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     

A Positive Mass Theorem on Asymptotically Hyperbolic Manifolds with Corners along a Hypersurface

In this paper we take an approach similar to that in [13] to establish a positive mass theorem for spin asymptotically hyperbolic manifolds admitting corners along a hypersurface. The main analysis uses an integral representation of a solution to a perturbed eigenfunction equation to obtain an asymptotic expansion of the solution in the right order. This allows us to understand the change of the mass aspect of a conformal change of asymptotically hyperbolic metrics. Vincent Bonini: The first named author supported by MSRI Postdoctoral Fellowship. Jie Qing: The second named author supported partially by NSF grant DMS 0402294. Submitted: April 6, 2007. Accepted: September 24, 2007.     

Rigidity of Geometrically Finite Hyperbolic Cone-Manifolds

In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three-dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically finite hyperbolic 3-manifolds.     

Scalar Curvature Rigidity for Asymptotically Locally Hyperbolic Manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the four dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.     

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.     

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     

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.     

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.     

Strong Jordan Separation and Applications to Rigidity

We prove that simple, thick hyperbolic P-manifolds of dimensionat least three exhibit Mostow rigidity. We also prove a quasi-isometryrigidity result for the fundamental groups of simple, thickhyperbolic P-manifolds of dimension at least three. The keytool in the proof of these rigidity results is a strong formof the Jordan separation theorem, for maps from Sⁿ Sⁿ⁺¹ whichare not necessarily injective.     

On the stability of Riemannian manifold with parallel spinors

Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU(m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results. Dedicated to Jeff Cheeger for his sixtieth birthday     

Scalar curvature rigidity and Ricci DeTurck flow on perturbations of Euclidean space

We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on \(\mathbb {R}^n\), which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, we extend known \(L^p\) bounds and decay rates for Ricci DeTurck flow and prove regularity of the flow at the initial data.     

A Characterization of Homogeneous Spaces with Positive Hyperbolic Rank

Here we present a necessary and sufficient condition for a negatively curved homogeneous space to be a rank one symmetric space. In particular, we show rigidity if such a space has positive hyperbolic rank greater than equal to that of its 'Abelian direction'. The notion of hyperbolic-rank extends the notion of rank in negatively curved spaces. The central theorem is an analogue of a result by U. Hamenstädt for compact negatively curved manifolds. We also provide an example of a nonsymmetric hyperbolic rank two homogeneous space, demonstrating the sharpness of the theorem.     

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.     

Conformal positive mass theorems for asymptotically flat manifolds with inner boundary

Inspired by Witten's insightful spinor proof of the positive mass theorem, in this paper, we use the spinor method to derive higher dimensional type conformal positive mass theorems on asymptotically flat spin manifolds with inner boundary, which states that under a condition about the plus (minus) relation between the scalar curvatures of the original and the conformal metrics in addition with some boundary condition, we will get the associated positivity of their ADM masses. The rigidity part of the plus part is used in the proof of black hole uniqueness theorems. They are related with quasi-local mass and the spectrum of Dirac operator.     

On cohomology of ACH Kähler manifolds

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is a sharp vanishing theorem for the second cohomology of such manifolds under certain assumptions. The borderline case characterizes a Kähler-Einstein manifold constructed by Calabi.

     

Rigidity of fiber bunched cocycles

We prove a rigidity theorem for fiber bunched matrix-valued Hölder cocycles over hyperbolic homeomorphisms. More precisely, we show that two such cocycles are cohomologous if and only if they have conjugated periodic data.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

Geometrically finite groups,Patterson-Sullivan measures and Ratner's ridigity theorem

Summary The rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated. Ratner's theorem generalizes to these foliations with respect to the Patterson-Sullivan measure. In the spirit of Mostow, we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups. Finally, we show that the frame flow on geometrically finite hyperbolic manifolds is Bernoulli.partially supported by NSF Grant No. DMS-820-04024partially supported by NSF Grant No. DMS-85-02319     

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.