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Evolving Compact Locally Convex Curves and Convex Hypersurfaces

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Abstract

A nonlocal curvature flow is investigated to evolve compact locally convex hypersurfaces in the Euclidean space \({\mathbb {E}}^{n+1}\). It is shown that the flow exists globally in all dimensions and deforms the evolving curve into an m-fold circle in the plane if \(n=1\) and drives the evolving hypersurface into a Euclidean sphere if \(n>1\).

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Acknowledgements

Laiyuan Gao is supported by National Natural Science Foundation of China (No.11801230). This work is partially completed when the first author visited University of California, San Diego. He thanks Prof. Lei Ni and Prof. Bennett Chow for their hospitality.

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  1. School of Mathematics and Statistics, Jiangsu Normal University, No.101, Shanghai Road, Xuzhou, Jiangsu, China

    Laiyuan Gao & Yuntao Zhang

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  1. Laiyuan Gao
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  2. Yuntao Zhang
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Correspondence to Yuntao Zhang.

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Gao, L., Zhang, Y. Evolving Compact Locally Convex Curves and Convex Hypersurfaces. manuscripta math. 167, 365–375 (2022). https://doi.org/10.1007/s00229-021-01278-7

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  • DOI: https://doi.org/10.1007/s00229-021-01278-7

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