are smooth positive w,phi on N
+with w=r^{-3-delta} on the end, such that [eq:strict-data]
+
+∂_nuphi=-1 on B,
+phi=O_2(r^{-1}),
-Delta_gphi-|K|_g|dphi|_gge w,
-$$
-and the outer flux $L_phi=lim_{rto∞}int_{S_r}∂_rphi,d A_delta$
-is finite and negative. For every $s>0$, the data $g_s=u_s^4g$,
-$K_s=u_s^2K$ with $u_s=1+sphi$ satisfy:
+
+and the outer flux L_phi=lim_{rto∞}int_{S_r}∂_rphi,d A_delta
+is finite and negative. For every s>0, the data g_s=u_s^4g,
+K_s=u_s^2K with u_s=1+sphi satisfy:
(i) DEC with strict gap
- $mu_s-|J_s|_{g_s}ge c_sr^{-3-delta}$ for $c_s>0$;
-(ii) $theta_+^{g_s,K_s}(B)<0$;
-(iii) decay class $min(q,1)>tfrac12$;
-(iv) $E_s=E+sA_phi$, $P_s=P$, $A_phi=-L_phi/(2pi)>0$, and
- $a_g(B)le a_{g_s}(B)le(1+s|phi|_∞)^4a_g(B)$.
+ mu_s-|J_s|_{g_s}ge c_sr^{-3-delta} for c_s>0;
+(ii) theta_+^{g_s,K_s}(B)<0;
+(iii) decay class min(q,1)>tfrac12;
+(iv) E_s=E+sA_phi, P_s=P, A_phi=-L_phi/(2pi)>0, and
+ a_g(B)le a_{g_s}(B)le(1+s|phi|_∞)^4a_g(B).
Proof.
-Take $bge|K|_g$ smooth and positive, equal to $b_0r^{-1-beta}$ on the
-end, with $b_0>0$. Let $zeta>0$ be a smooth regulariser equal to
-$r^{-2}$ on the end, and $w>0$ smooth on $N$ equal to $r^{-3-delta}$
+Take bge|K|_g smooth and positive, equal to b_0r^{-1-beta} on the
+end, with b_0>0. Let zeta>0 be a smooth regulariser equal to
+r^{-2} on the end, and w>0 smooth on N equal to r^{-3-delta}
on the end. The drift-Poisson equation [eq:drift-poisson]
-$$
+
-Delta_gphi=b√(|dphi|_g^2+zeta^2)+w,
-qquad
+
∂_nuphi=-1,quadphito0,
-$$
-is uniformly elliptic and smooth even at critical points of $phi$
-because of $zeta$. There is a smooth nonnegative solution with
-$phi=O_2(r^{-1})$.
-Existence. Truncate at a distant sphere $S_R$ with outer
+is uniformly elliptic and smooth even at critical points of phi
+because of zeta. There is a smooth nonnegative solution with
+phi=O_2(r^{-1}).
+
+Existence. Truncate at a distant sphere S_R with outer
Dirichlet and inner homogeneous Neumann data, and continue in the
-parameter $tin[0,1]$ with $b$ replaced by $tb$. At $t=0$ the mixed
+parameter tin[0,1] with b replaced by tb. At t=0 the mixed
Laplacian is invertible: the Dirichlet face supplies the Poincare
inequality, and a weak solution is smooth by boundary regularity. The
linearisation at any smooth solution carries a bounded drift of norm
-at most $b$; the strong maximum principle and the boundary Hopf
+at most b; the strong maximum principle and the boundary Hopf
principle force a trivial kernel, so the operator is an invertible
-Fredholm perturbation. Uniformity in $t$: if the solution supremum
+Fredholm perturbation. Uniformity in t: if the solution supremum
norms on a truncation are unbounded, divide by the supremum, and with
-$W^{2,p}$ estimates, derivative interpolation, and $p>3$ Sobolev
-compactness the limit $v$ solves $Delta_gv+Acdot dv=0$ with
-$|A|le b$ measurable, contradicting the maximum principles. The
-bounds are independent of $R$, yielding a smooth solution of
+W^{2,p} estimates, derivative interpolation, and p>3 Sobolev
+compactness the limit v solves Delta_gv+Acdot dv=0 with
+|A|le b measurable, contradicting the maximum principles. The
+bounds are independent of R, yielding a smooth solution of
[eq:drift-poisson] on the exterior.
Sign and end decay. The right side of [eq:drift-poisson]
is positive, so a negative minimum cannot lie in the interior, and the
-Hopf principle at a boundary minimum requires $∂_nuphile0$
-in the outer normal into $N$, whereas the present data give
-$∂_nuphi=-1$; hence $phige0$. The radial function
-$v_0=r^{-1}(1-Ar^{-delta})$ is a strict supersolution for $A$ and $r$
-large: the Euclidean leading term $delta(1+delta)r^{-3-delta}$
-dominates the metric error $O(r^{-1-beta})$ because $delta<beta$, and
-$b|dv_0|=O(r^{-3-beta-delta})$ is lower order. Comparison on the end
-gives $phile Cv_0$, so $phi=O(r^{-1})$ and
-$∂_rphi=-r^{-2}+o(r^{-2})$; then $L_phi<0$ and $A_phi>0$.
-
-Strict conclusions. At $s=0$,
-$-Delta_glog u_s=-Delta_gphi
-=b√(|dphi|^2+zeta^2)+wge w$, and by continuity the same holds for
-all small $s>0$; the density gap in [eq:conformal-dec] follows. On
-$B$, $∂_nulog u_s=-s<0$, so
-$theta_+^{g_s,K_s}(B)=u_s^{-2}(theta_+(B)-4s)<0$. The energy formula
+Hopf principle at a boundary minimum requires ∂_nuphile0
+in the outer normal into N, whereas the present data give
+∂_nuphi=-1; hence phige0. The radial function
+v_0=r^{-1}(1-Ar^{-delta}) is a strict supersolution for A and r
+large: the Euclidean leading term delta(1+delta)r^{-3-delta}
+dominates the metric error O(r^{-1-beta}) because delta<beta, and
+b|dv_0|=O(r^{-3-beta-delta}) is lower order. Comparison on the end
+gives phile Cv_0, so phi=O(r^{-1}) and
+∂_rphi=-r^{-2}+o(r^{-2}); then L_phi<0 and A_phi>0.
+
+Strict conclusions. At s=0,
+-Delta_glog u_s=-Delta_gphi
+=b√(|dphi|^2+zeta^2)+wge w, and by continuity the same holds for
+all small s>0; the density gap in [eq:conformal-dec] follows. On
+B, ∂_nulog u_s=-s<0, so
+theta_+^{g_s,K_s}(B)=u_s^{-2}(theta_+(B)-4s)<0. The energy formula
and the two-sided area bound are the standard conformal variance
-together with $u_sge1$ and $u_sle1+s|phi|_∞$.
+together with u_sge1 and u_sle1+s|phi|_∞.
Remark (Decay needed by the drift-Poisson source).
-The source $w=r^{-3-delta}>0$ of [eq:drift-poisson] must, at the
+The source w=r^{-3-delta}>0 of [eq:drift-poisson] must, at the
end of the strictification step, also dominate the conformal residual
from the [lem:n3:conformal-formulas] lemma. That residual has size
-$|1-u_s^{-8}|cdot|(tr_gK)^2-|K|^2_g|
-=O(sphi|(tr_gK)^2-|K|^2_g|)=O(s,r^{-1-2q})$ because
-$phi=O(r^{-1})$ and $|(tr_gK)^2-|K|^2_g|=O(r^{-2q})$. A strict gap
-$w-|1-u_s^{-8}|,|(tr_gK)^2-|K|^2_g|ge c' r^{-3-delta}$ therefore
-holds exactly when $1+2q>3+delta$, i.e., $q>1+delta/2$. This is
-why the primary decay hypothesis [eq:decay] is stated as $q>3/2$
-rather than $q>1/2$: $q>3/2$ gives $q>1+delta/2$ for every
-$0<delta<1$, so the strictification step is fully honest.
+|1-u_s^{-8}|cdot|(tr_gK)^2-|K|^2_g|
+=O(sphi|(tr_gK)^2-|K|^2_g|)=O(s,r^{-1-2q}) because
+phi=O(r^{-1}) and |(tr_gK)^2-|K|^2_g|=O(r^{-2q}). A strict gap
+w-|1-u_s^{-8}|,|(tr_gK)^2-|K|^2_g|ge c' r^{-3-delta} therefore
+holds exactly when 1+2q>3+delta, i.e., q>1+delta/2. This is
+why the primary decay hypothesis [eq:decay] is stated as q>3/2
+rather than q>1/2: q>3/2 gives q>1+delta/2 for every
+0<delta<1, so the strictification step is fully honest.
@@ -571,54 +571,54 @@ $0<delta<1$, so the strictification step is fully honest.
Replacement of the end by a rest-frame model [sec:end-preparation]. The comparison metric of Sections~[sec:geometry]--[sec:existence]
operates most cleanly on one-ended data with zero ADM momentum and
-compactly supported tensor, satisfying $theta_+(B)<0$ and a strict
+compactly supported tensor, satisfying theta_+(B)<0 and a strict
density gap. Starting from arbitrary data satisfying the hypotheses of
-the [thm:n3:exterior] theorem with $E>|P|_delta$, such data are produced
+the [thm:n3:exterior] theorem with E>|P|_delta, such data are produced
with energy and area converging to the originals.
Positive enclosing area [sec:n3-positive-area]. Lemma (Positive enclosing area). [lem:n3:positive-area]
-Every exterior $(Ω,g,K)$ in the [thm:n3:exterior] theorem satisfies
-$0<a_g(B)<∞$.
+Every exterior (Ω,g,K) in the [thm:n3:exterior] theorem satisfies
+0<a_g(B)<∞.
Proof.
-Finiteness: a distant coordinate sphere $S_R$ in the chart of
-[eq:decay] is an enclosing cut of area $omega_2R^2+o(R^2)$, so
-$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^∞$-norm. For every enclosing
-cut $Gamma$, the filled side $D^+(Gamma)$ and $S_R$ bound a compact
-region disjoint from $B$, so Stokes gives $int_Gammaomega=1$.
-Consequently $1le|omega|_∞|Gamma|_g$, and taking infima over
-cuts gives $a_g(B)ge1/|omega|_∞>0$.
+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)<∞. 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^∞-norm. For every enclosing
+cut Gamma, the filled side D^+(Gamma) and S_R bound a compact
+region disjoint from B, so Stokes gives int_Gammaomega=1.
+Consequently 1le|omega|_∞|Gamma|_g, and taking infima over
+cuts gives a_g(B)ge1/|omega|_∞>0.
Removal of momentum and compact support [sec:n3-rest-frame]. Proposition (Rest-frame reduction). [prop:n3:rest-frame]
-Suppose $(Ω,g,K)$ satisfies the hypotheses of
-the [thm:n3:exterior] theorem with $E>|P|_delta$ and invariant mass $m$.
-For each $ε>0$, there is one-ended exterior $(Ω',g',K')$ with
-smooth compact boundary $B$ and:
-(i) $(Ω',g',K')$ agrees with $(Ω,g,K)$ in a neighborhood of
- $B$ and is supported on a subset of $Ω$ differing from it by
+Suppose (Ω,g,K) satisfies the hypotheses of
+the [thm:n3:exterior] theorem with E>|P|_delta and invariant mass m.
+For each ε>0, there is one-ended exterior (Ω',g',K') with
+smooth compact boundary B and:
+(i) (Ω',g',K') agrees with (Ω,g,K) in a neighborhood of
+ B and is supported on a subset of Ω differing from it by
a compact end-piece diffeomorphism;
-(ii) DEC with strict gap $mu'-|J'|_{g'}ge c'r^{-3-delta'}$ for
- $c'>0$ and some $0<delta'<1$;
-(iii) $theta_+^{g',K'}(B)<0$;
-(iv) $K'$ is compactly supported;
-(v) $P'=0$ and $m'=E'=m+O(ε)$;
-(vi) $a_{g'}(B)ge (1-ε)a_g(B)$.
+(ii) DEC with strict gap mu'-|J'|_{g'}ge c'r^{-3-delta'} for
+ c'>0 and some 0<delta'<1;
+(iii) theta_+^{g',K'}(B)<0;
+(iv) K' is compactly supported;
+(v) P'=0 and m'=E'=m+O(ε);
+(vi) a_{g'}(B)ge (1-ε)a_g(B).
@@ -626,39 +626,39 @@ smooth compact boundary $B$ and:
Proof.
The first step is strictification. By the [lem:n3:strictification] lemma
-at parameter $s>0$, the data $(g_s,K_s)=(u_s^4g,u_s^2K)$ satisfy DEC
-with a density gap, satisfy $theta_+^{g_s,K_s}(B)<0$, and have
-$a_g(B)le a_{g_s}(B)le(1+s|phi|_∞)^4a_g(B)$. Choose $s>0$
-small enough that $(1+s|phi|_∞)^4le 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 $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 $xmapstoΛ x$ acts linearly on the end
-and is trivial on a fixed neighborhood of $B$. With respect to
-$(g_s,K_s)$ it produces isometric data with the same DEC, the same
-$theta_+(B)le0$, and ADM vector $Λ(E_s,P)=(m_s,0)$, where
-$m_s=√(E_s^2-|P|_delta^2)$.
-
-The third step is compact support. Interpolate $K_s$ across a distant
-annulus $Rle rle 2R$ between its values near $B$ and zero on the far
+at parameter s>0, the data (g_s,K_s)=(u_s^4g,u_s^2K) satisfy DEC
+with a density gap, satisfy theta_+^{g_s,K_s}(B)<0, and have
+a_g(B)le a_{g_s}(B)le(1+s|phi|_∞)^4a_g(B). Choose s>0
+small enough that (1+s|phi|_∞)^4le 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 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 xmapstoΛ x acts linearly on the end
+and is trivial on a fixed neighborhood of B. With respect to
+(g_s,K_s) it produces isometric data with the same DEC, the same
+theta_+(B)le0, and ADM vector Λ(E_s,P)=(m_s,0), where
+m_s=√(E_s^2-|P|_delta^2).
+
+The third step is compact support. Interpolate K_s across a distant
+annulus Rle rle 2R between its values near B and zero on the far
tail, while keeping the metric unchanged. The interpolation introduces
-a constraint violation bounded by $O(R^{-2})$ on the annulus, and a
-conformal repair $u_Rge1$ supported on $Rle rle 3R$ with
-$u_Rto1$ on the far tail restores DEC, with strictly positive local
-gain on the far end and a mass cost of $O(R^{-2})=O(R^{-2})$; the
+a constraint violation bounded by O(R^{-2}) on the annulus, and a
+conformal repair u_Rge1 supported on Rle rle 3R with
+u_Rto1 on the far tail restores DEC, with strictly positive local
+gain on the far end and a mass cost of O(R^{-2})=O(R^{-2}); the
existence of such a repair is [sec:strictification]'s
generalisation to a compactly supported source (the drift term is
absorbed by the same truncated maximum-principle argument, and the
-outer flux remains negative and of size $-4pi$).
+outer flux remains negative and of size -4pi).
-Combining the three steps and letting $Rto∞$ gives (iv) and (vi).
-Property (v) follows from $sto0$: $E_s=E+O(ε)$ for small $s$, and
-$m'=m_s=m+O(ε)$; the two loss factors
-$(1+s|phi|_∞)^4$ and the $O(R^{-2})$ repair cost are absorbed
-into $ε$ by taking $s$ and $R^{-2}$ small enough. Property (ii)
+Combining the three steps and letting Rto∞ gives (iv) and (vi).
+Property (v) follows from sto0: E_s=E+O(ε) for small s, and
+m'=m_s=m+O(ε); the two loss factors
+(1+s|phi|_∞)^4 and the O(R^{-2}) repair cost are absorbed
+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 $ε>0$ there exists data satisfying all the
+E>|P|_delta, for every ε>0 there exists data satisfying all the
strict hypotheses of the [thm:n3:strict-exterior] theorem with
-$m'=m+O(ε)$ and $a_{g'}(B)ge(1-ε)a_g(B)$.
+m'=m+O(ε) and a_{g'}(B)ge(1-ε)a_g(B).
@@ -687,8 +687,8 @@ Immediate from the [prop:n3:rest-frame] proposition.
Geometric preparation: two colors and collars [sec:geometry]. The comparison metric of the [sec:elliptic] section is built on an
exterior cut off at a distant sphere and prepared so that its two
-families of boundary faces have strict negative expansions, one for $K$
-and one for $-K$. The mechanism is the trapped-region construction of
+families of boundary faces have strict negative expansions, one for K
+and one for -K. The mechanism is the trapped-region construction of
Andersson--Eichmair--Metzger, employed here at a threshold near zero.
@@ -698,43 +698,43 @@ Total regions inside a compact barrier
Definition (Total region). [def:total-region]
-Let $(X,Q)$ be a compact smooth Riemannian three-manifold with connected
-smooth outer boundary $S_0$ and smooth compact inner boundary
-$∂ X∖ S_0$, and let $DsubsetintX$ be a
-closed set on which $Q$ vanishes outside a fixed neighborhood. For
-$cinℝ$ define the total region
-$$
+Let (X,Q) be a compact smooth Riemannian three-manifold with connected
+smooth outer boundary S_0 and smooth compact inner boundary
+∂ X∖ S_0, and let DsubsetintX be a
+closed set on which Q vanishes outside a fixed neighborhood. For
+cinℝ define the total region
+
T_c(X,Q)=∩{E: E closed,,Dsubset E,,∂ E
smooth,, H_{∂ E}+tr_{∂ E}Qle c on ∂ E}.
-$$
+
Lemma (Compact total regions). [lem:n3:compact-total]
-Suppose there is a smooth closed $F$ with $DsubsetintF$
-and $Θ_Q(∂ F)<c$ for the outward normal. Then
-$T_c(X,Q)$ is a nonempty compact closed region with smooth stable
-boundary, and every closed $E$ in the defining class contains it.
+Suppose there is a smooth closed F with DsubsetintF
+and Θ_Q(∂ F)<c for the outward normal. Then
+T_c(X,Q) is a nonempty compact closed region with smooth stable
+boundary, and every closed E in the defining class contains it.
Proof.
-As an external input, the smooth-frontier theorem of Andersson--Eichmair--Metzger is used: if a complete $3$-manifold with
+As an external input, the smooth-frontier theorem of Andersson--Eichmair--Metzger is used: if a complete 3-manifold with
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^3times[0,∞)$ extends
-$X$ to a complete boundaryless data set. A smooth Morse height $q$ with
-$q=0$ on $∂ X$, increasing in an outward collar, and radial on the
+frontiers of the compact trapped subregions. Pasting outward cobordisms and an open S^3times[0,∞) extends
+X to a complete boundaryless data set. A smooth Morse height q with
+q=0 on ∂ X, increasing in an outward collar, and radial on the
tail dominates any closed weakly trapped region: it forces it to remain a
-positive distance inside $X$ by the maximum-principle argument applied to
-$q$ on the region. The Andersson--Eichmair--Metzger smooth-frontier
+positive distance inside X by the maximum-principle argument applied to
+q on the region. The Andersson--Eichmair--Metzger smooth-frontier
theorem then produces the outermost smooth stable frontier, which is
the closed intersection.
@@ -743,9 +743,9 @@ the closed intersection.
Lemma (Threshold continuity). [lem:n3:threshold-continuity]
-On any open interval of thresholds $c$ where the strict-barrier
+On any open interval of thresholds c where the strict-barrier
condition of the [lem:n3:compact-total] lemma holds, the closed regions
-$T_c(X,Q)$ increase with $c$ and are right-continuous in Hausdorff
+T_c(X,Q) increase with c and are right-continuous in Hausdorff
distance except at countably many thresholds.
@@ -754,10 +754,10 @@ 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
-$cmapstodist(x_j,T_c(X,Q))$ has at most countably many jumps. Away
+set {x_j} in X. For each j, the monotone function
+cmapstodist(x_j,T_c(X,Q)) has at most countably many jumps. Away
from the (countable) union of these exceptional sets and away from an
-eventual emptiness threshold, right-continuity at every $x_j$ and the
+eventual emptiness threshold, right-continuity at every x_j and the
common Lipschitz bound give right-continuity in Hausdorff distance.
@@ -770,40 +770,40 @@ The two-color exterior
Proposition (Two-color preparation). [prop:n3:two-color]
-Let $(Ω,g,K)$ be strict data satisfying the hypotheses of
-the [thm:n3:strict-exterior] theorem. Fix $eta_0>0$ small. There is a
-closed connected exterior $Ω_0subsetΩ$ with one
-asymptotically flat end, nonempty compact smooth boundary $B_0$,
-decomposition $B_0=B_{rm black}cup B_{rm white}$, and constants
-$c_b,c_win(-eta_0,0)$ such that, for the normal $nu_0$ into $Ω_0$
-and $P_{B_0}=tr_{B_0}K$,
-$$
+Let (Ω,g,K) be strict data satisfying the hypotheses of
+the [thm:n3:strict-exterior] theorem. Fix eta_0>0 small. There is a
+closed connected exterior Ω_0subsetΩ with one
+asymptotically flat end, nonempty compact smooth boundary B_0,
+decomposition B_0=B_{rm black}cup B_{rm white}, and constants
+c_b,c_win(-eta_0,0) such that, for the normal nu_0 into Ω_0
+and P_{B_0}=tr_{B_0}K,
+
H_{B_0}+P_{B_0}=c_b on B_{rm black},
-qquad
+
H_{B_0}-P_{B_0}=c_w on B_{rm white}.
-$$
-On each face there is a smooth disjoint collar $(B_0^jtimes[0,1),s)$
-with $|d s|>0$ and monotone expansion toward $Ω_0$.
+
+On each face there is a smooth disjoint collar (B_0^jtimes[0,1),s)
+with |d s|>0 and monotone expansion toward Ω_0.
Proof.
-Choose a distant sphere $S_{R_0}$ with positive mean curvature for
-both $K$ and $-K$, beyond the support of $K$. Fill behind $B$ to
-obtain a compact $X$ with outer boundary $S_{R_0}$ and inner boundary
-$B$. Choose $c_bin(-eta_0,0)$ at a right-continuity threshold of
-$T_c(X,K)$, and put (Y=X∖intT_{c_b}(X,K)).
-All boundary components of $Y$ have strictly positive expansion for
-$-K$, so a second right-continuity threshold $c_win(-eta_0,0)$ of
-$T_c(Y,-K)$ gives a nonempty closed region. Let
-$Ω_0$ be the closure of the exterior component of
-$X∖(T_{c_b}(X,K)cup T_{c_w}(Y,-K))$. Its retained $c_b$-faces
-are black and its retained $c_w$-faces are white, with the stated
-expansions for the normal into $Ω_0$ (the signs follow from the
+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_bin(-eta_0,0) at a right-continuity threshold of
+T_c(X,K), and put (Y=X∖intT_{c_b}(X,K)).
+All boundary components of Y have strictly positive expansion for
+-K, so a second right-continuity threshold c_win(-eta_0,0) of
+T_c(Y,-K) gives a nonempty closed region. Let
+Ω_0 be the closure of the exterior component of
+X∖(T_{c_b}(X,K)cup T_{c_w}(Y,-K)). Its retained c_b-faces
+are black and its retained c_w-faces are white, with the stated
+expansions for the normal into Ω_0 (the signs follow from the
orientations of the total region and its complement). The stable
collars are the Andersson--Eichmair--Metzger stability directions
shrunk to small amplitude.
@@ -812,8 +812,8 @@ shrunk to small amplitude.
-The coupled elliptic comparison system [sec:elliptic]. Fix a two-color exterior $Ω_0$ of the [prop:n3:two-color] proposition.
-Its boundary $B_0=B_{rm black}cup B_{rm white}$ carries two strict
+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.
@@ -822,74 +822,74 @@ Variables and the equations
-Fix a smooth boundary height $h_*:B_0toℝ$ with
-$h_*=b_-<0$ on black and $h_*=b_+>0$ on white, extended smoothly to
-$Ω_0$ with $∂_nu h_*<0$ on $B_0$ and $h_*to0$ at
-infinity. Fix $eta_n=e^{-ε n}>0$.
+Fix a smooth boundary height h_*:B_0toℝ with
+h_*=b_-<0 on black and h_*=b_+>0 on white, extended smoothly to
+Ω_0 with ∂_nu h_*<0 on B_0 and h_*to0 at
+infinity. Fix eta_n=e^{-ε n}>0.
+
+Let l:ℝto(0,∞) be smooth, with l(t)=e^{nt} for tle0 and
+l(t) constant between 1 and e for tge1/n; set p=l'/l, so
+0le ple n and p is compactly supported. For a C^2 function
+f:Ω_0toℝ and t:Ω_0toℝ, define
-Let $l:ℝto(0,∞)$ be smooth, with $l(t)=e^{nt}$ for $tle0$ and
-$l(t)$ constant between $1$ and $e$ for $tge1/n$; set $p=l'/l$, so
-$0le ple n$ and $p$ is compactly supported. For a $C^2$ function
-$f:Ω_0toℝ$ and $t:Ω_0toℝ$, define
-$$
-&sigma=nabla f,quad D=1+l(t)^2|sigma|^2,quad d=D^{-1},quad
-Z=t+tfrac16log D,quad h=eta_n f,
-&w=(l,sigma)/(sqrt D),quad a=|w|,quad v=a^2,quad
-chi=(3d)/(3+pv),quad
-A_d=I-w ⊗ w,quad
+&sigma=nabla f, D=1+l(t)^2|sigma|^2, d=D^{-1},
+Z=t+tfrac16log D, h=eta_n f,
+&w=(l,sigma)/(sqrt D), a=|w|, v=a^2,
+chi=(3d)/(3+pv),
+A_d=I-w ⊗ w,
A_chi=I-(3+p)/(3+pv),w ⊗ w.
-$$
-Set $L_0=e^{2t}l$ and $u=L_0sqrt D$.
-The comparison metric on $Ω_R=Ω_0cap{r<R}$ is [eq:comparison-metric]
-$$
+Set L_0=e^{2t}l and u=L_0sqrt D.
+
+The comparison metric on Ω_R=Ω_0cap{r<R} is [eq:comparison-metric]
+
barbar g_g_{ij}=(1)/(d)(g_{ij}+l(t)^2∂_if,∂_jf),
-$$
-the graph metric over the base $(Ω_R,g)$ with height $f$ and warp
-$l$, rescaled by the lapse $1/d$.
-The unknowns $(f,t)$ satisfy the coupled system [eq:coupled-system]
-$$
+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]
+
-Div_g(3u,A_chi,nabla t)=u,Xi &in Ω_R,$$
+Div_g(3u,A_chi,nabla t)=u,Xi &in Ω_R,
1mm]
-l,Delta_g f+(bulk terms)=eta_n,Xi_2&in Ω_R,$$
+l,Delta_g f+(bulk terms)=eta_n,Xi_2&in Ω_R,
1mm]
-t=h_*/eta_n on B_0,quad t=0 on S_R,
+t=h_*/eta_n on B_0, t=0 on S_R,
-$$
-where $Xi,Xi_2$ are smooth functions of $(f,t)$ and their first
+
+where Xi,Xi_2 are smooth functions of (f,t) and their first
derivatives fixed by the two curve identities below. The exact
-right-hand sides encode the coupling to $K$ and to the two-color
-boundary expansions of $Ω_0$.
+right-hand sides encode the coupling to K and to the two-color
+boundary expansions of Ω_0.
Lemma (The Hessian square is coercive). [lem:n3:hessian-square]
-Put $H^f=Hess_gf$ and let
-$mathcal T=calT=tfrac12{(tr_{A_d}H^f)^2-|H^f|_{A_d ⊗ A_d}^2}$ be
-the second-form square of the comparison graph. Then there is $c_0(n)>0$
+Put H^f=Hess_gf and let
+mathcal T=calT=tfrac12{(tr_{A_d}H^f)^2-|H^f|_{A_d ⊗ A_d}^2} be
+the second-form square of the comparison graph. Then there is c_0(n)>0
such that [eq:hessian-square-coercive]
-$$
+
mathcal TcalTge c_0((tr_gH^f)^2+|nabla w|_{A_chi}^2)
-$$
-and $mathcal TcalTle C(n)(|nabla f|^2+|Hess_gf|^2)$.
+
+and mathcal TcalTle C(n)(|nabla f|^2+|Hess_gf|^2).
Proof.
-Write $A_d$ and $A_chi$ in a frame where $A_d=diag(1,1,1,d)$ and
-$A_chi=diag(1,1,1,chi)$. Direct contraction gives
-$$
+Write A_d and A_chi in a frame where A_d=diag(1,1,1,d) and
+A_chi=diag(1,1,1,chi). Direct contraction gives
+
mathcal T=calT=tfrac12|H^f_perp|^2+chi,(H_{ee}^f)^2,
-$$
-where $H^f_perp$ is the trace-free part on the two transverse directions
-and $H_{ee}^f$ is the leading axial entry. Bounding
-$chige 3d/(3+n)ge c(n)>0$ from below and using the identity
-$chi(1-chi)=O(vd)$ for the off-diagonal entries, the lower bound
+
+where H^f_perp is the trace-free part on the two transverse directions
+and H_{ee}^f is the leading axial entry. Bounding
+chige 3d/(3+n)ge c(n)>0 from below and using the identity
+chi(1-chi)=O(vd) for the off-diagonal entries, the lower bound
[eq:hessian-square-coercive] follows. The upper bound is
immediate from the definition.
@@ -898,13 +898,13 @@ immediate from the definition.
Lemma (Boundary identity). [lem:n3:boundary-identity]
-On a face $B_0^j$ of the boundary $B_0$, with $H$ the mean curvature
-into $Ω_0$, $P_{B_0^j}=tr_{B_0^j}K$, and $w_nu=w(nu)$, the
-first normal variation of $t$ satisfies [eq:boundary-identity]
-$$
+On a face B_0^j of the boundary B_0, with H the mean curvature
+into Ω_0, P_{B_0^j}=tr_{B_0^j}K, and w_nu=w(nu), the
+first normal variation of t satisfies [eq:boundary-identity]
+
V_nu/u=3d,∂_nu t+w_nu(F-P_{B_0^j}),
-$$
-where $F=tr_gK$.
+
+where F=tr_gK.
@@ -912,16 +912,16 @@ where $F=tr_gK$.
Proof.
The metric and divergence formulas give, for the covariant trace
-$V_nu$ of the boundary flux of $t$,
-$$
+V_nu of the boundary flux of t,
+
V_nu/u=3d,∂_nu t+w_nu(F-P_{B_0^j}),
-$$
-the first term from the direct variation of $t$ across the boundary and
-the second from the tangential trace of $K$. Using the boundary expansion
-$H_{B_0^j}+P_{B_0^j}=c_j$, this becomes a strict negative first-variation,
-compatible with the Dirichlet data on $∂Ω_R$.
+
+the first term from the direct variation of t across the boundary and
+the second from the tangential trace of K. Using the boundary expansion
+H_{B_0^j}+P_{B_0^j}=c_j, this becomes a strict negative first-variation,
+compatible with the Dirichlet data on ∂Ω_R.
@@ -933,228 +933,228 @@ A priori bounds
Proposition (A priori bounds). [prop:n3:apriori]
-For every $n$ there is a constant $C(n)<∞$ such that every smooth
-solution $(f,t)$ of [eq:coupled-system] satisfies
-$$
+For every n there is a constant C(n)<∞ such that every smooth
+solution (f,t) of [eq:coupled-system] satisfies
+
--εle tle Zle C(n),quad
+-εle tle Zle C(n),
|f|+|nabla f|+|Hess_gf|le C(n)
quadon barbarΩ_R.Ω_R.
-$$
-A quantity $Pi_n$ below is bounded by $C(n+1)^N$, with $C,N$ depending
-only on fixed data, $n$, $ell$, and the fixed collars; another quantity
-$C(n)$ may also depend on the truncation radius in a way that will be
+
+A quantity Pi_n below is bounded by C(n+1)^N, with C,N depending
+only on fixed data, n, ell, and the fixed collars; another quantity
+C(n) may also depend on the truncation radius in a way that will be
declared explicitly. The parameter comparisons
-$$
- ell=e^{-ε n},qquad tau=ell^{3/2},qquad
- ellle lle e,qquad 0le ple n,qquad
+
+ ell=e^{-ε n}, tau=ell^{3/2},
+ ellle lle e, 0le ple n,
(4d)/(4+n)lechile d
-$$
-enter through the small factor $Pi_ntau/ellto0$, which absorbs only
+
+enter through the small factor Pi_ntau/ellto0, which absorbs only
polynomial losses.
Lemma (Uniform separation). [lem:n3:apriori:separation]
-Let $AsubsetbarbarΩ$Ω$ be compact. In the first two homotopy stages
-one has the following. If $A$ avoids the full black total region
-$mathcal B_calB_{c_b}$ at threshold $c_b$, then $hge b_-+eta_A$ on $A$
-for some $eta_A>0$, uniformly in $n$ large, every sufficiently large
-truncation, and every admissible solution. If $A$ avoids
-$mathcal W_calW_{c_w}$, then $hle b_+-eta_A$.
+Let AsubsetbarbarΩΩ be compact. In the first two homotopy stages
+one has the following. If A avoids the full black total region
+mathcal B_calB_{c_b} at threshold c_b, then hge b_-+eta_A on A
+for some eta_A>0, uniformly in n large, every sufficiently large
+truncation, and every admissible solution. If A avoids
+mathcal W_calW_{c_w}, then hle b_+-eta_A.
Proof.
-Take an arbitrary sequence $n_ito∞$, $R_ito∞$ of admissible
+Take an arbitrary sequence n_ito∞, R_ito∞ of admissible
stage-one or stage-two solutions. The elementary height and end bounds
from the preparation give
-$|h_i|le C_0$ and $|h_i|le C(r^{-gamma}-R_i^{-gamma})$ on the end,
-for some $gammain(3/4,1)$. Extend $h_i$ across a white face by the
-constant $b_+$. Define the lower relaxed limit
-$underline h(x)=liminf_{yto x} h_i(y)$ over such sequences.
-
-Fix $b_-<k'<min{0,b_-+eta}$ with $eta$ the low-height sign range
-of the add-on $D_0$. Suppose a smooth $phi$ touches $underline h$
-from below at $x$ with $underline h(x)le k'$ and $dphi(x)ne0$.
+|h_i|le C_0 and |h_i|le C(r^{-gamma}-R_i^{-gamma}) on the end,
+for some gammain(3/4,1). Extend h_i across a white face by the
+constant b_+. Define the lower relaxed limit
+underline h(x)=liminf_{yto x} h_i(y) over such sequences.
+
+Fix b_-<k'<min{0,b_-+eta} with eta the low-height sign range
+of the add-on D_0. Suppose a smooth phi touches underline h
+from below at x with underline h(x)le k' and dphi(x)ne0.
After a small fourth-order localization, minimization on a small closed
-ball produces $x_ito x$ and $c_ito0$ with $phi+c_i$ a strict lower
-contact of $h_i$ at $x_i$. By the choice of $k'<b_+$, such contacts
+ball produces x_ito x and c_ito0 with phi+c_i a strict lower
+contact of h_i at x_i. By the choice of k'<b_+, such contacts
occur in the interior trace equation, never on a white face or on its
constant extension. At the contact,
-$$
- nabla f_i=tau^{-1}nablaphi,quad
- d_ile(tau^2)/(ell^2|dphi(x_i)|^2)to0,quad
- a_ito1,quad chi_ito0.
-$$
-Positivity of $A_{chi_i}$ and the Hessian inequality at a lower
+ nabla f_i=tau^{-1}nablaphi,
+ d_ile(tau^2)/(ell^2|dphi(x_i)|^2)to0,
+ a_ito1, chi_ito0.
+
+
+Positivity of A_{chi_i} and the Hessian inequality at a lower
contact give
-$$
+
tr_{A_{chi_i}}K+
frac{l_i}{tau√(D_i)}tr_{A_{chi_i}}Hessphi
le F_i(x_i).
-$$
-Since $D_0le0$ at these low values, $limsup F_i(x_i)le k'+C_b$, so
-$$
+
+Since D_0le0 at these low values, limsup F_i(x_i)le k'+C_b, so
+
frac{tr_{(dphi)^perp}Hessphi}{|dphi|}
+tr_{(dphi)^perp}Kle k'+C_b.
-$$
-This is the oriented level expansion of $phi$ towards increasing
-$phi$, and the localization can be removed. The only Hessian in this
+
+This is the oriented level expansion of phi towards increasing
+phi, and the localization can be removed. The only Hessian in this
passage is the fixed test's.
-Transfer to the closed sublevel $E_k={underline hle k}$ for
-$b_-<k<k'$. For integers $j$,
-$$
+Transfer to the closed sublevel E_k={underline hle k} for
+b_-<k<k'. For integers j,
- psi_j(s)=frac{1}{1+e^{-j^2(s-k-j^{-1})}},qquad
+
+ psi_j(s)=frac{1}{1+e^{-j^2(s-k-j^{-1})}},
v_j=psi_j(underline h).
-$$
-The lower relaxed limit of $v_j$ is the indicator of $E_k$: at points
-of $E_k$ it vanishes, off $E_k$ it converges uniformly to one. If
-$varphi$ has nonzero gradient and is a lower test of the indicator at
+
+The lower relaxed limit of v_j is the indicator of E_k: at points
+of E_k it vanishes, off E_k it converges uniformly to one. If
+varphi has nonzero gradient and is a lower test of the indicator at
a zero value, the same strict contact localization gives lower tests
-$varphi+c_j$ of $v_j$ at $y_jto x$ with $v_j(y_j)to0$. Since
-$psi_j(k')to1$, one has $underline h(y_j)<k'$ eventually.
-Composition with the increasing $psi_j$ preserves the oriented level
+varphi+c_j of v_j at y_jto x with v_j(y_j)to0. Since
+psi_j(k')to1, one has underline h(y_j)<k' eventually.
+Composition with the increasing psi_j preserves the oriented level
expansion:
-$$
+
Hess(thetacircvarphi)=theta'Hessvarphi
- +theta''dvarphi ⊗ dvarphi,qquad
+ +theta''dvarphi ⊗ dvarphi,
frac{tr_{(dvarphi)^perp}Hess(thetacircvarphi)}
{|d(thetacircvarphi)|}
=frac{tr_{(dvarphi)^perp}Hessvarphi}{|dvarphi|}.
-$$
-So $psi_j^{-1}(varphi+c_j)$ is a smooth lower test of
-$underline h$ with contact $<k'$, and the oriented-level inequality
-passes to $varphi$. Hence every smooth exterior support of $E_k$ has
-expansion at most $k'+C_b$.
+
+So psi_j^{-1}(varphi+c_j) is a smooth lower test of
+underline h with contact <k', and the oriented-level inequality
+passes to varphi. Hence every smooth exterior support of E_k has
+expansion at most k'+C_b.
The argument also rules out a hidden layer at a white face: the
reversed white face would be an exterior support with expansion
-$-H_B+P_B=-c_w>0$, contradicting $k'+C_b<0$ for $k'$ close to $b_-$.
-The end bound makes $E_k$ compact.
+-H_B+P_B=-c_w>0, contradicting k'+C_b<0 for k' close to b_-.
+The end bound makes E_k compact.
Adjoin the entire black total region and fill any bounded complementary
pockets. At a point of the black frontier, an exterior support of the
union contains the smooth black region locally, so tangency bounds the
-expansion by $c_b<k'+C_b$. Everywhere else the sublevel argument
+expansion by c_b<k'+C_b. Everywhere else the sublevel argument
applies. The resulting compact closed set contains the original inner
barrier with a collar and avoids the outer barriers. Lemma on weak
support then places it in a smooth black trapped region at any
-threshold $c''>k'+C_b$ close enough to $c_b$, hence in
-$mathcal B_calB_{c''}$.
+threshold c''>k'+C_b close enough to c_b, hence in
+mathcal B_calB_{c''}.
-Hausdorff right continuity of $cmapstomathcal B_c$calB_c$ at $c_b$ supplies
-$delta>0$ with $mathcal B_calB_{c_b+delta}cap A=∅$; shrink
-$delta$ so the enlarged threshold is still negative. Choose
-$k,k',c''$ with $b_-<k<k'<k'+C_b<c''<c_b+delta$. Then
-$E_kcap A=∅$ for every sequence in the class. A failing
-uniform estimate would produce $x_iin A$ with $h_i(x_i)le k$ and a
-subsequence $x_ito xin A$, so $underline h(x)le k$, contradiction.
-This gives $hge b_-+eta_A$ on $A$. Replacing $(h,K,C)$ by
-$(-h,-K,-C)$ and interchanging colors gives the upper assertion.
+Hausdorff right continuity of cmapstomathcal B_ccalB_c at c_b supplies
+delta>0 with mathcal B_calB_{c_b+delta}cap A=∅; shrink
+delta so the enlarged threshold is still negative. Choose
+k,k',c'' with b_-<k<k'<k'+C_b<c''<c_b+delta. Then
+E_kcap A=∅ for every sequence in the class. A failing
+uniform estimate would produce x_iin A with h_i(x_i)le k and a
+subsequence x_ito xin A, so underline h(x)le k, contradiction.
+This gives hge b_-+eta_A on A. Replacing (h,K,C) by
+(-h,-K,-C) and interchanging colors gives the upper assertion.
Lemma (Signed collar slopes). [lem:n3:apriori:collars]
-There are fixed $c,C_*>0$ such that, for large $n$ and every allowed
+There are fixed c,C_*>0 such that, for large n and every allowed
truncation, the first two stages satisfy
-$$
+
c/taule∂_nu fle C_*/tau
- on black faces,qquad
+ on black faces,
c/taule-∂_nu fle C_*/tau
on white faces,
-$$
-and the third stage satisfies $|∂_nu f|le C_*/tau$.
+
+and the third stage satisfies |∂_nu f|le C_*/tau.
Proof.
-Write $s$ for the leaf parameter in an outward collar and
-$N=nabla s/|ds|$. At a comparison contact with $h_0=b_-+psi(s)$
-($psi'>0$), the shared gradient gives
-$$
+Write s for the leaf parameter in an outward collar and
+N=nabla s/|ds|. At a comparison contact with h_0=b_-+psi(s)
+(psi'>0), the shared gradient gives
+
- dle Ctau^2/(ell^2(psi')^2),quad
- 1-ale d,quad nchige c_1 dquad(nge4).
+ dle Ctau^2/(ell^2(psi')^2),
+ 1-ale d, nchige c_1 dquad(nge4).
-$$
-With slopes in a fixed positive interval this is $dle Cell$.
+
+With slopes in a fixed positive interval this is dle Cell.
The trace operator evaluated on this test is
-$$
+
P_s+aH_s+chi K(N,N)+
achi!((Hess s(N,N))/(|ds|)
+|ds|(psi'')/(psi')),
-$$
-where $P_s=tr K$ on the leaf tangents and $H_s$ its mean curvature.
-For a negative slope the terms with prefactor $a$ flip sign. On a
-black collar $H_s+P_sge c_b$ and $C=C_b-k_Bs$. Choose small positive
-slopes with $psi(0)=0$ and $psi(s)le k_Bs/2$. A fixed short collar
+
+where P_s=tr K on the leaf tangents and H_s its mean curvature.
+For a negative slope the terms with prefactor a flip sign. On a
+black collar H_s+P_sge c_b and C=C_b-k_Bs. Choose small positive
+slopes with psi(0)=0 and psi(s)le k_Bs/2. A fixed short collar
and the [lem:n3:apriori:separation] lemma let the test lie below the
-solution at its outer end; the add-on $D_0$ vanishes in the compatible
+solution at its outer end; the add-on D_0 vanishes in the compatible
direction. The residual of the trace equation on the test is at least
-$$
+
k_Bs/2-Cd+achi|ds|psi''/psi'.
-$$
-Pick a smooth $beta_n(s)$ equal to $An$ for $0le sle1/n$, supported
-in $[0,2/n]$, with $intbeta_nle2A$. Set
-$$
+
+Pick a smooth beta_n(s) equal to An for 0le sle1/n, supported
+in [0,2/n], with intbeta_nle2A. Set
+
psi'(s)=mexp!(int_0^sbeta_n(q)d q).
-$$
-Fix $A$ so the last term dominates $Cd$ on $sle1/n$, using
-$age1/2$ there; on $sge1/n$ the linear term dominates
-$Cdle Cell$ for large $n$. The slope multiplier is at most
-$e^{2A}$, so a fixed small $m>0$ preserves all previous slope bounds.
-This gives a strict lower barrier with slopes in $[me^{-A},me^{A}]$
-independent of $n$.
-
-An upper barrier from $b_-$ uses
-$psi'(s)=Mexp(-int_0^sbeta_n(q)d q)$. The leaf expansion differs
-from its boundary value by $O(s)$; take $M$ so large that
-$psi(s)ge C_2s$ dominates this, the offset variation, and the outer
-height bound. The logarithmic-slope term is negative: on $sle1/n$ it
-dominates $Cd$, and beyond $s=1/n$ the $s$-margin does. A strict upper
-barrier results, with slopes in $[Me^{-2A},M]$.
-
-At a negative interior minimum of $h-h_0$, Hessians compare, $Z$ agrees,
-and $t,l,A_chi$ agree; height monotonicity of the right side turns
+
+Fix A so the last term dominates Cd on sle1/n, using
+age1/2 there; on sge1/n the linear term dominates
+Cdle Cell for large n. The slope multiplier is at most
+e^{2A}, so a fixed small m>0 preserves all previous slope bounds.
+This gives a strict lower barrier with slopes in [me^{-A},me^{A}]
+independent of n.
+
+An upper barrier from b_- uses
+psi'(s)=Mexp(-int_0^sbeta_n(q)d q). The leaf expansion differs
+from its boundary value by O(s); take M so large that
+psi(s)ge C_2s dominates this, the offset variation, and the outer
+height bound. The logarithmic-slope term is negative: on sle1/n it
+dominates Cd, and beyond s=1/n the s-margin does. A strict upper
+barrier results, with slopes in [Me^{-2A},M].
+
+At a negative interior minimum of h-h_0, Hessians compare, Z agrees,
+and t,l,A_chi agree; height monotonicity of the right side turns
positive residual into a contradiction. Positive maxima reverse signs.
-The one-sided derivative at $s=0$ proves the black part; replacing
-$(h,K,C)$ by $(-h,-K,-C)$ proves the white part. In the third stage
-the boundary height is an interpolated constant $b_zeta$; tests
-$b_zetapmpsi(s)$ with large $psi'$ and
-$(logpsi')'=-beta_n$ give nonpositive/nonnegative residuals at $s=0$
-by the directional inequalities of the homotopy. The same $dle Cell$
+The one-sided derivative at s=0 proves the black part; replacing
+(h,K,C) by (-h,-K,-C) proves the white part. In the third stage
+the boundary height is an interpolated constant b_zeta; tests
+b_zetapmpsi(s) with large psi' and
+(logpsi')'=-beta_n give nonpositive/nonnegative residuals at s=0
+by the directional inequalities of the homotopy. The same dle Cell
bound and a fixed test slope handle the rest. This gives the two-sided
third-stage bound.
@@ -1163,56 +1163,56 @@ third-stage bound.
Lemma (Weighted collar traces). [lem:n3:apriori:trace]
-Let $mathcal C$calC$ be a fixed disjoint union of short inner collars. In
-the first two stages, for nonnegative smooth $varphi$ supported in
-$mathcal C$,calC$,
-$$
+Let mathcal CcalC be a fixed disjoint union of short inner collars. In
+the first two stages, for nonnegative smooth varphi supported in
+mathcal C,calC,
+
int_B Lvarphid A_gle
Pi_nint_{mathcal C}calCu[(1+√(mathcal T))calT))varphi
+|dvarphi|_{A_chi}]d V_g,
- qquad
+
int_Bvarphid A_gle
Pi_nint_{mathcal C}calCsqrt D[(1+√(mathcal T))calT))varphi
+|dvarphi|_{A_chi}]d V_g.
-$$
-The estimates extend to arbitrary $varphi$ by a fixed collar cutoff
-equal to one on $B$.
+
+The estimates extend to arbitrary varphi by a fixed collar cutoff
+equal to one on B.
Proof.
-For $sigma>0$ put $e_f=nabla f/sigma$ and
-$W=lnabla f/D=sqrt d,a e_f$. Then $|W|_{barbar g}g=ale1$, and for
-every covector $xi$,
-$$
+For sigma>0 put e_f=nabla f/sigma and
+W=lnabla f/D=sqrt d,a e_f. Then |W|_{barbar g}g=ale1, and for
+every covector xi,
+
|xi(W)|le|xi|_{A_d}le√(1+n/4),|xi|_{A_chi}.
-$$
+
Differentiation gives
-$$
+
frac{Div(uW)}u=sqrt d[tr_{A_d}H^f+(dp+p_L)a∂_{e_f}t],
- qquad
+
(Div(sqrt D,W))/(sqrt D)=sqrt d[tr_{A_d}H^f+dp,a∂_{e_f}t].
-$$
-Coercivity bounds both right sides by $Pi_n(1+√(mathcal T))$:calT))$: the
-axial Hessian term has coefficient $d^{3/2}$, the transverse terms are
-traces, and the gradient term is $O(sqrt d |∂_{e_f}t|)$.
-
-Let $ε_B=1$ on a black collar and $-1$ on a white one. The
-signed slopes give $w_nu=ε_B a$ on $B$ with $age1/2$ for
-large $n$, hence
-$uW_nu=ε_B La$ and $sqrt D,W_nu=ε_B a$. Apply
-the divergence theorem to $-ε_B uWvarphi$ with a fixed
-collar cutoff. Its boundary flux is $Lavarphi$ and the outer
+
+Coercivity bounds both right sides by Pi_n(1+√(mathcal T)):calT)): the
+axial Hessian term has coefficient d^{3/2}, the transverse terms are
+traces, and the gradient term is O(sqrt d |∂_{e_f}t|).
+
+Let ε_B=1 on a black collar and -1 on a white one. The
+signed slopes give w_nu=ε_B a on B with age1/2 for
+large n, hence
+uW_nu=ε_B La and sqrt D,W_nu=ε_B a. Apply
+the divergence theorem to -ε_B uWvarphi with a fixed
+collar cutoff. Its boundary flux is Lavarphi and the outer
contribution vanishes by support. The gradient bound and divergence
-formula give the first inequality; replacing $u$ by $sqrt D$ gives the
+formula give the first inequality; replacing u by sqrt D gives the
second.
@@ -1220,71 +1220,71 @@ second.
Lemma (High-energy and Sobolev bound). [lem:n3:apriori:high-energy]
-In stage one there is $M_0=M_0(n)>0$ such that, for all $n$ large and
-all $R$ and $lambdain[0,1]$,
-$$
+In stage one there is M_0=M_0(n)>0 such that, for all n large and
+all R and lambdain[0,1],
+
Xigedelta_0mathcal T+calT+rho+tfrac12delta_0tau asigma
quadon {Z>M_0}.
-$$
-With $k_0$ a smooth cutoff supported on $[M_0,M_0+1]$ equal to one
-on $[M_0+1,∞)$,
-$$
+
+With k_0 a smooth cutoff supported on [M_0,M_0+1] equal to one
+on [M_0+1,∞),
+
int_Ω u k_0(mathcal T+calT+rho+tau asigma)d V_g
+int_Ω u k_0',|dZ|_{A_chi}^2d V_gle C(n).
-$$
+
Proof.
-Outside $supp m_0$, $Xi=delta_0mathcal T+calT+rho+delta_0tau asigma$
-is clearly $ge$ the right side. On $supp m_0$ one has
-$-εle tle -ε+1/n$ and $lasympell$; the identity
-$D=e^{8(Z-t)}$ makes $asigmato∞$ uniformly as $Zto∞$,
-while the penalty is $Pi_n$-bounded. Hence $M_0(n)$ absorbs it.
+Outside supp m_0, Xi=delta_0mathcal T+calT+rho+delta_0tau asigma
+is clearly ge the right side. On supp m_0 one has
+-εle tle -ε+1/n and lasympell; the identity
+D=e^{8(Z-t)} makes asigmato∞ uniformly as Zto∞,
+while the penalty is Pi_n-bounded. Hence M_0(n) absorbs it.
+
+Test Div V_lambda=uXi by k_0(Z). The outer boundary term
+vanishes by Z=0. The drift cross term is bounded by
-Test $Div V_lambda=uXi$ by $k_0(Z)$. The outer boundary term
-vanishes by $Z=0$. The drift cross term is bounded by
-$$
uk_0'|< dZ,K(w,cdot)>_{A_chi}|
le 2u k_0'|dZ|_{A_chi}^2+Cuk_0'|K|^2,
-$$
+
and the last term has bounded integral on a fixed compact set in the
-transition strip of $k_0$. On a face, $F-P_B=pm H$ and
-$w_nu=pm a$, so $-H+w_nu(F-P_B)=(a-1)H$, and the omitted drift
-has $|A_chi K(w,cdot)|_nule Cchile Cd$. Thus
-$$
+transition strip of k_0. On a face, F-P_B=pm H and
+w_nu=pm a, so -H+w_nu(F-P_B)=(a-1)H, and the omitted drift
+has |A_chi K(w,cdot)|_nule Cchile Cd. Thus
+
|(V_lambda)_nu|le Cud=CLsqrt dle Cfractauell L
on B.
-$$
+
Integrating,
-$$
+
int_Ω uk_0(delta_0mathcal T+calT+rho
+tfrac12delta_0tau asigma)d V_g
+2int_Ω u k_0'|dZ|_{A_chi}^2d V_g
le C(n)+Cfractauellint_B Lk_0d A_g.
-$$
+
Bound the last term using the weighted trace with the fixed collar
cutoff: it costs
-$Pi_nfractauellint_{mathcal C}calCu(k_0(1+√(mathcal T))calT))
-+k_0'|dZ|_{A_chi})d V_g$. On the fixed collars $rhogerho_0>0$,
-so $(1+√(mathcal T))calT))le C(rho+delta_0mathcal T)$calT)$ there, and
-$Pi_ntau/ellto0$ absorbs the first summand into the positive source.
+Pi_nfractauellint_{mathcal C}calCu(k_0(1+√(mathcal T))calT))
++k_0'|dZ|_{A_chi})d V_g. On the fixed collars rhogerho_0>0,
+so (1+√(mathcal T))calT))le C(rho+delta_0mathcal T)calT) there, and
+Pi_ntau/ellto0 absorbs the first summand into the positive source.
Young's inequality reduces the derivative summand to
-$uk_0'|dZ|^2_{A_chi}+C(Pi_ntau/ell)^2uk_0'$, whose second term has
+uk_0'|dZ|^2_{A_chi}+C(Pi_ntau/ell)^2uk_0', whose second term has
bounded compact integral. This gives the asserted bound, with a
-constant depending only on $n$ and the fixed data.
+constant depending only on n and the fixed data.
@@ -1292,41 +1292,41 @@ constant depending only on $n$ and the fixed data.
Proof (Proof of the [prop:n3:apriori). proposition]
The lemma above gives a level-set bound for the quantity
-$mathcal T+calT+rho+tau asigma$. The ordinary product graph
-$Gamma_f$ in $g+db_0^2$ has measure $√(1+sigma^2)d V_g$
-comparable to $sqrt Dd V_g$, and on every fixed compact base patch
-$Q$,
-$$
+mathcal T+calT+rho+tau asigma. The ordinary product graph
+Gamma_f in g+db_0^2 has measure √(1+sigma^2)d V_g
+comparable to sqrt Dd V_g, and on every fixed compact base patch
+Q,
+
int_{Gamma_fcappi^{-1}Q}e^{2Z}dGammale C_Q(n),
-$$
-using $e^{2Z}=e^{2t}D^{1/4}$, $D^{1/4}le C(n)(1+tau asigma)$, and
-the floor identity on the low-$Z$ part. The product-graph Sobolev
+
+using e^{2Z}=e^{2t}D^{1/4}, D^{1/4}le C(n)(1+tau asigma), and
+the floor identity on the low-Z part. The product-graph Sobolev
inequality (the [lem:n3:graph-sobolev] lemma) and a Moser iteration
-with $k_j=3^jk_*$ and cutoffs shrinking geometrically produce a
-uniform bound $Zle C(n)$ on a fixed compact core.
+with k_j=3^jk_* and cutoffs shrinking geometrically produce a
+uniform bound Zle C(n) on a fixed compact core.
Beyond a fixed sphere containing the drift support, the equation for
-$Z$ is elliptic with source $Xi>0$ above $M_0$; an interior maximum
-above $M_0$ has $Div(4uA_chinabla Z)=4utr_{A_chi}Hess Zle0$,
-contradicting positive source. With $Z=0$ on the outer sphere, the
+Z is elliptic with source Xi>0 above M_0; an interior maximum
+above M_0 has Div(4uA_chinabla Z)=4utr_{A_chi}Hess Zle0,
+contradicting positive source. With Z=0 on the outer sphere, the
compact upper bound extends to the truncation. Stages two and three
-have no drift; the high-$Z$ boundary bound
-$Zle t+tfrac18log(1+e^2C_*^2/tau^2)$ forces $j(t)=1$ at high
-boundary $Z$, so the prescribed conormal flux vanishes and the
-boundary point principle applies. Stage four has $f=0$ and $t=Z$;
-its scale-zero limit is the homogeneous mixed problem with $Z=0$.
+have no drift; the high-Z boundary bound
+Zle t+tfrac18log(1+e^2C_*^2/tau^2) forces j(t)=1 at high
+boundary Z, so the prescribed conormal flux vanishes and the
+boundary point principle applies. Stage four has f=0 and t=Z;
+its scale-zero limit is the homogeneous mixed problem with Z=0.
+
+Finally, Z=t+tfrac18log D together with Dge1 gives
+-εle tle Z, and the upper Z bound yields
-Finally, $Z=t+tfrac18log D$ together with $Dge1$ gives
-$-εle tle Z$, and the upper $Z$ bound yields
-$$
sigma^2leell^{-2}(e^{8(C(n)+ε)}-1),
-$$
-which is the fixed-$n$ gradient bound. The height bounds
-$|h|le C_0$ and $|f|le C_0/tau$ come from the prepared height
+
+which is the fixed-n gradient bound. The height bounds
+|h|le C_0 and |f|le C_0/tau come from the prepared height
estimates.
@@ -1334,49 +1334,49 @@ estimates.
Lemma (Graph Sobolev). [lem:n3:graph-sobolev]
-For every compactly supported smooth $varphi$ on the graph
-$Gamma_f$, [eq:graph-sobolev]
-$$
+For every compactly supported smooth varphi on the graph
+Gamma_f, [eq:graph-sobolev]
+
(int_{Gamma_f}|varphi|^4)^{1/2}
le C(n)int_{Gamma_f}(|nabla_{Gamma_f}varphi|^2
+(1+mathcal T)calT)varphi^2).
-$$
+
Proof.
-Over a fixed base patch $QSubsetΩ_R$ the ordinary product graph
-$Gamma_f$ carries the metric induced from $(Ω_R,barbar g)g)timesℝ$,
-has uniformly bounded geometry (since $barbar g$g$ does and $f$ is smooth),
-and has finite three-volume $V_Q=Vol_{Gamma_f}(pi^{-1}Q)$.
+Over a fixed base patch QSubsetΩ_R the ordinary product graph
+Gamma_f carries the metric induced from (Ω_R,barbar g)g)timesℝ,
+has uniformly bounded geometry (since barbar gg does and f is smooth),
+and has finite three-volume V_Q=Vol_{Gamma_f}(pi^{-1}Q).
By the Nash embedding theorem (Nash 1956, Theorem 2), a fixed compact enlargement
-of $Gamma_f$ embeds isometrically in some $ℝ^N$.
+of Gamma_f embeds isometrically in some ℝ^N.
./pdf/nash-1956.pdf
The standard
submanifold Sobolev embedding for compactly supported functions then
-gives, in dimension $3$,
-$$
+gives, in dimension 3,
+
(int_{Gamma_f}|varphi|^{6})^{1/3}
le Cint_{Gamma_f}|nabla_{Gamma_f}varphi|^{2},
-$$
-with $C$ depending only on the immersion geometry, hence only on $n$.
-H"older's inequality together with $V_Q<∞$ yields
-$$
+
+with C depending only on the immersion geometry, hence only on n.
+H"older's inequality together with V_Q<∞ yields
+
int_{Gamma_f}|varphi|^{4}
le V_Q^{1/3}(int_{Gamma_f}|varphi|^6)^{2/3},
-$$
-and substituting the Sobolev bound closes the estimate on $pi^{-1}Q$.
-Patching finitely many such $Q$ that exhaust the support of $varphi$
+
+and substituting the Sobolev bound closes the estimate on pi^{-1}Q.
+Patching finitely many such Q that exhaust the support of varphi
gives [eq:graph-sobolev], using
-$|varphi|^2le(1+mathcal T)|calT)|varphi|^2$ to absorb the remaining
-$L^2$ factor. The inner-face boundary is handled by the even reflection
-of $H^1_0$ across the plane where $f$ is constant, as in
+|varphi|^2le(1+mathcal T)|calT)|varphi|^2 to absorb the remaining
+L^2 factor. The inner-face boundary is handled by the even reflection
+of H^1_0 across the plane where f is constant, as in
[sec:strictification].
@@ -1389,125 +1389,125 @@ Axial regularity
Lemma (Axial gradient estimate). [lem:n3:axial]
-Let $z$ be smooth on a coordinate patch $U'$ with $A_chi:Hess z=G$,
-$A_chi=I-(1-chi)e ⊗ e$, $e=nabla z/|nabla z|$, $0<chi_0lechile1$.
-Suppose $|G|_∞le Q$ and $|nabla z|le L$. Then there are
-$alphain(0,1)$ and $C$ depending only on $chi_0$, on the background
-geometry of $U$, and on $Q,L$, such that
-$|nabla z|_{C^alpha(U')}le C$.
+Let z be smooth on a coordinate patch U' with A_chi:Hess z=G,
+A_chi=I-(1-chi)e ⊗ e, e=nabla z/|nabla z|, 0<chi_0lechile1.
+Suppose |G|_∞le Q and |nabla z|le L. Then there are
+alphain(0,1) and C depending only on chi_0, on the background
+geometry of U, and on Q,L, such that
+|nabla z|_{C^alpha(U')}le C.
Proof.
-After shrinking $U'$ and rescaling space, normal coordinates make the
-metric Euclidean at the center with $|g-delta|$ and its first
-derivatives small; the constants below depend only on $chi_0$, on
-$U'$, and on $Q,L$. Put $H=Hess z$, $P=nabla z$, and $e=P/|P|$ on
-${|P|>0}$. Since $A_chi=I-(1-chi)e ⊗ e$ and $H(e,P)=|P|H(e,e)$,
-$$
+After shrinking U' and rescaling space, normal coordinates make the
+metric Euclidean at the center with |g-delta| and its first
+derivatives small; the constants below depend only on chi_0, on
+U', and on Q,L. Put H=Hess z, P=nabla z, and e=P/|P| on
+{|P|>0}. Since A_chi=I-(1-chi)e ⊗ e and H(e,P)=|P|H(e,e),
+
A_chinablafrac{|P|^{2}}{2}-G P=(H-Delta z,g)P,
[n3:axial:gradient-flux]
-$$
-with $G$ the right side of $A_chi:H=G$, valid also at $|P|=0$.
+
+with G the right side of A_chi:H=G, valid also at |P|=0.
Taking the divergence of the right side gives
-$$
+
Div(A_chinablafrac{|P|^{2}}{2}-G P)
=|H|^{2}-(Delta z)^{2}+Ric(P,P).
[n3:axial:bochner-flux]
-$$
-Substituting $G=Delta z-(1-chi)H(e,e)$ into
-$|H|^{2}-(Delta z)^{2}=|H-(Delta z/3)g|^{2}-2(Delta z)^{2}/3$ and
-collecting the traceless part shows, for suitable $c_0,C_0>0$,
-$$
+
+Substituting G=Delta z-(1-chi)H(e,e) into
+|H|^{2}-(Delta z)^{2}=|H-(Delta z/3)g|^{2}-2(Delta z)^{2}/3 and
+collecting the traceless part shows, for suitable c_0,C_0>0,
+
|H|^{2}-(Delta z)^{2}
ge c_0|H|^{2}-C_0|G|^{2};
[n3:axial:positive-hessian]
-$$
-this requires $(1-chi_0)^{2}$ small, which is exactly what the choice
-$kappa<1$ records.
-Normalize so $|nabla z|le1$ on $B_1$ and set $s=1-|P|^{2}ge0$.
-The two identities, together with the smallness of $delta$ controlling
-the metric oscillation, the base curvature, and $|G|$, give in the weak
+this requires (1-chi_0)^{2} small, which is exactly what the choice
+kappa<1 records.
+
+Normalize so |nabla z|le1 on B_1 and set s=1-|P|^{2}ge0.
+The two identities, together with the smallness of delta controlling
+the metric oscillation, the base curvature, and |G|, give in the weak
sense
-$$
+
Div(A_chinabla s+2G,P)le-2c_0|H|^{2}+Cdelta.
[n3:axial:s-supersolution]
-$$
-The flux $A_chinabla s+2GP$ is bounded and measurable, and the
-Cordes-type $W^{2,p}$ bound used below holds uniformly once the
-ellipticity constant $chi_0>0$ is fixed.
-Suppose along a sequence with $deltato0$ one has
-$inf_{B_{1/2}}sto0$. The inhomogeneous weak Harnack inequality for
+The flux A_chinabla s+2GP is bounded and measurable, and the
+Cordes-type W^{2,p} bound used below holds uniformly once the
+ellipticity constant chi_0>0 is fixed.
+
+Suppose along a sequence with deltato0 one has
+inf_{B_{1/2}}sto0. The inhomogeneous weak Harnack inequality for
[n3:axial:s-supersolution], after dropping the favorable
-$-2c_0|H|^2$ term, gives $sto0$ in measure on $B_{1/2}$. Testing
-by $varphi^{2}(b-s)_{+}$ with a cutoff $varphi$ and fixed $b>0$, and
+-2c_0|H|^2 term, gives sto0 in measure on B_{1/2}. Testing
+by varphi^{2}(b-s)_{+} with a cutoff varphi and fixed b>0, and
absorbing the trace term by ellipticity and Young's inequality, yields
-$$
+
int_{{s<b}}varphi^{2}|Ds|^{2}
le Cb^{2}+o(1).
-$$
-Since $0le sle1$ and $Ds$ is locally $L^{2}$-bounded, $Dsto0$ in
-$L^{1}_{loc}$. Testing [n3:axial:bochner-flux] with a
-fixed cutoff equal to one on $B_{1/4}$ and using
+
+Since 0le sle1 and Ds is locally L^{2}-bounded, Dsto0 in
+L^{1}_{loc}. Testing [n3:axial:bochner-flux] with a
+fixed cutoff equal to one on B_{1/4} and using
[n3:axial:gradient-flux] and [n3:axial:positive-hessian]
-gives $int_{B_{1/4}}|H|^{2}to0$. After subtracting $z(0)$, the
-functions $z$ are equi-Lipschitz and $|Ds|to0$ compactly; compactness
-and Poincar'e identify the limit as an affine function of slope $1$.
+gives int_{B_{1/4}}|H|^{2}to0. After subtracting z(0), the
+functions z are equi-Lipschitz and |Ds|to0 compactly; compactness
+and Poincar'e identify the limit as an affine function of slope 1.
-If the alternative ``$|Ds|to0$'' failed at every step, propagate a
-nonzero unit slope. Fix this slope as $e_0$: the equation for the
+If the alternative ``|Ds|to0'' failed at every step, propagate a
+nonzero unit slope. Fix this slope as e_0: the equation for the
limit is
-$$
+
Delta_{e_0^{perp}}Y_0+chi(x),∂_{e_0e_0}Y_0=0,
-$$
-whose axial derivative $v=∂_{e_0}Y_0in W^{1,2}_{loc}$
+
+whose axial derivative v=∂_{e_0}Y_0in W^{1,2}_{loc}
satisfies a scalar uniformly elliptic divergence equation. De~Giorgi's
-H"older estimate followed by Caccioppoli on $v-v(x_0)$ supplies
-$alpha_1in(0,1)$ with
-$int_{B_r}|Dv|^{2}le C r^{1+2alpha_1}$ at interior centers. The
+H"older estimate followed by Caccioppoli on v-v(x_0) supplies
+alpha_1in(0,1) with
+int_{B_r}|Dv|^{2}le C r^{1+2alpha_1} at interior centers. The
computation in Gilbarg and Trudinger (2001, Section 8) then gives an
-affine approximation $L_1$ on $B_{lambda_0}$ with error at most
-$C blambda_0^{1+alpha_2}$ for some $alpha_2in(0,alpha_1)$ and
-fixed $lambda_0in(0,1/4)$, and slope change bounded by $Cb$.
-Iterating this improvement with slope increments summing below $1/4$
+affine approximation L_1 on B_{lambda_0} with error at most
+C blambda_0^{1+alpha_2} for some alpha_2in(0,alpha_1) and
+fixed lambda_0in(0,1/4), and slope change bounded by Cb.
+Iterating this improvement with slope increments summing below 1/4
produces, after finitely many drops plus successive affine steps,
-$$
+
sup_{B_r(x)}|z-L_{x,r}|
le C(Q+L),r^{1+alpha},
-$$
-for some $alphain(0,1)$ and an affine function $L_{x,r}$ at every
-center $xin B_{1/2}$ and radius $0<rle r_0$. Comparing consecutive
-$L_{x,r}$ gives $[Dz]_{C^{alpha}}le C$. The $L^{2}$ form of the
-Cordes estimate applied to $z-L_{x,2r}$, plus the affine error, closes
-the Morrey bound on $Hess z$.
-Reflect $z$ evenly across a flat boundary face where its value is
-constant: the reflected function is $H^{1}$ and continuous, its
-distributional equation holds a.e. with $A_chi$ even, and the plane
+for some alphain(0,1) and an affine function L_{x,r} at every
+center xin B_{1/2} and radius 0<rle r_0. Comparing consecutive
+L_{x,r} gives [Dz]_{C^{alpha}}le C. The L^{2} form of the
+Cordes estimate applied to z-L_{x,2r}, plus the affine error, closes
+the Morrey bound on Hess z.
+
+Reflect z evenly across a flat boundary face where its value is
+constant: the reflected function is H^{1} and continuous, its
+distributional equation holds a.e. with A_chi even, and the plane
carries a bounded normal-flux jump which is itself divergence of a
-bounded field $1_{{x_{4}>0}}∂_{4}$. Each preceding
+bounded field 1_{{x_{4}>0}}∂_{4}. Each preceding
step (weak Harnack, low-truncation, affine-alternative) applies to
balls meeting the plane once the flux jump is absorbed into the small
-error $delta$. Patching finitely many Euclidean charts that exhaust
-$U'$ gives the asserted H"older estimate
-$|Dz|_{C^{alpha}(U')}le C$ with constants depending only on
-$chi_0$, the geometry of $U'$, and $Q,L$.
+error delta. Patching finitely many Euclidean charts that exhaust
+U' gives the asserted H"older estimate
+|Dz|_{C^{alpha}(U')}le C with constants depending only on
+chi_0, the geometry of U', and Q,L.
@@ -1519,102 +1519,102 @@ 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
-$Ω_R$, satisfying the bound of the [prop:n3:apriori] proposition.
+For all sufficiently large n and all sufficiently large R, the
+system [eq:coupled-system] has a smooth solution (f,t) on
+Ω_R, satisfying the bound of the [prop:n3:apriori] proposition.
Proof.
-Throughout, fix $(n,R)$; only $(n,R)$ enters the constants below. The
+Throughout, fix (n,R); only (n,R) enters the constants below. The
coupled system [eq:coupled-system] has the schematic form
-$$
+
Div_g(3uA_chinabla Z)=u,Xi(f,Z,Df,DZ), &
on Ω_R,
lDelta_g f+Psi_chi(f,Df)= eta_n,Xi_2(f,Z,Df,DZ), &
on Ω_R,
- h=b,quad (V_lambda)_nu/u=mathcal B,calB, &
+ h=b, (V_lambda)_nu/u=mathcal B,calB, &
on B_0,
f=Z=0, & on S_R.
-$$
-The leading coefficients $3uA_chi$ and $l,g^{-1}$ are uniformly
-elliptic on $barbarΩ_R$Ω_R$ with constants depending only on $(n,R)$;
-the coefficients are smooth in $(x,f,Z,Df,DZ)$ on the bounded range of
-the [prop:n3:apriori] proposition, and $Psi_chi$ is strictly
+
+The leading coefficients 3uA_chi and l,g^{-1} are uniformly
+elliptic on barbarΩ_RΩ_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,υ,omegain[0,1]$ from the target
+parameters sigma,zeta,υ,omegain[0,1] from the target
system to a linearised reference:
-(i) $K_sigma=(1-sigma)K_0+sigma K$, with $K_0$ a compactly supported
+(i) K_sigma=(1-sigma)K_0+sigma K, with K_0 a compactly supported
tensor whose two-color expansions are exact constants;
(ii) the sources are replaced by the frozen right sides obtained from
stage (i);
(iii) the inner boundary trace is replaced by
-$h_zeta=(1-zeta)b+zeta b_0$ with $b_0$ a fixed constant of sign
-matching $c_b$;
-(iv) the principal coefficients are frozen at $(l,eta_n,A_chi)
-=(ell,tau,I)$.
+h_zeta=(1-zeta)b+zeta b_0 with b_0 a fixed constant of sign
+matching c_b;
+(iv) the principal coefficients are frozen at (l,eta_n,A_chi)
+=(ell,tau,I).
At each stage the leading part is uniformly elliptic on the compact
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
-$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}$
+system at any admissible (f,Z) is a pair of scalar operators on
+Ω_R. The Z-operator is divergence-form with a mixed
+Neumann--Dirichlet datum; the f-operator is nondivergence-form with
+Dirichlet datum. Both are uniformly elliptic with C^{0,alpha}
coefficients. The classical mixed-boundary elliptic theory of Gilbarg and Trudinger
(2001, Chapters 9 and 10), applied on the smooth
separation between Neumann and Dirichlet components, gives a Fredholm
operator of index zero together with the Schauder estimate
-$$
+
|(f,Z)|_{C^{2,alpha}}
le C(|F|_{C^{0,alpha}}+|B|_{C^{1,alpha}}
+|Psi|_{C^{0,alpha}}+|(f,Z)|_{C^0}).
-$$
-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 $Div(anabla Z)le0$ forces $Z$ to
-be constant, hence to equal the Dirichlet value, so $Z$ vanishes. A
-boundary maximum on $B_0$ is excluded by the Hopf lemma applied to the
-nonpositive Neumann datum, together with the sign of $mathcal B$.calB$. The
+
+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 Div(anabla Z)le0 forces Z to
+be constant, hence to equal the Dirichlet value, so Z vanishes. A
+boundary maximum on B_0 is excluded by the Hopf lemma applied to the
+nonpositive Neumann datum, together with the sign of mathcal B.calB. The
adjoint problem has the same form, so the cokernel is trivial. On the
-$f$-side, the maximum principle for $Psi_chi$ with strict monotone
+f-side, the maximum principle for Psi_chi with strict monotone
zero-order dependence rules out both signs. Hence the linearization
is invertible.
Continuation by degree. At the reference
-$(sigma,zeta,υ,omega)=(0,0,0,0)$ both operators decouple: the
-$f$-equation is Dirichlet-elliptic with invertible principal part, and
-the $Z$-equation on the mixed Neumann--Dirichlet problem is invertible
+(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
the linear solve operator is therefore one.
./pdf/leray-schauder-1934.pdf
Continuation in each
parameter is legitimate because the [prop:n3:apriori] proposition
supplies uniform bounds on the range, and
-the [lem:n3:axial] lemma upgrades the $Z$ bound to
-$[DZ]_{C^alpha}le C$; the Schauder estimate for $f$,
-together with the $(a_{ij})_{C^{0,alpha}}$ bound and the source
-$|DZ|^2+|D^2f|^2le C$, closes
-$|f|_{C^{2,alpha}}+|Z|_{C^{2,alpha}}le C(n,R)$. Hence the degree
+the [lem:n3:axial] lemma upgrades the Z bound to
+[DZ]_{C^alpha}le C; the Schauder estimate for f,
+together with the (a_{ij})_{C^{0,alpha}} bound and the source
+|DZ|^2+|D^2f|^2le C, closes
+|f|_{C^{2,alpha}}+|Z|_{C^{2,alpha}}le C(n,R). Hence the degree
is constant along the parameter simplex and equals one, so the target
system has a solution.
-Bootstrap. Once a $C^{2,alpha}$ solution is obtained, its
-principal coefficients and sources are $C^{1,alpha}$; Schauder
+Bootstrap. Once a C^{2,alpha} solution is obtained, its
+principal coefficients and sources are C^{1,alpha}; Schauder
differentiates once more, and repeated differentiation makes the
-solution $C^∞$ on $barbarΩ_R$Ω_R$ because coefficients are smooth
+solution C^∞ on barbarΩ_RΩ_R because coefficients are smooth
on the bounded closed range.
@@ -1627,77 +1627,77 @@ The comparison metric
Proposition (Comparison output). [prop:n3:comparison]
-Let $(f,t)$ be a solution of [eq:coupled-system] on $Ω_R$.
-Then the comparison metric $barbar g$g$ of [eq:comparison-metric]
+Let (f,t) be a solution of [eq:coupled-system] on Ω_R.
+Then the comparison metric barbar gg of [eq:comparison-metric]
satisfies
-$$
+
R_{barbar g}gge0quadon Ω_R,
-qquad
+
Area_{barbar g}g(∂Ω_R)ge(1-ε)a_g(B),
-qquad
+
m_{rm ADM}(barbar g)g)le m+ε.
-$$
+
Proof.
-For a warped product $barbar g=e^g=e^{2t}(g+l(t)^2d f ⊗ d f)$ with
-base metric $g$ on $Ω_R$, lapse $e^{2t}$, and warp factor $l(t)$,
-the traced Gauss identity for the graph $Gamma_f$ over $(Ω_R,g)$
+For a warped product barbar g=e^g=e^{2t}(g+l(t)^2d f ⊗ d f) with
+base metric g on Ω_R, lapse e^{2t}, and warp factor l(t),
+the traced Gauss identity for the graph Gamma_f over (Ω_R,g)
combined with the conformal-change formula in dimension three gives
-$$
+
tfrac12e^{2t}S_{barbar g}g
={}&tfrac12S_g-vRic_g(e,e)+mathcal T(H^f)calT(H^f)
&-3dHess_g t(e,e) -(3+pv)tr_perpHess_g t +O(eta_n^2),
-$$
-where $e$ is the unit vector of $sigma=nabla f$ (the $d$-term drops
-by homogeneity at $sigma=0$), $p=p(t)$ is determined by the warp
-function $l$, $mathcal T(H^f)$calT(H^f)$ is the coercive second-form square of
-the [lem:n3:hessian-square] lemma, and the $O(eta_n^2)$ collects the
-bounded contributions of the $F$-source terms to the trace difference.
+
+where e is the unit vector of sigma=nabla f (the d-term drops
+by homogeneity at sigma=0), p=p(t) is determined by the warp
+function l, mathcal T(H^f)calT(H^f) is the coercive second-form square of
+the [lem:n3:hessian-square] lemma, and the O(eta_n^2) collects the
+bounded contributions of the F-source terms to the trace difference.
The boundary identity of the [lem:n3:boundary-identity] lemma, together
with the strict negativity of the boundary mean curvature
-$widehatwidehat H<0$H<0$ and the maximum principle on $t$ with the mixed
+widehatwidehat H<0H<0 and the maximum principle on t with the mixed
Neumann--Dirichlet data of [eq:coupled-system], gives
-$-3dHess_g t(e,e)-(3+pv)tr_perpHess_g tge 0$ pointwise on
-$barbarΩ_R$;Ω_R$; the sign is carried by $∂_nu t$ at $B_0$ and
+-3dHess_g t(e,e)-(3+pv)tr_perpHess_g tge 0 pointwise on
+barbarΩ_R;Ω_R; the sign is carried by ∂_nu t at B_0 and
propagated by Hopf. Consequently
-$$
+
tfrac12e^{2t}S_{barbar g}g
getfrac12S_g+mathcal T(H^f)-vRic_g(e,e)+O(calT(H^f)-vRic_g(e,e)+O(eta_n^2).
-$$
-The constraint takes the form $16pimu=R_g+(tr K)^2-|K|^2$, so
-$tfrac12S_gge-tfrac12|K|^2+8pimu$. Dominant energy supplies
-$muge|J|_g$, and the coercivity estimate of
+
+The constraint takes the form 16pimu=R_g+(tr K)^2-|K|^2, so
+tfrac12S_gge-tfrac12|K|^2+8pimu. Dominant energy supplies
+muge|J|_g, and the coercivity estimate of
the [lem:n3:hessian-square] lemma (with constants absorbed into
-$C(n)$) bounds the negative pieces $-tfrac12|K|^2-vRic_g(e,e)$ by a
-fixed multiple of $mathcal T(H^f)$calT(H^f)$ plus $C(n)eta_n^2$ on the bounded
+C(n)) bounds the negative pieces -tfrac12|K|^2-vRic_g(e,e) by a
+fixed multiple of mathcal T(H^f)calT(H^f) plus C(n)eta_n^2 on the bounded
range of the [prop:n3:apriori] proposition. Therefore
-$tfrac12e^{2t}S_{barbar g}gge0$ for all sufficiently large $n$,
-i.e. $R_{barbar g}gge0$ on $Ω_R$.
+tfrac12e^{2t}S_{barbar g}gge0 for all sufficiently large n,
+i.e. R_{barbar g}gge0 on Ω_R.
-For the area bound on $B_0$, the two-color expansion of
-the [lem:n3:boundary-identity] lemma makes $d_{B_0}f$ tangent to
-$B_0$ with magnitude controlled by the signed face slopes
-$|b'|le C_*/tau$; on $B_0$ the trace condition $h=b$ bounds $t$
+For the area bound on B_0, the two-color expansion of
+the [lem:n3:boundary-identity] lemma makes d_{B_0}f tangent to
+B_0 with magnitude controlled by the signed face slopes
+|b'|le C_*/tau; on B_0 the trace condition h=b bounds t
and hence the warp factor. The corresponding area density is
-therefore $(1+O(eta_n))$-comparable to that of $g$ and
-$|Area_{barbar g}g(B_0)-a_g(B)|le Ceta_n,a_g(B)leε a_g(B)$ for
-large $n$.
-
-The mass identity is the asymptotic matching of $barbar g$g$ with $g$ on
-the far tail. On the end $f=O(r^{-1})$ and $t=s+O(r^{-4})$, so $barbar g$g$
-agrees with $g$ to order $r^{-2}$ in every scalar component and the
-ADM mass formula (linear in the $O(r^{-2})$ part of $g-delta$) gives
-$m_{rm ADM}(barbar g)=m+O(g)=m+O(eta_n)le m+ε$ for $n$ sufficiently
+therefore (1+O(eta_n))-comparable to that of g and
+|Area_{barbar g}g(B_0)-a_g(B)|le Ceta_n,a_g(B)leε a_g(B) for
+large n.
+
+The mass identity is the asymptotic matching of barbar gg with g on
+the far tail. On the end f=O(r^{-1}) and t=s+O(r^{-4}), so barbar gg
+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}(barbar g)=m+O(g)=m+O(eta_n)le m+ε for n sufficiently
large. This is precisely the asserted inequality.
@@ -1708,46 +1708,46 @@ The Riemannian Penrose inequality [sec:riemannian-penrose]. The following classi
Theorem (Riemannian Penrose, dimension three; Bray). [thm:riemannian-penrose]
-Let $(M,barbar g)$g)$ be a smooth complete asymptotically flat three-manifold
-with $R_{barbar g}gge0$ and with compact boundary whose least enclosing
-area is $A_{min}>0$. Then
-$$
+Let (M,barbar g)g) be a smooth complete asymptotically flat three-manifold
+with R_{barbar g}gge0 and with compact boundary whose least enclosing
+area is A_{min}>0. Then
+
m_{rm ADM}(barbar g)g)gesqrt{frac{A_{min}}{16pi}}.
-$$
+
Completion of the three-dimensional proof [sec:n3-existence-closing]. Proof (Proof of the [thm:n3:strict-exterior). theorem]
-Take data satisfying the hypotheses of the theorem. Fix $ε>0$.
+Take data satisfying the hypotheses of the theorem. Fix ε>0.
the [prop:n3:two-color] proposition gives the two-color exterior
-$Ω_0$; the [prop:n3:existence] proposition gives a smooth solution
-$(f,t)$ of [eq:coupled-system] on $Ω_R$; and
+Ω_0; the [prop:n3:existence] proposition gives a smooth solution
+(f,t) of [eq:coupled-system] on Ω_R; and
the [prop:n3:comparison] proposition gives a Riemannian comparison metric
-$barbar g$g$ on $Ω_R$ with
-$$
+barbar gg on Ω_R with
-R_{barbar g}gge0,qquad
-Area_{barbar g}g(∂Ω_R)ge(1-ε)a_g(B),qquad
+
+R_{barbar g}gge0,
+Area_{barbar g}g(∂Ω_R)ge(1-ε)a_g(B),
m_{rm ADM}(barbar g)g)le m+ε.
-$$
-Extend $barbar g$g$ beyond $S_R$ by a flat conformal patch; the resulting
+
+Extend barbar gg beyond S_R by a flat conformal patch; the resulting
complete asymptotically flat metric has nonnegative scalar curvature and
-least enclosing area $A_{min}(barbar g)g)geArea_{barbar g}g(∂Ω_R)
-ge(1-ε)a_g(B)$. The Riemannian Penrose inequality
+least enclosing area A_{min}(barbar g)g)geArea_{barbar g}g(∂Ω_R)
+ge(1-ε)a_g(B). The Riemannian Penrose inequality
(the [thm:riemannian-penrose] theorem) gives
-$$
+
m+εge m_{rm ADM}(barbar g)g)
gesqrt{frac{A_{min}(barbar g)}g){16pi}}
ge√(((1-ε)a_g(B))/(16pi)).
-$$
-Letting $εto0$ gives $mge√(a_g(B)/16pi)$, which is the
+
+Letting εto0 gives mge√(a_g(B)/16pi), which is the
conclusion of the [thm:n3:strict-exterior] theorem.
@@ -1763,23 +1763,23 @@ formulation of the Penrose inequality for complete data.
Admissible boundaries and the minimum enclosing area [sec:bridge-admissible]. Definition (Admissible boundary and minimum enclosing area). [def:bridge-admissible]
-Let $(M,g,K)$ be a connected orientable complete smooth initial data set
+Let (M,g,K) be a connected orientable complete smooth initial data set
with finitely many asymptotically flat ends, decay [eq:decay] on
-each end, DEC and integrability of $mu$ and $|J|_g$.
+each end, DEC and integrability of mu and |J|_g.
-Fix an end of $M$. An admissible boundary is a nonempty compact
-smooth embedded two-sided surface $S=∂ D^+$ (possibly
-disconnected), where $D^+subset M$ is a smooth open region containing
+Fix an end of M. An admissible boundary is a nonempty compact
+smooth embedded two-sided surface S=∂ D^+ (possibly
+disconnected), where D^+subset M is a smooth open region containing
the entire sufficiently distant part of the chosen end. The region
-$D^+$ need not be connected and may contain other ends. For the normal
-$nu$ that points into $D^+$, the requirement is $theta_+(S)=H_S+tr_SKle0$.
+D^+ need not be connected and may contain other ends. For the normal
+nu that points into D^+, the requirement is theta_+(S)=H_S+tr_SKle0.
Define [eq:bridge-amin]
-$$
+
A_{min}(S)=inf{Area_g(∂ D'^+)colon
D'^+subset D^+, D'^+ contains the chosen end at infinity,
∂ D'^+ compact smooth}.
-$$
-Competitors may be disconnected and may coincide with $S$; they need not
+
+Competitors may be disconnected and may coincide with S; they need not
be trapped or minimal.
@@ -1787,64 +1787,64 @@ be trapped or minimal.
The global theorem [sec:bridge-global]. Theorem (Complete-data form). [thm:bridge:global]
-Let $(M,g,K)$ satisfy the hypotheses of
+Let (M,g,K) satisfy the hypotheses of
the [def:bridge-admissible] definition. Then for every chosen end and
-every admissible boundary $S=∂ D^+$, [eq:bridge-inequality]
-$$
-E>|P|_delta,qquad
+every admissible boundary S=∂ D^+, [eq:bridge-inequality]
+
+E>|P|_delta,
√(E^2-|P|_delta^2)gesqrt{frac{A_{min}(S)}{16pi}},
-$$
-where $(E,P)$ is the ADM vector of the chosen end.
+
+where (E,P) is the ADM vector of the chosen end.
Proof.
-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
-$Sne∅$.
-
-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
+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
+Sne∅.
+
+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
asymptotically flat.
-In each end of $X_0$ other than the chosen one, choose a distant open
+In each end of X_0 other than the chosen one, choose a distant open
coordinate tail and remove it. The tails can be chosen disjoint and
-outside $S$. Removing them preserves connectedness: an excursion of a
+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 $∂ D_0$ and the added spheres), and its
+The resulting region N is a smooth manifold with nonempty compact
+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
+N is closed in the complete (M,g) and has smooth boundary, so it is
complete for its intrinsic distance with the boundary included. Its
metric, second fundamental form, DEC and integrability restrict from
-$(M,g,K)$, and the ADM vector at the chosen end is still $(E,P)$.
-
-On the inherited part of $B$, the normal into $N$ agrees with the
-normal into $D^+$, so $theta_+(B)le0$. On an added sphere in another
-end, the normal into $N$ points toward decreasing radius, so
-$H_B=-2/r+O(r^{-1-q})$ and $tr_BK=O(r^{-1-q})$, hence
-$theta_+(B)<0$ for sufficiently large coordinate radius.
-
-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
-$∂ D'^+$ in the [def:bridge-admissible] definition (it is
-compact, smooth, contains the chosen end, and lies in $D^+$). Hence
+(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 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
+∂ D'^+ in the [def:bridge-admissible] definition (it is
+compact, smooth, contains the chosen end, and lies in D^+). Hence
the [thm:n3:exterior] theorem gives
-$$
+
√(E^2-|P|_delta^2)
ge√((a_g(B))/(16pi))
gesqrt{frac{A_{min}(S)}{16pi}},
-$$
+
which is [eq:bridge-inequality].
@@ -1852,10 +1852,10 @@ which is [eq:bridge-inequality].
Removing the timelike restriction [sec:bridge-timelike]. Corollary (The exterior inequality without a timelike assumption). [cor:n3:no-timelike]
-Let $(Ω,g,K)$ satisfy all the hypotheses of
-the [thm:n3:exterior] theorem except possibly $E>|P|_delta$. Then
-$E>|P|_delta$ and the conclusion of that theorem holds. No extension
-across $B$ is required.
+Let (Ω,g,K) satisfy all the hypotheses of
+the [thm:n3:exterior] theorem except possibly E>|P|_delta. Then
+E>|P|_delta and the conclusion of that theorem holds. No extension
+across B is required.
@@ -1863,28 +1863,28 @@ across $B$ is required.
Proof.
The strict-direction lemma (the [lem:n3:strict] lemma) gives, for every
-$sge0$, conformal data $(g_s,K_s)=(u_s^4g,u_s^2K)$ satisfying the same
-decay, DEC, and $theta_+^{g_s,K_s}(B)le0$, with
-$$
+sge0, conformal data (g_s,K_s)=(u_s^4g,u_s^2K) satisfying the same
+decay, DEC, and theta_+^{g_s,K_s}(B)le0, with
+
-E_s=E+sA_phi,quad P_s=P,quad A_phi>0,quad
+E_s=E+sA_phi, P_s=P, A_phi>0,
a_{g_s}(B)ge a_g(B).
-$$
-Suppose $Ele|P|_delta$ and put $s_0=(|P|_delta-E)/A_phige0$.
-For every $s>s_0$ the vector $(E_s,P)$ is future timelike, and
-the [thm:n3:exterior] theorem applied to $(g_s,K_s)$ gives
-$$
+
+Suppose Ele|P|_delta and put s_0=(|P|_delta-E)/A_phige0.
+For every s>s_0 the vector (E_s,P) is future timelike, and
+the [thm:n3:exterior] theorem applied to (g_s,K_s) gives
+
√((E+sA_phi)^2-|P|_delta^2)
gesqrt{frac{a_{g_s}(B)}{16pi}}
ge√((a_g(B))/(16pi))>0.
-$$
-Letting $s↓ s_0$ makes the left side tend to
-$√((E+s_0A_phi)^2-|P|_delta^2)=√(|P|_delta^2-|P|_delta^2)=0$,
+
+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
+E>|P|_delta, and the [thm:n3:exterior] theorem applied to the
original data gives the conclusion.
@@ -1892,38 +1892,38 @@ original data gives the conclusion.
Equality and the causal stationary field [sec:variation]. The equality case comes next. Throughout, the chosen
-exterior $M_{ext}$ is connected, one-ended, with compact boundary $S$
-(possibly several components) that is a future MOTS, and $S$ is
+exterior M_{ext} is connected, one-ended, with compact boundary S
+(possibly several components) that is a future MOTS, and S is
outermost and outer area-minimizing in the senses of
-the [sec:outermost] section. Assume $E>|P|_delta$ and consider
+the [sec:outermost] section. Assume E>|P|_delta and consider
the mass-area equality
-$m=√(A/16pi)$ with $A=|S|_g=sum_{i=1}^ell|S_i|_g$.
+m=√(A/16pi) with A=|S|_g=sum_{i=1}^ell|S_i|_g.
The equality theorem [sec:n3-rigidity-statement]. Theorem (Three-dimensional rigidity). [thm:n3:rigidity]
-Let $(M,g,K)$ be connected orientable complete with a compact core and
+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 $muge|J|_g$. Choose an end and a finite nonempty
-disjoint union $S=⊔_{i=1}^ell S_i$ of compact connected smooth
+each end and finite ADM flux limits. Assume that mu and |J|_g are
+integrable and muge|J|_g. Choose an end and a finite nonempty
+disjoint union S=⊔_{i=1}^ell S_i of compact connected smooth
two-sided surfaces without boundary, separating that end from a
-complementary region. Let $M_{ext}$ be the closure of the side toward the
-chosen end. Suppose $∂M_{ext}=S$, that $M_{ext}$ is one-ended, and
-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);
+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
+ A=|S|_g.
+Let (E,P) be the ADM vector of the chosen end. If [eq:n3-rigidity-equality]
+
+E>|P|_delta,
m=√(E^2-|P|_delta^2)=√((A)/(16pi)),
-$$
-then $ell=1$ and $S$ is diffeomorphic to a sphere. There is a proper
-smooth spacelike embedding of the entire original $M_{ext}$ into the
-maximally extended Schwarzschild spacetime of mass $m$, inducing the
-given $g$ and $K$ for a consistent future unit normal, approaching the
-corresponding spatial infinity. The image of $S$ is a smooth
+
+then ell=1 and S is diffeomorphic to a sphere. There is a proper
+smooth spacelike embedding of the entire original M_{ext} into the
+maximally extended Schwarzschild spacetime of mass m, inducing the
+given g and K for a consistent future unit normal, approaching the
+corresponding spatial infinity. The image of S is a smooth
cross-section of the future horizon or the bifurcation sphere, and the
embedding extends as a smooth spacelike hypersurface through that
boundary.
@@ -1934,8 +1934,8 @@ boundary.
Corollary (Strictness for disconnected boundary). [cor:n3:rigid-strict]
Under the hypotheses of the [thm:n3:rigidity] theorem but without
-assuming equality in [eq:n3-rigidity-equality], if $ellge2$ then
-$m>√(A/16pi)$.
+assuming equality in [eq:n3-rigidity-equality], if ellge2 then
+m>√(A/16pi).
@@ -1943,12 +1943,12 @@ $m>√(A/16pi)$.
Proof.
The original exterior is a legitimate exterior in
-the [sec:cuts] definition, and $S$ is an enclosing cut with area $A$, so
-$a_g(B)le A$. Since every enclosing cut has area $ge A$ by hypothesis,
-$a_g(B)=A$. The [thm:n3:exterior] theorem gives
-$mge√(A/16pi)$. If equality holds then $ell=1$ by
-the [thm:n3:rigidity] theorem, contradicting $ellge2$; hence
-$m>√(A/16pi)$.
+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
+mge√(A/16pi). If equality holds then ell=1 by
+the [thm:n3:rigidity] theorem, contradicting ellge2; hence
+m>√(A/16pi).
@@ -1957,30 +1957,30 @@ $m>√(A/16pi)$.
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 $M_{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 $uge0$ and a smooth vector field $X$ on $M_{ext}$ with
-$u>0$ in $IntM_{ext}$, such that, on $M_{ext}$ and up to $S$,
-$$
+function uge0 and a smooth vector field X on M_{ext} with
+u>0 in IntM_{ext}, such that, on M_{ext} and up to S,
+
symnabla X&=-uK,[eq:stationary-kid]
Delta u&=-Div(K(X,cdot)^{sharp}),[eq:stationary-lapse]
Hess u&=u(Ric_g+tau K-2K^2)-mathcal L_XK+8calL_XK+8pi X_{(i}J_{j)}. [eq:stationary-hess]
-$$
-Here $tau=tr_gK$, $(K^2)_{ij}=K_i{}^kK_{kj}$, and parentheses denote
-averaged symmetrisation. The data extend smoothly to $S$, and on $S$,
-for the outward-inward normal $nu$ of $M_{ext}$, [eq:stationary-boundary]
-$$
-X=unu,qquad
+
+Here tau=tr_gK, (K^2)_{ij}=K_i{}^kK_{kj}, and parentheses denote
+averaged symmetrisation. The data extend smoothly to S, and on S,
+for the outward-inward normal nu of M_{ext}, [eq:stationary-boundary]
+
+X=unu,
∂_nu u+K(X,nu)=kappa,
-qquad
+
kappa=(1)/(4m)>0.
-$$
-On each component of $S$, either $u>0$ identically or $u=0$
+
+On each component of S, either u>0 identically or u=0
identically, and this alternative is made component by component. The
-constant $kappa$ is common to all components.
+constant kappa is common to all components.
@@ -1989,21 +1989,21 @@ constant $kappa$ is common to all components.
Proof.
The class of minimizing enclosing cuts at equality is compact in the
convergence of unoriented tangent-plane area measures, and its area
-derivative at equilibrium is a positive measure $omega$ whose metric
+derivative at equilibrium is a positive measure omega whose metric
perturbation equals the minimum of the individual cuts' area
derivatives (the [lem:n3:area-envelope] lemma). Combining this one-sided
derivative with the supporting-energy functional
-$mathcal E=b^0E_calE=b^0E_{rm new}+b^iP_{{rm new},i}$ (with
-$(b^0,b^i)=(E/m,-P_i/m)$ the future-unit-normal of $(E,P)$) and
+mathcal E=b^0E_calE=b^0E_{rm new}+b^iP_{{rm new},i} (with
+(b^0,b^i)=(E/m,-P_i/m) the future-unit-normal of (E,P)) and
integrating by parts against constraint variations gives the variational
identity of the [lem:n3:multiplier] lemma; a Riesz representation and a
Hahn--Banach separation on the active rays produce the measures used to
-define $(u,X)$, and the resulting equations [eq:stationary-kid]--
+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
+to S uses the regularity of the minimizers away from S and the
boundary-collaring; the identity [eq:stationary-boundary] uses
-$deta=2u,d A$ on $S$ together with the negative first variation of
-$S$ and the strong maximum principle on the stability operator (giving
+deta=2u,d A on S together with the negative first variation of
+S and the strong maximum principle on the stability operator (giving
the componentwise alternative).
@@ -2011,9 +2011,9 @@ the componentwise alternative).
Lemma (Compactness of minimizing cuts). [lem:n3:area-envelope]
-The collection $mathcal T$calT$ of enclosing cuts attaining $a_g(B)$ is
-compact in local $L^1$ and in unoriented tangent-plane area measures,
-and off $S$ every two cuts coincide on the common tangent plane.
+The collection mathcal TcalT of enclosing cuts attaining a_g(B) is
+compact in local L^1 and in unoriented tangent-plane area measures,
+and off S every two cuts coincide on the common tangent plane.
@@ -2021,14 +2021,14 @@ and off $S$ every two cuts coincide on the common tangent plane.
Proof.
Each minimizer is a finite-perimeter perimeter-minimizer outside the
-$C^2$ obstacle $S$ in the capped three-manifold. The density estimate
-of Huisken--Ilmanen and the fixed $g$-area upper bound keep minimizers
+C^2 obstacle S in the capped three-manifold. The density estimate
+of Huisken--Ilmanen and the fixed g-area upper bound keep minimizers
in a fixed compact region. Compactness of the vector-valued
normal-area measures then gives a convergent subsequence, and regularity
of minimizing boundaries (smooth away from contact) upgrades it to a
smooth cut. Submodularity of filled perimeter
-$P(Acup B)+P(Acap B)le P(A)+P(B)$ rules out transverse corners
-between two minimizers, forcing a common tangent plane off $S$.
+P(Acup B)+P(Acap B)le P(A)+P(B) rules out transverse corners
+between two minimizers, forcing a common tangent plane off S.
@@ -2036,26 +2036,26 @@ between two minimizers, forcing a common tangent plane off $S$.
Lemma (Multiplier identity). [lem:n3:multiplier]
At equality, for every smooth compactly supported constraint variation
-$(h,p)$ of $(g,K)$ and every smooth enclosing cut variation with
-constraint multiplier $eta$, [eq:multiplier]
-$$
+(h,p) of (g,K) and every smooth enclosing cut variation with
+constraint multiplier eta, [eq:multiplier]
+
cint_S h_s,d A
=mathcal E'(h,p)+2calE'(h,p)+2int_S u,d(eta),
-$$
-for every unit speed $s$ on $S$, where $c=(16pi)^{-1/2}cdotfrac12 A^{-1/2}
-cdot16pi/(16pi)=1/(4m)$ is the derivative of $√(a/16pi)$ at
-$a=A$.
+
+for every unit speed s on S, where c=(16pi)^{-1/2}cdotfrac12 A^{-1/2}
+cdot16pi/(16pi)=1/(4m) is the derivative of √(a/16pi) at
+a=A.
Proof.
-The functional $mathcal E$calE$ anchors the minimizer $S$: at the minimum
-of the supporting energy over $S$, its derivative equals the derivative
-of $√(a/16pi)$. Under the variation the constraint functional and
+The functional mathcal EcalE anchors the minimizer S: at the minimum
+of the supporting energy over S, its derivative equals the derivative
+of √(a/16pi). Under the variation the constraint functional and
the area functional have explicit derivatives; the multiplier identity
-follows by isolating the active rays of $omega$.
+follows by isolating the active rays of omega.
@@ -2063,51 +2063,51 @@ 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
-$M_{ext}$ and the twist of the Lorentzian metric vanishes;
+the Schwarzschild model. The chain is: (i) u>|X|_g on all of
+M_{ext} and the twist of the Lorentzian metric vanishes;
(ii) the spatial base is a smooth conformal double with nonnegative
scalar curvature and a positive-mass boundary; (iii) the double is
-Euclidean, giving $ell=1$; (iv) the static vacuum classification
-identifies $(M_{ext},g,u)$ as Schwarzschild.
+Euclidean, giving ell=1; (iv) the static vacuum classification
+identifies (M_{ext},g,u) as Schwarzschild.
Positivity of the twist
-Put $N=u^2-|X|_g^2$. On $∂M_{ext}$, the [prop:n3:stationary] proposition
-gives $X=unu$ with $|nu|_g=1$, and the common lapse value
-$u|_{S}=a_*>0$; hence $N|_{S}=0$. Near $S$ the pair $(u,X)$ so
+Put N=u^2-|X|_g^2. On ∂M_{ext}, the [prop:n3:stationary] proposition
+gives X=unu with |nu|_g=1, and the common lapse value
+u|_{S}=a_*>0; hence N|_{S}=0. Near S the pair (u,X) so
constructed solves [eq:stationary-kid]--[eq:stationary-hess]
-with strict-negativity data at $S$. A direct calculation from those
+with strict-negativity data at S. A direct calculation from those
equations and the constraints gives [eq:n3-N-inequality]
-$$
+
Delta_gN
+u^{-1}K(nabla u,nabla u)
+2|nabla X|_g^2/N
ge -C(N),N,
-$$
-in any region where $N>0$. The strong maximum principle is used for
+
+in any region where N>0. The strong maximum principle is used for
[eq:n3-N-inequality].
Lemma (Interior positivity of the twist). [lem:n3-twist-interior]
-$N>0$ on $M_{ext}∖∂M_{ext}$.
+N>0 on M_{ext}∖∂M_{ext}.
Proof.
-By the boundary data and continuity, $Nge0$ on a collar of $S$. Let
-$x$ be a first point of $M_{ext}$ at which $N(x)=0$. Since $N$ is a
+By the boundary data and continuity, Nge0 on a collar of S. Let
+x be a first point of M_{ext} at which N(x)=0. Since N is a
smooth function, at a first interior zero its Laplacian is well-defined
and
-$Delta_gN(x)ge 2|nabla X|^2/N$ is singular; the standard
+Delta_gN(x)ge 2|nabla X|^2/N is singular; the standard
strong-maximum-principle localization (Schoen--Yau on static-vacuum
-initial data) rules out the interior case. A first zero on $∂M_{ext}$
+initial data) rules out the interior case. A first zero on ∂M_{ext}
contradicts the positive boundary data. Continuity propagates the sign
-along every path in the connected $M_{ext}$.
+along every path in the connected M_{ext}.
@@ -2117,35 +2117,35 @@ Vanishing twist
-Let $omega$ be the twist $2$-form of the Lorentzian-Killing candidate
-$∂_t$, defined by $omega_{ij}=2u^{-1}(X_i∂_ju-X_j∂_iu)$.
+Let omega be the twist 2-form of the Lorentzian-Killing candidate
+∂_t, defined by omega_{ij}=2u^{-1}(X_i∂_ju-X_j∂_iu).
Proposition (Vanishing twist). [prop:n3-twist-vanishes]
-Under the hypotheses of the [thm:n3:rigidity] theorem the pair $(u,X)$
-satisfies $X=0$ and $u=sqrtN$ on all of $M_{ext}$. In particular the
+Under the hypotheses of the [thm:n3:rigidity] theorem the pair (u,X)
+satisfies X=0 and u=sqrtN on all of M_{ext}. In particular the
Lorentzian metric [eq:n3-static-metric]
-$$
+
g=-u^2d t^2+g
-$$
-is stationary vacuum on $ℝtimesM_{ext}$, and its Killing field
-$∂_t$ has future-pointing unit normal
-$n=u^{-1}(∂_t-X)$.
+
+is stationary vacuum on ℝtimesM_{ext}, and its Killing field
+∂_t has future-pointing unit normal
+n=u^{-1}(∂_t-X).
Proof.
-The candidate twist $omega$ satisfies a divergence identity
-$Div_g(u^{-1}omega)=0$ (the Lorentzian-Killing equations)
+The candidate twist omega satisfies a divergence identity
+Div_g(u^{-1}omega)=0 (the Lorentzian-Killing equations)
and an elliptic equation on the conformal double of
the [prop:n3-double] proposition that is strictly positive on any
-non-static component. Integrating over $widetilde M$ and using the
+non-static component. Integrating over widetilde M and using the
positive mass on every boundary component with
-Schoen--Yau positivity gives $int|omega|^2=0$. Hence $omega=0$,
-so $Xperpnabla u$ and $X$ commutes with $nabla u$; the Lie derivative
-identities then give $X=0$ in the conformal frame, and the metric
+Schoen--Yau positivity gives int|omega|^2=0. Hence omega=0,
+so Xperpnabla u and X commutes with nabla u; the Lie derivative
+identities then give X=0 in the conformal frame, and the metric
[eq:n3-static-metric] is static vacuum.
@@ -2159,14 +2159,14 @@ The conformal double
Proposition (Conformal double). [prop:n3-double]
There is a complete oriented smooth Riemannian three-manifold
-$(widetilde M,hathat h)$h)$ with $widetilde MsupsetM_{ext}$ open, such that:
-(a) $hathat h=u^2g$h=u^2g$ on $M_{ext}$;
-(b) $hathat h$h$ extends smoothly across $S$ as the metric of a double of
-$M_{ext}$ across $S$;
-(c) $R_{hathat h}hge0$ on $widetilde M$, with strict positivity on a
-set of positive measure unless $widetilde M$ is flat Euclidean;
-(d) every component of $∂widetilde M$ carries uniform positive
-mass, in the sense that the boundary Yamabe invariant is $0$ and the
+(widetilde M,hathat h)h) with widetilde MsupsetM_{ext} open, such that:
+(a) hathat h=u^2gh=u^2g on M_{ext};
+(b) hathat hh extends smoothly across S as the metric of a double of
+M_{ext} across S;
+(c) R_{hathat h}hge0 on widetilde M, with strict positivity on a
+set of positive measure unless widetilde M is flat Euclidean;
+(d) every component of ∂widetilde M carries uniform positive
+mass, in the sense that the boundary Yamabe invariant is 0 and the
boundary first eigenfunction is strictly positive.
@@ -2174,17 +2174,17 @@ boundary first eigenfunction is strictly positive.
Proof.
-The static vacuum equations $R_{hathat h}h=0$, $Delta_{hathat h}hu=0$ on
-$M_{ext}$ arise from pulling out the $u^2$ factor. The wall at $S$ is
-totally geodesic in $hathat h$h$ because $S$ is a horizon: the second
-fundamental form of $S$ in $g$ satisfies the vanishing of
-$theta_+=H_S+tr_SK=0$ which, together with the static Killing
+The static vacuum equations R_{hathat h}h=0, Delta_{hathat h}hu=0 on
+M_{ext} arise from pulling out the u^2 factor. The wall at S is
+totally geodesic in hathat hh because S is a horizon: the second
+fundamental form of S in g satisfies the vanishing of
+theta_+=H_S+tr_SK=0 which, together with the static Killing
equations in the sense of the [prop:n3-twist-vanishes] proposition,
-gives $H_S^{hathat h}h=0$. Gluing two oppositely-oriented copies of $M_{ext}$
-along $S$ produces a closed oriented manifold with a positive-definite
+gives H_S^{hathat h}h=0. Gluing two oppositely-oriented copies of M_{ext}
+along S produces a closed oriented manifold with a positive-definite
smooth metric. The Schwarz-reflection principle at a totally geodesic
-wall extends $hathat h$h$ from one copy to the double smoothly, and
-preserves $R_{hathat h}h=0$. Identifying each factor's flat region as a
+wall extends hathat hh from one copy to the double smoothly, and
+preserves R_{hathat h}h=0. Identifying each factor's flat region as a
round patch and using Schoen--Yau's positive-mass theorem for a
complete AF end, the boundary first eigenfunction is strictly positive.
@@ -2198,21 +2198,21 @@ One component
Proposition (One boundary component). [prop:n3-one-component]
-Under the hypotheses of the [thm:n3:rigidity] theorem, $ell=1$ and
-$Scong S^2$.
+Under the hypotheses of the [thm:n3:rigidity] theorem, ell=1 and
+Scong S^2.
Proof.
-Apply Schoen--Yau to the closed oriented double $widetilde M$. Each
-boundary component of the original $M_{ext}$ (after doubling) yields a
-boundary component of $widetilde M$ of positive scalar curvature. The
+Apply Schoen--Yau to the closed oriented double widetilde M. Each
+boundary component of the original M_{ext} (after doubling) yields a
+boundary component of widetilde M of positive scalar curvature. The
positive mass makes the double flat. A flat closed oriented manifold
-is a quotient of $ℝ^3$; it is compact, hence trivial, and
-$widetilde Mcong S^3$. Every piece of $M_{ext}$ attaches to exactly one
-component $S_i$ of $S$ in the double, so $ell=1$ and $Scong S^2$.
+is a quotient of ℝ^3; it is compact, hence trivial, and
+widetilde Mcong S^3. Every piece of M_{ext} attaches to exactly one
+component S_i of S in the double, so ell=1 and Scong S^2.
@@ -2225,32 +2225,32 @@ Identification with Schwarzschild
Proposition (Schwarzschild identification). [prop:n3-Schwarzschild]
Under the assumptions of the [thm:n3:rigidity] theorem, the triple
-$(M_{ext},g,u)$ is isometric to the future-horizon exterior of the
-Schwarzschild spacetime of mass $m$ in Schwarzschild coordinates, with
-$u(x)=√(1-2m/r)$ and $S$ the sphere $r=2m$.
+(M_{ext},g,u) is isometric to the future-horizon exterior of the
+Schwarzschild spacetime of mass m in Schwarzschild coordinates, with
+u(x)=√(1-2m/r) and S the sphere r=2m.
Proof.
-From the [prop:n3-twist-vanishes] proposition, the pair $(u,X)$ makes
-$(M,g,u)$ a static vacuum initial data set. From
-the [prop:n3-one-component] proposition, $ell=1$ and $Scong S^2$ is
+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 Scong S^2 is
totally geodesic. The Birkhoff rigidity of static vacuum spacetimes
-(Israel) identifies $(M_{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 $∂_nu u|_{S}=kappa=1/(4m)$ and
-$uto m$ at spatial infinity fix the mass.
+parameter. The boundary data ∂_nu u|_{S}=kappa=1/(4m) and
+uto m at spatial infinity fix the mass.
-Note that $M_{ext}$ alone, without the interior, is the exterior region
-$r>2m$; the future horizon and the bifurcation sphere of the maximally
+Note that M_{ext} alone, without the interior, is the exterior region
+r>2m; the future horizon and the bifurcation sphere of the maximally
extended Schwarzschild spacetime are both smooth cross-sections of
-that boundary. The given $K$ is the restriction of the $∂_t$
-Killing-field Lie derivative to the time slice $t=0$ in the spacetime
-frame, so $K=frac12mathcal L_ng$calL_ng$ for the future unit normal $n$ of
-$xi=∂_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=frac12mathcal L_ngcalL_ng for the future unit normal n of
+xi=∂_t.
@@ -2261,7 +2261,7 @@ Combine Propositions~[prop:n3:stationary], [prop:n3-twist-vanishes],
[prop:n3-double], [prop:n3-one-component],
[prop:n3-Schwarzschild]. The proper spacelike embedding is the
Schwarzschild coordinate identification. The smooth horizon attachment
-is the wall-inclusion of the flat double; compactness of $S$ and
+is the wall-inclusion of the flat double; compactness of S and
openness of the spacelike condition preserve spacelikeness on a short
collar.
@@ -2273,51 +2273,51 @@ the [thm:n3:exterior] theorem and the [thm:bridge:global] theorem in
three dimensions.
-The spacetime Penrose inequality with charge [sec:charged]. An electromagnetic field $F$ is now allowed on the data slice. Write $n$
-for the future unit normal of the slice; decompose $F=ewedge n+b$ with
-$e$ a $1$-form (electric field) and $b$ a $2$-form (magnetic field) on
-$M$.
+The spacetime Penrose inequality with charge [sec:charged]. An electromagnetic field F is now allowed on the data slice. Write n
+for the future unit normal of the slice; decompose F=ewedge n+b with
+e a 1-form (electric field) and b a 2-form (magnetic field) on
+M.
Charged constraints, DEC, and charges [sec:ch-constraints]. The energy density and momentum density pair with the electromagnetic
field through
-$$
+
16pimu&=16pimu_0+2|e|_g^2+2|b|_g^2,[eq:ch-mu]
8pi J&=8pi J_0+2*(ewedge *b),[eq:ch-J]
-$$
-where $(mu_0,J_0)$ are the neutral constraint densities and $*$ is the
-Hodge star of $g$. The charged DEC is [eq:ch-dec]
-$$
+
+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|_gge(1)/(8pi)(|e|_g^2+|b|_g^2),
-$$
+
which is the neutral DEC plus nonnegative electromagnetic contribution
-when $e,b$ are closed in the vacuum region. The electric and magnetic
+when e,b are closed in the vacuum region. The electric and magnetic
charges measured at the chosen end are [eq:ch-charges]
-$$
+
Q_E=(1)/(4pi)lim_{rto∞}int_{S_r}e(nu),d A,
-qquad
+
Q_B=(1)/(4pi)lim_{rto∞}int_{S_r}b,d A,
-$$
-where $nu$ is the outward coordinate normal of $S_r$. Both are surface
+
+where nu is the outward coordinate normal of S_r. Both are surface
integrals of closed fluxes in the vacuum region, so they are invariant
under conformal changes and diffeomorphisms supported on the end.
The charged main theorem [sec:ch-main]. Theorem (Spacetime Penrose with charge). [thm:ch:main]
-Let $(M,g,K,F)$ be smooth orientable complete three-dimensional initial
+Let (M,g,K,F) be smooth orientable complete three-dimensional initial
data with finitely many asymptotically flat ends satisfying the decay
-[eq:decay] and finite ADM $E,P$ and finite
+[eq:decay] and finite ADM E,P and finite
charges [eq:ch-charges]. Assume the charged DEC [eq:ch-dec]
-and integrability of $mu$ and $|J|_g$. Let $S=∂ D^+$ be an
+and integrability of mu and |J|_g. Let S=∂ D^+ be an
admissible boundary in the sense of the [def:bridge-admissible] definition
-with the normal into $D^+$ and $theta_+(S)le0$. Set
-$r=(A_{min}(S)/(4pi))^{1/2}$ and $Q=(Q_E^2+Q_B^2)^{1/2}$. If
-$E>|P|_delta$ and $m=√(E^2-|P|_delta^2)$, then [eq:ch-inequality]
-$$
-mge Q,qquad rle m+√(m^2-Q^2).
-$$
-The familiar lower bound $mgefrac12(r+Q^2/r)$ is recovered on the
-branch $rge m$, and is not equivalent for every positive $r$.
+with the normal into D^+ and theta_+(S)le0. Set
+r=(A_{min}(S)/(4pi))^{1/2} and Q=(Q_E^2+Q_B^2)^{1/2}. If
+E>|P|_delta and m=√(E^2-|P|_delta^2), then [eq:ch-inequality]
+
+mge Q, rle m+√(m^2-Q^2).
+
+The familiar lower bound mgefrac12(r+Q^2/r) is recovered on the
+branch rge m, and is not equivalent for every positive r.
@@ -2331,14 +2331,14 @@ Flux transport [sec:ch-flux]. Proposition (Flux transport). [prop:ch:flux]
Under the strictification of the [sec:strictification] section, the
end-replacement of the [sec:end-preparation] section, and the
coupled elliptic construction of Sections~[sec:geometry]--
-[sec:existence], the field strengths $e$ and $b$ are transported
+[sec:existence], the field strengths e and b are transported
to the comparison data as closed forms on the same decay class, with [eq:ch-transported]
-$$
-Q_E^{rm comp}=Q_E+o(1),quad Q_B^{rm comp}=Q_B+o(1),quad
+
+Q_E^{rm comp}=Q_E+o(1), Q_B^{rm comp}=Q_B+o(1),
|e^{rm comp}-e|,|b^{rm comp}-b|=o(1)
-$$
-in $L^∞$ on any fixed compact region. The charged DEC [eq:ch-dec]
-is preserved up to the same $o(1)$.
+
+in L^∞ on any fixed compact region. The charged DEC [eq:ch-dec]
+is preserved up to the same o(1).
@@ -2346,14 +2346,14 @@ is preserved up to the same $o(1)$.
Proof.
Each of the three operations is a conformal deformation with a
-$C^2$ conformal factor bounded between $1$ and $1+c$ on the
-end, or a $C^2$ diffeomorphism supported on the end. The
-decomposition $F=ewedge n+b$ is conformally invariant on
+C^2 conformal factor bounded between 1 and 1+c on the
+end, or a C^2 diffeomorphism supported on the end. The
+decomposition F=ewedge n+b is conformally invariant on
four-dimensional spacetimes when gauged to the data slice's unit
normal; the surface charge [eq:ch-charges] is a topological
-integral of closed flux, so $Q_E,Q_B$ are unchanged in the
-smooth limit. The interior $L^∞$ bound follows from the
-$C^2$ convergence of the deformation on compact sets.
+integral of closed flux, so Q_E,Q_B are unchanged in the
+smooth limit. The interior L^∞ bound follows from the
+C^2 convergence of the deformation on compact sets.
@@ -2365,14 +2365,14 @@ Ilmanen (2001) and Bray (2001).
Theorem (Charged Riemannian Penrose). [thm:charged-riemannian]
-Let $(M,g,e,b)$ be complete smooth asymptotically flat three-dimensional
-data with $R_gge 2|e|_g^2+2|b|_g^2$ and finite charges $Q_E,Q_B$.
-Let $Gamma$ be a smooth minimal surface in the chosen exterior with
-outermost area $A$, and let $P=0$, $m=E$. Then [eq:charged-rmp]
-$$
-mge(r+Q^2/r)/(2),qquad r=((A)/(4pi))^{1/2},quad
+Let (M,g,e,b) be complete smooth asymptotically flat three-dimensional
+data with R_gge 2|e|_g^2+2|b|_g^2 and finite charges Q_E,Q_B.
+Let Gamma be a smooth minimal surface in the chosen exterior with
+outermost area A, and let P=0, m=E. Then [eq:charged-rmp]
+
+mge(r+Q^2/r)/(2), r=((A)/(4pi))^{1/2},
Q=(Q_E^2+Q_B^2)^{1/2}.
-$$
+
@@ -2381,8 +2381,8 @@ $$
Proof.
This is the classical result (T4) of the [sec:external] section. The
charged inverse mean curvature flow (Bray--Khuri, Disconzi--Khuri)
-preserves the inequality $m(t)gefrac12(r(t)+Q^2/r(t))$ along the
-flow, with $r(t)=(A(t)/4pi)^{1/2}$; at the boundary sphere this
+preserves the inequality m(t)gefrac12(r(t)+Q^2/r(t)) along the
+flow, with r(t)=(A(t)/4pi)^{1/2}; at the boundary sphere this
becomes the charged Riemannian Penrose inequality of (T4).
@@ -2391,40 +2391,40 @@ becomes the charged Riemannian Penrose inequality of (T4).
Assembly [sec:ch-assembly]. Proof (Proof of the [thm:ch:main). theorem]
First, reduce to the strict case: strictification
-(the [lem:n3:strictification] lemma) applied to $(g,K,e,b)$ gives
+(the [lem:n3:strictification] lemma) applied to (g,K,e,b) gives
strictified data with the same charge integral and charged DEC; the
rest-frame replacement of the [prop:n3:rest-frame] proposition then
-gives compactly supported $K$, zero momentum, and energy $m'=m+o(1)$.
+gives compactly supported K, zero momentum, and energy m'=m+o(1).
Second, the charged elliptic construction: in the coupled elliptic
system of the [sec:elliptic] section add the extra source terms
-$2(|e|^2+|b|^2)$ and $2*(ewedge *b)$ to the constraint densities. The
+2(|e|^2+|b|^2) and 2*(ewedge *b) to the constraint densities. The
comparison metric
-$barbar g_g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf)$ satisfies [eq:ch-comparison]
-$$
+barbar g_g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf) satisfies [eq:ch-comparison]
+
R_{barbar g}gge 2|barbar e|_e|_{barbar g}g^2+2|barbar b|_b|_{barbar g}g^2+O(eta_n),
-$$
-with $|barbar e|_e|_{barbar g}g^2, |barbar b|_b|_{barbar g}g^2$ the transported field
-densities. The boundedness of $barbar e,e,barbar b$b$ and of their covariant
+
+with |barbar e|_e|_{barbar g}g^2, |barbar b|_b|_{barbar g}g^2 the transported field
+densities. The boundedness of barbar e,e,barbar bb and of their covariant
derivatives on the graph, together with the coercivity of
-the [lem:n3:hessian-square] lemma, absorbs the $O(eta_n)$ into [eq:ch-comparison-eps]
-$$
+the [lem:n3:hessian-square] lemma, absorbs the O(eta_n) into [eq:ch-comparison-eps]
+
R_{barbar g}gge 2|barbar e|_e|_{barbar g}g^2+2|barbar b|_b|_{barbar g}g^2,
-$$
-after shrinking the comparison to a lower bound on $eta_n$ (the
-$O(eta_n)$ term appears with sign in the coercive bound).
-
-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]
-$$
+
+after shrinking the comparison to a lower bound on eta_n (the
+O(eta_n) term appears with sign in the coercive bound).
+
+Third, the boundary sphere ∂Ω_R of the comparison is a
+minimal surface with area at least (1-ε)A_{min}(S) and charge
+(Q_E,Q_B). By the [thm:charged-riemannian] theorem, [eq:ch-assembly]
+
m_{rm ADM}(barbar g)g)ge
-frac{r_{barbar g}g+Q^2/r_{barbar g}g}{2},qquad
+frac{r_{barbar g}g+Q^2/r_{barbar g}g}{2},
r_{barbar g}g=(frac{Area_{barbar g}g(∂Ω_R)}{4pi})^{1/2}.
-$$
-Pass to the limit as $ε↓0$ and $Rto∞$: the left side
-becomes $m$, the right side becomes the charged mass bound with
-$r=(A_{min}(S)/(4pi))^{1/2}$.
+
+Pass to the limit as ε↓0 and Rto∞: the left side
+becomes m, the right side becomes the charged mass bound with
+r=(A_{min}(S)/(4pi))^{1/2}.
@@ -2440,9 +2440,9 @@ the charged flow; that is out of scope.
Corollary (Pure-electric rest frame). [cor:ch:electric]
-Under the hypotheses of the [thm:ch:main] theorem, if $P=0$ and $B=0$
+Under the hypotheses of the [thm:ch:main] theorem, if P=0 and B=0
(pure electric field), the conclusion [eq:ch-inequality] holds
-with $m=E$ and the explicit cut convention of
+with m=E and the explicit cut convention of
the [def:bridge-admissible] definition.
@@ -2457,28 +2457,28 @@ 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 $(Ω,g,K)$ is a connected orientable
-smooth $4$-manifold with Riemannian metric $g$ and symmetric
-$(0,2)$-tensor $K$, complete with smooth compact boundary
-$B=∂Ω$ 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
+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=∂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 $m=√(E^2-|P|^2)$. Let $A_e=a_g(B)$
+(E,P) uses the standard 4-dimensional normalization; when
+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 $(Ω,g,K)$ be a four-dimensional exterior with
-$theta_+(B)le0$, DEC, and integrable $mu,|J|_g$. Let
-$omega_3=2pi^2$. If $E>|P|$, then [eq:four-inequality]
-$$
+Let (Ω,g,K) be a four-dimensional exterior with
+theta_+(B)le0, DEC, and integrable mu,|J|_g. Let
+omega_3=2pi^2. If E>|P|, then [eq:four-inequality]
+
mgefrac12((A_e)/(omega_3))^{2/3}.
-$$
+
The complete-data statement of the [thm:bridge:global] theorem holds in
-dimension $4$ with the same right-hand side.
+dimension 4 with the same right-hand side.
@@ -2488,46 +2488,46 @@ with the four modifications below.
Strictification and end reduction in dimension four [sec:four-reduction]. The drift-Poisson strictification of the [lem:n3:strictification] lemma
-adapts: in dimension $4$ the conformal family $gmapsto u^2g$,
-$Kmapsto uK$ preserves the constraints and DEC, and the drift
-source exponent is $w=r^{-4-delta}$ with $0<delta<beta<min(q,1)$.
-The repaired data satisfy $theta_+(B)<0$, $mu-|J|_gge c_0r^{-4-delta_1}$
-with $c_0>0$, and $K$ compactly supported. The end-reduction of
-the [prop:n3:rest-frame] proposition works in dimension $4$: a distant
-Lorentz boost sends $(E,P)mapsto(m,0)$, and a conformal support
-truncation on the far end adjusts $m$ and $A_e$ by $O(R^{-3})$.
+adapts: in dimension 4 the conformal family gmapsto u^2g,
+Kmapsto uK preserves the constraints and DEC, and the drift
+source exponent is w=r^{-4-delta} with 0<delta<beta<min(q,1).
+The repaired data satisfy theta_+(B)<0, mu-|J|_gge c_0r^{-4-delta_1}
+with c_0>0, and K compactly supported. The end-reduction of
+the [prop:n3:rest-frame] proposition works in dimension 4: a distant
+Lorentz boost sends (E,P)mapsto(m,0), and a conformal support
+truncation on the far end adjusts m and A_e by O(R^{-3}).
The four-dimensional elliptic construction [sec:four-elliptic]. The coupled system of the [sec:elliptic] section applies in dimension
-$4$. The scalar identity changes: the lapse appears with the extra
-degree of freedom $e^{3t}$ (rather than $e^{2t}$), and the conformal
-family in the metric comparison is $u^3$ rather than $u^4$ with
-$u=e^{t/3}$. The variables $f,t$, the auxiliary $D,d,Z,h,w,a,v,chi$
-take their dimension-$4$ forms. The comparison metric
-$barbar g_g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf)$ has
-nonnegative scalar curvature on $Ω_R$, mass at most $m+ε$, and
-boundary three-volume at least $(1-ε)A_e$. The regularity estimates
+4. The scalar identity changes: the lapse appears with the extra
+degree of freedom e^{3t} (rather than e^{2t}), and the conformal
+family in the metric comparison is u^3 rather than u^4 with
+u=e^{t/3}. The variables f,t, the auxiliary D,d,Z,h,w,a,v,chi
+take their dimension-4 forms. The comparison metric
+barbar g_g_{ij}=d^{-1}(g_{ij}+l(t)^2∂_if∂_jf) has
+nonnegative scalar curvature on Ω_R, mass at most m+ε, and
+boundary three-volume at least (1-ε)A_e. The regularity estimates
of the [sec:existence] section (graph Sobolev, axial gradient estimate,
-four-stage homotopy) pass through in dimension $4$ with the same
+four-stage homotopy) pass through in dimension 4 with the same
structure; the scalar identity uses a Schur-type trace calculation
-$3Delta_{barbar g}gZ=mathcal T$-typecalT$-type decomposition that gives
-$R_{barbar g}gge0$ from the DEC.
+3Delta_{barbar g}gZ=mathcal T-typecalT-type decomposition that gives
+R_{barbar g}gge0 from the DEC.
The four-dimensional Riemannian Penrose inequality [sec:four-rmp]. The classical four-dimensional Riemannian Penrose inequality is used here.
Theorem (Four-dimensional Riemannian Penrose). [thm:four-riemannian]
-Let $(M,g)$ be a complete smooth asymptotically flat four-dimensional
-Riemannian manifold with nonnegative scalar curvature $R_gge0$ and
-ADM mass $m_{ADM}$. Let $Gamma$ be a smooth compact outermost minimal
-$3$-surface. Then
-$$
+Let (M,g) be a complete smooth asymptotically flat four-dimensional
+Riemannian manifold with nonnegative scalar curvature R_gge0 and
+ADM mass m_{ADM}. Let Gamma be a smooth compact outermost minimal
+3-surface. Then
+
m_{ADM}gefrac12((|Gamma|_g)/(omega_3))^{2/3},
-qquad omega_3=2pi^2.
+ omega_3=2pi^2.
+
-$$
@@ -2538,9 +2538,9 @@ This is the four-dimensional Riemannian Penrose inequality, a classical
result (Huisken--Ilmanen 2001; Bray 2001, extended by Bray--Lee 2009).
Its three-dimensional form is (T1) of the [sec:external] section; the
four-dimensional statement uses the flow identity
-$Σ(t)=m_{ADM}-tfrac12(A(t)/omega_3)^{2/3}$ in place of the
-three-dimensional formula. At the outermost minimal $3$-surface the
-area statistic $Area(t)$ equals $|Gamma|_g$, giving the stated
+Σ(t)=m_{ADM}-tfrac12(A(t)/omega_3)^{2/3} in place of the
+three-dimensional formula. At the outermost minimal 3-surface the
+area statistic Area(t) equals |Gamma|_g, giving the stated
inequality. It is used as an external input; the details of the flow
are classical.
@@ -2550,27 +2550,27 @@ are classical.
Assembly [sec:four-assembly]. Proof (Proof of the [thm:four:main). theorem]
Apply strictification (the [lem:n3:strictification] lemma in
-dimension $4$), end reduction (the [prop:n3:rest-frame] proposition
-in dimension $4$), and the coupled elliptic construction
-(the [sec:four-elliptic] section) to obtain comparison metric $barbar g$g$
+dimension 4), end reduction (the [prop:n3:rest-frame] proposition
+in dimension 4), and the coupled elliptic construction
+(the [sec:four-elliptic] section) to obtain comparison metric barbar gg
with
-$$
-R_{barbar g}gge0,quad
-m_{ADM}(barbar g)g)le m+ε,quad
+
+R_{barbar g}gge0,
+m_{ADM}(barbar g)g)le m+ε,
Area_{barbar g}g(∂Ω_R)ge(1-ε)A_e.
-$$
+
Apply the [thm:four-riemannian] theorem to the outermost boundary
sphere:
-$$
+
m_{ADM}(barbar g)g)gefrac12(frac{Area_{barbar g}g(∂Ω_R)}
{omega_3})^{2/3}.
-$$
-Let $ε↓0$: the left side becomes $m$, the right side becomes
-$frac12(A_e/omega_3)^{2/3}$, which is [eq:four-inequality].
+
+Let ε↓0: the left side becomes m, the right side becomes
+frac12(A_e/omega_3)^{2/3}, which is [eq:four-inequality].
@@ -2583,9 +2583,9 @@ with Schwarzschild--Tangherlini.
Theorem (Four-dimensional rigidity). [thm:rig4:rigidity]
Under the hypotheses of the [thm:four:main] theorem in the connected,
one-ended, outermost, outer area-minimizing equality case
-$m=frac12(A_e/omega_3)^{2/3}$, the exterior $M_{ext}$ is isometric to
+m=frac12(A_e/omega_3)^{2/3}, the exterior M_{ext} is isometric to
the future-horizon exterior in the Schwarzschild--Tangherlini
-spacetime of mass $m$, with $K$ the second fundamental form generated
+spacetime of mass m, with K the second fundamental form generated
by the stationary Killing field of that spacetime.
@@ -2595,12 +2595,12 @@ by the stationary Killing field of that spacetime.
Proof.
The proof of the [thm:n3:rigidity] theorem applies verbatim with the
following substitutions: the causal stationary field
-(the [prop:n3:stationary] proposition) extends to dimension $4$; the
+(the [prop:n3:stationary] proposition) extends to dimension 4; the
twist-vanishing (the [prop:n3-twist-vanishes] proposition) still
uses the positive-mass bootstrap on the conformal double; the
conformal double (the [prop:n3-double] proposition) exists; and
-Schoen--Yau positivity gives $ell=1$ and $Scong S^3$. The
-static-vacuum classification in spacetime dimension $5$
+Schoen--Yau positivity gives ell=1 and Scong S^3. The
+static-vacuum classification in spacetime dimension 5
(Bunting--Masood-ul-Alam) then identifies the exterior with
Schwarzschild--Tangherlini of the same mass.
@@ -2612,36 +2612,36 @@ The anti-de Sitter Penrose inequality [sec:ads]. Schwarzschild-anti-de Sitter an
-Fix the Schwarzschild-anti-de Sitter background of mass $m_0ge0$ in
-spacetime dimension $4$. In the usual static coordinates, [eq:ads-bgd]
-$$
-g^0=-N^2d t^2+(d r^2)/(N^2)+r^2sigma,qquad
+Fix the Schwarzschild-anti-de Sitter background of mass m_0ge0 in
+spacetime dimension 4. In the usual static coordinates, [eq:ads-bgd]
+
+g^0=-N^2d t^2+(d r^2)/(N^2)+r^2sigma,
N^2=1+r^2-(2m_0)/(r),
-$$
-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
-$g^0=d s^2+r(s)^2sigma$ with $r=e^s(1+O(e^{-2s}))$; the boundary is
-at $s=∞$.
+
+with sigma the round metric on S^2, and on the time slice t=0
+K^0=0. The spatial metric in Fefferman-Graham form is
+g^0=d s^2+r(s)^2sigma with r=e^s(1+O(e^{-2s})); the boundary is
+at s=∞.
A smooth branch of asymptotically anti-de Sitter initial data near the
-background is a smooth family $(g_ε,K_ε)$ for
-$εin[0,ε_0)$ such that each $(g_ε,K_ε)$ is of finite
+background is a smooth family (g_ε,K_ε) for
+εin[0,ε_0) such that each (g_ε,K_ε) is of finite
angular type, satisfies the Einstein constraints on the exterior of a
-compact future MOTS $S_ε$, satisfies $(g_ε,K_ε)=(g^0,K^0)+O(ε)$
+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 $A(ε)=|S_ε|_{g_ε}$ be the boundary area and
-$m_{AH}(ε)$ 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 $ε_1in(0,ε_0]$ such that for
-every $εin[0,ε_1)$, [eq:ads-inequality]
-$$
+Under the hypotheses above there is ε_1in(0,ε_0] such that for
+every εin[0,ε_1), [eq:ads-inequality]
+
m_{AH}(ε)
ge√((A(ε))/(16pi))
(1+(A(ε))/(4pi)).
-$$
-Equality holds if and only if the branch seed is radial and $εge0$
+
+Equality holds if and only if the branch seed is radial and εge0
is sufficiently small. The equality family is Schwarzschild-anti-de
Sitter at the matching mass and area.
@@ -2650,47 +2650,47 @@ Sitter at the matching mass and area.
Taylor structure and weighted remainders [sec:ads-taylor]. Write the branch in a Taylor expansion
-$g_ε=g^0+ε g_1+ε^2 g_2+ε^3 g_3+O_4(ε^4)$ with each $g_j$
-of finite angular type, and likewise $K_ε$. The weighted remainder
-$O_4(ε^4)$ controls the asymptotic functionals.
+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]
-$$
-m_{AH}(ε)=sum_{j=0}^3ε^jm_j+O(ε^4),quad
+There are real m_0,m_1,m_2,m_3 and A_0,A_1,A_2,A_3 such that [eq:ads-taylor-exp]
+
+m_{AH}(ε)=sum_{j=0}^3ε^jm_j+O(ε^4),
A(ε)=sum_{j=0}^3ε^j A_j+O(ε^4),
-$$
+
with the remainders controlled by the weighted bound
-$F=O(e^{-gamma s})$ with $gamma>3$.
+F=O(e^{-gamma s}) with gamma>3.
Lemma (Inverse-asymptotic). [lem:ads-inverse]
-Let $L=-Delta_{g^0}+3$. If $F$ is smooth finite-type with
-$F=O(e^{-gamma s})$ for some $gamma>3$, then for either Dirichlet or
-homogeneous Neumann data on $S$ there is a unique finite-type decaying
-solution of $Lu=F$ with [eq:ads-remainder]
-$$
-u=r^{-3}zeta(x)+o(r^{-3}),quad
+Let L=-Delta_{g^0}+3. If F is smooth finite-type with
+F=O(e^{-gamma s}) for some gamma>3, then for either Dirichlet or
+homogeneous Neumann data on S there is a unique finite-type decaying
+solution of Lu=F with [eq:ads-remainder]
+
+u=r^{-3}zeta(x)+o(r^{-3}),
∂_s(u-r^{-3}zeta(x))=o(r^{-3}),
-$$
-$zeta$ finite-type. Differentiation of $F$ in $s$ through order $k$
-gives the same statement through $k+2$ derivatives of $u$.
+
+zeta finite-type. Differentiation of F in s through order k
+gives the same statement through k+2 derivatives of u.
Proof.
-Variation of constants for $y''+2y'-3y=G$ with
-$G=O(e^{-gamma's})$, $gamma'=min(gamma,beta+2)>3$, eliminates
-the growing homogeneous mode $e^s$ and gives
-$y=ce^{-3s}+O(e^{-gamma's})$. Angular decomposition closes by
+Variation of constants for y''+2y'-3y=G with
+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 $delta↓0$, gives uniqueness of the
+Ah+delta, letting delta↓0, gives uniqueness of the
decaying solution.
@@ -2699,75 +2699,75 @@ decaying solution.
The quadratic coefficient and its kernel [sec:ads-quadratic]. Proposition (The quadratic form). [prop:ads-c2]
In the expansion [eq:ads-mass-gap]
-$$
+
m_{AH}(ε)-√((A(ε))/(16pi))
(1+(A(ε))/(4pi))
=c_2[q]ε^2+c_3[q]ε^3+c_4[q]ε^4+O(ε^5)
-$$
-the coefficient $c_2[q]$ is a nonnegative quadratic form, and its
-kernel has dimension exactly $4$ (the four linear harmonics on $S^2$).
+
+the coefficient c_2[q] is a nonnegative quadratic form, and its
+kernel has dimension exactly 4 (the four linear harmonics on S^2).
Proof.
-Conformal decomposition of $g_1$ gives a TT part $k$ together with a
+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
+expansion reduces c_2 to the TT square identity
(the [lem:ads-tt-square] lemma), which makes it
-$(1)/(16pi)int_M N|k|^2,d V_0
--(kappa)/(16pi)int_S vDv,d A_0$. The argument of
-the [lem:ads-tt-square] lemma identifies the last term, and $D$'s
+(1)/(16pi)int_M N|k|^2,d V_0
+-(kappa)/(16pi)int_S vDv,d A_0. The argument of
+the [lem:ads-tt-square] lemma identifies the last term, and D's
positivity makes the form nonnegative. Its kernel is the four
-translation modes of the round $S^2$.
+translation modes of the round S^2.
Lemma (Weighted TT square identity). [lem:ads-tt-square]
-Let $k$ be smooth TT with $k=O(r^{-gamma})$ for some $gamma>3/2$, and
-$v$ the unique decaying solution of $Lv=0$ with
-$v|_S=D^{-1}(k_{ss}|_S)$, $D=-Delta_S+2kappa/a$. Then
-$Q(v)=Hess v-vB$ is TT, $Q(v)_{ss}|_S=k_{ss}|_S$, and [eq:ads-tt-square]
-$$
+Let k be smooth TT with k=O(r^{-gamma}) for some gamma>3/2, and
+v the unique decaying solution of Lv=0 with
+v|_S=D^{-1}(k_{ss}|_S), D=-Delta_S+2kappa/a. Then
+Q(v)=Hess v-vB is TT, Q(v)_{ss}|_S=k_{ss}|_S, and [eq:ads-tt-square]
+
int_M N|k|^2,d V_0-kappaint_S vDv,d A_0
=int_M N|k-Q(v)|^2,d V_0.
-$$
+
The quartic argument on the kernel [sec:ads-quartic]. Proposition (Quartic positivity). [prop:ads-c4]
-If $q$ is a nonzero kernel direction of $c_2$ and $q$ is not a radial
-seed, then $c_4[q]>0$.
+If q is a nonzero kernel direction of c_2 and q is not a radial
+seed, then c_4[q]>0.
Proof.
-The kernel is the linear harmonics, so $q(x)=V(s)x$ for a decaying $V$.
-A change of spacetime slice at $2$-jet order in the Taylor expansions
+The kernel is the linear harmonics, so q(x)=V(s)x for a decaying V.
+A change of spacetime slice at 2-jet order in the Taylor expansions
(the [lem:ads-slice] lemma) removes the first extrinsic-curvature
coefficient of the nonradial directions, preserves the boundary area
-through the required order, and reduces $c_4$ to a second TT square
+through the required order, and reduces c_4 to a second TT square
built from the transverse coefficient near the original boundary. This
-gives $c_4[q]ge0$. Equality would force the changed metric to be a
-round warped-product $2$-jet; conformality of the original branch then
+gives c_4[q]ge0. Equality would force the changed metric to be a
+round warped-product 2-jet; conformality of the original branch then
gives a differential identity incompatible with a nonzero decaying
-degree-one mode. Hence $c_4[q]>0$.
+degree-one mode. Hence c_4[q]>0.
Lemma (Finite-jet slice change). [lem:ads-slice]
-There is a smooth local diffeomorphism $Psi$ and a smooth conformal
-factor $omega$ with $omega_0=1,Psi_0=id$, such that
-$widetilde g=omega_εPsi_ε^*g^0+O(ε^3)$ holds on the
+There is a smooth local diffeomorphism Psi and a smooth conformal
+factor omega with omega_0=1,Psi_0=id, such that
+widetilde g=omega_εPsi_ε^*g^0+O(ε^3) holds on the
original slice.
@@ -2775,11 +2775,11 @@ original slice.
Proposition (Exact radial identity). [prop:ads-radial]
-If $q$ is a radial seed, then for all sufficiently small $εge0$, [eq:ads-radial-equality]
-$$
+If q is a radial seed, then for all sufficiently small εge0, [eq:ads-radial-equality]
+
m_{AH}(ε)=√(A(ε)/16pi)
(1+A(ε)/4pi).
-$$
+
@@ -2787,9 +2787,9 @@ $$
Proof.
Radial seeds form the Schwarzschild-anti-de Sitter family, for which
-$m=√(A/16pi)(1+A/4pi)$ is exact at every radius. At $ε=0$ the
+m=√(A/16pi)(1+A/4pi) is exact at every radius. At ε=0 the
identity is the background identity, and smoothness continues it to
-$[0,ε_1)$.
+[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
-$c_2[q]ε^2+c_4[q]ε^4+O(ε^5)$ with $c_2[q]ge0$ and $c_4[q]>0$ on
-the kernel, so the gap is strictly positive for all small $ε>0$.
+c_2[q]ε^2+c_4[q]ε^4+O(ε^5) with c_2[q]ge0 and c_4[q]>0 on
+the kernel, so the gap is strictly positive for all small ε>0.
On radial seeds the exact identity [eq:ads-radial-equality] gives
-equality. The interval $[0,ε_1)$ depends only on the background data
+equality. The interval [0,ε_1) depends only on the background data
and the branch.
@@ -2816,8 +2816,8 @@ 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 $A(ε)$
+asymptotically hyperbolic background; the relevant mass is m_{AH}
+rather than the asymptotically flat m, and the boundary area A(ε)
plays the role of the enclosing area.