## View abstract

### Invited talk

Tuesday, July 20, 12:15 ~ 13:15 UTC-3

## Compactness of conformally compact Einstein manifolds

### Sun-Yung Alice Chang

Given a manifold $(M^n; [h])$, when is it the boundary of a conformally compact Einstein manifold $(X^{n+1}; g+)$ with $r^2 g+ |_{M} = h$ for some defining function $r$ on $X^{n+1}$? This problem of finding "conformal filling in" is motivated by problems in the AdS/CFT correspondence in quantum gravity (proposed by Maldacena in 1998) and from the geometric considerations to study the structure of non-compact asymptotically hyperbolic Einstein manifolds.
In this talk, instead of addressing the existence problem of conformal filling in, we will discuss the compactness problem. That is, given a sequence of conformally compact Einstein manifold with boundary, we are interested to study the compactness of the sequence class under some local and non-local boundary constraints. I will survey some recent development in this research area. As an application, I will report some recent work on the global uniqueness of a conformally compact Einstein metric defined on the (n+1)-dimensional ball with its boundary metric with its boundary metric infinity a metric near the canonical metric on the n-dimensional sphere when $n \geq 3$, where the existence of conformally filling in metric was constructed in the earlier work of Graham-Lee.