On compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, III

TL;DR

This paper proves a compactness theorem for conformally compact Einstein metrics in dimensions four and higher, demonstrating uniqueness of Graham-Lee metrics on the ball and exploring conformal invariants.

Contribution

It establishes a compactness result for conformally compact Einstein metrics and proves the global uniqueness of Graham-Lee metrics in higher dimensions.

Findings

Proved a compactness theorem for conformally compact Einstein metrics.

Established the global uniqueness of Graham-Lee metrics on the ball.

Identified gap phenomena for certain conformal invariants.

Abstract

In this paper, we establish a compactness result for a class of conformally compact Einstein metrics defined on manifolds of dimension $d\ge 4$. As an application, we derive the global uniqueness of a class of conformally compact Einstein metric defined on the $d$-dimensional ball constructed in the earlier work of Graham-Lee with $d\ge 4$. As a second application, we establish some gap phenomenon for a class of conformal invariants.