Quantitative maximal diameter rigidity of positive Ricci curvature

Tianyin Ren; Xiaochun Rong

doi:10.1515/crelle-2024-0015

Article

Quantitative maximal diameter rigidity of positive Ricci curvature

Tianyin Ren and Xiaochun Rong

Published/Copyright: June 26, 2024

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2024 Issue 813

Abstract

In Riemannian geometry, the Cheng’s maximal diameter rigidity theorem says that if a complete n-manifold M of Ricci curvature, Ric M ≥ ( n - 1 ) , has the maximal diameter π, then M is isometric to the unit sphere S 1 n . The main result in this paper is a quantitative maximal diameter rigidity: if M satisfies that Ric M ≥ n - 1 , diam ⁡ ( M ) ≈ π , and the Riemannian universal cover of every metric ball in M of a definite radius satisfies a Reifenberg condition, then M is diffeomorphic and bi-Hölder close to S 1 n .

Funding statement: Partially supported by NSFC 11821101, BNSF Z190003, and a research fund from Capital Normal University.

Acknowledgements

The authors would like to thank Shicheng Xu for helpful comments on Corollaries 0.6 and 0.7, and thanks for some comments from a referee that helps in improving some exposition in this paper.

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Received: 2023-08-16

Revised: 2024-02-21

Published Online: 2024-06-26

Published in Print: 2024-08-01

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https://doi.org/10.1515/crelle-2024-0015