Title: Conformal Geometry on 4-manifolds
Speaker: Sun-Yung Alice Chang
Speaker Info: Princeton University
Brief Description:
Special Note:
Abstract:
In the lectures, I will report on the study of a class of integral conformal invariants on 4-manifolds and some geometric applications.Date: Thursday, March 30, 2023In the first lecture, I will survey some known results and PDE techniques in conformal geometry.
On 4-manifolds, one of the key ingredients is the study of the integrand of the Chern-Gauss-Bonnet formula. The part that involves the Ricci tensor gives a fully non-linear PDE known as the secondary elementary $\sigma_2$ equation. It turns out this PDE can be studied via a 4th order linear operator (part of the family o GJMS operator) and its associated 4th order curvature called the Q-curvature. I will explain the connection and give some applications describing the diffeomorphism type for a class of 4-manifolds.
As another application, we will study the problem of "conformal filling in" in ADS/CFT theory. Namely, given a manifold $(M^n, [h])$, when is it the boundary of a conformally compact Einstein manifold $(X^{n+1}, g^+)$ with $r^2g^+|_M=h$ for some defining function $r$ on $X^{n+1}$? The model example is the $n$-sphere as the conformal infinity of the hyperbolic $(n+1)$ ball.
In the second lecture, I will briefly survey some recent progress made regarding the "existence" and "uniqueness" part of the problem. Then report some joint work with Yuxin Ge, in which we study the "compactness" part. That is, given a sequence of conformally compact Einstein manifolds with boundary, we study conditions under which the compactness of the sequence of metrics in the interior follows from the compactness of the restriction of the metrics on the boundary. We will describe the conditions in terms of integral conformal invariants.