In this paper, we consider conformal metrics on a unit 4-disc with an asymptotically hyperbolic end and possible isolated conic singularities. We define a mass term of the AH end. If the $σ_{2}$ curvature has lower bound $σ_{2}\geq\frac{3}{2}$, we prove a Penrose type inequality relating the mass and contributions from singularities. We also classify sharp cases, which is the standard hyperbolic 4-space $\mathbb{H}^{4}$ when no singularity occurs. It is worth noting that our curvature condition implies non-positive energy density.
We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we deve…
We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for a class of hypersurfaces in certain locally hyperbolic manifolds. As an application we derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension $n\geq 3$. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chruściel and Simon on the validity of a Penrose-type inequality for black hole solutions carrying a higher genus horizon.
In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over an inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resultin…
We establish versions of the Positive Mass and Penrose inequalities for a class of asymptotically hyperbolic hypersurfaces. In particular, under the usual dominant energy condition, we prove in all dimensions $n\geq 3$ an optimal Penrose inequality for certain graphs in hyperbolic space $\mathbb H^{n+1}$ whose boundary has constant mean curvature $n-1$.
In this paper we prove a rigidity result for the equality case of the Penrose inequality on $3$-dimensional asymptotically flat manifolds with nonnegative scalar curvature and corners. Our result also has deep connections with the equality cases of Theorem 1 in \cite{Miao2} and Theorem 1.1 in \cite{LM}.
We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data.
In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes from a previous result of P. Chruściel and W. Simon, which states that the Penrose inequality we prove implies a static uniqueness theorem for negative mass Kottler metrics.
In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation scheme for the weak free boundary inverse mean curvature flow, introduced by Marquardt in 2012, and establish the monotonicity of a free boundary version of the Hawking mass. Our result also implies a non-optimal Penrose inequality for asymptotically flat support surfaces in $\mathb…
We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically hyperbolic manifold $M^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically hyperbolic manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and if $\partial U_T^i = Σ_0^i \cup Σ_T^i$ and $m_H(Σ_T^i) \rightarrow 0$ then $U_T^i$ converges to a topological annulus portion of hyperbolic space with…
On an asymptotically flat manifold $M^n$ with nonnegative scalar curvature, with outer minimizing boundary $Σ$, we prove a Penrose-like inequality in dimensions $ n < 8$, under suitable assumptions on the mean curvature and the scalar curvature of $ Σ$.
We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi--Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on $S^2$.
Building on previous works of H. L. Bray, of P. Miao, and of S. Almaraz, E. Barbosa, and L. L. de Lima, we develop a doubling procedure for asymptotically flat half-spaces $(M,g)$ with horizon boundary $Σ\subset M$ and mass $m\in\mathbb{R}$. If $3\leq \dim(M)\leq 7$, $(M,g)$ has non-negative scalar curvature, and the boundary $\partial M$ is mean-convex, we obtain the Riemannian Penrose-type inequality $$ m\geq\left(\frac{1}{2}\right)^{\frac{n}{n-1}}\,\left(\frac{|Σ|}{ω_{n-1}}\right)^{\frac{n-2}{n-1}} $$ as a corollary. Moreover, in the case where $\partial M$ is not totally geodesic, we show …
We prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\geq 3$. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter-Schwarzschild solution. This sharpens previous results by Dahl-Gicquaud-Sakovich and settles, for this class of initial data sets, the conjectured Penrose i…
The positive mass theorem is one of the fundamental results in general relativity. It states that, assuming the dominant energy condition, the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the cosmic censorship conjecture in general relativity. In [14], Lu and Miao proved a quasi-local Penrose inequality for the quasi-local energy with reference in the Schwarzschild manifold. In this article, we prove a quasi-local Penrose inequality for the quasi-local energy with re…
We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to $\R^{n}\setminus Ω, n\ge 3$, and so that their boundary is a minimal hypersurface. (Here, $Ω\subset \R^{n}$ is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by $(V/β_{n})^{(n-2)/n}$, where $V$ is the Euclidean volume of $Ω$ and $β_{n}$ is the volume of the Euclidean unit $n$-ball. This gives a partial proof to a conjecture of Bray and Iga \cite{brayiga}. Surprisingly, we do not require the boundary to be outermost.
In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Kottler space.
In this note we prove a global rigidity result for asymptotically flat, scalar flat Euclidean hypersurfaces with a minimal horizon lying in a hyperplane, under a natural ellipticity condition. As a consequence we obtain, in the context of the Riemannian Penrose conjecture, a local rigidity result for the family of exterior Schwarzschild solutions (viewed as graphs in Euclidean space).
We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfies the inequality: m - 1/2(R + Q^2/R) < 0, where m is the total mass, R=sqrt(A/4) is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality to charged black holes.
We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus proving the Riemannian positive mass theorem in this case. If the graph has convex horizons, we also prove the Riemannian Penrose inequality by giving a lower bound to the boundary integrals using the Aleksandrov-Fenchel inequality.
The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding Penrose type inequality for this mass and weighted Alexandrov-Fenchel inequalities in the hyperbolic space $\H^n$. In the second part, we esta…
We prove a version the Penrose inequality for black hole space-times which are perturbations of the Schwarzschild exterior in a slab around a null hypersurface $\underline{\mathcal{N}}_0$. $\underline{\mathcal{N}}_0$ terminates at past null infinity $\mathcal{I}^-$ and $\mathcal{S}_0:=\partial\underline{\mathcal{N}}_0$ is chosen to be a marginally outer trapped sphere. We show that the area of $\mathcal{S}_0$ yields a lower bound for the Bondi energy of sections of past null infinity, thus also for the total ADM energy. Our argument is perturbative, and rests on suitably deforming the initial …
In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Reissner-Nordström-anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds in which can be realized as graphs over Reissner-Nordström-anti-deSitter manifold.
The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, H. Bray extended this result to the general case of mul…
We present a proof of the Riemannian Penrose inequality with charge in the context of asymptotically flat initial data sets for the Einstein-Maxwell equations, having possibly multiple black holes with no charged matter outside the horizon, and satisfying the relevant dominant energy condition. The proof is based on a generalization of Hubert Bray's conformal flow of metrics adapted to this setting.
We present a proof of the Riemannian Penrose inequality with charge $r\leq m + \sqrt{m^2-q^2}$, where $A=4πr^2$ is the area of the outermost apparent horizon with possibly multiple connected components, $m$ is the total ADM mass, and $q$ the total charge of a strongly asymptotically flat initial data set for the Einstein-Maxwell equations, satisfying the charged dominant energy condition, with no charged matter outside the horizon.
The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic a…
In 2015, Mantoulidis and Schoen constructed $3$-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be ``far away'' from being round. The resulting manifolds, called extensions, are geometrically not ``close'' to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality. Their construction was later adapted to $n+1$ dimensions by Cabrera Pacheco and Miao. In …
We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominat energy condition. The appropriate existence and regularity results are established in the special case of spherically symmetric Cauchy data, and are applied to give a new proof of the general Penrose Inequality for these data sets. When appropriately coupled with an inverse mean curvature flow, analogous existence and regularity results for the associated system of equations in the…
In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay order $τ$ satisfies $τ> \frac {n-4}{3}.$ Then we show a positive mass theorem for asymptotically flat graphs over ${\mathbb R}^n$. Moreover we obtain also Penrose type inequalities in this case.
The Riemannian Penrose inequality is a remarkable geometric inequality between the ADM mass of an asymptotically flat manifold with non-negative scalar curvature and the area of its outermost minimal surface. A version of the Riemannian Penrose inequality has also been established for the Einstein-Maxwell equations, where the lower bound on the mass also depends on the electric charge. In the context of quasi-local mass, one is interested in determining if, and for which quasi-local mass definitions, a quasi-local version of these inequalities also holds. It is known that the Brown-York quasi-…
This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the …
For asymptotically flat initial data of Einstein's equations satisfying an energy condition, we show that the Penrose inequality holds between the ADM mass and the area of an outermost apparent horizon, if the data are restricted suitably. We prove this by generalizing Geroch's proof of monotonicity of the Hawking mass under a smooth inverse mean curvature flow, for data with non-negative Ricci scalar. Unlike Geroch we need not confine ourselves to minimal surfaces as horizons. Modulo smoothness issues we also show that our restrictions on the data can locally be fulfilled by a suitable choice…
We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically flat manifold $M^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically flat manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and if $\partial U_T^i = Σ_0^i \cup Σ_T^i$ and $m_H(Σ_T^i) \rightarrow 0$ then $U_T^i$ converges to a flat annulus with respect to $L^2$ metric convergence. If instea…
We formulate and prove the stability statement associated with the spacetime Penrose inequality for $n$-dimensional spherically symmetric, asymptotically flat initial data satisfying the dominant energy condition. We assume that the ADM mass is close to the half area radius of the outermost apparent horizon and, following the generalized Jang equation approach, show that the initial data must arise from an isometric embedding into a static spacetime close to to the exterior region of a Schwarzschild spacetime in the following sense. Namely, the time slice is close to the Schwarzschild time sli…
Based on scale critical initial data, we construct smooth asymptotically flat Cauchy initial data for the Einstein vacuum system that does not contain Marginally Outer Trapped Surfaces (MOTS) but whose future evolution contains a trapped region, which itself is bounded by an apparent horizon (a smooth hypersurface foliated by MOTS). Although the long time behaviour of these solutions is unknown, a statement of Kerr Stability would yield a dynamical, scale critical, non-spherically symmetric class of vacuum examples for the conjectures of Weak Cosmic Censorship and Final State. Owing to estimat…
We note an area-charge inequality orignially due to Gibbons: if the outermost horizon $S$ in an asymptotically flat electrovacuum initial data set is connected then $|q|\leq r$, where $q$ is the total charge and $r=\sqrt{A/4π}$ is the area radius of $S$. A consequence of this inequality is that for connected black holes the following lower bound on the area holds: $r\geq m-\sqrt{m^2-q^2}$. In conjunction with the upper bound $r\leq m + \sqrt{m^2-q^2}$ which is expected to hold always, this implies the natural generalization of the Riemannian Penrose inequality: $m\geq 1/2(r+q^2/r)$.
The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant. Following the ideas of Mantoulidis and Schoen [20…
We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a space-time in the context of general relativity, then outermost minimal spheres with total area $A$ correspond to apparent horizons of black holes contributing a mass $\sqrt{A/16π}$, scalar curvature correspo…
In this paper we show positive mass theorems and Penrose type inequalities for the Gauss-Bonnet-Chern mass, which was introduced recently in \cite{GWW}, for asymptotically flat CF manifolds and its rigidity.
We give an explicit formula for the Gauss-Bonnet-Chern mass of an asymptotically flat graphical manifold of arbitrary codimension and use it to prove the positive mass theorem and the Penrose inequality for graphs with flat normal bundle.
We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically hyperbolic Penrose inequality. Our results require that the scalar curvature of the metric satisfies $R_{g}\geq -6$, and we construct an example of a compact perturbation of Schwarzschild-anti-deSitter without $R_{g}\geq -6$ so that large centered coordinate spheres are not isope…
In a paper \cite{P} in 1973, R. Penrose made a physical argument that the total mass of a spacetime which contains black holes with event horizons of total area $A$ should be at least $\sqrt{A/16π}$. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemannian geometry known as the Riemannian Penrose inequality. One particularly geometric aspect of this problem is the fact that apparent horizons of black holes in this setting correspond to minimal surfaces in Riemannian 3-manifolds. The Riemannian Penrose inequality was first prove…
