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Nonlinear Analysis

Compactness of asymptotically hyperbolic Einstein manifolds in dimension 4 and applications

Speaker: Yuxion Ge, Institut de Mathematique de Toulouse, University Paul Sabatier

Location:  HILL 705
Date & time: Wednesday, 03 May 2023 at 10:00AM - 11:00AM

Joint with Complex Analysis and Geometry

Abstract: Given a closed riemannian manfiold of dimension 3 (M^3,[h]), when will we fill in an asymptotically hyperbolic Einstein manifold of dimension 4 (X4,g+) such that h = r2g+ on the boundary M^3 = \del X^4 for some defining function r on X4? This problem is motivated by the correspondance AdS/CFT in quantum gravity proposed by Maldacena in 1998 et comes also from the study of the structure of asymptotically hyperbolic Einstein manifolds. In this talk, I discuss the compactness issue of asymptotically hyperbolic Einstein mani- folds in dimension 4, that is, how the compactness on conformal infinity leads to the compactness of the compactification of such manifolds under the suitable conditions on the topology and on some conformal invariants. As application, I discuss the uniqueness problem and non-existence result. It is based on the joint works with Alice Chang.

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