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Convexity of 2-Convex Translating and Expanding Solitons to the Mean Curvature Flow in \({{\mathbb {R}}}^{n+1}\)

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Abstract

In this paper, inspired by the work of Spruck–Xiao (Am J Math 142(3):993–1015, 2020) and based partly on a result of Derdziński (Math Z 172(3):273–280, 1980), we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in \(\mathbb {R}^{n+1}\). More precisely, for \(n\ge 3\), we show that any n-dimensional complete 2-convex translating solitons are convex, and any n-dimensional complete 2-convex self-expanders asymptotic to (strictly) mean convex cones are convex.

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Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Notes

  1. In early April 2021, we became aware of the work of Singley [24] and found that his work suits our needs more directly.

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Acknowledgements

Junming Xie and Jiangtao Yu are very grateful to their Ph.D. advisor Professor Huai-Dong Cao for suggesting this project in Summer 2018 and for his invaluable suggestions, constant support and encouragement. The first author would also like to thank Professor Lu Wang for helpful discussions, and Professor Knut Smoczyk for answering questions related to his paper [25].

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  1. Junming Xie and Jiangtao Yu have contributed equally to this work.

Authors and Affiliations

  1. Department of Math, Lehigh University, Bethelem, PA, 18015, USA

    Junming Xie

  2. Institute of Geometry and Physics, University of Science and Technology of China, Hefei, 230026, Anhui, China

    Jiangtao Yu

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  1. Junming Xie
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  2. Jiangtao Yu
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These authors contributed equally to this work.

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Correspondence to Jiangtao Yu.

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Xie, J., Yu, J. Convexity of 2-Convex Translating and Expanding Solitons to the Mean Curvature Flow in \({{\mathbb {R}}}^{n+1}\). J Geom Anal 33, 252 (2023). https://doi.org/10.1007/s12220-023-01260-7

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