Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

Gap phenomena and curvature estimates for conformally compact Einstein manifolds


Authors: Gang Li, Jie Qing and Yuguang Shi
Journal: Trans. Amer. Math. Soc. 369 (2017), 4385-4413
MSC (2010): Primary 53C25; Secondary 58J05
DOI: https://doi.org/10.1090/tran/6925
Published electronically: February 13, 2017
MathSciNet review: 3624414
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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

$\displaystyle \left (\frac {Y(\partial X, [\hat {g}])}{Y(\mathbb{S}^{n-1}, [g_{... ...0, t))} \leq \frac {Vol(B_{g^+}(p, t))} {Vol(B_{g_{\mathbb{H}}}(0, t))}\leq 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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C25, 58J05

Retrieve articles in all journals with MSC (2010): 53C25, 58J05


Additional Information

Gang Li
Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China – and – Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, People’s Republic of China
Email: runxing3@gmail.com

Jie Qing
Affiliation: Department of Mathematics, University of California Santa Cruz, Santa Cruz, California 95064
Email: qing@ucsc.edu

Yuguang Shi
Affiliation: Key Laboratory of Pure and Applied Mathematics, School of Mathematics Science, Peking University, Beijing, 100871, People’s Republic of China
Email: ygshi@math.pku.edu.cn

DOI: https://doi.org/10.1090/tran/6925
Keywords: Conformally compact Einstein manifolds, gap phenomena, rigidity, curvature estimates, renormalized volumes, Yamabe constants
Received by editor(s): March 24, 2015
Received by editor(s) in revised form: June 26, 2015, November 4, 2015, and January 3, 2016
Published electronically: February 13, 2017
Additional Notes: The research of the first author was supported by China Postdoctoral Science Foundation grant 2014M550540
The research of the second author was supported by NSF grant DMS-1303543.
The research of the third author was supported by NSF of China grant 10990013.
Article copyright: © Copyright 2017 American Mathematical Society