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A note on the topological stability theorem from \({{\,\textrm{RCD}\,}}\) spaces to Riemannian manifolds

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Abstract

Inspired by a recent work of Wang-Zhao in [72], in this note we prove that for a fixed n-dimensional closed Riemannian manifold \((M^n, g)\), if an \({{\,\textrm{RCD}\,}}(K, n)\) space \((X, {\textsf{d}}, {\mathfrak {m}})\) is Gromov-Hausdorff close to \(M^n\), then there exists a regular homeomorphism F from X to \(M^n\) such that F is Lipschitz continuous and that \(F^{-1}\) is Hölder continuous, where the Lipschitz constant of F, the Hölder exponent and the Hölder constant of \(F^{-1}\) can be chosen arbitrary close to 1. This is sharp in the sense that in general such a map cannot be improved to being bi-Lipschitz. Moreover if X is smooth, then such a homeomorphism can be chosen as a diffeomorphism. It is worth mentioning that the Lipschitz-Hölder continuity of F improves the intrinsic Reifenberg theorem for closed manifolds with Ricci curvature bounded below established by Cheeger-Colding. The Nash embedding theorem plays a key role in the proof.

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Acknowledgements

The both authors wish to thank Elia Bruè and Daniele Semola for fruitful discussions on the paper. They also thank Zhangkai Huang for giving us comments on the paper. Moreover they are grateful to the referee for his/her careful reading on the first version and for giving us very helpful comments. The first named author acknowledges supports of the Grant-in-Aid for Scientific Research (B) of 20H01799, the Grant-in-Aid for Scientific Research (B) of 21H00977 and Grant-in-Aid for Transformative Research Areas (A) of 22H05105.

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  1. Mathematical Institute, Tohoku University, Sendai, Japan

    Shouhei Honda & Yuanlin Peng

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  1. Shouhei Honda
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Honda, S., Peng, Y. A note on the topological stability theorem from \({{\,\textrm{RCD}\,}}\) spaces to Riemannian manifolds. manuscripta math. 172, 971–1007 (2023). https://doi.org/10.1007/s00229-022-01418-7

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