EVENT DETAILS AND ABSTRACT

Title: A new view of asymptotically hyperbolic spaces and some Lorentzian spaces via microlocal analysis I
Speaker: Andras Vasy
Speaker Info: Stanford University
Brief Description:
Special Note:
Abstract:

In these lectures I describe a new approach to analysis of the resolvent of the Laplacian on asymptotically hyperbolic spaces, as well the asymptotic behavior of solutions of the wave or Klein-Gordon equations on Lorentzian spaces such as Minkowski-like spaces, asymptotically de Sitter spaces and Kerr-de Sitter space. This approach relies on 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 resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis.

The simplest application is to the analysis on Riemannian or Lorentzian 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. In this case, one continues a suitably modified version of a conjugate of the spectral family of the Laplacian across the conformal boundary to obtain a family of operators on a compact manifold directly; in the case of hyperbolic space, it is connected to de Sitter space via this extension.

Other 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.