Gap phenomena and curvature estimates for conformally compact Einstein manifolds
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- by Gang Li, Jie Qing and Yuguang Shi PDF
- Trans. Amer. Math. Soc. 369 (2017), 4385-4413 Request permission
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 \[ \left (\frac {Y(\partial X, [\hat {g}])}{Y(\mathbb {S}^{n-1}, [g_{\mathbb {S}}])}\right )^{\frac {n-1}{2}}\leq \frac {Vol(\partial B_{g^+}(p, t))} {Vol(\partial B_{g_{\mathbb {H}}}(0, t))} \leq \frac {Vol(B_{g^+}(p, t))} {Vol(B_{g_{\mathbb {H}}}(0, t))}\leq 1, \] 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
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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
- MR Author ID: 1064338
- Email: runxing3@gmail.com
- Jie Qing
- Affiliation: Department of Mathematics, University of California Santa Cruz, Santa Cruz, California 95064
- MR Author ID: 268101
- 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
- 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. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4385-4413
- MSC (2010): Primary 53C25; Secondary 58J05
- DOI: https://doi.org/10.1090/tran/6925
- MathSciNet review: 3624414