Abstract:

The study of conformally compact Einstein manifolds is a fundamental problem in both geometry analysis and theoretical physics. In this talk, I will introduce some basic facts of conformally compact Einstein manifolds, including the definition, examples, basic properties and some problems. Then I will talk about my recent work on the boundary regularity of 4-dimensional conformally compact Einstein manifolds. We show that, a C^{2,σ} conformally compact Einstein manifold is in fact C^{m,α} compact if its boundary metric is in C^{m,α}.