An introduction to the Penrose inequality III
Levi Lopes de Lima
Department of MathematicsFederal University of Ceará
Gelosp2013 - July, 2013
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 1 / 23
Joint work with Fred Girão (UFC/Fortaleza).
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequalityfor strictly mean convex hypersurfaces in a certain class of locally hyperbolic manifolds ofdimension n ≥ 3.
This provides natural generalizations of the classical Minkowski inequality for convexhypersufaces in Rn.
As an application we establish an optimal Penrose inequality for asymptotically locallyhyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of whathappens in the equality case.
This provides a large class of examples of initial data sets (corresponding to time-symmetricsolutions of Einstein equations in General Relativity with a negative cosmological constant)for which an optimal Penrose inequality holds true.
In particular, in the physical dimension n = 3 we obtain, for this class of IDS, a proof of aPenrose-type inequality for exotic black holes solutions first conjectured by Gibbons andChrusciel-Simon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 2 / 23
Joint work with Fred Girão (UFC/Fortaleza).
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequalityfor strictly mean convex hypersurfaces in a certain class of locally hyperbolic manifolds ofdimension n ≥ 3.
This provides natural generalizations of the classical Minkowski inequality for convexhypersufaces in Rn.
As an application we establish an optimal Penrose inequality for asymptotically locallyhyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of whathappens in the equality case.
This provides a large class of examples of initial data sets (corresponding to time-symmetricsolutions of Einstein equations in General Relativity with a negative cosmological constant)for which an optimal Penrose inequality holds true.
In particular, in the physical dimension n = 3 we obtain, for this class of IDS, a proof of aPenrose-type inequality for exotic black holes solutions first conjectured by Gibbons andChrusciel-Simon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 2 / 23
Joint work with Fred Girão (UFC/Fortaleza).
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequalityfor strictly mean convex hypersurfaces in a certain class of locally hyperbolic manifolds ofdimension n ≥ 3.
This provides natural generalizations of the classical Minkowski inequality for convexhypersufaces in Rn.
As an application we establish an optimal Penrose inequality for asymptotically locallyhyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of whathappens in the equality case.
This provides a large class of examples of initial data sets (corresponding to time-symmetricsolutions of Einstein equations in General Relativity with a negative cosmological constant)for which an optimal Penrose inequality holds true.
In particular, in the physical dimension n = 3 we obtain, for this class of IDS, a proof of aPenrose-type inequality for exotic black holes solutions first conjectured by Gibbons andChrusciel-Simon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 2 / 23
Joint work with Fred Girão (UFC/Fortaleza).
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequalityfor strictly mean convex hypersurfaces in a certain class of locally hyperbolic manifolds ofdimension n ≥ 3.
This provides natural generalizations of the classical Minkowski inequality for convexhypersufaces in Rn.
As an application we establish an optimal Penrose inequality for asymptotically locallyhyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of whathappens in the equality case.
This provides a large class of examples of initial data sets (corresponding to time-symmetricsolutions of Einstein equations in General Relativity with a negative cosmological constant)for which an optimal Penrose inequality holds true.
In particular, in the physical dimension n = 3 we obtain, for this class of IDS, a proof of aPenrose-type inequality for exotic black holes solutions first conjectured by Gibbons andChrusciel-Simon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 2 / 23
Joint work with Fred Girão (UFC/Fortaleza).
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequalityfor strictly mean convex hypersurfaces in a certain class of locally hyperbolic manifolds ofdimension n ≥ 3.
This provides natural generalizations of the classical Minkowski inequality for convexhypersufaces in Rn.
As an application we establish an optimal Penrose inequality for asymptotically locallyhyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of whathappens in the equality case.
This provides a large class of examples of initial data sets (corresponding to time-symmetricsolutions of Einstein equations in General Relativity with a negative cosmological constant)for which an optimal Penrose inequality holds true.
In particular, in the physical dimension n = 3 we obtain, for this class of IDS, a proof of aPenrose-type inequality for exotic black holes solutions first conjectured by Gibbons andChrusciel-Simon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 2 / 23
Joint work with Fred Girão (UFC/Fortaleza).
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequalityfor strictly mean convex hypersurfaces in a certain class of locally hyperbolic manifolds ofdimension n ≥ 3.
This provides natural generalizations of the classical Minkowski inequality for convexhypersufaces in Rn.
As an application we establish an optimal Penrose inequality for asymptotically locallyhyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of whathappens in the equality case.
This provides a large class of examples of initial data sets (corresponding to time-symmetricsolutions of Einstein equations in General Relativity with a negative cosmological constant)for which an optimal Penrose inequality holds true.
In particular, in the physical dimension n = 3 we obtain, for this class of IDS, a proof of aPenrose-type inequality for exotic black holes solutions first conjectured by Gibbons andChrusciel-Simon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 2 / 23
Joint work with Fred Girão (UFC/Fortaleza).
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequalityfor strictly mean convex hypersurfaces in a certain class of locally hyperbolic manifolds ofdimension n ≥ 3.
This provides natural generalizations of the classical Minkowski inequality for convexhypersufaces in Rn.
As an application we establish an optimal Penrose inequality for asymptotically locallyhyperbolic (ALH) graphs carrying a minimal horizon, with a precise description of whathappens in the equality case.
This provides a large class of examples of initial data sets (corresponding to time-symmetricsolutions of Einstein equations in General Relativity with a negative cosmological constant)for which an optimal Penrose inequality holds true.
In particular, in the physical dimension n = 3 we obtain, for this class of IDS, a proof of aPenrose-type inequality for exotic black holes solutions first conjectured by Gibbons andChrusciel-Simon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 2 / 23
The reference metrics (Chrusciel-Herzlich-Nagy)
Fix n ≥ 3, ε = 0,±1 and let (Nn−1, h) be a closed space form with curvature ε.
In the product manifold Pε = Iε × N, consider the metric
gε =dr2
ρε(r)2+ r2h, r ∈ Iε,
whereρε(r) =
√r2 + ε.
Here, I−1 = (1,+∞) and I0 = I1 = (0,+∞).
The metric gε is locally hyperbolic (Kgε ≡ −1).
For instance, if ε = 1 and (N, h) is a round sphere then (P1, g1) is hyperbolic space Hn.
Also, if ε = 0 and (N2, h) is a torus then (P0, g0) is a cusp manifold.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 3 / 23
The reference metrics (Chrusciel-Herzlich-Nagy)
Fix n ≥ 3, ε = 0,±1 and let (Nn−1, h) be a closed space form with curvature ε.
In the product manifold Pε = Iε × N, consider the metric
gε =dr2
ρε(r)2+ r2h, r ∈ Iε,
whereρε(r) =
√r2 + ε.
Here, I−1 = (1,+∞) and I0 = I1 = (0,+∞).
The metric gε is locally hyperbolic (Kgε ≡ −1).
For instance, if ε = 1 and (N, h) is a round sphere then (P1, g1) is hyperbolic space Hn.
Also, if ε = 0 and (N2, h) is a torus then (P0, g0) is a cusp manifold.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 3 / 23
The reference metrics (Chrusciel-Herzlich-Nagy)
Fix n ≥ 3, ε = 0,±1 and let (Nn−1, h) be a closed space form with curvature ε.
In the product manifold Pε = Iε × N, consider the metric
gε =dr2
ρε(r)2+ r2h, r ∈ Iε,
whereρε(r) =
√r2 + ε.
Here, I−1 = (1,+∞) and I0 = I1 = (0,+∞).
The metric gε is locally hyperbolic (Kgε ≡ −1).
For instance, if ε = 1 and (N, h) is a round sphere then (P1, g1) is hyperbolic space Hn.
Also, if ε = 0 and (N2, h) is a torus then (P0, g0) is a cusp manifold.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 3 / 23
The reference metrics (Chrusciel-Herzlich-Nagy)
Fix n ≥ 3, ε = 0,±1 and let (Nn−1, h) be a closed space form with curvature ε.
In the product manifold Pε = Iε × N, consider the metric
gε =dr2
ρε(r)2+ r2h, r ∈ Iε,
whereρε(r) =
√r2 + ε.
Here, I−1 = (1,+∞) and I0 = I1 = (0,+∞).
The metric gε is locally hyperbolic (Kgε ≡ −1).
For instance, if ε = 1 and (N, h) is a round sphere then (P1, g1) is hyperbolic space Hn.
Also, if ε = 0 and (N2, h) is a torus then (P0, g0) is a cusp manifold.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 3 / 23
The reference metrics (Chrusciel-Herzlich-Nagy)
Fix n ≥ 3, ε = 0,±1 and let (Nn−1, h) be a closed space form with curvature ε.
In the product manifold Pε = Iε × N, consider the metric
gε =dr2
ρε(r)2+ r2h, r ∈ Iε,
whereρε(r) =
√r2 + ε.
Here, I−1 = (1,+∞) and I0 = I1 = (0,+∞).
The metric gε is locally hyperbolic (Kgε ≡ −1).
For instance, if ε = 1 and (N, h) is a round sphere then (P1, g1) is hyperbolic space Hn.
Also, if ε = 0 and (N2, h) is a torus then (P0, g0) is a cusp manifold.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 3 / 23
The reference metrics (Chrusciel-Herzlich-Nagy)
Fix n ≥ 3, ε = 0,±1 and let (Nn−1, h) be a closed space form with curvature ε.
In the product manifold Pε = Iε × N, consider the metric
gε =dr2
ρε(r)2+ r2h, r ∈ Iε,
whereρε(r) =
√r2 + ε.
Here, I−1 = (1,+∞) and I0 = I1 = (0,+∞).
The metric gε is locally hyperbolic (Kgε ≡ −1).
For instance, if ε = 1 and (N, h) is a round sphere then (P1, g1) is hyperbolic space Hn.
Also, if ε = 0 and (N2, h) is a torus then (P0, g0) is a cusp manifold.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 3 / 23
The reference metrics (Chrusciel-Herzlich-Nagy)
Fix n ≥ 3, ε = 0,±1 and let (Nn−1, h) be a closed space form with curvature ε.
In the product manifold Pε = Iε × N, consider the metric
gε =dr2
ρε(r)2+ r2h, r ∈ Iε,
whereρε(r) =
√r2 + ε.
Here, I−1 = (1,+∞) and I0 = I1 = (0,+∞).
The metric gε is locally hyperbolic (Kgε ≡ −1).
For instance, if ε = 1 and (N, h) is a round sphere then (P1, g1) is hyperbolic space Hn.
Also, if ε = 0 and (N2, h) is a torus then (P0, g0) is a cusp manifold.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 3 / 23
Asymptotically locally hyperbolic manifolds (Chrusciel-Herzlich-Nagy)
Definition
Fix ε and (N, h) as above. A complete n-dimensional manifold (M, g), possibly carrying a compactinner boundary Σ, is said to be asymptotically locally hyperbolic (ALH) if there exist subsetsK ⊂ M and K0 ⊂ Pε, with K compact, and a diffeomorphism Ψ : M − K → Pε − K0 such that
‖Ψ∗g − gε‖gε = O(r−τ ), ‖DΨ∗g‖gε = O(r−τ ), r → +∞,
for some τ > n/2. We also assume that Rg + n(n − 1) ∈ L1.
For this class of manifolds, a mass-like invariant m(M,g) ∈ R can be defined as
m(M,g) = limr→+∞
cn
ˆNr
(ρε(divgεe − d trgεe)− i∇gερεe + (trgεdρε)
)(νr )dNr ,
where e = Ψ∗g − gε, Nr = {r} × N, νr is the outward unit vector to Nr and
cn =1
2(n − 1)τn−1, τn−1 = arean−1(N, h).
This invariant measures the rate of the convergence g → g0,ε as r → +∞.
Here, we are leaving aside the cases where N = Sn/Γ, Γ 6= {Id}.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 4 / 23
Asymptotically locally hyperbolic manifolds (Chrusciel-Herzlich-Nagy)
Definition
Fix ε and (N, h) as above. A complete n-dimensional manifold (M, g), possibly carrying a compactinner boundary Σ, is said to be asymptotically locally hyperbolic (ALH) if there exist subsetsK ⊂ M and K0 ⊂ Pε, with K compact, and a diffeomorphism Ψ : M − K → Pε − K0 such that
‖Ψ∗g − gε‖gε = O(r−τ ), ‖DΨ∗g‖gε = O(r−τ ), r → +∞,
for some τ > n/2. We also assume that Rg + n(n − 1) ∈ L1.
For this class of manifolds, a mass-like invariant m(M,g) ∈ R can be defined as
m(M,g) = limr→+∞
cn
ˆNr
(ρε(divgεe − d trgεe)− i∇gερεe + (trgεdρε)
)(νr )dNr ,
where e = Ψ∗g − gε, Nr = {r} × N, νr is the outward unit vector to Nr and
cn =1
2(n − 1)τn−1, τn−1 = arean−1(N, h).
This invariant measures the rate of the convergence g → g0,ε as r → +∞.
Here, we are leaving aside the cases where N = Sn/Γ, Γ 6= {Id}.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 4 / 23
Asymptotically locally hyperbolic manifolds (Chrusciel-Herzlich-Nagy)
Definition
Fix ε and (N, h) as above. A complete n-dimensional manifold (M, g), possibly carrying a compactinner boundary Σ, is said to be asymptotically locally hyperbolic (ALH) if there exist subsetsK ⊂ M and K0 ⊂ Pε, with K compact, and a diffeomorphism Ψ : M − K → Pε − K0 such that
‖Ψ∗g − gε‖gε = O(r−τ ), ‖DΨ∗g‖gε = O(r−τ ), r → +∞,
for some τ > n/2. We also assume that Rg + n(n − 1) ∈ L1.
For this class of manifolds, a mass-like invariant m(M,g) ∈ R can be defined as
m(M,g) = limr→+∞
cn
ˆNr
(ρε(divgεe − d trgεe)− i∇gερεe + (trgεdρε)
)(νr )dNr ,
where e = Ψ∗g − gε, Nr = {r} × N, νr is the outward unit vector to Nr and
cn =1
2(n − 1)τn−1, τn−1 = arean−1(N, h).
This invariant measures the rate of the convergence g → g0,ε as r → +∞.
Here, we are leaving aside the cases where N = Sn/Γ, Γ 6= {Id}.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 4 / 23
Asymptotically locally hyperbolic manifolds (Chrusciel-Herzlich-Nagy)
Definition
Fix ε and (N, h) as above. A complete n-dimensional manifold (M, g), possibly carrying a compactinner boundary Σ, is said to be asymptotically locally hyperbolic (ALH) if there exist subsetsK ⊂ M and K0 ⊂ Pε, with K compact, and a diffeomorphism Ψ : M − K → Pε − K0 such that
‖Ψ∗g − gε‖gε = O(r−τ ), ‖DΨ∗g‖gε = O(r−τ ), r → +∞,
for some τ > n/2. We also assume that Rg + n(n − 1) ∈ L1.
For this class of manifolds, a mass-like invariant m(M,g) ∈ R can be defined as
m(M,g) = limr→+∞
cn
ˆNr
(ρε(divgεe − d trgεe)− i∇gερεe + (trgεdρε)
)(νr )dNr ,
where e = Ψ∗g − gε, Nr = {r} × N, νr is the outward unit vector to Nr and
cn =1
2(n − 1)τn−1, τn−1 = arean−1(N, h).
This invariant measures the rate of the convergence g → g0,ε as r → +∞.
Here, we are leaving aside the cases where N = Sn/Γ, Γ 6= {Id}.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 4 / 23
Asymptotically locally hyperbolic manifolds (Chrusciel-Herzlich-Nagy)
Definition
Fix ε and (N, h) as above. A complete n-dimensional manifold (M, g), possibly carrying a compactinner boundary Σ, is said to be asymptotically locally hyperbolic (ALH) if there exist subsetsK ⊂ M and K0 ⊂ Pε, with K compact, and a diffeomorphism Ψ : M − K → Pε − K0 such that
‖Ψ∗g − gε‖gε = O(r−τ ), ‖DΨ∗g‖gε = O(r−τ ), r → +∞,
for some τ > n/2. We also assume that Rg + n(n − 1) ∈ L1.
For this class of manifolds, a mass-like invariant m(M,g) ∈ R can be defined as
m(M,g) = limr→+∞
cn
ˆNr
(ρε(divgεe − d trgεe)− i∇gερεe + (trgεdρε)
)(νr )dNr ,
where e = Ψ∗g − gε, Nr = {r} × N, νr is the outward unit vector to Nr and
cn =1
2(n − 1)τn−1, τn−1 = arean−1(N, h).
This invariant measures the rate of the convergence g → g0,ε as r → +∞.
Here, we are leaving aside the cases where N = Sn/Γ, Γ 6= {Id}.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 4 / 23
Asymptotically locally hyperbolic manifolds (Chrusciel-Herzlich-Nagy)
Definition
Fix ε and (N, h) as above. A complete n-dimensional manifold (M, g), possibly carrying a compactinner boundary Σ, is said to be asymptotically locally hyperbolic (ALH) if there exist subsetsK ⊂ M and K0 ⊂ Pε, with K compact, and a diffeomorphism Ψ : M − K → Pε − K0 such that
‖Ψ∗g − gε‖gε = O(r−τ ), ‖DΨ∗g‖gε = O(r−τ ), r → +∞,
for some τ > n/2. We also assume that Rg + n(n − 1) ∈ L1.
For this class of manifolds, a mass-like invariant m(M,g) ∈ R can be defined as
m(M,g) = limr→+∞
cn
ˆNr
(ρε(divgεe − d trgεe)− i∇gερεe + (trgεdρε)
)(νr )dNr ,
where e = Ψ∗g − gε, Nr = {r} × N, νr is the outward unit vector to Nr and
cn =1
2(n − 1)τn−1, τn−1 = arean−1(N, h).
This invariant measures the rate of the convergence g → g0,ε as r → +∞.
Here, we are leaving aside the cases where N = Sn/Γ, Γ 6= {Id}.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 4 / 23
The black hole solutions I
Fix ε = 0,±1, m > 0 and consider the interval
Im,ε = {r > rm,ε},
where rm,ε is the positive root of
r2 + ε−2m
rn−2= 0.
If (Nn−1, h) is a compact space form with curvature ε, in the product manifoldPm,ε = Im,ε × N define the metric
gm,ε =dr2
ρm,ε(r)2+ r2h,
where
ρm,ε(r) =
√r2 + ε−
2mrn−2
.
We note that gm,ε extends smoothly to Pm,ε = [rm,ε,+∞)× N and the slice defined byr = rm,ε is called the horizon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 5 / 23
The black hole solutions I
Fix ε = 0,±1, m > 0 and consider the interval
Im,ε = {r > rm,ε},
where rm,ε is the positive root of
r2 + ε−2m
rn−2= 0.
If (Nn−1, h) is a compact space form with curvature ε, in the product manifoldPm,ε = Im,ε × N define the metric
gm,ε =dr2
ρm,ε(r)2+ r2h,
where
ρm,ε(r) =
√r2 + ε−
2mrn−2
.
We note that gm,ε extends smoothly to Pm,ε = [rm,ε,+∞)× N and the slice defined byr = rm,ε is called the horizon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 5 / 23
The black hole solutions I
Fix ε = 0,±1, m > 0 and consider the interval
Im,ε = {r > rm,ε},
where rm,ε is the positive root of
r2 + ε−2m
rn−2= 0.
If (Nn−1, h) is a compact space form with curvature ε, in the product manifoldPm,ε = Im,ε × N define the metric
gm,ε =dr2
ρm,ε(r)2+ r2h,
where
ρm,ε(r) =
√r2 + ε−
2mrn−2
.
We note that gm,ε extends smoothly to Pm,ε = [rm,ε,+∞)× N and the slice defined byr = rm,ε is called the horizon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 5 / 23
The black hole solutions I
Fix ε = 0,±1, m > 0 and consider the interval
Im,ε = {r > rm,ε},
where rm,ε is the positive root of
r2 + ε−2m
rn−2= 0.
If (Nn−1, h) is a compact space form with curvature ε, in the product manifoldPm,ε = Im,ε × N define the metric
gm,ε =dr2
ρm,ε(r)2+ r2h,
where
ρm,ε(r) =
√r2 + ε−
2mrn−2
.
We note that gm,ε extends smoothly to Pm,ε = [rm,ε,+∞)× N and the slice defined byr = rm,ε is called the horizon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 5 / 23
The black hole solutions I
Fix ε = 0,±1, m > 0 and consider the interval
Im,ε = {r > rm,ε},
where rm,ε is the positive root of
r2 + ε−2m
rn−2= 0.
If (Nn−1, h) is a compact space form with curvature ε, in the product manifoldPm,ε = Im,ε × N define the metric
gm,ε =dr2
ρm,ε(r)2+ r2h,
where
ρm,ε(r) =
√r2 + ε−
2mrn−2
.
We note that gm,ε extends smoothly to Pm,ε = [rm,ε,+∞)× N and the slice defined byr = rm,ε is called the horizon.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 5 / 23
The black hole solutions II
If (θ1, · · · , θn−1) are orthonormal coordinates in N then the sectional curvatures of gm,ε are
Kgm,ε (∂r , ∂θi ) = −1− (n − 2)mrn
andKgm,ε (∂θi , ∂θj ) = −1 +
2mrn,
so that the scalar curvature of gm,ε is Rgm,ε = −n(n − 1).
Moreover, each gm,ε is a static metric in the sense that ρm,ε satisfies
(∆ρm,ε)gm,ε − Hessgm,ερm,ε + ρm,εRicgm,ε = 0,
which means that the Lorentzian metric
gm,ε = −ρ2m,εdt2 + gm,ε,
defined on Qm,ε = R× Pm,ε, is a solution to the vacuum Einstein field equations withnegative cosmological constant:
Ricgm,ε= −ngm,ε.
Thus, gm,ε defines an initial data set for a time-symmetric (actually, static) vacuum solution ofEinstein equations carrying a black hole.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 6 / 23
The black hole solutions II
If (θ1, · · · , θn−1) are orthonormal coordinates in N then the sectional curvatures of gm,ε are
Kgm,ε (∂r , ∂θi ) = −1− (n − 2)mrn
andKgm,ε (∂θi , ∂θj ) = −1 +
2mrn,
so that the scalar curvature of gm,ε is Rgm,ε = −n(n − 1).
Moreover, each gm,ε is a static metric in the sense that ρm,ε satisfies
(∆ρm,ε)gm,ε − Hessgm,ερm,ε + ρm,εRicgm,ε = 0,
which means that the Lorentzian metric
gm,ε = −ρ2m,εdt2 + gm,ε,
defined on Qm,ε = R× Pm,ε, is a solution to the vacuum Einstein field equations withnegative cosmological constant:
Ricgm,ε= −ngm,ε.
Thus, gm,ε defines an initial data set for a time-symmetric (actually, static) vacuum solution ofEinstein equations carrying a black hole.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 6 / 23
The black hole solutions II
If (θ1, · · · , θn−1) are orthonormal coordinates in N then the sectional curvatures of gm,ε are
Kgm,ε (∂r , ∂θi ) = −1− (n − 2)mrn
andKgm,ε (∂θi , ∂θj ) = −1 +
2mrn,
so that the scalar curvature of gm,ε is Rgm,ε = −n(n − 1).
Moreover, each gm,ε is a static metric in the sense that ρm,ε satisfies
(∆ρm,ε)gm,ε − Hessgm,ερm,ε + ρm,εRicgm,ε = 0,
which means that the Lorentzian metric
gm,ε = −ρ2m,εdt2 + gm,ε,
defined on Qm,ε = R× Pm,ε, is a solution to the vacuum Einstein field equations withnegative cosmological constant:
Ricgm,ε= −ngm,ε.
Thus, gm,ε defines an initial data set for a time-symmetric (actually, static) vacuum solution ofEinstein equations carrying a black hole.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 6 / 23
The black hole solutions II
If (θ1, · · · , θn−1) are orthonormal coordinates in N then the sectional curvatures of gm,ε are
Kgm,ε (∂r , ∂θi ) = −1− (n − 2)mrn
andKgm,ε (∂θi , ∂θj ) = −1 +
2mrn,
so that the scalar curvature of gm,ε is Rgm,ε = −n(n − 1).
Moreover, each gm,ε is a static metric in the sense that ρm,ε satisfies
(∆ρm,ε)gm,ε − Hessgm,ερm,ε + ρm,εRicgm,ε = 0,
which means that the Lorentzian metric
gm,ε = −ρ2m,εdt2 + gm,ε,
defined on Qm,ε = R× Pm,ε, is a solution to the vacuum Einstein field equations withnegative cosmological constant:
Ricgm,ε= −ngm,ε.
Thus, gm,ε defines an initial data set for a time-symmetric (actually, static) vacuum solution ofEinstein equations carrying a black hole.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 6 / 23
The black hole solutions III
One easily verifies that, as r → +∞,
‖gm,ε − gε‖gε = O(mr−n) ,
where gε is the corresponding reference metric.
Thus, each gm,ε, m > 0, is asymptotically locally hyperbolic (ALH).
Physical reasoning allows us to interpret m as the total mass of the black hole solution gm,ε.
Indeed, a computation shows that m(Pm,ε,gm,ε) = m.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 7 / 23
The black hole solutions III
One easily verifies that, as r → +∞,
‖gm,ε − gε‖gε = O(mr−n) ,
where gε is the corresponding reference metric.
Thus, each gm,ε, m > 0, is asymptotically locally hyperbolic (ALH).
Physical reasoning allows us to interpret m as the total mass of the black hole solution gm,ε.
Indeed, a computation shows that m(Pm,ε,gm,ε) = m.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 7 / 23
The black hole solutions III
One easily verifies that, as r → +∞,
‖gm,ε − gε‖gε = O(mr−n) ,
where gε is the corresponding reference metric.
Thus, each gm,ε, m > 0, is asymptotically locally hyperbolic (ALH).
Physical reasoning allows us to interpret m as the total mass of the black hole solution gm,ε.
Indeed, a computation shows that m(Pm,ε,gm,ε) = m.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 7 / 23
The black hole solutions III
One easily verifies that, as r → +∞,
‖gm,ε − gε‖gε = O(mr−n) ,
where gε is the corresponding reference metric.
Thus, each gm,ε, m > 0, is asymptotically locally hyperbolic (ALH).
Physical reasoning allows us to interpret m as the total mass of the black hole solution gm,ε.
Indeed, a computation shows that m(Pm,ε,gm,ε) = m.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 7 / 23
The black hole solutions III
One easily verifies that, as r → +∞,
‖gm,ε − gε‖gε = O(mr−n) ,
where gε is the corresponding reference metric.
Thus, each gm,ε, m > 0, is asymptotically locally hyperbolic (ALH).
Physical reasoning allows us to interpret m as the total mass of the black hole solution gm,ε.
Indeed, a computation shows that m(Pm,ε,gm,ε) = m.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 7 / 23
The black hole solutions III
One easily verifies that, as r → +∞,
‖gm,ε − gε‖gε = O(mr−n) ,
where gε is the corresponding reference metric.
Thus, each gm,ε, m > 0, is asymptotically locally hyperbolic (ALH).
Physical reasoning allows us to interpret m as the total mass of the black hole solution gm,ε.
Indeed, a computation shows that m(Pm,ε,gm,ε) = m.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 7 / 23
The black hole solutions IV
It turns out that each gm,ε can be isometrically embedded as a graph in (Qε, gε), whereQε = R× Pε and
gε = ρε(r)2dt2 +dr2
ρε(r)2+ r2dθ2.
Notice that (Qε, gε) is locally hyperbolic!
The radial function defining the graph, u = um,ε(r), satisfies u(rm,ε) = 0 and
ρε(r)2(
dudr
)2=
1ρm,ε(r)2
−1
ρε(r)2, r ≥ rm,ε.
It follows that the graph realization of the black hole solution meets the slice t = 0orthogonally along the minimal ‘horizon’ H defined by r = rm,ε.
Notice also that the mass m relates to the area |H| of the black hole horizon by
m =12
( |H|τn−1
) nn−1
+ ε
(|H|τn−1
) n−2n−1
, τn−1 = arean−1(N).
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 8 / 23
The black hole solutions IV
It turns out that each gm,ε can be isometrically embedded as a graph in (Qε, gε), whereQε = R× Pε and
gε = ρε(r)2dt2 +dr2
ρε(r)2+ r2dθ2.
Notice that (Qε, gε) is locally hyperbolic!
The radial function defining the graph, u = um,ε(r), satisfies u(rm,ε) = 0 and
ρε(r)2(
dudr
)2=
1ρm,ε(r)2
−1
ρε(r)2, r ≥ rm,ε.
It follows that the graph realization of the black hole solution meets the slice t = 0orthogonally along the minimal ‘horizon’ H defined by r = rm,ε.
Notice also that the mass m relates to the area |H| of the black hole horizon by
m =12
( |H|τn−1
) nn−1
+ ε
(|H|τn−1
) n−2n−1
, τn−1 = arean−1(N).
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 8 / 23
The black hole solutions IV
It turns out that each gm,ε can be isometrically embedded as a graph in (Qε, gε), whereQε = R× Pε and
gε = ρε(r)2dt2 +dr2
ρε(r)2+ r2dθ2.
Notice that (Qε, gε) is locally hyperbolic!
The radial function defining the graph, u = um,ε(r), satisfies u(rm,ε) = 0 and
ρε(r)2(
dudr
)2=
1ρm,ε(r)2
−1
ρε(r)2, r ≥ rm,ε.
It follows that the graph realization of the black hole solution meets the slice t = 0orthogonally along the minimal ‘horizon’ H defined by r = rm,ε.
Notice also that the mass m relates to the area |H| of the black hole horizon by
m =12
( |H|τn−1
) nn−1
+ ε
(|H|τn−1
) n−2n−1
, τn−1 = arean−1(N).
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 8 / 23
The black hole solutions IV
It turns out that each gm,ε can be isometrically embedded as a graph in (Qε, gε), whereQε = R× Pε and
gε = ρε(r)2dt2 +dr2
ρε(r)2+ r2dθ2.
Notice that (Qε, gε) is locally hyperbolic!
The radial function defining the graph, u = um,ε(r), satisfies u(rm,ε) = 0 and
ρε(r)2(
dudr
)2=
1ρm,ε(r)2
−1
ρε(r)2, r ≥ rm,ε.
It follows that the graph realization of the black hole solution meets the slice t = 0orthogonally along the minimal ‘horizon’ H defined by r = rm,ε.
Notice also that the mass m relates to the area |H| of the black hole horizon by
m =12
( |H|τn−1
) nn−1
+ ε
(|H|τn−1
) n−2n−1
, τn−1 = arean−1(N).
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 8 / 23
The black hole solutions IV
It turns out that each gm,ε can be isometrically embedded as a graph in (Qε, gε), whereQε = R× Pε and
gε = ρε(r)2dt2 +dr2
ρε(r)2+ r2dθ2.
Notice that (Qε, gε) is locally hyperbolic!
The radial function defining the graph, u = um,ε(r), satisfies u(rm,ε) = 0 and
ρε(r)2(
dudr
)2=
1ρm,ε(r)2
−1
ρε(r)2, r ≥ rm,ε.
It follows that the graph realization of the black hole solution meets the slice t = 0orthogonally along the minimal ‘horizon’ H defined by r = rm,ε.
Notice also that the mass m relates to the area |H| of the black hole horizon by
m =12
( |H|τn−1
) nn−1
+ ε
(|H|τn−1
) n−2n−1
, τn−1 = arean−1(N).
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 8 / 23
The black hole solutions IV
It turns out that each gm,ε can be isometrically embedded as a graph in (Qε, gε), whereQε = R× Pε and
gε = ρε(r)2dt2 +dr2
ρε(r)2+ r2dθ2.
Notice that (Qε, gε) is locally hyperbolic!
The radial function defining the graph, u = um,ε(r), satisfies u(rm,ε) = 0 and
ρε(r)2(
dudr
)2=
1ρm,ε(r)2
−1
ρε(r)2, r ≥ rm,ε.
It follows that the graph realization of the black hole solution meets the slice t = 0orthogonally along the minimal ‘horizon’ H defined by r = rm,ε.
Notice also that the mass m relates to the area |H| of the black hole horizon by
m =12
( |H|τn−1
) nn−1
+ ε
(|H|τn−1
) n−2n−1
, τn−1 = arean−1(N).
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 8 / 23
The Penrose conjecture for ALH manifolds
Let (M, g) be an ALH manifold (relative to the reference metric gε). Assume thatRg ≥ −n(n − 1) and that M carries an outermost minimal horizon Σ. Then,
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality occurring if and only if (M, g) is (isometric to) the corresponding black holesolution.
In the physical dimension n = 3, this appears as a conjectured Penrose-type inequality inpapers by Gibbons and Chrusciel-Simon.
In the following we establish this inequality for ALH graphs in any dimension n ≥ 3, includingthe corresponding rigidity statement.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 9 / 23
The Penrose conjecture for ALH manifolds
Let (M, g) be an ALH manifold (relative to the reference metric gε). Assume thatRg ≥ −n(n − 1) and that M carries an outermost minimal horizon Σ. Then,
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality occurring if and only if (M, g) is (isometric to) the corresponding black holesolution.
In the physical dimension n = 3, this appears as a conjectured Penrose-type inequality inpapers by Gibbons and Chrusciel-Simon.
In the following we establish this inequality for ALH graphs in any dimension n ≥ 3, includingthe corresponding rigidity statement.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 9 / 23
The Penrose conjecture for ALH manifolds
Let (M, g) be an ALH manifold (relative to the reference metric gε). Assume thatRg ≥ −n(n − 1) and that M carries an outermost minimal horizon Σ. Then,
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality occurring if and only if (M, g) is (isometric to) the corresponding black holesolution.
In the physical dimension n = 3, this appears as a conjectured Penrose-type inequality inpapers by Gibbons and Chrusciel-Simon.
In the following we establish this inequality for ALH graphs in any dimension n ≥ 3, includingthe corresponding rigidity statement.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 9 / 23
The Penrose conjecture for ALH manifolds
Let (M, g) be an ALH manifold (relative to the reference metric gε). Assume thatRg ≥ −n(n − 1) and that M carries an outermost minimal horizon Σ. Then,
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality occurring if and only if (M, g) is (isometric to) the corresponding black holesolution.
In the physical dimension n = 3, this appears as a conjectured Penrose-type inequality inpapers by Gibbons and Chrusciel-Simon.
In the following we establish this inequality for ALH graphs in any dimension n ≥ 3, includingthe corresponding rigidity statement.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 9 / 23
The Penrose conjecture for ALH manifolds
Let (M, g) be an ALH manifold (relative to the reference metric gε). Assume thatRg ≥ −n(n − 1) and that M carries an outermost minimal horizon Σ. Then,
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality occurring if and only if (M, g) is (isometric to) the corresponding black holesolution.
In the physical dimension n = 3, this appears as a conjectured Penrose-type inequality inpapers by Gibbons and Chrusciel-Simon.
In the following we establish this inequality for ALH graphs in any dimension n ≥ 3, includingthe corresponding rigidity statement.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 9 / 23
ALH hypersurfaces in Qε
DefinitionA complete, isometrically immersed hypersurface (M, g)# (Qε, gε), possibly with an innerboundary Σ, is asymptotically locally hyperbolic (ALH) if there exist subsets K ⊂ M, K0 ⊂ Pε suchthat M − K , the end of M, can be written as a vertical graph over Pε − K0, with the graph beingassociated to a smooth function u : Pε − K0 → R such the previous asymptotic conditions holdsfor the nonparametric chart Ψu(x , u(x)) = x , x ∈ K0. Moreover, we assume thatRΨu∗g + n(n − 1) is integrable.
Under these conditions, the mass of (M, g) is well defined and can be computed by using Ψu .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 10 / 23
ALH hypersurfaces in Qε
DefinitionA complete, isometrically immersed hypersurface (M, g)# (Qε, gε), possibly with an innerboundary Σ, is asymptotically locally hyperbolic (ALH) if there exist subsets K ⊂ M, K0 ⊂ Pε suchthat M − K , the end of M, can be written as a vertical graph over Pε − K0, with the graph beingassociated to a smooth function u : Pε − K0 → R such the previous asymptotic conditions holdsfor the nonparametric chart Ψu(x , u(x)) = x , x ∈ K0. Moreover, we assume thatRΨu∗g + n(n − 1) is integrable.
Under these conditions, the mass of (M, g) is well defined and can be computed by using Ψu .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 10 / 23
ALH hypersurfaces in Qε
DefinitionA complete, isometrically immersed hypersurface (M, g)# (Qε, gε), possibly with an innerboundary Σ, is asymptotically locally hyperbolic (ALH) if there exist subsets K ⊂ M, K0 ⊂ Pε suchthat M − K , the end of M, can be written as a vertical graph over Pε − K0, with the graph beingassociated to a smooth function u : Pε − K0 → R such the previous asymptotic conditions holdsfor the nonparametric chart Ψu(x , u(x)) = x , x ∈ K0. Moreover, we assume thatRΨu∗g + n(n − 1) is integrable.
Under these conditions, the mass of (M, g) is well defined and can be computed by using Ψu .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 10 / 23
ALH hypersurfaces in Qε
DefinitionA complete, isometrically immersed hypersurface (M, g)# (Qε, gε), possibly with an innerboundary Σ, is asymptotically locally hyperbolic (ALH) if there exist subsets K ⊂ M, K0 ⊂ Pε suchthat M − K , the end of M, can be written as a vertical graph over Pε − K0, with the graph beingassociated to a smooth function u : Pε − K0 → R such the previous asymptotic conditions holdsfor the nonparametric chart Ψu(x , u(x)) = x , x ∈ K0. Moreover, we assume thatRΨu∗g + n(n − 1) is integrable.
Under these conditions, the mass of (M, g) is well defined and can be computed by using Ψu .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 10 / 23
The integral formula for the mass
For any hypersurface M ⊂ Qε = R× Pε endowed with a unit normal N, an old formula byReilly says that
divM (G(A)X) = 2σ2(A)Θ,
where G(A) = σ1(A)I − A is the Newton tensor of the shape operator A, X is the tangentialcomponent of ∂/∂t and Θ = 〈N, ∂/∂t〉. This uses that ∂/∂t is Killing and that Kgε
≡ −1.
Assume from now on that M ⊂ Qε is ALH and its inner boundary Σ lies on a horizontal totallygeodesic hypersurface, say P ' Pε. Assume further that M meets P orthogonally along Σ(which implies that Σ ⊂ M is minimal and hence a horizon).
TheoremUnder the above conditions,
m(M,g) = cn
ˆM
Θ (Rg + n(n − 1)) dM + cn
ˆΣρεHdΣ,
where H is the mean curvature of Σ ⊂ P and ρε(r) =√
r2 + ε. In particular, if Rg ≥ −n(n − 1)and M is a graph (Θ > 0) then
m(M,g) ≥ cn
ˆΣρεHdΣ.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 11 / 23
The integral formula for the mass
For any hypersurface M ⊂ Qε = R× Pε endowed with a unit normal N, an old formula byReilly says that
divM (G(A)X) = 2σ2(A)Θ,
where G(A) = σ1(A)I − A is the Newton tensor of the shape operator A, X is the tangentialcomponent of ∂/∂t and Θ = 〈N, ∂/∂t〉. This uses that ∂/∂t is Killing and that Kgε
≡ −1.
Assume from now on that M ⊂ Qε is ALH and its inner boundary Σ lies on a horizontal totallygeodesic hypersurface, say P ' Pε. Assume further that M meets P orthogonally along Σ(which implies that Σ ⊂ M is minimal and hence a horizon).
TheoremUnder the above conditions,
m(M,g) = cn
ˆM
Θ (Rg + n(n − 1)) dM + cn
ˆΣρεHdΣ,
where H is the mean curvature of Σ ⊂ P and ρε(r) =√
r2 + ε. In particular, if Rg ≥ −n(n − 1)and M is a graph (Θ > 0) then
m(M,g) ≥ cn
ˆΣρεHdΣ.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 11 / 23
The integral formula for the mass
For any hypersurface M ⊂ Qε = R× Pε endowed with a unit normal N, an old formula byReilly says that
divM (G(A)X) = 2σ2(A)Θ,
where G(A) = σ1(A)I − A is the Newton tensor of the shape operator A, X is the tangentialcomponent of ∂/∂t and Θ = 〈N, ∂/∂t〉. This uses that ∂/∂t is Killing and that Kgε
≡ −1.
Assume from now on that M ⊂ Qε is ALH and its inner boundary Σ lies on a horizontal totallygeodesic hypersurface, say P ' Pε. Assume further that M meets P orthogonally along Σ(which implies that Σ ⊂ M is minimal and hence a horizon).
TheoremUnder the above conditions,
m(M,g) = cn
ˆM
Θ (Rg + n(n − 1)) dM + cn
ˆΣρεHdΣ,
where H is the mean curvature of Σ ⊂ P and ρε(r) =√
r2 + ε. In particular, if Rg ≥ −n(n − 1)and M is a graph (Θ > 0) then
m(M,g) ≥ cn
ˆΣρεHdΣ.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 11 / 23
The integral formula for the mass
For any hypersurface M ⊂ Qε = R× Pε endowed with a unit normal N, an old formula byReilly says that
divM (G(A)X) = 2σ2(A)Θ,
where G(A) = σ1(A)I − A is the Newton tensor of the shape operator A, X is the tangentialcomponent of ∂/∂t and Θ = 〈N, ∂/∂t〉. This uses that ∂/∂t is Killing and that Kgε
≡ −1.
Assume from now on that M ⊂ Qε is ALH and its inner boundary Σ lies on a horizontal totallygeodesic hypersurface, say P ' Pε. Assume further that M meets P orthogonally along Σ(which implies that Σ ⊂ M is minimal and hence a horizon).
TheoremUnder the above conditions,
m(M,g) = cn
ˆM
Θ (Rg + n(n − 1)) dM + cn
ˆΣρεHdΣ,
where H is the mean curvature of Σ ⊂ P and ρε(r) =√
r2 + ε. In particular, if Rg ≥ −n(n − 1)and M is a graph (Θ > 0) then
m(M,g) ≥ cn
ˆΣρεHdΣ.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 11 / 23
The integral formula for the mass
For any hypersurface M ⊂ Qε = R× Pε endowed with a unit normal N, an old formula byReilly says that
divM (G(A)X) = 2σ2(A)Θ,
where G(A) = σ1(A)I − A is the Newton tensor of the shape operator A, X is the tangentialcomponent of ∂/∂t and Θ = 〈N, ∂/∂t〉. This uses that ∂/∂t is Killing and that Kgε
≡ −1.
Assume from now on that M ⊂ Qε is ALH and its inner boundary Σ lies on a horizontal totallygeodesic hypersurface, say P ' Pε. Assume further that M meets P orthogonally along Σ(which implies that Σ ⊂ M is minimal and hence a horizon).
TheoremUnder the above conditions,
m(M,g) = cn
ˆM
Θ (Rg + n(n − 1)) dM + cn
ˆΣρεHdΣ,
where H is the mean curvature of Σ ⊂ P and ρε(r) =√
r2 + ε. In particular, if Rg ≥ −n(n − 1)and M is a graph (Θ > 0) then
m(M,g) ≥ cn
ˆΣρεHdΣ.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 11 / 23
The Alexandrov-Fenchel inequality
We have seen thatm(M,g) ≥ cn
ˆΣρ0,εHdΣ.
In order to proceed, we need a new Alexandrov-Fenchel inequality for a class ofhypersurfaces in (Pε, gε)!
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex (H > 0) then
cn
ˆΣρεHdΣ ≥
12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if Σ is a slice.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 12 / 23
The Alexandrov-Fenchel inequality
We have seen thatm(M,g) ≥ cn
ˆΣρ0,εHdΣ.
In order to proceed, we need a new Alexandrov-Fenchel inequality for a class ofhypersurfaces in (Pε, gε)!
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex (H > 0) then
cn
ˆΣρεHdΣ ≥
12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if Σ is a slice.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 12 / 23
The Alexandrov-Fenchel inequality
We have seen thatm(M,g) ≥ cn
ˆΣρ0,εHdΣ.
In order to proceed, we need a new Alexandrov-Fenchel inequality for a class ofhypersurfaces in (Pε, gε)!
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex (H > 0) then
cn
ˆΣρεHdΣ ≥
12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if Σ is a slice.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 12 / 23
The Alexandrov-Fenchel inequality
We have seen thatm(M,g) ≥ cn
ˆΣρ0,εHdΣ.
In order to proceed, we need a new Alexandrov-Fenchel inequality for a class ofhypersurfaces in (Pε, gε)!
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex (H > 0) then
cn
ˆΣρεHdΣ ≥
12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if Σ is a slice.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 12 / 23
The Alexandrov-Fenchel inequality
We have seen thatm(M,g) ≥ cn
ˆΣρ0,εHdΣ.
In order to proceed, we need a new Alexandrov-Fenchel inequality for a class ofhypersurfaces in (Pε, gε)!
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex (H > 0) then
cn
ˆΣρεHdΣ ≥
12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if Σ is a slice.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 12 / 23
The optimal Penrose inequality
This proves the first part of our main result.
TheoremIf M ⊂ Q0,ε is an ALH graph as above, with Σ ⊂ P = P0,ε being mean convex (H ≥ 0) andstar-shaped, then
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if (M, g) is (congruent to) the graph realization of thecorresponding black hole solution.
For ε = 1, this sharpens previous results by Dahl-Gicquaud-Sakovich.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 13 / 23
The optimal Penrose inequality
This proves the first part of our main result.
TheoremIf M ⊂ Q0,ε is an ALH graph as above, with Σ ⊂ P = P0,ε being mean convex (H ≥ 0) andstar-shaped, then
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if (M, g) is (congruent to) the graph realization of thecorresponding black hole solution.
For ε = 1, this sharpens previous results by Dahl-Gicquaud-Sakovich.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 13 / 23
The optimal Penrose inequality
This proves the first part of our main result.
TheoremIf M ⊂ Q0,ε is an ALH graph as above, with Σ ⊂ P = P0,ε being mean convex (H ≥ 0) andstar-shaped, then
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if (M, g) is (congruent to) the graph realization of thecorresponding black hole solution.
For ε = 1, this sharpens previous results by Dahl-Gicquaud-Sakovich.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 13 / 23
The optimal Penrose inequality
This proves the first part of our main result.
TheoremIf M ⊂ Q0,ε is an ALH graph as above, with Σ ⊂ P = P0,ε being mean convex (H ≥ 0) andstar-shaped, then
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if (M, g) is (congruent to) the graph realization of thecorresponding black hole solution.
For ε = 1, this sharpens previous results by Dahl-Gicquaud-Sakovich.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 13 / 23
The optimal Penrose inequality
This proves the first part of our main result.
TheoremIf M ⊂ Q0,ε is an ALH graph as above, with Σ ⊂ P = P0,ε being mean convex (H ≥ 0) andstar-shaped, then
m(M,g) ≥12
( |Σ|τn−1
) nn−1
+ ε
(|Σ|τn−1
) n−2n−1
,
with the equality holding if and only if (M, g) is (congruent to) the graph realization of thecorresponding black hole solution.
For ε = 1, this sharpens previous results by Dahl-Gicquaud-Sakovich.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 13 / 23
The proof of AF I
The proof uses the IMCF:∂X∂t
= −ξ
H,
where ξ is the inward unit normal to Σ.
It is convenient to use the parameter s satisfying ds = dr/ρε(r), which gives
s =
arcsinh r ε = 1
log r , ε = 0log(2
√r2 − 1 + 2r), ε = −1
In terms of this parameter,gε = ds2 + λε(s)2h,
where
λε(s) =
sinh s ε = 1
es, ε = 0es
4 + e−s, ε = −1
Notice that λ2ε = λ2
ε + ε.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 14 / 23
The proof of AF I
The proof uses the IMCF:∂X∂t
= −ξ
H,
where ξ is the inward unit normal to Σ.
It is convenient to use the parameter s satisfying ds = dr/ρε(r), which gives
s =
arcsinh r ε = 1
log r , ε = 0log(2
√r2 − 1 + 2r), ε = −1
In terms of this parameter,gε = ds2 + λε(s)2h,
where
λε(s) =
sinh s ε = 1
es, ε = 0es
4 + e−s, ε = −1
Notice that λ2ε = λ2
ε + ε.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 14 / 23
The proof of AF I
The proof uses the IMCF:∂X∂t
= −ξ
H,
where ξ is the inward unit normal to Σ.
It is convenient to use the parameter s satisfying ds = dr/ρε(r), which gives
s =
arcsinh r ε = 1
log r , ε = 0log(2
√r2 − 1 + 2r), ε = −1
In terms of this parameter,gε = ds2 + λε(s)2h,
where
λε(s) =
sinh s ε = 1
es, ε = 0es
4 + e−s, ε = −1
Notice that λ2ε = λ2
ε + ε.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 14 / 23
The proof of AF I
The proof uses the IMCF:∂X∂t
= −ξ
H,
where ξ is the inward unit normal to Σ.
It is convenient to use the parameter s satisfying ds = dr/ρε(r), which gives
s =
arcsinh r ε = 1
log r , ε = 0log(2
√r2 − 1 + 2r), ε = −1
In terms of this parameter,gε = ds2 + λε(s)2h,
where
λε(s) =
sinh s ε = 1
es, ε = 0es
4 + e−s, ε = −1
Notice that λ2ε = λ2
ε + ε.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 14 / 23
The proof of AF I
The proof uses the IMCF:∂X∂t
= −ξ
H,
where ξ is the inward unit normal to Σ.
It is convenient to use the parameter s satisfying ds = dr/ρε(r), which gives
s =
arcsinh r ε = 1
log r , ε = 0log(2
√r2 − 1 + 2r), ε = −1
In terms of this parameter,gε = ds2 + λε(s)2h,
where
λε(s) =
sinh s ε = 1
es, ε = 0es
4 + e−s, ε = −1
Notice that λ2ε = λ2
ε + ε.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 14 / 23
The proof of AF I
The proof uses the IMCF:∂X∂t
= −ξ
H,
where ξ is the inward unit normal to Σ.
It is convenient to use the parameter s satisfying ds = dr/ρε(r), which gives
s =
arcsinh r ε = 1
log r , ε = 0log(2
√r2 − 1 + 2r), ε = −1
In terms of this parameter,gε = ds2 + λε(s)2h,
where
λε(s) =
sinh s ε = 1
es, ε = 0es
4 + e−s, ε = −1
Notice that λ2ε = λ2
ε + ε.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 14 / 23
The proof of AF II
It is shown that if the initial hypersurface Σ0 ⊂ P0,ε is star-shaped and strictly mean convex(H > 0) then the evolving hypersurface Σt is defined for all t > 0, remains star-shaped andstrictly mean convex and expands to infinity in the sense that the principal curvaturesconverge exponentially to 1 as t → +∞.
Moreover, there exists α ∈ R so that if u = u(t , θ) is the graphing function then the rescaling
u(t , θ) = u(t , θ)−t
n − 1
converges to α in the sense that
|∇u|+ |∇2u| = o(1).
In particular,
λε(u) ∼ λε(u) ∼ et
n−1 .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 15 / 23
The proof of AF II
It is shown that if the initial hypersurface Σ0 ⊂ P0,ε is star-shaped and strictly mean convex(H > 0) then the evolving hypersurface Σt is defined for all t > 0, remains star-shaped andstrictly mean convex and expands to infinity in the sense that the principal curvaturesconverge exponentially to 1 as t → +∞.
Moreover, there exists α ∈ R so that if u = u(t , θ) is the graphing function then the rescaling
u(t , θ) = u(t , θ)−t
n − 1
converges to α in the sense that
|∇u|+ |∇2u| = o(1).
In particular,
λε(u) ∼ λε(u) ∼ et
n−1 .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 15 / 23
The proof of AF II
It is shown that if the initial hypersurface Σ0 ⊂ P0,ε is star-shaped and strictly mean convex(H > 0) then the evolving hypersurface Σt is defined for all t > 0, remains star-shaped andstrictly mean convex and expands to infinity in the sense that the principal curvaturesconverge exponentially to 1 as t → +∞.
Moreover, there exists α ∈ R so that if u = u(t , θ) is the graphing function then the rescaling
u(t , θ) = u(t , θ)−t
n − 1
converges to α in the sense that
|∇u|+ |∇2u| = o(1).
In particular,
λε(u) ∼ λε(u) ∼ et
n−1 .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 15 / 23
The proof of AF II
It is shown that if the initial hypersurface Σ0 ⊂ P0,ε is star-shaped and strictly mean convex(H > 0) then the evolving hypersurface Σt is defined for all t > 0, remains star-shaped andstrictly mean convex and expands to infinity in the sense that the principal curvaturesconverge exponentially to 1 as t → +∞.
Moreover, there exists α ∈ R so that if u = u(t , θ) is the graphing function then the rescaling
u(t , θ) = u(t , θ)−t
n − 1
converges to α in the sense that
|∇u|+ |∇2u| = o(1).
In particular,
λε(u) ∼ λε(u) ∼ et
n−1 .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 15 / 23
The proof of hyperbolic AF III
DefineJ (Σ) = −
ˆΣ
pdΣ, p = 〈Dρε, ξ〉,
andK(Σ) = τn−1A(Σ)
nn−1 , A(Σ) = A/τn−1.
These quantities appear in the following preliminary result.
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex then
J (Σ) ≤ K(Σ),
with the equality holding if and only if Σ is totally umbilical.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 16 / 23
The proof of hyperbolic AF III
DefineJ (Σ) = −
ˆΣ
pdΣ, p = 〈Dρε, ξ〉,
andK(Σ) = τn−1A(Σ)
nn−1 , A(Σ) = A/τn−1.
These quantities appear in the following preliminary result.
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex then
J (Σ) ≤ K(Σ),
with the equality holding if and only if Σ is totally umbilical.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 16 / 23
The proof of hyperbolic AF III
DefineJ (Σ) = −
ˆΣ
pdΣ, p = 〈Dρε, ξ〉,
andK(Σ) = τn−1A(Σ)
nn−1 , A(Σ) = A/τn−1.
These quantities appear in the following preliminary result.
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex then
J (Σ) ≤ K(Σ),
with the equality holding if and only if Σ is totally umbilical.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 16 / 23
The proof of hyperbolic AF III
DefineJ (Σ) = −
ˆΣ
pdΣ, p = 〈Dρε, ξ〉,
andK(Σ) = τn−1A(Σ)
nn−1 , A(Σ) = A/τn−1.
These quantities appear in the following preliminary result.
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex then
J (Σ) ≤ K(Σ),
with the equality holding if and only if Σ is totally umbilical.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 16 / 23
The proof of hyperbolic AF III
DefineJ (Σ) = −
ˆΣ
pdΣ, p = 〈Dρε, ξ〉,
andK(Σ) = τn−1A(Σ)
nn−1 , A(Σ) = A/τn−1.
These quantities appear in the following preliminary result.
TheoremIf Σ ⊂ Pε is star-shaped and strictly mean convex then
J (Σ) ≤ K(Σ),
with the equality holding if and only if Σ is totally umbilical.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 16 / 23
The proof of hyperbolic AF IV
Letting Σ flow under the IMCF, we have
dJdt
= nˆ
Σ
ρε
HdΣ
(∗)
≥n
n − 1J ,
where (∗) is a recent inequality by Brendle.
On the other hand,dAdt
= A ⇒dKdt
=n
n − 1K,
and this immediately yieldsddtJ −K
An
n−1≥ 0.
But the asymptotics gives
limt→+∞
J −K
An
n−1= 0,
and the theorem follows.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 17 / 23
The proof of hyperbolic AF IV
Letting Σ flow under the IMCF, we have
dJdt
= nˆ
Σ
ρε
HdΣ
(∗)
≥n
n − 1J ,
where (∗) is a recent inequality by Brendle.
On the other hand,dAdt
= A ⇒dKdt
=n
n − 1K,
and this immediately yieldsddtJ −K
An
n−1≥ 0.
But the asymptotics gives
limt→+∞
J −K
An
n−1= 0,
and the theorem follows.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 17 / 23
The proof of hyperbolic AF IV
Letting Σ flow under the IMCF, we have
dJdt
= nˆ
Σ
ρε
HdΣ
(∗)
≥n
n − 1J ,
where (∗) is a recent inequality by Brendle.
On the other hand,dAdt
= A ⇒dKdt
=n
n − 1K,
and this immediately yieldsddtJ −K
An
n−1≥ 0.
But the asymptotics gives
limt→+∞
J −K
An
n−1= 0,
and the theorem follows.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 17 / 23
The proof of hyperbolic AF IV
Letting Σ flow under the IMCF, we have
dJdt
= nˆ
Σ
ρε
HdΣ
(∗)
≥n
n − 1J ,
where (∗) is a recent inequality by Brendle.
On the other hand,dAdt
= A ⇒dKdt
=n
n − 1K,
and this immediately yieldsddtJ −K
An
n−1≥ 0.
But the asymptotics gives
limt→+∞
J −K
An
n−1= 0,
and the theorem follows.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 17 / 23
The proof of hyperbolic AF IV
Letting Σ flow under the IMCF, we have
dJdt
= nˆ
Σ
ρε
HdΣ
(∗)
≥n
n − 1J ,
where (∗) is a recent inequality by Brendle.
On the other hand,dAdt
= A ⇒dKdt
=n
n − 1K,
and this immediately yieldsddtJ −K
An
n−1≥ 0.
But the asymptotics gives
limt→+∞
J −K
An
n−1= 0,
and the theorem follows.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 17 / 23
The proof of hyperbolic AF V
We now considerI(Σ) =
ˆΣρεHdΣ.
If K is the extrinsic scalar curvalure of Σ, then
dIdt
= 2ˆ
Σ
ρεKH
dΣ + 2J
≤n − 2n − 1
I + 2J ,
so that the previous theorem gives
ddt
(I − (n − 1)K) ≤n − 2n − 1
(I − (n − 1)K) + 2 (J −K)
≤n − 2n − 1
(I − (n − 1)K) .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 18 / 23
The proof of hyperbolic AF V
We now considerI(Σ) =
ˆΣρεHdΣ.
If K is the extrinsic scalar curvalure of Σ, then
dIdt
= 2ˆ
Σ
ρεKH
dΣ + 2J
≤n − 2n − 1
I + 2J ,
so that the previous theorem gives
ddt
(I − (n − 1)K) ≤n − 2n − 1
(I − (n − 1)K) + 2 (J −K)
≤n − 2n − 1
(I − (n − 1)K) .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 18 / 23
The proof of hyperbolic AF V
We now considerI(Σ) =
ˆΣρεHdΣ.
If K is the extrinsic scalar curvalure of Σ, then
dIdt
= 2ˆ
Σ
ρεKH
dΣ + 2J
≤n − 2n − 1
I + 2J ,
so that the previous theorem gives
ddt
(I − (n − 1)K) ≤n − 2n − 1
(I − (n − 1)K) + 2 (J −K)
≤n − 2n − 1
(I − (n − 1)K) .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 18 / 23
The proof of hyperbolic AF V
We now considerI(Σ) =
ˆΣρεHdΣ.
If K is the extrinsic scalar curvalure of Σ, then
dIdt
= 2ˆ
Σ
ρεKH
dΣ + 2J
≤n − 2n − 1
I + 2J ,
so that the previous theorem gives
ddt
(I − (n − 1)K) ≤n − 2n − 1
(I − (n − 1)K) + 2 (J −K)
≤n − 2n − 1
(I − (n − 1)K) .
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 18 / 23
The proof of hyperbolic AF VI
The previous inequality can be rewritten as
dLdt≤ 0,
whereL(Σ) = A(Σ)
− n−2n−1 (I(Σ)− (n − 1)K(Σ)) .
But, as we shall see below, the asymptotics also gives
lim inft→+∞
L(t) ≥ (n − 1)τn−1ε,
so thatL(0) ≥ (n − 1)τn−1ε,
as desired.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 19 / 23
The proof of hyperbolic AF VI
The previous inequality can be rewritten as
dLdt≤ 0,
whereL(Σ) = A(Σ)
− n−2n−1 (I(Σ)− (n − 1)K(Σ)) .
But, as we shall see below, the asymptotics also gives
lim inft→+∞
L(t) ≥ (n − 1)τn−1ε,
so thatL(0) ≥ (n − 1)τn−1ε,
as desired.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 19 / 23
The proof of hyperbolic AF VI
The previous inequality can be rewritten as
dLdt≤ 0,
whereL(Σ) = A(Σ)
− n−2n−1 (I(Σ)− (n − 1)K(Σ)) .
But, as we shall see below, the asymptotics also gives
lim inft→+∞
L(t) ≥ (n − 1)τn−1ε,
so thatL(0) ≥ (n − 1)τn−1ε,
as desired.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 19 / 23
The proof of hyperbolic AF VI
The previous inequality can be rewritten as
dLdt≤ 0,
whereL(Σ) = A(Σ)
− n−2n−1 (I(Σ)− (n − 1)K(Σ)) .
But, as we shall see below, the asymptotics also gives
lim inft→+∞
L(t) ≥ (n − 1)τn−1ε,
so thatL(0) ≥ (n − 1)τn−1ε,
as desired.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 19 / 23
The proof of hyperbolic AF VII (the lower bound for L)
A computation using the asymptotics of the flow gives
A(Σt ) =
λn−1ε + o(e
(n−3)tn−1 ),
and ˆΣt
ρεHdΣt = (n − 1)
ˆλ2ελ
n−2ε + o(e
(n−2)t(n−1) ).
Hence, if we use the characteristic equation λ2ε = λ2
ε + ε,
lim inft→+∞
L(Σt ) = (n − 1)τn−1 lim inft→+∞
fflλ2ελ
n−2ε −
(fflλn−1ε
) nn−1
+ o(e(n−2)t
n−1 )(fflλn−1ε )
) n−2n−1
+ o(e(n−4)t
n−1 )
≥ (n − 1)τn−1ε lim inft→+∞
fflλn−2ε(ffl
λn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
+
+(n − 1)τn−1 lim inft→+∞
fflλnε −
(fflλn−1ε
) nn−1
(fflλn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
+
+(n − 1)τn−1 lim inft→+∞
o(e(n−2)t
n−1 )(fflλn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
. �
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 20 / 23
The proof of hyperbolic AF VII (the lower bound for L)A computation using the asymptotics of the flow gives
A(Σt ) =
λn−1ε + o(e
(n−3)tn−1 ),
and ˆΣt
ρεHdΣt = (n − 1)
ˆλ2ελ
n−2ε + o(e
(n−2)t(n−1) ).
Hence, if we use the characteristic equation λ2ε = λ2
ε + ε,
lim inft→+∞
L(Σt ) = (n − 1)τn−1 lim inft→+∞
fflλ2ελ
n−2ε −
(fflλn−1ε
) nn−1
+ o(e(n−2)t
n−1 )(fflλn−1ε )
) n−2n−1
+ o(e(n−4)t
n−1 )
≥ (n − 1)τn−1ε lim inft→+∞
fflλn−2ε(ffl
λn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
+
+(n − 1)τn−1 lim inft→+∞
fflλnε −
(fflλn−1ε
) nn−1
(fflλn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
+
+(n − 1)τn−1 lim inft→+∞
o(e(n−2)t
n−1 )(fflλn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
. �
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 20 / 23
The proof of hyperbolic AF VII (the lower bound for L)A computation using the asymptotics of the flow gives
A(Σt ) =
λn−1ε + o(e
(n−3)tn−1 ),
and ˆΣt
ρεHdΣt = (n − 1)
ˆλ2ελ
n−2ε + o(e
(n−2)t(n−1) ).
Hence, if we use the characteristic equation λ2ε = λ2
ε + ε,
lim inft→+∞
L(Σt ) = (n − 1)τn−1 lim inft→+∞
fflλ2ελ
n−2ε −
(fflλn−1ε
) nn−1
+ o(e(n−2)t
n−1 )(fflλn−1ε )
) n−2n−1
+ o(e(n−4)t
n−1 )
≥ (n − 1)τn−1ε lim inft→+∞
fflλn−2ε(ffl
λn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
+
+(n − 1)τn−1 lim inft→+∞
fflλnε −
(fflλn−1ε
) nn−1
(fflλn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
+
+(n − 1)τn−1 lim inft→+∞
o(e(n−2)t
n−1 )(fflλn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
. �
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 20 / 23
The proof of hyperbolic AF VII (the lower bound for L)A computation using the asymptotics of the flow gives
A(Σt ) =
λn−1ε + o(e
(n−3)tn−1 ),
and ˆΣt
ρεHdΣt = (n − 1)
ˆλ2ελ
n−2ε + o(e
(n−2)t(n−1) ).
Hence, if we use the characteristic equation λ2ε = λ2
ε + ε,
lim inft→+∞
L(Σt ) = (n − 1)τn−1 lim inft→+∞
fflλ2ελ
n−2ε −
(fflλn−1ε
) nn−1
+ o(e(n−2)t
n−1 )(fflλn−1ε )
) n−2n−1
+ o(e(n−4)t
n−1 )
≥ (n − 1)τn−1ε lim inft→+∞
fflλn−2ε(ffl
λn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
+
+(n − 1)τn−1 lim inft→+∞
fflλnε −
(fflλn−1ε
) nn−1
(fflλn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
+
+(n − 1)τn−1 lim inft→+∞
o(e(n−2)t
n−1 )(fflλn−1ε
) n−2n−1
+ o(e(n−4)t
n−1 )
. �
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 20 / 23
The proof of hyperbolic AF VI (rigidity)
The analysis here is based on recent work by Huang and Wu.
If the equality holds in Penrose then σ2(A) = 0, which implies that the graph M ⊂ Hn+1 ismean convex (σ1(A) ≥ 0).
An elementary algebraic inequality then implies that
G(A) := σ1(A)I − A ≥ 0,
which means by a classical computation that the graph M is an elliptic solution of σ2(A) = 0.
Since the black hole solution is elliptic as well, the rigidity follows by applying a suitableversion of the Maximum Principle.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 21 / 23
The proof of hyperbolic AF VI (rigidity)
The analysis here is based on recent work by Huang and Wu.
If the equality holds in Penrose then σ2(A) = 0, which implies that the graph M ⊂ Hn+1 ismean convex (σ1(A) ≥ 0).
An elementary algebraic inequality then implies that
G(A) := σ1(A)I − A ≥ 0,
which means by a classical computation that the graph M is an elliptic solution of σ2(A) = 0.
Since the black hole solution is elliptic as well, the rigidity follows by applying a suitableversion of the Maximum Principle.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 21 / 23
The proof of hyperbolic AF VI (rigidity)
The analysis here is based on recent work by Huang and Wu.
If the equality holds in Penrose then σ2(A) = 0, which implies that the graph M ⊂ Hn+1 ismean convex (σ1(A) ≥ 0).
An elementary algebraic inequality then implies that
G(A) := σ1(A)I − A ≥ 0,
which means by a classical computation that the graph M is an elliptic solution of σ2(A) = 0.
Since the black hole solution is elliptic as well, the rigidity follows by applying a suitableversion of the Maximum Principle.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 21 / 23
The proof of hyperbolic AF VI (rigidity)
The analysis here is based on recent work by Huang and Wu.
If the equality holds in Penrose then σ2(A) = 0, which implies that the graph M ⊂ Hn+1 ismean convex (σ1(A) ≥ 0).
An elementary algebraic inequality then implies that
G(A) := σ1(A)I − A ≥ 0,
which means by a classical computation that the graph M is an elliptic solution of σ2(A) = 0.
Since the black hole solution is elliptic as well, the rigidity follows by applying a suitableversion of the Maximum Principle.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 21 / 23
The proof of hyperbolic AF VI (rigidity)
The analysis here is based on recent work by Huang and Wu.
If the equality holds in Penrose then σ2(A) = 0, which implies that the graph M ⊂ Hn+1 ismean convex (σ1(A) ≥ 0).
An elementary algebraic inequality then implies that
G(A) := σ1(A)I − A ≥ 0,
which means by a classical computation that the graph M is an elliptic solution of σ2(A) = 0.
Since the black hole solution is elliptic as well, the rigidity follows by applying a suitableversion of the Maximum Principle.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 21 / 23
The proof of hyperbolic AF VI (rigidity)
The analysis here is based on recent work by Huang and Wu.
If the equality holds in Penrose then σ2(A) = 0, which implies that the graph M ⊂ Hn+1 ismean convex (σ1(A) ≥ 0).
An elementary algebraic inequality then implies that
G(A) := σ1(A)I − A ≥ 0,
which means by a classical computation that the graph M is an elliptic solution of σ2(A) = 0.
Since the black hole solution is elliptic as well, the rigidity follows by applying a suitableversion of the Maximum Principle.
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 21 / 23
Further comments
If N is a surface of genus γ ≥ 1, we obtain
m(M,g) ≥(
4πτ2
)3/2√|Σ|16π
(1− γ +
|Σ|4π
),
where we should take τ2 = 4π if γ = 1. This appears as a conjectured ineuqality in papers byGibbons and Chrusciel-Simon. Also, it is related to recent work by Lee-Neves.
Also, our AF inequality is related to recent work by Brendle-Hung-Wang, where a similarinequality is proved for hypersurfaces in adSS-space by essentially the same method.
If we take Λ→ 0, we recover the Minkowski inequality in Rn, first proved by Guan-Li:
cn
ˆΣ
HdΣ ≥12
(|Σ|ωn−1
) n−2n−1
.
There is by now a lot of activity on ‘higher order’ AF inequalities in space forms with potentialapplications to Penrose-type inequalities. See, for instance, papers by Ge-Wang-Wu, dealingwith the hyperbolic case and by Makowski-Scheuer, dealing with the spherical case andposted last tuesday in the arXiv!
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 22 / 23
Further comments
If N is a surface of genus γ ≥ 1, we obtain
m(M,g) ≥(
4πτ2
)3/2√|Σ|16π
(1− γ +
|Σ|4π
),
where we should take τ2 = 4π if γ = 1. This appears as a conjectured ineuqality in papers byGibbons and Chrusciel-Simon. Also, it is related to recent work by Lee-Neves.
Also, our AF inequality is related to recent work by Brendle-Hung-Wang, where a similarinequality is proved for hypersurfaces in adSS-space by essentially the same method.
If we take Λ→ 0, we recover the Minkowski inequality in Rn, first proved by Guan-Li:
cn
ˆΣ
HdΣ ≥12
(|Σ|ωn−1
) n−2n−1
.
There is by now a lot of activity on ‘higher order’ AF inequalities in space forms with potentialapplications to Penrose-type inequalities. See, for instance, papers by Ge-Wang-Wu, dealingwith the hyperbolic case and by Makowski-Scheuer, dealing with the spherical case andposted last tuesday in the arXiv!
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 22 / 23
Further comments
If N is a surface of genus γ ≥ 1, we obtain
m(M,g) ≥(
4πτ2
)3/2√|Σ|16π
(1− γ +
|Σ|4π
),
where we should take τ2 = 4π if γ = 1. This appears as a conjectured ineuqality in papers byGibbons and Chrusciel-Simon. Also, it is related to recent work by Lee-Neves.
Also, our AF inequality is related to recent work by Brendle-Hung-Wang, where a similarinequality is proved for hypersurfaces in adSS-space by essentially the same method.
If we take Λ→ 0, we recover the Minkowski inequality in Rn, first proved by Guan-Li:
cn
ˆΣ
HdΣ ≥12
(|Σ|ωn−1
) n−2n−1
.
There is by now a lot of activity on ‘higher order’ AF inequalities in space forms with potentialapplications to Penrose-type inequalities. See, for instance, papers by Ge-Wang-Wu, dealingwith the hyperbolic case and by Makowski-Scheuer, dealing with the spherical case andposted last tuesday in the arXiv!
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 22 / 23
Further comments
If N is a surface of genus γ ≥ 1, we obtain
m(M,g) ≥(
4πτ2
)3/2√|Σ|16π
(1− γ +
|Σ|4π
),
where we should take τ2 = 4π if γ = 1. This appears as a conjectured ineuqality in papers byGibbons and Chrusciel-Simon. Also, it is related to recent work by Lee-Neves.
Also, our AF inequality is related to recent work by Brendle-Hung-Wang, where a similarinequality is proved for hypersurfaces in adSS-space by essentially the same method.
If we take Λ→ 0, we recover the Minkowski inequality in Rn, first proved by Guan-Li:
cn
ˆΣ
HdΣ ≥12
(|Σ|ωn−1
) n−2n−1
.
There is by now a lot of activity on ‘higher order’ AF inequalities in space forms with potentialapplications to Penrose-type inequalities. See, for instance, papers by Ge-Wang-Wu, dealingwith the hyperbolic case and by Makowski-Scheuer, dealing with the spherical case andposted last tuesday in the arXiv!
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 22 / 23
Further comments
If N is a surface of genus γ ≥ 1, we obtain
m(M,g) ≥(
4πτ2
)3/2√|Σ|16π
(1− γ +
|Σ|4π
),
where we should take τ2 = 4π if γ = 1. This appears as a conjectured ineuqality in papers byGibbons and Chrusciel-Simon. Also, it is related to recent work by Lee-Neves.
Also, our AF inequality is related to recent work by Brendle-Hung-Wang, where a similarinequality is proved for hypersurfaces in adSS-space by essentially the same method.
If we take Λ→ 0, we recover the Minkowski inequality in Rn, first proved by Guan-Li:
cn
ˆΣ
HdΣ ≥12
(|Σ|ωn−1
) n−2n−1
.
There is by now a lot of activity on ‘higher order’ AF inequalities in space forms with potentialapplications to Penrose-type inequalities. See, for instance, papers by Ge-Wang-Wu, dealingwith the hyperbolic case and by Makowski-Scheuer, dealing with the spherical case andposted last tuesday in the arXiv!
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 22 / 23
Further comments
If N is a surface of genus γ ≥ 1, we obtain
m(M,g) ≥(
4πτ2
)3/2√|Σ|16π
(1− γ +
|Σ|4π
),
where we should take τ2 = 4π if γ = 1. This appears as a conjectured ineuqality in papers byGibbons and Chrusciel-Simon. Also, it is related to recent work by Lee-Neves.
Also, our AF inequality is related to recent work by Brendle-Hung-Wang, where a similarinequality is proved for hypersurfaces in adSS-space by essentially the same method.
If we take Λ→ 0, we recover the Minkowski inequality in Rn, first proved by Guan-Li:
cn
ˆΣ
HdΣ ≥12
(|Σ|ωn−1
) n−2n−1
.
There is by now a lot of activity on ‘higher order’ AF inequalities in space forms with potentialapplications to Penrose-type inequalities. See, for instance, papers by Ge-Wang-Wu, dealingwith the hyperbolic case and by Makowski-Scheuer, dealing with the spherical case andposted last tuesday in the arXiv!
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 22 / 23
THANKS FOR YOUR ATTENTION!!!
Levi Lopes de Lima (DM–UFC) A Penrose inequality São Paulo, July 2013 23 / 23