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Positive energy theorem for asymptotically hyperbolic manifolds

Friday 16th December 2005 - 14:30 to 15:00
INI Seminar Room 1

General Relativity is a geometrical theory of gravity which asserts that the geometry of space-time is closely related to matter. There exists some consistent definition for total energy (and momentum) of isolated systems which by definition are manifolds whose metric approaches a background metric (Euclidean or hyperbolic). The positive mass theorem can be considered as attempts at understanding the relationship between the local energy density (namely the stress-energy tensor) and the total energy of a space-time. On one hand P. T. Chrusciel and G. Nagy rigorously defined in a recent work notions of mass and momentum for manifolds which are asymptotic to a standard hyperbolic slice of Minkowski space-time. On the other hand P. T. Chrusciel and M. Herzlich proved a positive mass theorem for Riemannian asymptotically hyperbolic manifolds. My work extends this result for orientable 3-dimensional manifolds which are asymptotic to a standard hyperbolic slice of anti-de Sitter space-time in the following way: we define a sesquilinear form Q which is closely related to the energy-momentum and prove, under the relevant energy condition, that Q is a nonnegative Hermitian form which is in fact definite unless our manifold is isometrically embeddable in anti-de Sitter.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons