Abstract. We present a set of global invariants, called "mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the "boundary at infinity" has spherical topology one single invariant is obtained, called the mass; we show positivity thereof. We apply the definition to conformally compactifiable manifolds, and show that the mass is completion-independent. We also prove the result, closely related to the problem at hand, that conformal completions of conformally compactifiable manifolds are unique.Introduction.

We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These constants are shown to be optimal and are computed explicitly in most practical cases. Academic Press

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss-Codazzi constraint equations under deformations are also discussed.Comment: 32 pages, supercedes math.DG/0501067; final published versio

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We define a renormalized characteristic class for Einstein asymptotically complex hyperbolic (ache) manifolds of dimension 4: for any such manifold, the polynomial in the curvature associated to the characteristic class χ − 3τ is shown to converge. This extends a work of Burns and Epstein in the Kähler-Einstein case.We also define a new global invariant for any compact 3-dimensional pseudoconvex CR manifold, by a renormalization procedure of the η invariant of a sequence of metrics which approximate the CR structure.Finally, we get a formula relating the renormalized characteristic class to the topological number χ − 3τ and the invariant of the CR structure arising at infinity.1991 Mathematics Subject Classification. 53C55, 58J37, 58J60 (primary), 32V15, 58J28 (secondary).

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Abstract. 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.

We show that complete conformally flat manifolds of dimension n 3 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line; or are globally conformally equivalent to R n or to a spherical spaceform S n /Γ. This extends previous results due to Cheng, Noronha, Chen, Zhu and Zhu.