diff f6bf760..6fbd3ac spacetime_penrose_area_charge_v2.txt
diff --git a/spacetime_penrose_area_charge_v2.txt b/spacetime_penrose_area_charge_v2.txt
index a576bab..db2bb31 100644
--- a/spacetime_penrose_area_charge_v2.txt
+++ b/spacetime_penrose_area_charge_v2.txt
@@ -3,10 +3,10 @@ Introduction [sec:intro]. When a massive star collapses, it may form a black hol
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
-cosmic censorship. This is the Penrose inequality. The theorems
+cosmic censorship. This is the Penrose ineuality. 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
+classical Riemannian Penrose ineuality for a metric of nonnegative
scalar curvature.
An initial data set for the Einstein equations is a triple (M,g,K): a
@@ -30,7 +30,7 @@ appears as such a surface in a spacelike slice.
Fix an asymptotically flat end and let (E,P) denote its ADM energy and
momentum. Once E>|P|, the invariant mass of the end is
-m=√(E^2-|P|^2). The spacetime Penrose inequalities bound m below by
+m=√(E^2-|P|^2). The spacetime Penrose ineualities 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
@@ -109,7 +109,7 @@ trapped-region boundary and disjoint collars then prepare a fixed exterior
for a coupled elliptic deformation. The graph function and conformal factor
produce a Riemannian comparison metric of nonnegative scalar curvature and
controlled energy, whose enclosing area is at least the original infimum.
-The Riemannian Penrose inequality finishes the comparison. The charged
+The Riemannian Penrose ineuality finishes the comparison. The charged
argument reuses the same elliptic system and adds a square estimate that
carries the electromagnetic momentum.
@@ -135,7 +135,7 @@ External inputs [sec:external]. Three classical results are used without proof.
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
+(T1) Riemannian Penrose ineuality (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
@@ -155,7 +155,7 @@ contributions end and the classical machinery begins.
to the Schwarzschild exterior {r>2M}, with mass
M=|∂ N|_h^{1/2}/√(16pi).
-(T4) Charged Riemannian Penrose inequality
+(T4) Charged Riemannian Penrose ineuality
(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
@@ -184,7 +184,7 @@ momentum density J are determined by the constraint equations [eq:constraints]
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.
+is the pointwise ineuality 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]
@@ -193,7 +193,7 @@ An end of M is a neighborhood of infinity diffeomorphic to
K_{ij}=O_1(r^{-1-q}),
- q>tfrac32,
+ q>3/2,
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]
@@ -269,10 +269,10 @@ 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)}.
+spacetime of mass m satisfies m=1/2(A/omega_{n-1})^{(n-2)/(n-1)}.
-The three-dimensional exterior inequality [sec:n3]. Statement [sec:n3-statement]
+The three-dimensional exterior ineuality [sec:n3]. Statement [sec:n3-statement]
@@ -301,7 +301,7 @@ The proof assembles three ingredients: a conformal deformation that
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
+elliptic comparison with the Riemannian Penrose ineuality
([sec:geometry]--[sec:riemannian-penrose]).
@@ -328,7 +328,7 @@ 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
+the Green kernel of L_g=-Delta_g+1/8R_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
@@ -368,11 +368,11 @@ Taking infima gives a_{g_s}(B)ge a_g(B).
-The strictified exterior inequality [sec:n3-strictified]. The strict case is also needed, proved in full in Sections~[sec:end-preparation]
+The strictified exterior ineuality [sec:n3-strictified]. The strict case is also needed, proved in full in Sections~[sec:end-preparation]
through~[sec:existence].
-Theorem (Strictified exterior inequality). [thm:n3:strict-exterior]
+Theorem (Strictified exterior ineuality). [thm:n3:strict-exterior]
Let (Ω,g,K) satisfy all the hypotheses of
the [thm:n3:exterior] theorem together with
- K compactly supported;
@@ -397,12 +397,12 @@ metric g_star on the strictified exterior with [eq:comparison]
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 ineuality
(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
+Completion of the exterior ineuality [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
@@ -417,8 +417,8 @@ m'gesqrt{a_{g'}(B)/16pi}, so
√((a_g(B))/(16pi))
- lefrac{m'}{√(1-ε)}
- lefrac{m+ε}{√(1-ε)}.
+ lem'/√(1-ε)
+ lem+ε/√(1-ε).
Let ε↓0 to conclude.
@@ -465,7 +465,7 @@ three gives
u^2J_u=J+2pi(nablalog u,·)-(tr_gpi)dlog u
=J+4K(nablalog u,·),
which is [eq:conformal-J]. Norms in g_u add a factor u^{-2},
-and the triangle inequality yields [eq:conformal-dec]. Finally
+and the triangle ineuality 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].
@@ -474,7 +474,7 @@ 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
+Let (N,g,K) satisfy [eq:decay] with q>3/2, 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]
@@ -489,7 +489,7 @@ 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;
+(iii) decay class min(q,1)>1/2;
(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).
@@ -515,7 +515,7 @@ 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
Laplacian is invertible: the Dirichlet face supplies the Poincare
-inequality, and a weak solution is smooth by boundary regularity. The
+ineuality, 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
principle force a trivial kernel, so the operator is an invertible
@@ -774,14 +774,14 @@ 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
+decomposition B_0=B_{black}cup B_{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},
+H_{B_0}+P_{B_0}=c_b on B_{black},
-H_{B_0}-P_{B_0}=c_w on B_{rm white}.
+H_{B_0}-P_{B_0}=c_w on B_{white}.
On each face there is a smooth disjoint collar (B_0^jtimes[0,1),s)
@@ -813,7 +813,7 @@ shrunk to small amplitude.
The coupled elliptic comparison system [sec:elliptic]. Fix a two-color exterior Ω_0 of the [prop:n3:two-color] proposition.
-Its boundary B_0=B_{rm black}cup B_{rm white} carries two strict
+Its boundary B_0=B_{black}cup B_{white} carries two strict
negative boundary expansions; and its far end is asymptotically flat at
infinity.
@@ -833,7 +833,7 @@ l(t) constant between 1 and e for tge1/n; set p=l'/l, so
f:Ω_0toℝ and t:Ω_0toℝ, define
&sigma=nabla f, D=1+l(t)^2|sigma|^2, d=D^{-1},
-Z=t+tfrac16log D, h=eta_n f,
+Z=t+1/6log D, h=eta_n f,
&w=(l,sigma)/(sqrt D), a=|w|, v=a^2,
chi=(3d)/(3+pv),
A_d=I-w ⊗ w,
@@ -997,12 +997,12 @@ constant extension. At the contact,
a_ito1, chi_ito0.
-Positivity of A_{chi_i} and the Hessian inequality at a lower
+Positivity of A_{chi_i} and the Hessian ineuality at a lower
contact give
tr_{A_{chi_i}}K+
- frac{l_i}{tau√(D_i)}tr_{A_{chi_i}}Hessphi
+ l_i/tau√(D_i)tr_{A_{chi_i}}Hessphi
le F_i(x_i).
@@ -1043,7 +1043,7 @@ expansion:
So psi_j^{-1}(phi+c_j) is a smooth lower test of
-underline h with contact <k', and the oriented-level inequality
+underline h with contact <k', and the oriented-level ineuality
passes to phi. Hence every smooth exterior support of E_k has
expansion at most k'+C_b.
@@ -1154,7 +1154,7 @@ The one-sided derivative at s=0 proves the black part; replacing
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
+by the directional ineualities of the homotopy. The same dle Cell
bound and a fixed test slope handle the rest. This gives the two-sided
third-stage bound.
@@ -1186,7 +1186,7 @@ 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
+W=lnabla f/D=sqrt d,a e_f. Then |W|_{gbarbar}=ale1, and for
every covector xi,
@@ -1201,7 +1201,7 @@ Differentiation gives
(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+√(T)):calT)): the
+Coercivity bounds both right sides by Pi_n(1+√(T)):T)): 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|).
@@ -1212,7 +1212,7 @@ uW_nu=ε_B La and sqrt D,W_nu=ε_B a. Apply
the divergence theorem to -ε_B uWphi with a fixed
collar cutoff. Its boundary flux is Laphi 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 ineuality; replacing u by sqrt D gives the
second.
@@ -1224,7 +1224,7 @@ 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_0T+calT+rho+tfrac12delta_0tau asigma
+ Xigedelta_0T+T+rho+1/2delta_0tau asigma
quadon {Z>M_0}.
@@ -1232,7 +1232,7 @@ 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(T+calT+rho+tau asigma)d V_g
+ int_Ω u k_0(T+T+rho+tau asigma)d V_g
+int_Ω u k_0',|dZ|_{A_chi}^2d V_gle C(n).
@@ -1242,7 +1242,7 @@ on [M_0+1,∞),
Proof.
-Outside supp m_0, Xi=delta_0T+calT+rho+delta_0tau asigma
+Outside supp m_0, Xi=delta_0T+T+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∞,
@@ -1269,8 +1269,8 @@ has |A_chi K(w,·)|_nule Cchile Cd. Thus
Integrating,
- int_Ω uk_0(delta_0T+calT+rho
- +tfrac12delta_0tau asigma)d V_g
+ int_Ω uk_0(delta_0T+T+rho
+ +1/2delta_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.
@@ -1281,7 +1281,7 @@ Pi_nfractauellint_{C}+√(T))
+k_0'|dZ|_{A_chi})d V_g. On the fixed collars rhogerho_0>0,
so (1+√(T)) C(rho+delta_0T) there, and
Pi_ntau/ellto0 absorbs the first summand into the positive source.
-Young's inequality reduces the derivative summand to
+Young's ineuality reduces the derivative summand to
uk_0'|dZ|^2_{A_chi}+C(Pi_ntau/ell)^2uk_0', whose second term has
bounded compact integral. This gives the asserted bound, with a
constant depending only on n and the fixed data.
@@ -1292,7 +1292,7 @@ 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
-T+calT+rho+tau asigma. The ordinary product graph
+T+T+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,
@@ -1303,7 +1303,7 @@ Q,
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
+ineuality (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.
@@ -1313,12 +1313,12 @@ 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
+Zle t+1/8log(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
+Finally, Z=t+1/8log D together with Dge1 gives
-εle tle Z, and the upper Z bound yields
@@ -1348,7 +1348,7 @@ le C(n)int_{Gamma_f}(|nabla_{Gamma_f}phi|^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,
+Gamma_f carries the metric induced from (Ω_R,gbarbar))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
@@ -1364,7 +1364,7 @@ gives, in dimension 3,
with C depending only on the immersion geometry, hence only on n.
-H"older's inequality together with V_Q<∞ yields
+H"older's ineuality together with V_Q<∞ yields
int_{Gamma_f}|phi|^{4}
@@ -1374,7 +1374,7 @@ H"older's inequality together with V_Q<∞ yields
and substituting the Sobolev bound closes the estimate on pi^{-1}Q.
Patching finitely many such Q that exhaust the support of phi
gives [eq:graph-sobolev], using
-|phi|^2le(1+T)|calT)|phi|^2 to absorb the remaining
+|phi|^2le(1+T)|T)|phi|^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].
@@ -1447,11 +1447,11 @@ 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
+inf_{B_{1/2}}sto0. The inhomogeneous weak Harnack ineuality for
[n3:axial:s-supersolution], after dropping the favorable
-2c_0|H|^2 term, gives sto0 in measure on B_{1/2}. Testing
by phi^{2}(b-s)_{+} with a cutoff phi and fixed b>0, and
-absorbing the trace term by ellipticity and Young's inequality, yields
+absorbing the trace term by ellipticity and Young's ineuality, yields
int_{{s<b}}phi^{2}|Ds|^{2}
@@ -1632,11 +1632,11 @@ Then the comparison metric barbar gg of [eq:comparison-metric]
satisfies
-R_{barbar g}gge0quadon Ω_R,
+R_{gbarbar}ge0quadon Ω_R,
-Area_{barbar g}g(∂Ω_R)ge(1-ε)a_g(B),
+Area_{gbarbar}(∂Ω_R)ge(1-ε)a_g(B),
-m_{rm ADM}(barbar g)g)le m+ε.
+m_{ADM}(gbarbar))le m+ε.
@@ -1650,8 +1650,8 @@ 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)+T(H^f)
+1/2e^{2t}S_{gbarbar}
+={}&1/2S_g-vRic_g(e,e)+T(H^f)
&-3dHess_g t(e,e) -(3+pv)tr_perpHess_g t +O(eta_n^2),
where e is the unit vector of sigma=nabla f (the d-term drops
@@ -1662,27 +1662,27 @@ 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<0H<0 and the maximum principle on t with the mixed
+Hhat<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
propagated by Hopf. Consequently
-tfrac12e^{2t}S_{barbar g}g
-getfrac12S_g+T(H^f)-vRic_g(e,e)+O(calT(H^f)-vRic_g(e,e)+O(eta_n^2).
+1/2e^{2t}S_{gbarbar}
+ge1/2S_g+T(H^f)-vRic_g(e,e)+O(T(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
+1/2S_gge-1/2|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
+C(n)) bounds the negative pieces -1/2|K|^2-vRic_g(e,e) by a
fixed multiple of T(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.
+1/2e^{2t}S_{gbarbar}ge0 for all sufficiently large n,
+i.e. R_{gbarbar}ge0 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
@@ -1690,30 +1690,30 @@ 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
+|Area_{gbarbar}(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.
+m_{ADM}(barbar g)=m+O(g)=m+O(eta_n)le m+ε for n sufficiently
+large. This is precisely the asserted ineuality.
-The Riemannian Penrose inequality [sec:riemannian-penrose]. The following classical result is used without proof.
+The Riemannian Penrose ineuality [sec:riemannian-penrose]. The following classical result is used without proof.
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
+Let (M,gbarbar)) be a smooth complete asymptotically flat three-manifold
+with R_{gbarbar}ge0 and with compact boundary whose least enclosing
area is A_{min}>0. Then
-m_{rm ADM}(barbar g)g)gesqrt{frac{A_{min}}{16pi}}.
+m_{ADM}(gbarbar))gesqrt{frac{A_{min}}{16pi}}.
@@ -1730,19 +1730,19 @@ the [prop:n3:comparison] proposition gives a Riemannian comparison metric
barbar gg on Ω_R with
-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.
+R_{gbarbar}ge0,
+Area_{gbarbar}(∂Ω_R)ge(1-ε)a_g(B),
+m_{ADM}(gbarbar))le m+ε.
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}(gbarbar))geArea_{gbarbar}(∂Ω_R)
+ge(1-ε)a_g(B). The Riemannian Penrose ineuality
(the [thm:riemannian-penrose] theorem) gives
-m+εge m_{rm ADM}(barbar g)g)
+m+εge m_{ADM}(gbarbar))
gesqrt{frac{A_{min}(barbar g)}g){16pi}}
ge√(((1-ε)a_g(B))/(16pi)).
@@ -1759,7 +1759,7 @@ complete-data formulation packages the same conclusion for a connected
initial data set with several ends, an admissible boundary that may be
disconnected or may contain other ends on the inner side, and an
enclosing infimum that can move into other ends. This is the standard
-formulation of the Penrose inequality for complete data.
+formulation of the Penrose ineuality for complete data.
Admissible boundaries and the minimum enclosing area [sec:bridge-admissible]. Definition (Admissible boundary and minimum enclosing area). [def:bridge-admissible]
@@ -1789,7 +1789,7 @@ be trapped or minimal.
The global theorem [sec:bridge-global]. Theorem (Complete-data form). [thm:bridge:global]
Let (M,g,K) satisfy the hypotheses of
the [def:bridge-admissible] definition. Then for every chosen end and
-every admissible boundary S=∂ D^+, [eq:bridge-inequality]
+every admissible boundary S=∂ D^+, [eq:bridge-ineuality]
E>|P|_delta,
√(E^2-|P|_delta^2)gesqrt{frac{A_{min}(S)}{16pi}},
@@ -1845,13 +1845,13 @@ the [thm:n3:exterior] theorem gives
gesqrt{frac{A_{min}(S)}{16pi}},
-which is [eq:bridge-inequality].
+which is [eq:bridge-ineuality].
-Removing the timelike restriction [sec:bridge-timelike]. Corollary (The exterior inequality without a timelike assumption). [cor:n3:no-timelike]
+Removing the timelike restriction [sec:bridge-timelike]. Corollary (The exterior ineuality 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
@@ -1993,7 +1993,7 @@ 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
-E=b^0E_{rm new}+b^iP_{{rm new},i} (with
+E=b^0E_{new}+b^iP_{{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
@@ -2080,7 +2080,7 @@ gives X=unu with |nu|_g=1, and the common lapse value
u|_{S}=a_*>0; hence N|_{S}=0. Near S the pair (u,X) so
constructed solves [eq:stationary-kid]--[eq:stationary-hess]
with strict-negativity data at S. A direct calculation from those
-equations and the constraints gives [eq:n3-N-inequality]
+equations and the constraints gives [eq:n3-N-ineuality]
Delta_gN
+u^{-1}K(nabla u,nabla u)
@@ -2088,7 +2088,7 @@ Delta_gN
ge -C(N),N,
in any region where N>0. The strong maximum principle is used for
-[eq:n3-N-inequality].
+[eq:n3-N-ineuality].
Lemma (Interior positivity of the twist). [lem:n3-twist-interior]
@@ -2199,7 +2199,7 @@ 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.
+S~= S^2.
@@ -2211,8 +2211,8 @@ 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.
+widetilde M~= S^3. Every piece of M_{ext} attaches to exactly one
+component S_i of S in the double, so ell=1 and S~= S^2.
@@ -2236,7 +2236,7 @@ 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
+the [prop:n3-one-component] proposition, ell=1 and S~= S^2 is
totally geodesic. The Birkhoff rigidity of static vacuum spacetimes
(Israel) identifies (M_{ext},g,u) with Schwarzschild: a static vacuum
spacetime containing an apparent horizon in its analytic completion
@@ -2273,7 +2273,7 @@ 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
+The spacetime Penrose ineuality 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.
@@ -2312,7 +2312,7 @@ 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]
+E>|P|_delta and m=√(E^2-|P|_delta^2), then [eq:ch-ineuality]
mge Q, rle m+√(m^2-Q^2).
@@ -2324,7 +2324,7 @@ branch rge m, and is not equivalent for every positive r.
The proof parallels the neutral case with two extra inputs: the charged
elliptic comparison and the classical charged Riemannian Penrose
-inequality. The two pieces are presented separately and then assembled.
+ineuality. The two pieces are presented separately and then assembled.
Flux transport [sec:ch-flux]. Proposition (Flux transport). [prop:ch:flux]
@@ -2334,8 +2334,8 @@ coupled elliptic construction of Sections~[sec:geometry]--
[sec:existence], the field strengths e and b are transported
to the comparison data as closed forms on the same decay class, with [eq:ch-transported]
-Q_E^{rm comp}=Q_E+o(1), Q_B^{rm comp}=Q_B+o(1),
-|e^{rm comp}-e|,|b^{rm comp}-b|=o(1)
+Q_E^{comp}=Q_E+o(1), Q_B^{comp}=Q_B+o(1),
+|e^{comp}-e|,|b^{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).
@@ -2359,7 +2359,7 @@ C^2 convergence of the deformation on compact sets.
-The charged Riemannian Penrose inequality [sec:ch-rmp]. The following classical result is used, proved by extension of the
+The charged Riemannian Penrose ineuality [sec:ch-rmp]. The following classical result is used, proved by extension of the
inverse mean curvature flow / conformal flow arguments of Huisken--
Ilmanen (2001) and Bray (2001).
@@ -2381,9 +2381,9 @@ Q=(Q_E^2+Q_B^2)^{1/2}.
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
+preserves the ineuality 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).
+becomes the charged Riemannian Penrose ineuality of (T4).
@@ -2402,14 +2402,14 @@ system of the [sec:elliptic] section add the extra source terms
comparison metric
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),
+R_{gbarbar}ge 2|barbar e|_e|_{gbarbar}^2+2|barbar b|_b|_{gbarbar}^2+O(eta_n),
-with |barbar e|_e|_{barbar g}g^2, |barbar b|_b|_{barbar g}g^2 the transported field
+with |barbar e|_e|_{gbarbar}^2, |barbar b|_b|_{gbarbar}^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]
-R_{barbar g}gge 2|barbar e|_e|_{barbar g}g^2+2|barbar b|_b|_{barbar g}g^2,
+R_{gbarbar}ge 2|barbar e|_e|_{gbarbar}^2+2|barbar b|_b|_{gbarbar}^2,
after shrinking the comparison to a lower bound on eta_n (the
O(eta_n) term appears with sign in the coercive bound).
@@ -2418,9 +2418,9 @@ 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},
-r_{barbar g}g=(frac{Area_{barbar g}g(∂Ω_R)}{4pi})^{1/2}.
+m_{ADM}(gbarbar))ge
+frac{r_{gbarbar}+Q^2/r_{gbarbar}}{2},
+r_{gbarbar}=(frac{Area_{gbarbar}(∂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
@@ -2441,7 +2441,7 @@ 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
-(pure electric field), the conclusion [eq:ch-inequality] holds
+(pure electric field), the conclusion [eq:ch-ineuality] holds
with m=E and the explicit cut convention of
the [def:bridge-admissible] definition.
@@ -2470,10 +2470,10 @@ be the four-dimensional enclosing infimum in the sense of
the [def:bridge-admissible] definition.
-Theorem (Four-dimensional exterior inequality). [thm:four:main]
+Theorem (Four-dimensional exterior ineuality). [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]
+omega_3=2pi^2. If E>|P|, then [eq:four-ineuality]
mgefrac12((A_e)/(omega_3))^{2/3}.
@@ -2510,11 +2510,11 @@ boundary three-volume at least (1-ε)A_e. The regularity estimates
of the [sec:existence] section (graph Sobolev, axial gradient estimate,
four-stage homotopy) pass through in dimension 4 with the same
structure; the scalar identity uses a Schur-type trace calculation
-3Delta_{barbar g}gZ=T-typecalT-type decomposition that gives
-R_{barbar g}gge0 from the DEC.
+3Delta_{gbarbar}Z=T-typecalT-type decomposition that gives
+R_{gbarbar}ge0 from the DEC.
-The four-dimensional Riemannian Penrose inequality [sec:four-rmp]. The classical four-dimensional Riemannian Penrose inequality is used here.
+The four-dimensional Riemannian Penrose ineuality [sec:four-rmp]. The classical four-dimensional Riemannian Penrose ineuality is used here.
Theorem (Four-dimensional Riemannian Penrose). [thm:four-riemannian]
@@ -2534,14 +2534,14 @@ m_{ADM}gefrac12((|Gamma|_g)/(omega_3))^{2/3},
Proof.
-This is the four-dimensional Riemannian Penrose inequality, a classical
+This is the four-dimensional Riemannian Penrose ineuality, a classical
result (Huisken--Ilmanen 2001; Bray 2001, extended by Bray--Lee 2009).
Its three-dimensional form is (T1) of the [sec:external] section; the
four-dimensional statement uses the flow identity
-Σ(t)=m_{ADM}-tfrac12(A(t)/omega_3)^{2/3} in place of the
+Σ(t)=m_{ADM}-1/2(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
+ineuality. It is used as an external input; the details of the flow
are classical.
@@ -2556,27 +2556,27 @@ in dimension 4), and the coupled elliptic construction
with
-R_{barbar g}gge0,
-m_{ADM}(barbar g)g)le m+ε,
-Area_{barbar g}g(∂Ω_R)ge(1-ε)A_e.
+R_{gbarbar}ge0,
+m_{ADM}(gbarbar))le m+ε,
+Area_{gbarbar}(∂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)}
+m_{ADM}(gbarbar))gefrac12(frac{Area_{gbarbar}(∂Ω_R)}
{omega_3})^{2/3}.
Let ε↓0: the left side becomes m, the right side becomes
-frac12(A_e/omega_3)^{2/3}, which is [eq:four-inequality].
+frac12(A_e/omega_3)^{2/3}, which is [eq:four-ineuality].
-Four-dimensional rigidity [sec:four-rigidity]. The equality case of [eq:four-inequality] identifies the exterior
+Four-dimensional rigidity [sec:four-rigidity]. The equality case of [eq:four-ineuality] identifies the exterior
with Schwarzschild--Tangherlini.
@@ -2599,7 +2599,7 @@ following substitutions: the causal stationary field
twist-vanishing (the [prop:n3-twist-vanishes] proposition) still
uses the positive-mass bootstrap on the conformal double; the
conformal double (the [prop:n3-double] proposition) exists; and
-Schoen--Yau positivity gives ell=1 and Scong S^3. The
+Schoen--Yau positivity gives ell=1 and S~= 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.
@@ -2608,7 +2608,7 @@ Schwarzschild--Tangherlini of the same mass.
-The anti-de Sitter Penrose inequality [sec:ads]. Schwarzschild-anti-de Sitter and the branch [sec:ads-bgd]
+The anti-de Sitter Penrose ineuality [sec:ads]. Schwarzschild-anti-de Sitter and the branch [sec:ads-bgd]
@@ -2635,7 +2635,7 @@ 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]
+every εin[0,ε_1), [eq:ads-ineuality]
m_{AH}(ε)
ge√((A(ε))/(16pi))
@@ -2795,7 +2795,7 @@ identity is the background identity, and smoothness continues it to
-Proof of the anti-de Sitter inequality
+Proof of the anti-de Sitter ineuality