Compactness of conformally compact Einstein 4-manifolds II

TL;DR

This paper proves new compactness results for conformally compact Einstein 4-manifolds, improving previous findings and establishing global uniqueness under certain curvature conditions.

Contribution

It advances the understanding of the geometric structure of conformally compact Einstein 4-manifolds by improving compactness results and proving global uniqueness in specific cases.

Findings

Improved compactness results for conformally compact Einstein 4-manifolds.

Derived compactness under small L^2-norm of Weyl curvature.

Proved global uniqueness of certain Einstein metrics on the 4-Ball.

Abstract

In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee.