Scalar curvature rigidity of hyperbolic product manifolds

This paper presents a scalar curvature rigidity result of real hyperbolic product manifolds in analogy to M. Min–Oos result in [14]. In order to prove this, we consider Dirac bundles obtained from the spinor bundle, and we derive Killing equations trivializing these Dirac bundles. Moreover, an integrated Bochner–Weitzenböck formula is shown which allows the usage of the non–compact Bochner technique.Mathematics Subject Classification (2000): Primary 53C24, Sec. 53C21in final form: 11 August 2003     

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.     

Scattering poles for asymptotically hyperbolic manifolds

For a class of manifolds that includes quotients of real hyperbolic -dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for . In order to carry out the proof, we use Shmuel Agmon's perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations.

     

Generalized scattering phases for asymptotically hyperbolic manifolds

We prove asymptotic expansions of generalized scattering phases asssociated to pairs of Laplacians, for a class of noncompact manifolds with infinite volume and negative curvature near infinity. We use one of these expansions to define relative determinants which appear naturally in this context. To cite this article: J.-M. Bouclet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Fractal Weyl laws for asymptotically hyperbolic manifolds

For asymptotically hyperbolic manifolds with hyperbolic trapped sets we prove a fractal upper bound on the number of resonances near the essential spectrum, with power determined by the dimension of the trapped set. This covers the case of general convex cocompact quotients (including the case of connected trapped sets) where our result implies a bound on the number of zeros of the Selberg zeta function in disks of arbitrary size along the imaginary axis. Although no sharp fractal lower bounds are known, the case of quasifuchsian groups, included here, is most likely to provide them.     

On asymptotically harmonic manifolds of negative curvature

We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of visual and harmonic measures on the ideal boundary. Then, we prove an asymptotic analogue of the classical mean value property of harmonic manifolds, and we characterize asymptotically harmonic manifolds, among Cartan–Hadamard spaces of strictly negative curvature, by the existence of an asymptotic equivalent \(\tau (u)\mathrm {e}^{Er}\) for the volume-density of geodesic spheres (with \(\tau \) constant in case \(DR_M\) is bounded). Finally, we show the existence of a Margulis function, and explicitly compute it, for all asymptotically harmonic manifolds.     

Conformal Ricci flow on asymptotically hyperbolic manifolds

In this article, we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.     

Islands at infinity on manifolds of asymptotically nonnegative curvature

We introduce an invariant which measures the R-eccentricity of a point in a complete Riemannian manifold M and show that it goes to zero when the point goes to infinity, if M has asymptotically nonnegative curvature. As a consequence we show that the isometry group is compact if M has asymptotically nonnegative curvature and a point with positive sectional curvature. Both authors were partially supported by CNPq of Brazil and the second author was also partially supported by FAPERJ of Brazil.     

The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates)

The Hamiltonian formulation of the Einstein equations is achieved by means of a foliation of the background Lorentz Manifold. The usage of maximal surfaces is the frequently applied gauge for numerical research of asymptotically flat manifolds. In this paper we construct a foliation of asymptotically hyperbolic 3-surfaces through 2-surfaces (with constant mean curvature) homeomorphic to spheres. This is established by using the volume preserving mean curvature flow. These spheres define a geometric intrinsic radius coordinate near infinity and therefore define a center of mass for the Bondi case.This paper was founded by the Deutschen Foschungsgemeinschaft, Sonderforschungsbereich 382 of the Universities Tübingen and Stuttgart.     

Complete manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry

In this paper, we study complete Riemannian n-manifolds (n ≥ 3) with asymptotically nonnegative Ricci curvature and weak bounded geometry. We show among other things that the total Betti number of such a manifold has polynomial growth of degree n ² + n. Further more, such a manifold is of finite topological type if the volume growth rate of the metric ball around the base point is less than This work is partly supported by the National Natural Science Foundation (10371047) of China. Received: 13 June 2006     

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.     

Hermitian metrics of negative holomorphic sectional curvature on some hyperbolic manifolds

Research partially supported by National Science Foundation