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

Sun-Yung A. Chang, Yuxin Ge, Xiaoshang Jin, Jie Qing

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.

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

Abstract

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

Paper Structure

This paper contains 19 sections, 33 theorems, 184 equations.

Table of Contents

  1. Introduction and Statement of results
  2. Introduction
  3. Preliminaries
  4. Basic properties of adapted metrics $g^*$.
  5. Elliptic estimates for the adapted metrics
  6. Some results in Riemannian geometry for CCE manifolds
  7. Interior regularity in all dimensions
  8. Boundary regularity when dimension is even
  9. Boundary regularity in all dimensions
  10. Gauged Einstein equation
  11. Local gauge
  12. $\varepsilon$-regularity
  13. Regularity of $g^*$
  14. Proof of Theorems \ref{['maintheorem3bis']} and \ref{['maintheorem3']}
  15. Proof of Theorem \ref{['maintheorem3bis']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose that $X$ is a smooth oriented $d$-dimensional manifold with compact boundary with $d$ even and $d \ge 4$. Let $\{g_i^+\}$ be a set of conformally compact Einstein metrics on $X$. Assume that the corresponding metrics $\{h_i\}$ at conformal infinity have non-negative scalar curvature, and hav then the set $\{g_i^*\}$ of the adapted metrics (after diffeomorphisms that fix the boundary) is co

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Theorem 1.3
  • Corollary 1.4
  • Remark 2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 50 more