Colloquium / Seminars
Topic：Conformal Geometry on 4-manifolds
Speaker：Prof. Sun-Yung Alice Chang
(Princeton University, USA)
Date time：Dec. 15, 2023 11:00 - 12:00
Abstract：In this talk, I will give a survey talk on 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 second elementary symmetric equation. It turns out this PDE can be studied
via a 4-th order linear operator (part of the family of GJMS operator) and its associated 4-th
order curvature called the Q-curvature. I will explain the connection and give an application
of using this concept to compute the renormalized volume.
As another application, we will study the problem of ”conformal filling in” in ADS/CFT
theory. Namely, given a manifold (Mn; [h]), when is it the boundary of a conformally compact
Einstein manifold ( Xn+1; g+) with r2
g+|M = h for some defining function r on Xn+1?
The model example is the n-sphere as the conformal infinity of the hyperbolic (n+1) ball.