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Triharmonic CMC Hypersurfaces in Space Forms with at Most 3 Distinct Principal Curvatures

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Abstract

A k-harmonic map is a critical point of the k-energy defined on the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if \(M^{n} (n\ge 3)\) is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form \(\mathbb {R}^{n+1}(c)\), then M has constant scalar curvature. This supports the generalized Chen’s conjecture when \(c\le 0\). When \(c=1\), we give an optimal upper bound of the mean curvature H for a non-totally umbilical proper CMC k-harmonic hypersurface with constant scalar curvature in a sphere. As an application, we give the complete classification of the 3-dimensional complete proper CMC triharmonic hypersurfaces in \(\mathbb {S}^{4}\).

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Acknowledgements

The authors would like to thank Professor Haizhong Li for bringing the question to our attention and his valuable suggestions and comments. We are also grateful to Professor C. Oniciuc for giving helpful comments on the first version of the paper and bringing some references to our attention, which help us to add Corollary 1.9 and improve Theorem 1.10. We also thank the anonymous referee for the careful reading and pointing out the issues.

Funding

The first author was partially supported by Natural Science Foundation of Shaanxi Province Grant No. 2020JQ-101 and the Fundamental Research Funds for the Central Universities Grant No. 310201911cx013. The second author was partially supported by NSFC Grant No. 11831005 and No. 11671224.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China

    Hang Chen

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Zhida Guan

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  1. Hang Chen
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  2. Zhida Guan
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Correspondence to Hang Chen.

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Chen, H., Guan, Z. Triharmonic CMC Hypersurfaces in Space Forms with at Most 3 Distinct Principal Curvatures. Results Math 77, 155 (2022). https://doi.org/10.1007/s00025-022-01698-1

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