An introduction to the Penrose inequality III

An introduction to the Penrose inequality III

Levi Lopes de Lima

Department of Mathematics Federal University of Ceará

Gelosp2013 - July, 2013

Joint work with Fred Girão (UFC/Fortaleza).

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for strictly mean convex hypersurfaces in a certain class oflocallyhyperbolic manifolds of dimensionn≥3.

This provides natural generalizations of the classical Minkowski inequality for convex hypersufaces inRⁿ.

As an application we establish an optimal Penrose inequality for asymptotically locally hyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of what happens in the equality case.

This provides a large class of examples of initial data sets (corresponding to time-symmetric solutions of Einstein equations in General Relativity with a negative cosmological constant) for which an optimal Penrose inequality holds true.

In particular, in the physical dimensionn=3 we obtain, for this class of IDS, a proof of a Penrose-type inequality for exotic black holes solutions first conjectured by Gibbons and Chru´sciel-Simon.

Joint work with Fred Girão (UFC/Fortaleza).

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for strictly mean convex hypersurfaces in a certain class oflocallyhyperbolic manifolds of dimensionn≥3.

This provides natural generalizations of the classical Minkowski inequality for convex hypersufaces inRⁿ.

As an application we establish an optimal Penrose inequality for asymptotically locally hyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of what happens in the equality case.

This provides a large class of examples of initial data sets (corresponding to time-symmetric solutions of Einstein equations in General Relativity with a negative cosmological constant) for which an optimal Penrose inequality holds true.

In particular, in the physical dimensionn=3 we obtain, for this class of IDS, a proof of a Penrose-type inequality for exotic black holes solutions first conjectured by Gibbons and Chru´sciel-Simon.

Joint work with Fred Girão (UFC/Fortaleza).

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for strictly mean convex hypersurfaces in a certain class oflocallyhyperbolic manifolds of dimensionn≥3.

This provides natural generalizations of the classical Minkowski inequality for convex hypersufaces inRⁿ.

As an application we establish an optimal Penrose inequality for asymptotically locally hyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of what happens in the equality case.

This provides a large class of examples of initial data sets (corresponding to time-symmetric solutions of Einstein equations in General Relativity with a negative cosmological constant) for which an optimal Penrose inequality holds true.

In particular, in the physical dimensionn=3 we obtain, for this class of IDS, a proof of a Penrose-type inequality for exotic black holes solutions first conjectured by Gibbons and Chru´sciel-Simon.

Joint work with Fred Girão (UFC/Fortaleza).

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for strictly mean convex hypersurfaces in a certain class oflocallyhyperbolic manifolds of dimensionn≥3.

This provides natural generalizations of the classical Minkowski inequality for convex hypersufaces inRⁿ.

As an application we establish an optimal Penrose inequality for asymptotically locally hyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of what happens in the equality case.

This provides a large class of examples of initial data sets (corresponding to time-symmetric solutions of Einstein equations in General Relativity with a negative cosmological constant) for which an optimal Penrose inequality holds true.

In particular, in the physical dimensionn=3 we obtain, for this class of IDS, a proof of a Penrose-type inequality for exotic black holes solutions first conjectured by Gibbons and Chru´sciel-Simon.

Joint work with Fred Girão (UFC/Fortaleza).

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for strictly mean convex hypersurfaces in a certain class oflocallyhyperbolic manifolds of dimensionn≥3.

This provides natural generalizations of the classical Minkowski inequality for convex hypersufaces inRⁿ.

As an application we establish an optimal Penrose inequality for asymptotically locally hyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of what happens in the equality case.

This provides a large class of examples of initial data sets (corresponding to time-symmetric solutions of Einstein equations in General Relativity with a negative cosmological constant) for which an optimal Penrose inequality holds true.

In particular, in the physical dimensionn=3 we obtain, for this class of IDS, a proof of a Penrose-type inequality for exotic black holes solutions first conjectured by Gibbons and Chru´sciel-Simon.

Joint work with Fred Girão (UFC/Fortaleza).

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for strictly mean convex hypersurfaces in a certain class oflocallyhyperbolic manifolds of dimensionn≥3.

This provides natural generalizations of the classical Minkowski inequality for convex hypersufaces inRⁿ.

As an application we establish an optimal Penrose inequality for asymptotically locally hyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of what happens in the equality case.

This provides a large class of examples of initial data sets (corresponding to time-symmetric solutions of Einstein equations in General Relativity with a negative cosmological constant) for which an optimal Penrose inequality holds true.

In particular, in the physical dimensionn=3 we obtain, for this class of IDS, a proof of a Penrose-type inequality for exotic black holes solutions first conjectured by Gibbons and Chru´sciel-Simon.

Joint work with Fred Girão (UFC/Fortaleza).

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for strictly mean convex hypersurfaces in a certain class oflocallyhyperbolic manifolds of dimensionn≥3.

This provides natural generalizations of the classical Minkowski inequality for convex hypersufaces inRⁿ.

As an application we establish an optimal Penrose inequality for asymptotically locally hyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of what happens in the equality case.

This provides a large class of examples of initial data sets (corresponding to time-symmetric solutions of Einstein equations in General Relativity with a negative cosmological constant) for which an optimal Penrose inequality holds true.

In particular, in the physical dimensionn=3 we obtain, for this class of IDS, a proof of a Penrose-type inequality for exotic black holes solutions first conjectured by Gibbons and Chru´sciel-Simon.

The reference metrics (Chru´sciel-Herzlich-Nagy)

Fixn≥3,=0,±1 and let(Nⁿ⁻¹,h)be a closed space form with curvature.

In the product manifoldP=I×N, consider the metric g= dr²

ρ(r)²+r²h, r∈I, where

ρ(r) =p r²+. Here,I−1= (1,+∞)andI₀=I₁= (0,+∞).

The metricgislocally hyperbolic(Kg≡ −1).

For instance, if=1 and(N,h)is a round sphere then(P1,g1)is hyperbolic spaceHⁿ.

Also, if=0 and(N²,h)is a torus then(P0,g0)is a cusp manifold.

The reference metrics (Chru´sciel-Herzlich-Nagy)

Fixn≥3,=0,±1 and let(Nⁿ⁻¹,h)be a closed space form with curvature.

In the product manifoldP=I×N, consider the metric g= dr²

ρ(r)²+r²h, r∈I, where

ρ(r) =p r²+. Here,I−1= (1,+∞)andI₀=I₁= (0,+∞).

The metricgislocally hyperbolic(Kg≡ −1).

For instance, if=1 and(N,h)is a round sphere then(P1,g1)is hyperbolic spaceHⁿ.

Also, if=0 and(N²,h)is a torus then(P0,g0)is a cusp manifold.

The reference metrics (Chru´sciel-Herzlich-Nagy)

Fixn≥3,=0,±1 and let(Nⁿ⁻¹,h)be a closed space form with curvature.

In the product manifoldP=I×N, consider the metric

g= dr²

ρ(r)²+r²h, r∈I, where

ρ(r) =p r²+. Here,I−1= (1,+∞)andI₀=I₁= (0,+∞).

The metricgislocally hyperbolic(Kg≡ −1).

For instance, if=1 and(N,h)is a round sphere then(P1,g1)is hyperbolic spaceHⁿ.

Also, if=0 and(N²,h)is a torus then(P0,g0)is a cusp manifold.

The reference metrics (Chru´sciel-Herzlich-Nagy)

Fixn≥3,=0,±1 and let(Nⁿ⁻¹,h)be a closed space form with curvature.

In the product manifoldP=I×N, consider the metric g= dr²

ρ(r)²+r²h, r∈I, where

ρ(r) =p r²+. Here,I−1= (1,+∞)andI₀=I₁= (0,+∞).

The metricgislocally hyperbolic(Kg≡ −1).

For instance, if=1 and(N,h)is a round sphere then(P1,g1)is hyperbolic spaceHⁿ.

Also, if=0 and(N²,h)is a torus then(P0,g0)is a cusp manifold.

The reference metrics (Chru´sciel-Herzlich-Nagy)

Fixn≥3,=0,±1 and let(Nⁿ⁻¹,h)be a closed space form with curvature.

In the product manifoldP=I×N, consider the metric g= dr²

ρ(r)²+r²h, r∈I, where

ρ(r) =p r²+. Here,I−1= (1,+∞)andI₀=I₁= (0,+∞).

The metricgislocally hyperbolic(Kg≡ −1).

For instance, if=1 and(N,h)is a round sphere then(P1,g1)is hyperbolic spaceHⁿ.

Also, if=0 and(N²,h)is a torus then(P0,g0)is a cusp manifold.

The reference metrics (Chru´sciel-Herzlich-Nagy)

Fixn≥3,=0,±1 and let(Nⁿ⁻¹,h)be a closed space form with curvature.

In the product manifoldP=I×N, consider the metric g= dr²

ρ(r)²+r²h, r∈I, where

ρ(r) =p r²+. Here,I−1= (1,+∞)andI₀=I₁= (0,+∞).

The metricgislocally hyperbolic(Kg≡ −1).

For instance, if=1 and(N,h)is a round sphere then(P1,g1)is hyperbolic spaceHⁿ.

Also, if=0 and(N²,h)is a torus then(P0,g0)is a cusp manifold.

The reference metrics (Chru´sciel-Herzlich-Nagy)

Fixn≥3,=0,±1 and let(Nⁿ⁻¹,h)be a closed space form with curvature.

In the product manifoldP=I×N, consider the metric g= dr²

ρ(r)²+r²h, r∈I, where

ρ(r) =p r²+. Here,I−1= (1,+∞)andI₀=I₁= (0,+∞).

The metricgislocally hyperbolic(Kg≡ −1).

For instance, if=1 and(N,h)is a round sphere then(P1,g1)is hyperbolic spaceHⁿ.

Also, if=0 and(N²,h)is a torus then(P0,g0)is a cusp manifold.

Asymptotically locally hyperbolic manifolds (Chru´sciel-Herzlich-Nagy)

Definition

Fixand(N,h)as above. A completen-dimensional manifold(M,g), possibly carrying a compact inner boundaryΣ, is said to beasymptotically locally hyperbolic (ALH)if there exist subsets K⊂MandK₀⊂P, withKcompact, and a diffeomorphismΨ :M−K→P−K₀such that

kΨ∗g−gkg=O(r^−τ), kDΨ∗gkg=O(r^−τ), r→+∞,

for someτ >n/2. We also assume thatRg+n(n−1)∈L¹.

For this class of manifolds, a mass-like invariantm_(M,g)∈Rcan be defined as m_(M,g)= lim

r→+∞cn

ˆ

N_r

ρ(divge−dtrge)−i∇^gρe+ (trgdρ) (νr)dNr,

wheree= Ψ∗g−g,Nr ={r} ×N,νris theoutwardunit vector toNrand

cn= 1

2(n−1)τn−1

, τn−1=arean−1(N,h).

This invariant measures the rate of the convergenceg→g0,asr→+∞. Here, we are leaving aside the cases whereN=Sⁿ/Γ,Γ6={Id}.

Asymptotically locally hyperbolic manifolds (Chru´sciel-Herzlich-Nagy) Definition

Fixand(N,h)as above. A completen-dimensional manifold(M,g), possibly carrying a compact inner boundaryΣ, is said to beasymptotically locally hyperbolic (ALH)if there exist subsets K⊂MandK₀⊂P, withKcompact, and a diffeomorphismΨ :M−K→P−K₀such that

kΨ∗g−gkg=O(r^−τ), kDΨ∗gkg=O(r^−τ), r→+∞,

for someτ >n/2. We also assume thatRg+n(n−1)∈L¹.

For this class of manifolds, a mass-like invariantm_(M,g)∈Rcan be defined as m_(M,g)= lim

r→+∞cn

ˆ

N_r

ρ(divge−dtrge)−i∇^gρe+ (trgdρ) (νr)dNr,

wheree= Ψ∗g−g,Nr ={r} ×N,νris theoutwardunit vector toNrand

cn= 1

2(n−1)τn−1

, τn−1=arean−1(N,h).

This invariant measures the rate of the convergenceg→g0,asr→+∞. Here, we are leaving aside the cases whereN=Sⁿ/Γ,Γ6={Id}.

Asymptotically locally hyperbolic manifolds (Chru´sciel-Herzlich-Nagy) Definition

Fixand(N,h)as above. A completen-dimensional manifold(M,g), possibly carrying a compact inner boundaryΣ, is said to beasymptotically locally hyperbolic (ALH)if there exist subsets K⊂MandK₀⊂P, withKcompact, and a diffeomorphismΨ :M−K→P−K₀such that

kΨ∗g−gkg=O(r^−τ), kDΨ∗gkg=O(r^−τ), r→+∞,

for someτ >n/2. We also assume thatRg+n(n−1)∈L¹.

For this class of manifolds, a mass-like invariantm_(M,g)∈Rcan be defined as

m_(M,g)= lim

r→+∞cn

ˆ

N_r

ρ(divge−dtrge)−i∇^gρe+ (trgdρ) (νr)dNr,

wheree= Ψ∗g−g,Nr ={r} ×N,νris theoutwardunit vector toNrand

cn= 1

2(n−1)τn−1

, τn−1=arean−1(N,h).

This invariant measures the rate of the convergenceg→g0,asr→+∞. Here, we are leaving aside the cases whereN=Sⁿ/Γ,Γ6={Id}.

Asymptotically locally hyperbolic manifolds (Chru´sciel-Herzlich-Nagy) Definition

Fixand(N,h)as above. A completen-dimensional manifold(M,g), possibly carrying a compact inner boundaryΣ, is said to beasymptotically locally hyperbolic (ALH)if there exist subsets K⊂MandK₀⊂P, withKcompact, and a diffeomorphismΨ :M−K→P−K₀such that

kΨ∗g−gkg=O(r^−τ), kDΨ∗gkg=O(r^−τ), r→+∞,

for someτ >n/2. We also assume thatRg+n(n−1)∈L¹.

For this class of manifolds, a mass-like invariantm_(M,g)∈Rcan be defined as m_(M,g)= lim

r→+∞cn

ˆ

N_r

ρ(divge−dtrge)−i∇^gρe+ (trgdρ) (νr)dNr,

wheree= Ψ∗g−g,Nr ={r} ×N,νris theoutwardunit vector toNrand

cn= 1

2(n−1)τn−1

, τn−1=arean−1(N,h).

This invariant measures the rate of the convergenceg→g0,asr→+∞.

Here, we are leaving aside the cases whereN=Sⁿ/Γ,Γ6={Id}.

Asymptotically locally hyperbolic manifolds (Chru´sciel-Herzlich-Nagy) Definition

Fixand(N,h)as above. A completen-dimensional manifold(M,g), possibly carrying a compact inner boundaryΣ, is said to beasymptotically locally hyperbolic (ALH)if there exist subsets K⊂MandK₀⊂P, withKcompact, and a diffeomorphismΨ :M−K→P−K₀such that

kΨ∗g−gkg=O(r^−τ), kDΨ∗gkg=O(r^−τ), r→+∞,

for someτ >n/2. We also assume thatRg+n(n−1)∈L¹.

For this class of manifolds, a mass-like invariantm_(M,g)∈Rcan be defined as m_(M,g)= lim

r→+∞cn

ˆ

N_r

ρ(divge−dtrge)−i∇^gρe+ (trgdρ) (νr)dNr,

wheree= Ψ∗g−g,Nr ={r} ×N,νris theoutwardunit vector toNrand

cn= 1

2(n−1)τn−1

, τn−1=arean−1(N,h).

This invariant measures the rate of the convergenceg→g0,asr→+∞.

Here, we are leaving aside the cases whereN=Sⁿ/Γ,Γ6={Id}.

Asymptotically locally hyperbolic manifolds (Chru´sciel-Herzlich-Nagy) Definition

Fixand(N,h)as above. A completen-dimensional manifold(M,g), possibly carrying a compact inner boundaryΣ, is said to beasymptotically locally hyperbolic (ALH)if there exist subsets K⊂MandK₀⊂P, withKcompact, and a diffeomorphismΨ :M−K→P−K₀such that

kΨ∗g−gkg=O(r^−τ), kDΨ∗gkg=O(r^−τ), r→+∞,

for someτ >n/2. We also assume thatRg+n(n−1)∈L¹.

For this class of manifolds, a mass-like invariantm_(M,g)∈Rcan be defined as m_(M,g)= lim

r→+∞cn

ˆ

N_r

ρ(divge−dtrge)−i∇^gρe+ (trgdρ) (νr)dNr,

wheree= Ψ∗g−g,Nr ={r} ×N,νris theoutwardunit vector toNrand

cn= 1

2(n−1)τn−1

, τn−1=arean−1(N,h).

This invariant measures the rate of the convergenceg→g0,asr→+∞.

The black hole solutions I

Fix=0,±1,m>0 and consider the interval

Im,={r>rm,},

whererm,is the positive root of

r²+− 2m rⁿ⁻² =0.

If(Nⁿ⁻¹,h)is a compact space form with curvature, in the product manifold Pm,=Im,×Ndefine the metric

gm,= dr²

ρm,(r)²+r²h, where

ρm,(r) = r

r²+− 2m rⁿ⁻².

We note thatgm,extends smoothly toPm,= [rm,,+∞)×Nand the slice defined by r=rm,is called thehorizon.

The black hole solutions I

Fix=0,±1,m>0 and consider the interval

Im,={r>rm,},

whererm,is the positive root of

r²+− 2m rⁿ⁻² =0.

If(Nⁿ⁻¹,h)is a compact space form with curvature, in the product manifold Pm,=Im,×Ndefine the metric

gm,= dr²

ρm,(r)²+r²h, where

ρm,(r) = r

r²+− 2m rⁿ⁻².

We note thatgm,extends smoothly toPm,= [rm,,+∞)×Nand the slice defined by r=rm,is called thehorizon.

The black hole solutions I

Fix=0,±1,m>0 and consider the interval

Im,={r>rm,},

whererm,is the positive root of

r²+− 2m rⁿ⁻² =0.

If(Nⁿ⁻¹,h)is a compact space form with curvature, in the product manifold Pm,=Im,×Ndefine the metric

gm,= dr²

ρm,(r)²+r²h, where

ρm,(r) = r

r²+− 2m rⁿ⁻².

We note thatgm,extends smoothly toPm,= [rm,,+∞)×Nand the slice defined by r=rm,is called thehorizon.

The black hole solutions I

Fix=0,±1,m>0 and consider the interval

Im,={r>rm,},

whererm,is the positive root of

r²+− 2m rⁿ⁻² =0.

If(Nⁿ⁻¹,h)is a compact space form with curvature, in the product manifold Pm,=Im,×Ndefine the metric

gm,= dr²

ρm,(r)²+r²h, where

ρm,(r) = r

r²+− 2m rⁿ⁻².

We note thatgm,extends smoothly toPm,= [rm,,+∞)×Nand the slice defined by r=r is called thehorizon.

The black hole solutions I

Fix=0,±1,m>0 and consider the interval

Im,={r>rm,},

whererm,is the positive root of

r²+− 2m rⁿ⁻² =0.

If(Nⁿ⁻¹,h)is a compact space form with curvature, in the product manifold Pm,=Im,×Ndefine the metric

gm,= dr²

ρm,(r)²+r²h, where

ρm,(r) = r

r²+− 2m rⁿ⁻².

We note thatgm,extends smoothly toPm,= [rm,,+∞)×Nand the slice defined by r=rm,is called thehorizon.

The black hole solutions II

If(θ1,· · ·, θn−1)are orthonormal coordinates inNthen the sectional curvatures ofgm,are Kgm,(∂r, ∂_θi) =−1−(n−2)m

rⁿ and

Kg_m,(∂θ_i, ∂θ_j) =−1+2m rⁿ , so that the scalar curvature ofgm,isRg_m,=−n(n−1).

Moreover, eachgm,is astaticmetric in the sense thatρm,satisfies (∆ρm,)gm,−Hessg_m,ρm,+ρm,Ricg_m,=0, which means that the Lorentzian metric

g_m,=−ρ²_m,dt²+gm,,

defined onQm,=R×Pm,, is a solution to the vacuum Einstein field equations with negative cosmological constant:

Ric_gm,=−ng_m,.

Thus,gm,defines an initial data set for a time-symmetric (actually, static) vacuum solution of Einstein equations carrying a black hole.

The black hole solutions II

If(θ1,· · ·, θn−1)are orthonormal coordinates inNthen the sectional curvatures ofgm,are Kgm,(∂r, ∂_θi) =−1−(n−2)m

rⁿ and

Kg_m,(∂θ_i, ∂θ_j) =−1+2m rⁿ , so that the scalar curvature ofgm,isRg_m,=−n(n−1).

Moreover, eachgm,is astaticmetric in the sense thatρm,satisfies (∆ρm,)gm,−Hessg_m,ρm,+ρm,Ricg_m,=0, which means that the Lorentzian metric

g_m,=−ρ²_m,dt²+gm,,

defined onQm,=R×Pm,, is a solution to the vacuum Einstein field equations with negative cosmological constant:

Ric_gm,=−ng_m,.

Thus,gm,defines an initial data set for a time-symmetric (actually, static) vacuum solution of Einstein equations carrying a black hole.

The black hole solutions II

If(θ1,· · ·, θn−1)are orthonormal coordinates inNthen the sectional curvatures ofgm,are Kgm,(∂r, ∂_θi) =−1−(n−2)m

rⁿ and

Kg_m,(∂θ_i, ∂θ_j) =−1+2m rⁿ , so that the scalar curvature ofgm,isRg_m,=−n(n−1).

Moreover, eachgm,is astaticmetric in the sense thatρm,satisfies (∆ρm,)gm,−Hessg_m,ρm,+ρm,Ricg_m,=0, which means that the Lorentzian metric

g_m,=−ρ²_m,dt²+gm,,

defined onQm,=R×Pm,, is a solution to the vacuum Einstein field equations with negative cosmological constant:

Ric_gm,=−ng_m,.

Thus,gm,defines an initial data set for a time-symmetric (actually, static) vacuum solution of Einstein equations carrying a black hole.

The black hole solutions II

If(θ1,· · ·, θn−1)are orthonormal coordinates inNthen the sectional curvatures ofgm,are Kgm,(∂r, ∂_θi) =−1−(n−2)m

rⁿ and

Kg_m,(∂θ_i, ∂θ_j) =−1+2m rⁿ , so that the scalar curvature ofgm,isRg_m,=−n(n−1).

Moreover, eachgm,is astaticmetric in the sense thatρm,satisfies (∆ρm,)gm,−Hessg_m,ρm,+ρm,Ricg_m,=0, which means that the Lorentzian metric

g_m,=−ρ²_m,dt²+gm,,

defined onQm,=R×Pm,, is a solution to the vacuum Einstein field equations with negative cosmological constant:

Ric_gm,=−ng_m,.

Thus,gm,defines an initial data set for a time-symmetric (actually, static) vacuum solution of Einstein equations carrying a black hole.

The black hole solutions III

One easily verifies that, asr→+∞,

kgm,−gkg=O mr⁻ⁿ , wheregis the corresponding reference metric.

Thus, eachgm,,m>0, isasymptotically locally hyperbolic(ALH).

Physical reasoning allows us to interpretmas thetotal massof the black hole solutiongm,.

Indeed, a computation shows thatm_(Pm,,gm,)=m.

The black hole solutions III

One easily verifies that, asr→+∞,

kgm,−gkg=O mr⁻ⁿ , wheregis the corresponding reference metric.

Thus, eachgm,,m>0, isasymptotically locally hyperbolic(ALH).

Physical reasoning allows us to interpretmas thetotal massof the black hole solutiongm,.

Indeed, a computation shows thatm_(Pm,,gm,)=m.

The black hole solutions III

One easily verifies that, asr→+∞,

kgm,−gkg=O mr⁻ⁿ , wheregis the corresponding reference metric.

Thus, eachgm,,m>0, isasymptotically locally hyperbolic(ALH).

Physical reasoning allows us to interpretmas thetotal massof the black hole solutiongm,.

Indeed, a computation shows thatm_(Pm,,gm,)=m.

The black hole solutions III

One easily verifies that, asr→+∞,

kgm,−gkg=O mr⁻ⁿ , wheregis the corresponding reference metric.

Thus, eachgm,,m>0, isasymptotically locally hyperbolic(ALH).

Physical reasoning allows us to interpretmas thetotal massof the black hole solutiongm,.

Indeed, a computation shows thatm_(Pm,,gm,)=m.

The black hole solutions III

One easily verifies that, asr→+∞,

kgm,−gkg=O mr⁻ⁿ , wheregis the corresponding reference metric.

Thus, eachgm,,m>0, isasymptotically locally hyperbolic(ALH).

Physical reasoning allows us to interpretmas thetotal massof the black hole solutiongm,.

Indeed, a computation shows thatm_(Pm,,gm,)=m.

The black hole solutions III

One easily verifies that, asr→+∞,

kgm,−gkg=O mr⁻ⁿ , wheregis the corresponding reference metric.

Thus, eachgm,,m>0, isasymptotically locally hyperbolic(ALH).

Physical reasoning allows us to interpretmas thetotal massof the black hole solutiongm,.

Indeed, a computation shows thatm_(Pm,,gm,)=m.

The black hole solutions IV

It turns out that eachgm,can be isometrically embedded as a graph in(Q,g), where Q=R×Pand

g=ρ(r)²dt²+ dr²

ρ(r)²+r²dθ². Notice that(Q,g)is locally hyperbolic!

The radial function defining the graph,u=um,(r), satisfiesu(rm,) =0 and ρ(r)²

du dr

2

= 1

ρm,(r)²− 1

ρ(r)², r≥rm,.

It follows that the graph realization of the black hole solution meets the slicet=0 orthogonallyalong the minimal ‘horizon’Hdefined byr=rm,.

Notice also that the massmrelates to the area|H|of the black hole horizon by

m=1 2



 |H|

τn−1

_n−1ⁿ +

|H| τn−1

ⁿ⁻²_n−1



, τn−1=arean−1(N).

The black hole solutions IV

It turns out that eachgm,can be isometrically embedded as a graph in(Q,g), where Q=R×Pand

g=ρ(r)²dt²+ dr²

ρ(r)²+r²dθ². Notice that(Q,g)is locally hyperbolic!

The radial function defining the graph,u=um,(r), satisfiesu(rm,) =0 and ρ(r)²

du dr

2

= 1

ρm,(r)²− 1

ρ(r)², r≥rm,.

It follows that the graph realization of the black hole solution meets the slicet=0 orthogonallyalong the minimal ‘horizon’Hdefined byr=rm,.

Notice also that the massmrelates to the area|H|of the black hole horizon by

m=1 2



 |H|

τn−1

_n−1ⁿ +

|H| τn−1

ⁿ⁻²_n−1



, τn−1=arean−1(N).

The black hole solutions IV

It turns out that eachgm,can be isometrically embedded as a graph in(Q,g), where Q=R×Pand

g=ρ(r)²dt²+ dr²

ρ(r)²+r²dθ². Notice that(Q,g)is locally hyperbolic!

The radial function defining the graph,u=um,(r), satisfiesu(rm,) =0 and

ρ(r)² du

dr 2

= 1

ρm,(r)²− 1

ρ(r)², r≥rm,.

It follows that the graph realization of the black hole solution meets the slicet=0 orthogonallyalong the minimal ‘horizon’Hdefined byr=rm,.

Notice also that the massmrelates to the area|H|of the black hole horizon by

m=1 2



 |H|

τn−1

_n−1ⁿ +

|H| τn−1

ⁿ⁻²_n−1



, τn−1=arean−1(N).

The black hole solutions IV

It turns out that eachgm,can be isometrically embedded as a graph in(Q,g), where Q=R×Pand

g=ρ(r)²dt²+ dr²

ρ(r)²+r²dθ². Notice that(Q,g)is locally hyperbolic!

The radial function defining the graph,u=um,(r), satisfiesu(rm,) =0 and ρ(r)²

du dr

2

= 1

ρm,(r)²− 1

ρ(r)², r≥rm,.

It follows that the graph realization of the black hole solution meets the slicet=0 orthogonallyalong the minimal ‘horizon’Hdefined byr=rm,.

Notice also that the massmrelates to the area|H|of the black hole horizon by

m=1 2



 |H|

τn−1

_n−1ⁿ +

|H| τn−1

ⁿ⁻²_n−1



, τn−1=arean−1(N).

The black hole solutions IV

It turns out that eachgm,can be isometrically embedded as a graph in(Q,g), where Q=R×Pand

g=ρ(r)²dt²+ dr²

ρ(r)²+r²dθ². Notice that(Q,g)is locally hyperbolic!

The radial function defining the graph,u=um,(r), satisfiesu(rm,) =0 and ρ(r)²

du dr

2

= 1

ρm,(r)²− 1

ρ(r)², r≥rm,.

It follows that the graph realization of the black hole solution meets the slicet=0 orthogonallyalong the minimal ‘horizon’Hdefined byr=rm,.

Notice also that the massmrelates to the area|H|of the black hole horizon by

m=1 2



 |H|

τn−1

_n−1ⁿ +

|H|

τn−1

ⁿ⁻²_n−1



, τn−1=arean−1(N).

The black hole solutions IV

It turns out that eachgm,can be isometrically embedded as a graph in(Q,g), where Q=R×Pand

g=ρ(r)²dt²+ dr²

ρ(r)²+r²dθ². Notice that(Q,g)is locally hyperbolic!

The radial function defining the graph,u=um,(r), satisfiesu(rm,) =0 and ρ(r)²

du dr

2

= 1

ρm,(r)²− 1

ρ(r)², r≥rm,.

It follows that the graph realization of the black hole solution meets the slicet=0 orthogonallyalong the minimal ‘horizon’Hdefined byr=rm,.

Notice also that the massmrelates to the area|H|of the black hole horizon by

m=1 2



 |H|

τn−1

_n−1ⁿ +

|H|

τn−1

ⁿ⁻²_n−1



, τn−1=arean−1(N).

The Penrose conjecture for ALH manifolds

Let(M,g)be an ALH manifold (relative to the reference metricg). Assume that Rg≥ −n(n−1)and thatMcarries an outermost minimal horizonΣ. Then,

m_(M,g)≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ| τn−1

ⁿ⁻²_n−1



,

with the equality occurring if and only if(M,g)is (isometric to) the corresponding black hole solution.

In the physical dimensionn=3, this appears as a conjectured Penrose-type inequality in papers by Gibbons and Chru´sciel-Simon.

In the following we establish this inequality for ALH graphs in any dimensionn≥3, including the corresponding rigidity statement.

The Penrose conjecture for ALH manifolds

Let(M,g)be an ALH manifold (relative to the reference metricg). Assume that Rg≥ −n(n−1)and thatMcarries an outermost minimal horizonΣ. Then,

m_(M,g)≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality occurring if and only if(M,g)is (isometric to) the corresponding black hole solution.

In the physical dimensionn=3, this appears as a conjectured Penrose-type inequality in papers by Gibbons and Chru´sciel-Simon.

In the following we establish this inequality for ALH graphs in any dimensionn≥3, including the corresponding rigidity statement.

The Penrose conjecture for ALH manifolds

Let(M,g)be an ALH manifold (relative to the reference metricg). Assume that Rg≥ −n(n−1)and thatMcarries an outermost minimal horizonΣ. Then,

m_(M,g)≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality occurring if and only if(M,g)is (isometric to) the corresponding black hole solution.

In the physical dimensionn=3, this appears as a conjectured Penrose-type inequality in papers by Gibbons and Chru´sciel-Simon.

In the following we establish this inequality for ALH graphs in any dimensionn≥3, including the corresponding rigidity statement.

The Penrose conjecture for ALH manifolds

Let(M,g)be an ALH manifold (relative to the reference metricg). Assume that Rg≥ −n(n−1)and thatMcarries an outermost minimal horizonΣ. Then,

m_(M,g)≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality occurring if and only if(M,g)is (isometric to) the corresponding black hole solution.

In the physical dimensionn=3, this appears as a conjectured Penrose-type inequality in papers by Gibbons and Chru´sciel-Simon.

In the following we establish this inequality for ALH graphs in any dimensionn≥3, including the corresponding rigidity statement.

The Penrose conjecture for ALH manifolds

Let(M,g)be an ALH manifold (relative to the reference metricg). Assume that Rg≥ −n(n−1)and thatMcarries an outermost minimal horizonΣ. Then,

m_(M,g)≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality occurring if and only if(M,g)is (isometric to) the corresponding black hole solution.

In the physical dimensionn=3, this appears as a conjectured Penrose-type inequality in papers by Gibbons and Chru´sciel-Simon.

In the following we establish this inequality for ALH graphs in any dimensionn≥3, including the corresponding rigidity statement.

ALH hypersurfaces in Q

Definition

A complete, isometrically immersed hypersurface(M,g)#(Q,g), possibly with an inner boundaryΣ, isasymptotically locally hyperbolic (ALH)if there exist subsetsK⊂M,K0⊂Psuch thatM−K, the end ofM, can be written as a vertical graph overP−K0, with the graph being associated to a smooth functionu:P−K₀→Rsuch the previous asymptotic conditions holds for the nonparametric chartΨu(x,u(x)) =x,x∈K₀. Moreover, we assume that

RΨu∗g+n(n−1)is integrable.

Under these conditions, the mass of(M,g)is well defined and can be computed by usingΨu.

ALH hypersurfaces in Q

Definition

A complete, isometrically immersed hypersurface(M,g)#(Q,g), possibly with an inner boundaryΣ, isasymptotically locally hyperbolic (ALH)if there exist subsetsK⊂M,K0⊂Psuch thatM−K, the end ofM, can be written as a vertical graph overP−K0, with the graph being associated to a smooth functionu:P−K₀→Rsuch the previous asymptotic conditions holds for the nonparametric chartΨu(x,u(x)) =x,x∈K₀. Moreover, we assume that

RΨu∗g+n(n−1)is integrable.

Under these conditions, the mass of(M,g)is well defined and can be computed by usingΨu.

ALH hypersurfaces in Q

Definition

A complete, isometrically immersed hypersurface(M,g)#(Q,g), possibly with an inner boundaryΣ, isasymptotically locally hyperbolic (ALH)if there exist subsetsK⊂M,K0⊂Psuch thatM−K, the end ofM, can be written as a vertical graph overP−K0, with the graph being associated to a smooth functionu:P−K₀→Rsuch the previous asymptotic conditions holds for the nonparametric chartΨu(x,u(x)) =x,x∈K₀. Moreover, we assume that

RΨu∗g+n(n−1)is integrable.

Under these conditions, the mass of(M,g)is well defined and can be computed by usingΨu.

ALH hypersurfaces in Q

Definition

A complete, isometrically immersed hypersurface(M,g)#(Q,g), possibly with an inner boundaryΣ, isasymptotically locally hyperbolic (ALH)if there exist subsetsK⊂M,K0⊂Psuch thatM−K, the end ofM, can be written as a vertical graph overP−K0, with the graph being associated to a smooth functionu:P−K₀→Rsuch the previous asymptotic conditions holds for the nonparametric chartΨu(x,u(x)) =x,x∈K₀. Moreover, we assume that

RΨu∗g+n(n−1)is integrable.

Under these conditions, the mass of(M,g)is well defined and can be computed by usingΨu.

The integral formula for the mass

ForanyhypersurfaceM⊂Q=R×Pendowed with a unit normalN, an old formula by Reilly says that

divM(G(A)X) =2σ2(A)Θ,

whereG(A) =σ1(A)I−Ais the Newton tensor of the shape operator A,Xis the tangential component of∂/∂tandΘ =hN, ∂/∂ti. This uses that∂/∂tis Killing and thatK_g

≡ −1. Assume from now on thatM⊂Qis ALH and its inner boundaryΣlies on a horizontal totally geodesic hypersurface, sayP'P. Assume further thatMmeetsP orthogonallyalongΣ (which implies thatΣ⊂Mis minimal and hence a horizon).

Theorem

Under the above conditions, m_(M,g)=cn

ˆ

M

Θ (Rg+n(n−1))dM+cn

ˆ

Σ

ρHdΣ,

where H is the mean curvature ofΣ⊂P andρ(r) =p

r²+.In particular, if Rg≥ −n(n−1) and M is a graph (Θ>0) then

m_(M,g)≥cn

ˆ

Σ

ρHdΣ.

The integral formula for the mass

ForanyhypersurfaceM⊂Q=R×Pendowed with a unit normalN, an old formula by Reilly says that

divM(G(A)X) =2σ2(A)Θ,

whereG(A) =σ1(A)I−Ais the Newton tensor of the shape operator A,Xis the tangential component of∂/∂tandΘ =hN, ∂/∂ti. This uses that∂/∂tis Killing and thatK_g

≡ −1.

Assume from now on thatM⊂Qis ALH and its inner boundaryΣlies on a horizontal totally geodesic hypersurface, sayP'P. Assume further thatMmeetsP orthogonallyalongΣ (which implies thatΣ⊂Mis minimal and hence a horizon).

Theorem

Under the above conditions, m_(M,g)=cn

ˆ

M

Θ (Rg+n(n−1))dM+cn

ˆ

Σ

ρHdΣ,

where H is the mean curvature ofΣ⊂P andρ(r) =p

r²+.In particular, if Rg≥ −n(n−1) and M is a graph (Θ>0) then

m_(M,g)≥cn

ˆ

Σ

ρHdΣ.

The integral formula for the mass

ForanyhypersurfaceM⊂Q=R×Pendowed with a unit normalN, an old formula by Reilly says that

divM(G(A)X) =2σ2(A)Θ,

whereG(A) =σ1(A)I−Ais the Newton tensor of the shape operator A,Xis the tangential component of∂/∂tandΘ =hN, ∂/∂ti. This uses that∂/∂tis Killing and thatK_g

≡ −1.

Assume from now on thatM⊂Qis ALH and its inner boundaryΣlies on a horizontal totally geodesic hypersurface, sayP'P. Assume further thatMmeetsP orthogonallyalongΣ (which implies thatΣ⊂Mis minimal and hence a horizon).

Theorem

Under the above conditions, m_(M,g)=cn

ˆ

M

Θ (Rg+n(n−1))dM+cn

ˆ

Σ

ρHdΣ,

where H is the mean curvature ofΣ⊂P andρ(r) =p

r²+.In particular, if Rg≥ −n(n−1) and M is a graph (Θ>0) then

m_(M,g)≥cn

ˆ

Σ

ρHdΣ.

The integral formula for the mass

ForanyhypersurfaceM⊂Q=R×Pendowed with a unit normalN, an old formula by Reilly says that

divM(G(A)X) =2σ2(A)Θ,

whereG(A) =σ1(A)I−Ais the Newton tensor of the shape operator A,Xis the tangential component of∂/∂tandΘ =hN, ∂/∂ti. This uses that∂/∂tis Killing and thatK_g

≡ −1.

Assume from now on thatM⊂Qis ALH and its inner boundaryΣlies on a horizontal totally geodesic hypersurface, sayP'P. Assume further thatMmeetsP orthogonallyalongΣ (which implies thatΣ⊂Mis minimal and hence a horizon).

Theorem

Under the above conditions, m_(M,g)=cn

ˆ

M

Θ (Rg+n(n−1))dM+cn

ˆ

Σ

ρHdΣ,

where H is the mean curvature ofΣ⊂P andρ(r) =p

r²+.In particular, if Rg≥ −n(n−1) and M is a graph (Θ>0) then

m_(M,g)≥cn

ˆ

Σ

ρHdΣ.

The integral formula for the mass

ForanyhypersurfaceM⊂Q=R×Pendowed with a unit normalN, an old formula by Reilly says that

divM(G(A)X) =2σ2(A)Θ,

whereG(A) =σ1(A)I−Ais the Newton tensor of the shape operator A,Xis the tangential component of∂/∂tandΘ =hN, ∂/∂ti. This uses that∂/∂tis Killing and thatK_g

≡ −1.

Assume from now on thatM⊂Qis ALH and its inner boundaryΣlies on a horizontal totally geodesic hypersurface, sayP'P. Assume further thatMmeetsP orthogonallyalongΣ (which implies thatΣ⊂Mis minimal and hence a horizon).

Theorem

Under the above conditions, m_(M,g)=cn

ˆ

M

Θ (Rg+n(n−1))dM+cn

ˆ

Σ

ρHdΣ,

where H is the mean curvature ofΣ⊂P andρ(r) =p

r²+.In particular, if Rg≥ −n(n−1) and M is a graph (Θ>0) then

m_(M,g)≥cn

ˆ

Σ

ρHdΣ.

The Alexandrov-Fenchel inequality

We have seen that

m_(M,g)≥cn

ˆ

Σ

ρ0,HdΣ.

In order to proceed, we need anewAlexandrov-Fenchel inequality for a class of hypersurfaces in(P,g)!

Theorem

IfΣ⊂Pis star-shaped and strictly mean convex (H>0) then

cn

ˆ

Σ

ρHdΣ≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ| τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only ifΣis a slice.

The Alexandrov-Fenchel inequality

We have seen that

m_(M,g)≥cn

ˆ

Σ

ρ0,HdΣ.

In order to proceed, we need anewAlexandrov-Fenchel inequality for a class of hypersurfaces in(P,g)!

Theorem

IfΣ⊂Pis star-shaped and strictly mean convex (H>0) then

cn

ˆ

Σ

ρHdΣ≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ| τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only ifΣis a slice.

The Alexandrov-Fenchel inequality

We have seen that

m_(M,g)≥cn

ˆ

Σ

ρ0,HdΣ.

In order to proceed, we need anewAlexandrov-Fenchel inequality for a class of hypersurfaces in(P,g)!

Theorem

IfΣ⊂Pis star-shaped and strictly mean convex (H>0) then

cn

ˆ

Σ

ρHdΣ≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ| τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only ifΣis a slice.

The Alexandrov-Fenchel inequality

We have seen that

m_(M,g)≥cn

ˆ

Σ

ρ0,HdΣ.

In order to proceed, we need anewAlexandrov-Fenchel inequality for a class of hypersurfaces in(P,g)!

Theorem

IfΣ⊂Pis star-shaped and strictly mean convex (H>0) then

cn

ˆ

Σ

ρHdΣ≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only ifΣis a slice.

The Alexandrov-Fenchel inequality

We have seen that

m_(M,g)≥cn

ˆ

Σ

ρ0,HdΣ.

In order to proceed, we need anewAlexandrov-Fenchel inequality for a class of hypersurfaces in(P,g)!

Theorem

IfΣ⊂Pis star-shaped and strictly mean convex (H>0) then

cn

ˆ

Σ

ρHdΣ≥ 1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only ifΣis a slice.

The optimal Penrose inequality

This proves the first part of our main result.

Theorem

If M⊂Q_0,is an ALH graph as above, withΣ⊂P=P_0,being mean convex (H≥0) and star-shaped, then

m_(M,g)≥1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ| τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only if(M,g)is (congruent to) the graph realization of the corresponding black hole solution.

For=1, this sharpens previous results by Dahl-Gicquaud-Sakovich.

The optimal Penrose inequality

This proves the first part of our main result.

Theorem

If M⊂Q_0,is an ALH graph as above, withΣ⊂P=P_0,being mean convex (H≥0) and star-shaped, then

m_(M,g)≥1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ| τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only if(M,g)is (congruent to) the graph realization of the corresponding black hole solution.

For=1, this sharpens previous results by Dahl-Gicquaud-Sakovich.

The optimal Penrose inequality

This proves the first part of our main result.

Theorem

If M⊂Q_0,is an ALH graph as above, withΣ⊂P=P_0,being mean convex (H≥0) and star-shaped, then

m_(M,g)≥1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only if(M,g)is (congruent to) the graph realization of the corresponding black hole solution.

For=1, this sharpens previous results by Dahl-Gicquaud-Sakovich.

The optimal Penrose inequality

This proves the first part of our main result.

Theorem

If M⊂Q_0,is an ALH graph as above, withΣ⊂P=P_0,being mean convex (H≥0) and star-shaped, then

m_(M,g)≥1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only if(M,g)is (congruent to) the graph realization of the corresponding black hole solution.

For=1, this sharpens previous results by Dahl-Gicquaud-Sakovich.

The optimal Penrose inequality

This proves the first part of our main result.

Theorem

If M⊂Q_0,is an ALH graph as above, withΣ⊂P=P_0,being mean convex (H≥0) and star-shaped, then

m_(M,g)≥1 2



 |Σ|

τn−1

_n−1ⁿ +

|Σ|

τn−1

ⁿ⁻²_n−1



,

with the equality holding if and only if(M,g)is (congruent to) the graph realization of the corresponding black hole solution.

For=1, this sharpens previous results by Dahl-Gicquaud-Sakovich.