Abstract
A triharmonic map is a critical point of the tri-energy functional defined on the space of smooth maps between two Riemannian manifolds. In this paper, we prove that any CMC proper triharmonic hypersurface in the 5-dimensional space form \({\mathbb {R}}^{5}(c)\) must have constant scalar curvature. Furthermore, we show that any CMC triharmonic hypersurface in \({\mathbb {R}}^5\) or \({\mathbb {H}}^5\) must be minimal, which supports the generalized Chen’s conjecture; we also give some characterizations of CMC proper triharmonic hypersurfaces in \({\mathbb {S}}^5\). Similar results are obtained in the higher dimension case under an additional assumption on the numbers of the distinct principal curvatures.
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Acknowledgements
The authors would like to thank Professor Haizhong Li for bringing the question to our attention and his very helpful suggestions and comments. 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 Nos. 11831005 and 11671224.
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Chen, H., Guan, Z. Triharmonic CMC hypersurfaces in \({\mathbb {R}}^{5}(c)\). manuscripta math. 172, 209–220 (2023). https://doi.org/10.1007/s00229-022-01409-8
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DOI: https://doi.org/10.1007/s00229-022-01409-8