Gap phenomena and curvature estimates for Conformally Compact Einstein Manifolds

TL;DR

This paper advances the understanding of conformally compact Einstein manifolds by removing previous curvature assumptions, establishing new gap theorems, and proving volume inequalities and curvature estimates for manifolds with large Yamabe constants.

Contribution

It removes the L^2 Weyl curvature assumption in gap theorems, proves a volume inequality, and provides curvature estimates for manifolds with large Yamabe constants.

Findings

Removed the L^2 boundedness assumption of Weyl curvature in gap theorems.

Established a volume inequality for conformally compact Einstein manifolds.

Derived curvature estimates for manifolds with large Yamabe constant at infinity.

Abstract

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of the relative volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore we obtain the complete proof of the rigidity theorem for conformally compact Einstein manifolds in general dimensions with no spin structure assumption (cf. $[29, 15]$) as well as the new curvature pinch estimates for conformally compact Einstein manifolds with conformal infinities of very large Yamabe constant. We also derive the curvature estimates for conformally compact Einstein manifolds with conformal infinities of large Yamabe constant.