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Closed Self-Similar Solutions to Flows by Negative Powers of Curvature

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Abstract

In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers of mean curvature, Gauss curvature, and other curvature functions with suitable properties. We prove such self-similar solutions, not necessarily strictly convex for some cases, must be slices of warped product manifolds. A new auxiliary function is the key of the proofs.

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Acknowledgements

The author would like to thank Professor Xianfeng Wang for her interest of the work and valuable comments. And the author was supported in part by the Natural Science Basic Research Program of Shaanxi Province (Program No. 2022JQ-065), Youth Innovation Team of Shaanxi Universities, Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSZ012) and the Fundamental Research Funds for the Central Universities (Grant Nos. GK202307001, GK202202007). The author is also grateful to the anonymous reviewers for helpful comments.

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  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, Shaanxi Province, People’s Republic of China

    Shanze Gao

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  1. Shanze Gao
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Gao, S. Closed Self-Similar Solutions to Flows by Negative Powers of Curvature. J Geom Anal 33, 370 (2023). https://doi.org/10.1007/s12220-023-01427-2

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