Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)

By András Vasy. Inventiones Mathematicae, 2010

286 citations recorded by the source corpus.

In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Sect. 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis.Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose’s b-structures. These include asymptotically de Sitter-type metrics on a blow-up of the natural compactification, Kerr-de Sitter-type metrics, as well as asymptotically Minkowski metrics.The simplest application is a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces), including a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. These results are also available in a follow-up paper which is more expository in nature (Vasy in Uhlmann, G. (ed.) Inverse Problems and Applications. Inside Out II, 2012).The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dc4aff0f53029a6b931879986618ed351413dcb4

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In Applied Mathematics

1. K. Datchev
2. Semyon Dyatlov

In Mathematical Physics

1. K. Datchev
2. Semyon Dyatlov
3. András Vasy

Citations in the corpus

• Fractal Weyl laws for asymptotically hyperbolic manifolds (2012, background): “oles called resonances, as shown by Mazzeo{Melrose [MaMe] and Guillarmou [Gu05a] for asymptotically hyperbolic manifolds, Guillope{Zworski [GuZw95a] when curvature near innity is constant, and Vasy [Va10] for asymptotically hyperbolic manifolds with g 1 even. When M= nHn, Bunke{Olbrich [BuOl] and Pa”