Gap phenomena and curvature estimates for Conformally Compact Einstein Manifolds
Gap phenomena and curvature estimates for Conformally Compact Einstein Manifolds
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DOI：
10.1090/tran/6925
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发表时间：
2014-10
影响因子：
1.3
通讯作者：
Gang Li;J. Qing;Yuguang Shi
中科院分区：
文献类型：
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作者：
Gang Li;J. Qing;Yuguang Shi
Author（s）：Li，Gang; Qing，Jie; Shi，Yuguang|摘要：本文首先利用文[12]中的结果去掉了文[9]中差距定理中关于Weyl曲率L^2 $有界的假设，从而得到了一类具有极大重整化体积的共形紧Einstein流形的间隙定理.我们还使用爆破方法来获得曲率估计共形紧爱因斯坦流形与大重整化体积。本文的第二部分是关于具有大Yamabe常数共形无穷大的共形紧Einstein流形。基于文[15]的思想，我们给出了共形紧Einstein流形上相对体积不等式（1.9）的完整证明。因此，我们得到了一般维数下共形紧爱因斯坦流形的刚性定理的完整证明，没有自旋结构的假设（参见。$[29，15]$）以及具有极大Yamabe常数的共形无穷大的共形紧Einstein流形的新的曲率箍缩估计。我们还得到了具有大Yamabe常数共形无穷大的共形紧Einstein流形的曲率估计。
Author(s): Li, Gang; Qing, Jie; Shi, Yuguang | 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.