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October 2019 Quantitative volume space form rigidity under lower Ricci curvature bound I
Lina Chen, Xiaochun Rong, Shicheng Xu
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J. Differential Geom. 113(2): 227-272 (October 2019). DOI: 10.4310/jdg/1571882427

Abstract

Let $M$ be a compact $n$-manifold of $\mathrm{Ric}_M \geq (n - 1) H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions:

(i) There is $ \rho \gt 0$ such that for any $x \in M$, the open $ \rho $-ball at $x^{\ast}$ in the (local) Riemannian universal covering space, $ (U^{\ast}_{\rho} , x^{\ast}) \to (B_{\rho} (x) , x)$, has the maximal volume, i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$.

(ii) For $H = -1$, the volume entropy of $M$ is maximal, i.e., $n - 1$ ([LW1]).

The main results of this paper are quantitative space form rigidity, i.e., statements that $M$ is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H = 1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].

Funding Statement

The first author is supported partially by a research fund from Capital Normal University.
The second author is supported partially by NSF Grant DMS 0203164 and by a research fund from Capital Normal University.
The third author is supported partially by NSFC Grant 11821101, 11871349 and by Youth Innovative Research Team of Capital Normal University.

Citation

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Lina Chen. Xiaochun Rong. Shicheng Xu. "Quantitative volume space form rigidity under lower Ricci curvature bound I." J. Differential Geom. 113 (2) 227 - 272, October 2019. https://doi.org/10.4310/jdg/1571882427

Information

Published: October 2019
First available in Project Euclid: 24 October 2019

zbMATH: 07122208
MathSciNet: MR4023292
Digital Object Identifier: 10.4310/jdg/1571882427

Rights: Copyright © 2019 Lehigh University

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Vol.113 • No. 2 • October 2019
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