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?)

  • [1] Michael T. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc. 2 (1989), no. 3, 455-490. MR 999661, https://doi.org/10.2307/1990939
  • [2] Michael T. Anderson, $ L^2$ curvature and volume renormalization of AHE metrics on 4-manifolds, Math. Res. Lett. 8 (2001), no. 1-2, 171-188. MR 1825268, https://doi.org/10.4310/MRL.2001.v8.n2.a6
  • [3] Lars Andersson and Mattias Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998), no. 1, 1-27. MR 1616570, https://doi.org/10.1023/A:1006547905892
  • [4] Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859
  • [5] Eric Bahuaud and Romain Gicquaud, Conformal compactification of asymptotically locally hyperbolic metrics, J. Geom. Anal. 21 (2011), no. 4, 1085-1118. MR 2836592, https://doi.org/10.1007/s12220-010-9179-3
  • [6] Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), no. 2, 313-349. MR 1001844, https://doi.org/10.1007/BF01389045
  • [7] Olivier Biquard, Asymptotically symmetric Einstein metrics, SMF/AMS Texts and Monographs, vol. 13, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2006. Translated from the 2000 French original by Stephen S. Wilson. MR 2260400
  • [8] Vincent Bonini, Pengzi Miao, and Jie Qing, Ricci curvature rigidity for weakly asymptotically hyperbolic manifolds, Comm. Anal. Geom. 14 (2006), no. 3, 603-612. MR 2260724
  • [9] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang, A conformally invariant sphere theorem in four dimensions, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 105-143. MR 2031200, https://doi.org/10.1007/s10240-003-0017-z
  • [10] Sun-Yung A. Chang, Jie Qing, and Paul Yang, On the topology of conformally compact Einstein 4-manifolds, Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math., vol. 350, Amer. Math. Soc., Providence, RI, 2004, pp. 49-61. MR 2082390, https://doi.org/10.1090/conm/350/06337
  • [11] P. Yang, Dzh. King, and S.-Yu. A. Chang, Renormalized volumes for conformally compact Einstein manifolds, Sovrem. Mat. Fundam. Napravl. 17 (2006), 129-142 (Russian, with Russian summary); English transl., J. Math. Sci. (N. Y.) 149 (2008), no. 6, 1755-1769. MR 2336463, https://doi.org/10.1007/s10958-008-0094-0
  • [12] Sun-Yung A. Chang, Jie Qing, and Paul Yang, On a conformal gap and finiteness theorem for a class of four-manifolds, Geom. Funct. Anal. 17 (2007), no. 2, 404-434. MR 2322490, https://doi.org/10.1007/s00039-007-0603-1
  • [13] Jeff Cheeger and Aaron Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. (2) 182 (2015), no. 3, 1093-1165. MR 3418535, https://doi.org/10.4007/annals.2015.182.3.5
  • [14] Piotr T. Chruściel, Erwann Delay, John M. Lee, and Dale N. Skinner, Boundary regularity of conformally compact Einstein metrics, J. Differential Geom. 69 (2005), no. 1, 111-136. MR 2169584
  • [15] Piotr T. Chruściel and Marc Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), no. 2, 231-264. MR 2038048, https://doi.org/10.2140/pjm.2003.212.231
  • [16] Satyaki Dutta and Mohammad Javaheri, Rigidity of conformally compact manifolds with the round sphere as the conformal infinity, Adv. Math. 224 (2010), no. 2, 525-538. MR 2609014, https://doi.org/10.1016/j.aim.2009.12.004
  • [17] Charles Fefferman and C. Robin Graham, Conformal invariants, The mathematical heritage of Élie Cartan (Lyon, 1984), Astérisque Numero Hors Serie (1985), 95-116. MR 837196
  • [18] C. Robin Graham, Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School ``Geometry and Physics'' (Srní, 1999), 2000, pp. 31-42. MR 1758076
  • [19] C. Robin Graham and John M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), no. 2, 186-225. MR 1112625, https://doi.org/10.1016/0001-8708(91)90071-E
  • [20] Emmanuel Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR 1481970
  • [21] M. Henningson and K. Skenderis, The holographic Weyl anomaly, J. High Energy Phys. 7 (1998), Paper 23, 12 pp. (electronic). MR 1644988, https://doi.org/10.1088/1126-6708/1998/07/023
  • [22] Xue Hu, Jie Qing, and Yuguang Shi, Regularity and rigidity of asymptotically hyperbolic manifolds, Adv. Math. 230 (2012), no. 4-6, 2332-2363. MR 2927372, https://doi.org/10.1016/j.aim.2012.04.013
  • [23] Jin-ichi Itoh and Minoru Tanaka, The dimension of a cut locus on a smooth Riemannian manifold, Tohoku Math. J. (2) 50 (1998), no. 4, 571-575. MR 1653438, https://doi.org/10.2748/tmj/1178224899
  • [24] John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37-91. MR 888880, https://doi.org/10.1090/S0273-0979-1987-15514-5
  • [25] Man Chun Leung, Pinching theorem on asymptotically hyperbolic spaces, Internat. J. Math. 4 (1993), no. 5, 841-857. MR 1245353, https://doi.org/10.1142/S0129167X93000388
  • [26] Rafe Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), no. 10, 1615-1664. MR 1133743, https://doi.org/10.1080/03605309108820815
  • [27] Maung Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285 (1989), no. 4, 527-539. MR 1027758, https://doi.org/10.1007/BF01452046
  • [28] V. Ozols, Cut loci in Riemannian manifolds, Tôhoku Math. J. (2) 26 (1974), 219-227. MR 0390967
  • [29] Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
  • [30] Jie Qing, On the rigidity for conformally compact Einstein manifolds, Int. Math. Res. Not. 21 (2003), 1141-1153. MR 1962123, https://doi.org/10.1155/S1073792803209193
  • [31] Yuguang Shi and Gang Tian, Rigidity of asymptotically hyperbolic manifolds, Comm. Math. Phys. 259 (2005), no. 3, 545-559. MR 2174416, https://doi.org/10.1007/s00220-005-1370-1
  • [32] Gang Tian and Jeff Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math. 160 (2005), no. 2, 357-415. MR 2138071, https://doi.org/10.1007/s00222-004-0412-1
  • [33] Xiaodong Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), no. 2, 273-299. MR 1879228

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

American Mathematical Society