The mass of asymptotically hyperbolic Riemannian manifolds

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.

arXiv comment27 pages, Latex2e with several style files; various misprints corrected, positivity theorem for black holes considerably strengthened, to appear in Pacific Jour. of Mathematics

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• The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times
• Positive mass theorems for asymptotically hyperbolic Riemannian manifolds with boundary
• Hilbert manifold structure for asymptotically hyperbolic relativistic initial data
• Hyperbolic positive energy theorems