标题：GAP PHENOMENA AND CURVATURE ESTIMATES FOR CONFORMALLY COMPACT EINSTEIN MANIFOLDS
作者：Li, Gang; Qing, Jie; Shi, Yuguang
作者机构：[Li, Gang] Shandong Univ, Dept Math, Jinan 250100, Shandong, Peoples R China.; [Li, Gang] Peking Univ, Beijing Int Ctr Math Res, Beijing 100871, Peo 更多
通讯作者：Li, G;Li, G
通讯作者地址：[Li, G]Shandong Univ, Dept Math, Jinan 250100, Shandong, Peoples R China;[Li, G]Peking Univ, Beijing Int Ctr Math Res, Beijing 100871, Peoples R China 更多
来源：TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
关键词：Conformally compact Einstein manifolds; gap phenomena; rigidity;; curvature estimates; renormalized volumes; Yamabe constants
摘要：In this paper we obtain first a gap theorem for a class of conformally compact Einstein manifolds with a renormalized volume that is close to its maximum value. We also use a blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The major part of this paper is the study of how a property of the conformal infinity influences the geometry of the interior of a conformally compact Einstein manifold. Specifically we are interested in conformally compact Einstein manifolds with conformal infinity whose Yamabe invariant is close to that of the round sphere. Based on the approach initiated by Dutta and Javaheri we present a complete proof of the relative volume inequality; (Y(partial derivative X,[(g) over cap])/Y(Sn-1 ,[gs]))(n-1) <= Vol(partial derivative B-g+(p,t))/Vol(partial derivative B-gH+(0,t)) <= Vol(B-g+(p,t))/Vol(B-gH+(0,t)) <= 1,; for conformally compact Einstein manifolds. This leads not only to the complete proof of the rigidity theorem for conformally compact Einstein manifolds in arbitrary dimension without spin assumption but also a new curvature pinching estimate for conformally compact Einstein manifolds with conformal infinities having large Yamabe invariant. We also derive curvature estimates for such manifolds.