Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action. Submitted: March 16, 2007. Accepted: June 14, 2007.     

Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3

By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C~2-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.     

Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality

This is a survey about our recent works on the Gauss-Bonnet-Chern (GBC) mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically flat and for asymptotically hyperbolic manifolds, respectively, by using a higher order scalar curvature. Then we prove its positivity and the Penrose inequality for graphical manifolds. One of the crucial steps in the proof of the Penrose inequality is the use of an Alexandrov-Fenchel inequality, which is a classical inequality in the Euclidean space. In the hyperbolic space, we have established this new Alexandrov-Fenchel inequality. We also have a similar work for asymptotically locally hyperbolic manifolds. At the end, we discuss the relation between the GBC mass and Chern’s magic form.     

New features of soliton dynamics in 2+1 dimensions

Exponentially localized soliton solutions have been found recently for all the equations of the hierarchy related to the Zakharov-Shabat hyperbolic spectral problem in the plane. In particular theN ²-soliton solution of the Davey-Stewartson I equation is considered. It is shown that the boundaries fix the kinematics of solitons, while the dynamics of their mutual interaction is determined by the chosen initial condition. The interacting solitons can have, asymptotically, zero mass and can simulate quantum effects as inelastic scattering, fusion and fission, creation and annihilation.Work supported in part by M.U.R.S.T.     

On Discretization Schemes for Stochastic Evolution Equations

Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved. Mathematics Subject Classifications (2000) Primary: 60H15; Secondary: 65M60.István Gyöngy: This paper was written while the first named author was visiting the University of Paris X. The research of this author is partially supported by EU Network HARP.Annie Millet: The research of the second named author is partially supported by the research project BMF2003-01345.     

Penrose Type Inequalities for Asymptotically Hyperbolic Graphs

In this paper, we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space ${\mathbb{H}^n}$ . The graphs are considered as unbounded hypersurfaces of ${\mathbb{H}^{n+1}}$ which carry the induced metric and have an interior boundary. 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 the 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 resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam’s article (The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions. http://arxiv.org/abs/1010.4256, 2010) concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu (The equality case of the penrose inequality for asymptotically flat graphs. http://arxiv.org/abs/1205.2061, 2012), we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric.     

Positive Energy-Momentum Theorem for AdS-Asymptotically Hyperbolic Manifolds

The aim of this paper is to prove a positive energy-momentum theorem under the (well known in general relativity) dominant energy condition, for AdS-asymptotically hyperbolic manifolds. These manifolds are by definition endowed with a Riemannian metric and a symmetric 2-tensor which respectively tend to the metric and second fundamental form of a standard hyperbolic slice in Anti-de Sitter space-time. There exists a positive mass theorem for asymptotically hyperbolic spin Riemannian manifolds (with zero extrinsic curvature), and we present an extension of this result for the non zero extrinsic curvature case. Communicated by Sergiu Klainerman Submitted: January 15, 2006 Accepted: January 15, 2006     

On the Uniqueness of AdS Space-Time in Higher Dimensions

In this paper, based on an intrinsic definition of asymptotically AdS space-times, we show that the standard anti-de Sitter space-time is the unique strictly stationary asymptotically AdS solution to the vacuum Einstein equations with negative cosmological constant in dimension 8. Instead of using the positive energy theorem for asymptotically hyperbolic spaces with spin our approach appeals to the classic positive mass theorem for asymptotically flat spaces. Communicated by Piotr T. ChruscielSubmitted 17/10/03, accepted 07/11/03     

Computing Asymptotic Invariants with the Ricci Tensor on Asymptotically Flat and Asymptotically Hyperbolic Manifolds

We prove in a simple and coordinate-free way the equivalence between the classical definitions of the mass or of the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.     

Invariant curves for variable step size integrators

The behaviour of one-step methods with variable step size applied to is investigated. Usually the step size for the current step depends on one or several previous steps. However, under some natural assumptions it can be shown that the step size asymptotically depends only on the locationx. This allows to introduce anx-dependent time transformation taking a variable step size method to a constant step-size method. By means of such a transformation general properties of constant step size methods carry over to variable step size methods. This is used to show that if the differential equation admits a hyperbolic periodic solution the variable step size method admits an invariant closed curve near the orbit of the periodic solution.The first author was partially supported by NSF Grant DMS87-19952 during his stay at UCLA.     

Ricci flow of conformally compact metrics

In this paper we prove that given a smoothly conformally compact asymptotically hyperbolic metric there is a short-time solution to the Ricci flow that remains smoothly conformally compact and asymptotically hyperbolic. We adapt recent results of Schnürer, Schulze and Simon to prove a stability result for conformally compact Einstein metrics sufficiently close to the hyperbolic metric.     

Quasi-Local Mass Integrals and the Total Mass

On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown–York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.     

On the Center of Mass of Asymptotically Hyperbolic Initial Data Sets

We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minkowskian spacetimes (isolated systems). In particular, we unite—an altered version of—the approach based on Hamiltonian charges with an approach based on CMC-foliations near infinity. The newly defined center of mass transforms appropriately under changes of the asymptotic coordinates and evolves in the direction of an appropriately defined linear momentum under the Einstein evolution equations.     

Dehn filling in relatively hyperbolic groups

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2π Theorem in the context of relatively hyperbolic groups. The first author was supported in part by NSF Grant DMS-0504251. The second author was supported in part by an NSF Mathematical Sciences Post-doctoral Research Fellowship. Both authors thank the NSF for their support. Most of this work was done while both authors were Taussky-Todd Fellows at Caltech.     

A System of Differential-Operator Equations of Different Orders in Hilbert Spaces

We consider an abstract Cauchy problem for a system of nonhomogeneous abstract differential equations in Hilbert spaces. The “main” equation is of the second order and “boundary” equations are of the first order. Existence of a solution is proved. Application to mixed (initial boundary-value) problems for one-dimensional second order hyperbolic equations and for fourth order PDEs with the time derivative in boundary conditions has been shown. The first author was partially supported by 60% funds of the University of Bologna and G.N.A.M.P.A. of INdAM; the second author was supported by the Israel Ministry of Absorption.     

Refined gluing for vacuum Einstein constraint equations

We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in every case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus.     

Constant angle surfaces in ℍ<Superscript>2</Superscript> × ℝ

We classify all surfaces in ℍ² × ℝ for which the unit normal makes a constant angle with the ℝ-direction. Here ℍ² is the hyperbolic plane. The author was supported by grants CEEX ET 5883/2006-2008 and PNII ID_ 398/2007-2010 ANCS (Romania).     

Structure and rigidity in hyperbolic groups I

We introduce certain classes of hyperbolic groups according to their possible actions on real trees. Using these classes and results from the theory of (small) group actions on real trees, we study the structure of hyperbolic groups and their automorphism group.The second author was partially supported by an NSF grant.     

Asymptotic expansions for hyperbolic–parabolic systems with a small parameter

In this paper we consider initial-boundary value problems for systems with a small parameter ?. The problems are mixed hyperbolic–parabolic when ? > 0 and hyperbolic when ? = 0. Often the solution can be expanded asymptotically in ? and to first approximation it consists of the solution of the corresponding hyperbolic problem and a boundary layer part. We prove sufficient conditions for the expansion to exist and give estimates of the remainder. We also examine how the boundary conditions should be choosen to avoid O(1) boundary layers.     

Aperiodic tilings of the hyperbolic plane by convex polygons

Several aperiodic hyperbolic tiling systems consisting of a single convex tile are constructed. Research partially supported by GIF grant no. G-454.213.06.95 and SFB 343 Bielefeld. The work of the first author was supported in part by NSF grant DMS-9424613. The work of the second author was supported in part by a grant of the Israel Academy of Sciences, and by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Federal Republic of Germany).