10:00 to 11:00 D Hafner ([Bordeaux])On scattering theory for field equations in the Kerr metricSession: Black hole mini-symposium We show asymptotic completeness for the massless Dirac field and the non- superradiant modes of the Klein-Gordon field in the Kerr metric. In the first part we treat massless Dirac fields. We introduce a new Newman- Penrose tetrad in which the expression of the equation contains no artificial long-range perturbations. The main technique used is then a Mourre estimate. The geometry near the horizon requires us to apply a unitary transformation before we find ourselves in a situation where the generator of dilations is a good conjugate operator. The results are reinterpreted to provide a solution to the Goursat problem on the Penrose compactified exterior. In the second part we treat Klein-Gordon fields. We start with an abstract Hilbert space result. From a Mourre estimate for a positive selfadjoint oparator one can deduce a Mourre estimate for its square root. Using this result and the techniques explained in the first part of the talk, we can establish an asymptotic completeness result for the non-superradiant modes of the Klein-Gordon field. Because of the mass of the field the wave operators have to be Dollard modified at infinity. INI 1 11:30 to 12:30 G Galloway ([Miami])On the topology of black holes in higher dimensionsSession: Black hole mini-symposium A basic result in the theory of black holes is Hawking's theorem on black hole topology which asserts that for 3+1 AF stationary black hole spacetimes obeying the dominant energy condition, cross sections S of the event must be spherical. The proof is a beautiful variational argument showing that if S is not spherical then it can be deformed to an outer trapped surface in the domain of outer communications, which is forbidden by basic results. The conclusion also holds, by a similar argument, for outermost apparent horizons in black hole spacetimes that are not necessarily stationary. Since Gauss-Bonnet is used, the results are restricted to 3+1. In this talk we present a recent result with Rick Schoen which extends Hawking's results to arbitrarily high dimensions, by showing that outermost apparent horizons must be of positive Yamabe type, i.e., must carry metrics of positive scalar curvature. In the time symmetric case, this follows from the minimal surface methodology of Schoen and Yau in their treatment of manifolds of positive scalar curvature. The present result, however, does not impose any restrictions on the extrinsic curvature of space. While the Jang equation is not used, a neat technique appearing in Schoen-Yau II proves useful. INI 1 14:00 to 15:00 H Reall ([California])Higher dimensional black holesSession: Black hole mini-symposium This talk will review black hole solutions of general relativity in more than four dimensions. Outline: motivation, vacuum solutions, supersymmetric solutions, open questions. INI 1 15:30 to 16:30 M Dafermos ([Cambridge])Decay of radiation in spacetimes with black holesSession: Black hole mini-symposium An important problem in general relativity is understanding radiation tails'' in the exterior regions of spacetimes containing black holes. The heuristic picture of what these tails should look like goes back to work of R. Price in 1972. These tails are related to stability properties of black hole exteriors, and also to the details of their inner structure, in particular, the generic presence of weak null singularities inside black holes. In this talk, I shall describe a rigorous proof of Price's power-law decay rates for the collapse of a spherically symmetric self-gravitating scalar field. Applications of these ideas to linear and non-linear wave equations on various fixed black hole backgrounds will also be discussed. This constitutes joint work with I. Rodnianski. INI 1
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