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Uniqueness of the Catenoid and the Delaunay Surface via a One-Parameter Family of Touching Spheres

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Abstract

We provide a new characterization of the catenoid and the Delaunay surface among constant mean curvature surfaces touching a one-parameter family of spheres along arcs in a 3-dimensional Euclidean space. Furthermore we prove a higher-dimensional analogue for constant mean curvature hypersurfaces. We also give a generalization of the classical theorem due to Joachimsthal.

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Acknowledgements

The authors would like to express their deep gratitude to the reviewer for his/her careful reading of our manuscript and helpful comments and suggestions.

Funding

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant Number: 2017R1D1A1B03036369). The second author was supported by the National Research Foundation of Korea (NRF-2021R1A2C1003365).

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Authors and Affiliations

  1. Department of Mathematics, Chungnam National University, Daehak-Ro 99, Yuseong-Gu, Daejeon, 34134, Korea

    Sung-Hong Min

  2. Department of Mathematics and Research Institute of Natural Science, Sookmyung Women’s University, Cheongpa-ro 47-gil 100, Yongsan-ku, Seoul, 04310, Korea

    Keomkyo Seo

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  1. Sung-Hong Min
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Min, SH., Seo, K. Uniqueness of the Catenoid and the Delaunay Surface via a One-Parameter Family of Touching Spheres. Results Math 78, 60 (2023). https://doi.org/10.1007/s00025-023-01837-2

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