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     

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.     

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.     

Resolvent Estimates for the Laplacian on Asymptotically Hyperbolic Manifolds

Combining results of Cardoso-Vodev [6] and Froese-Hislop [9], we use Mourre’s theory to prove high energy estimates for the boundary values of the weighted resolvent of the Laplacian on an asymptotically hyperbolic manifold. We derive estimates involving a class of pseudo-differential weights which are more natural in the asymptotically hyperbolic geometry than the weights used in [6]. submitted 28/04/05, accepted 26/09/05     

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.     

Sectional Curvature Rigidity of Asymptotically Locally Hyperbolic Manifolds

This paper presents a rigidity result of real hyperbolic quotients inanalogy to Min-Oo's result in Math. Ann. 285(4) (1989),527–539, but without the spin condition. In order to prove this,we use special Killing forms on the exterior form bundle. Moreover, wemake an assumption on the sectional curvature to obtain the necessaryeigenvalue estimates of the curvature endomorphism in theBochner–Weitzenböck formula of ^k M.     

Conformal CR Positive Mass Theorem

In this paper, we prove the following version of conformal CR positive mass theorem: Suppose that \((N, J,\theta )\) and \((N, J,\hat{\theta }=e^{2f}\theta )\) are three-dimensional asymptotically flat pseudohermitian manifolds such that their Tanaka-Webster curvatures satisfy \(e^{2f}\hat{R}-R\ge 0.\) Then the p-mass of \((N, J, \theta )\) and \((N, J, \hat{\theta })\) satisfy \( m(J, \hat{\theta })-m(J, \theta )\ge 0, \) and equality holds if and only if \(\hat{\theta }=\theta \). We also prove that the p-mass is independent of the choice of the sequence of coordinates spheres.     

The Index of Cusp Operators on Manifolds with Corners

We recall a Fredholm criterion for fully elliptic cusp(pseudo)differential operators on a compact manifold with corners ofarbitrary codimension, acting on suitable Sobolev spaces. Then we give aformula for the index in terms of regularized `trace' functionalssimilar to the residue trace of Wodzicki and Guillemin.     

A Frobenius Theorem on Convenient Manifolds

 A Frobenius Theorem for finite dimensional, involutive subbundles of the tangent bundle of a convenient manifold is proved. As first key applications Lie’s second fundamental theorem and Nelson’s theorem are treated in the convenient case. (Received 2 February 2001; in revised form 29 May 2001)     

A Frobenius Theorem on Convenient Manifolds

 A Frobenius Theorem for finite dimensional, involutive subbundles of the tangent bundle of a convenient manifold is proved. As first key applications Lie’s second fundamental theorem and Nelson’s theorem are treated in the convenient case.     

Radiation Fields on Asymptotically Euclidean Manifolds

Abstract

F.G. Friedlander introduced the notion of radiation fields for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. We also obtain a support theorem for the radiation fields.     

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.     

Counting Hyperbolic Manifolds with Bounded Diameter

Let ρ _n (V) be the number of complete hyperbolic manifolds of dimension n with volume less than V . Burger et al [Geom. Funct. Anal. 12(6) (2002), 1161–1173.] showed that when n ≥ 4 there exist a, b > 0 depending on the dimension such that aV log V ≤ log ρ _n (V) ≤ bV log V, for V ≫ 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number grows double-exponentially with volume. Additionally, this bound holds in dimension 3.     

Fundamental Theorems of Morse Theory for Optimization on Manifolds with Corners

This note extends the fundamental theorems of Morse theory for stable stationary solutions to optimization problems on manifolds with corners.     

A Slice Theorem for Open Manifolds

Let (M ⁿ , g ₀) be open with bounded geometry, K g ₀ – c ₀, c > 0, inf _e (₀(g ₀)) > 0 and denote by comp ^r (g ₀)K _s< 0 comp ^r (g ₀) the submanifold of metrics with strictly negative curvature, comp ^r g(₀) = component of g ₀ in the rth Sobolev completion of the space of metrics with bounded geometry. Denote by the unit component of the completed diffeomorphism group. Then acts properly on comp ^r (g ₀) _{K s < 0} and admits slices. We apply this to Teichmüller theory for open surfaces.     

A Groshev Type Theorem for Convergence on Manifolds

We deal with Diophantine approximation on the so-called non-degenerate manifolds and prove an analogue of the Khintchine–Groshev theorem. The problem we consider was first posed by A. Baker [1] for the rational normal curve. The non-degenerate manifolds form a large class including any connected analytic manifold which is not contained in a hyperplane. We also present a new approach which develops the ideas of Sprindzuk"s classical method of essential and inessential domains first used by him to solve Mahler"s problem [28].     

Commensurability of Hyperbolic Manifolds with Geodesic Boundary

Suppose , let M ₁, M ₂ be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π₁ (M ₁) is quasi-isometric to π₁ (M ₂) (with respect to the word metric). Also suppose that if n=3, then ∂M ₁ and ∂M ₂ are compact. We show that M ₁ is commensurable with M ₂. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M ₁ above and G is a finitely generated group which is quasi-isometric to π₁ (M), then there exists a hyperbolic manifold with geodesic boundary M′ with the following properties: M′ is commensurable with M, and G is a finite extension of a group which contains π₁ (M′) as a finite-index subgroupMathematics Subject Classification (2000). Primary: 20F65; secondary: 30C65, 57N16     

A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

A manifold with fibered cusp metrics X can be considered as a geometrical generalization of locally symmetric spaces of \mathbbQ{\mathbb{Q}}-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology H ^p (X). Similar to the situation of locally symmetric spaces, these representatives are computed by special values or residues of generalized eigenforms of the Hodge–Laplace operator on Ω ^p (X).     

Submanifolds in the Context of Infinite Dimensional Manifolds with Corners

In this paper we introduce new definitions of submanifold and immersion in the context of infinite dimensional manifolds with corners. We show that they are the natural concepts in this context by giving positive answers to the problems of transitivity of submanifolds, inverse image of submanifolds and transversality, the problem of good immersion in quadrants of Banach spaces and the relation between a map being differentiable and its graph being a submanifold.