Some Progress in Conformal Geometry

TL;DR

This survey reviews recent research in conformal geometry focusing on PDEs, including bubble tree structures, gap and finiteness theorems, and new compactification methods for Einstein metrics.

Contribution

Introduces a bubble tree structure for Yamabe metrics, establishes new theorems, and proposes an eigenfunction compactification in conformal geometry.

Findings

Established a gap theorem for Yamabe metrics

Proved a finiteness theorem for diffeomorphism types

Derived diameter bounds for $\sigma_2$-metrics

Abstract

This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the $\sigma_2$-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.