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Rigidity of minimal hypersurfaces with free boundary in a ball

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Abstract

In this paper, we give two rigidity results of free boundary hypersurfaces in a ball. First, we prove that any minimal hypersurface with free boundary in a closed geodesic ball in a round open hemisphere \(\mathbb S^{n+1}_+\) which is Killing-graphical is a geodesic disk. We note that we do not assume any topological condition on the hypersurface. We consider analogous result for self-shrinkers of the mean curvature flow. More precisely, we proved that any graphical self-shrinker with free boundary in a ball centered at the origin in \(\mathbb R^{n+1}\) is a flat disk passing through the origin.

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Acknowledgements

We warmly thank the referee for his/her valuable suggestions to improve our paper. The both authors were supported in part by the National Research Foundation of Korea (NRF-2020R1A2C1A01005698).

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Authors and Affiliations

  1. Department of Mathematics, Pusan National University, Busan, 46241, Korea

    Sangwoo Park & Juncheol Pyo

  2. KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Korea

    Juncheol Pyo

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  1. Sangwoo Park
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  2. Juncheol Pyo
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Correspondence to Juncheol Pyo.

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Park, S., Pyo, J. Rigidity of minimal hypersurfaces with free boundary in a ball. Geom Dedicata 216, 27 (2022). https://doi.org/10.1007/s10711-022-00688-5

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