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Transactions of the American Mathematical Society

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Quantitative volume space form rigidity under lower Ricci curvature bound II


Authors: Lina Chen, Xiaochun Rong and Shicheng Xu
Journal: Trans. Amer. Math. Soc. 370 (2018), 4509-4523
MSC (2010): Primary 53C21, 53C23, 53C24
DOI: https://doi.org/10.1090/tran/7279
Published electronically: November 16, 2017
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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.


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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
Email: rong@math.rutgers.edu

Shicheng Xu
Affiliation: Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China
Email: shichxu@foxmail.com

DOI: https://doi.org/10.1090/tran/7279
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.
Article copyright: © Copyright 2017 American Mathematical Society

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