## View abstract

### Invited talk

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

## Compactness of conformally compact Einstein manifolds

### Sun-Yung Alice Chang

#### Princeton University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakb715a937305e79a2d6b98be7bafff14a').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyb715a937305e79a2d6b98be7bafff14a = 'ch&#97;ng' + '&#64;'; addyb715a937305e79a2d6b98be7bafff14a = addyb715a937305e79a2d6b98be7bafff14a + 'm&#97;th' + '&#46;' + 'pr&#105;nc&#101;t&#111;n' + '&#46;' + '&#101;d&#117;'; var addy_textb715a937305e79a2d6b98be7bafff14a = 'ch&#97;ng' + '&#64;' + 'm&#97;th' + '&#46;' + 'pr&#105;nc&#101;t&#111;n' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakb715a937305e79a2d6b98be7bafff14a').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyb715a937305e79a2d6b98be7bafff14a + '\'>'+addy_textb715a937305e79a2d6b98be7bafff14a+'<\/a>';

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.

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