Quantitative volume space form rigidity under lower Ricci curvature bound II
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- by Lina Chen, Xiaochun Rong and Shicheng Xu PDF
- Trans. Amer. Math. Soc. 370 (2018), 4509-4523 Request permission
Abstract:
This is the second paper of two in a series under the same title; both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to an $H$-space form if for every ball of definite size on $M$, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of $M$ is bounded for $H\ne 1$.
In the first paper, we verified the conjecture for the case that the Riemannian universal covering space $\tilde M$ is not collapsed. In the present paper, we will verify this conjecture for the case that Ricci curvature is also bounded above, while the above non-collapsing condition on $\tilde M$ is not required.
References
- Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445. MR 1074481, DOI 10.1007/BF01233434
- Simon Brendle and Richard Schoen, Manifolds with $1/4$-pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009), no. 1, 287–307. MR 2449060, DOI 10.1090/S0894-0347-08-00613-9
- Jeff Cheeger and Tobias H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189–237. MR 1405949, DOI 10.2307/2118589
- Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480. MR 1484888
- Jeff Cheeger, Kenji Fukaya, and Mikhael Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 327–372. MR 1126118, DOI 10.1090/S0894-0347-1992-1126118-X
- Tobias H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996), no. 1-3, 193–214. MR 1369415, DOI 10.1007/s002220050050
- Tobias H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 (1997), no. 3, 477–501. MR 1454700, DOI 10.2307/2951841
- L. Chen, X. Rong, and S. Xu, Quantitive volume rigidity of space form under lower Ricci curvature bound I, to appear in J. Differential Geom.
- Xianzhe Dai, Guofang Wei, and Rugang Ye, Smoothing Riemannian metrics with Ricci curvature bounds, Manuscripta Math. 90 (1996), no. 1, 49–61. MR 1387754, DOI 10.1007/BF02568293
- Karsten Grove and Hermann Karcher, How to conjugate $C^{1}$-close group actions, Math. Z. 132 (1973), 11–20. MR 356104, DOI 10.1007/BF01214029
- Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
- M. Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978), no. 2, 231–241. MR 540942
- Ernst Heintze, Mannigfaltigkeiten negativer Krümmung, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 350, Universität Bonn, Mathematisches Institut, Bonn, 2002 (German). MR 1940403
- H. Huang, L. Kong, X. Rong, and S. Xu, Collapsed manifolds with local Ricci bounded covering geometry. In preparation.
- François Ledrappier and Xiaodong Wang, An integral formula for the volume entropy with applications to rigidity, J. Differential Geom. 85 (2010), no. 3, 461–477. MR 2739810
- G. Perelman, Manifolds of positive Ricci curvature with almost maximal volume, J. Amer. Math. Soc. 7 (1994), no. 2, 299–305. MR 1231690, DOI 10.1090/S0894-0347-1994-1231690-7
- Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
- Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301. MR 1001277
- Wan-Xiong Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989), no. 2, 303–394. MR 1010165
- Gang Tian and Bing Wang, On the structure of almost Einstein manifolds, J. Amer. Math. Soc. 28 (2015), no. 4, 1169–1209. MR 3369910, DOI 10.1090/jams/834
Additional Information
- Lina Chen
- Affiliation: Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China
- Address at time of publication: Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China
- Email: chenlina_mail@163.com
- Xiaochun Rong
- Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
- MR Author ID: 336377
- Email: rong@math.rutgers.edu
- Shicheng Xu
- Affiliation: Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China
- MR Author ID: 923312
- ORCID: 0000-0001-5088-4818
- Email: shichxu@foxmail.com
- Received by editor(s): June 18, 2016
- Received by editor(s) in revised form: May 13, 2017
- Published electronically: November 16, 2017
- Additional Notes: The first author was supported in part by a research fund from Capital Normal University.
The second author was supported partially by NSF Grant DMS 0203164 and by a research fund from Capital Normal University.
The third author was supported partially by NSFC Grant 11401398 and by the Youth Innovative Research Team of Capital Normal University. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4509-4523
- MSC (2010): Primary 53C21, 53C23, 53C24
- DOI: https://doi.org/10.1090/tran/7279
- MathSciNet review: 3811536