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Local rigidity of constant mean curvature hypersurfaces in space forms (II)

  • Xinxin Cheng
  • Yayun Chen
  • Tongzhu Li
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Abstract

This is the second article of a sequence of research on the local rigidity of constant mean curvature (CMC) hypersurfaces in space forms. In the previous one, we studied the local rigidity of CMC hypersurfaces whose the number of the distinct principal curvatures satisfies g≤ 3. In this paper, we study the local rigidity of CMC hypersurfaces with g≥ 4. When g>4, we prove that if the k-order mean curvatures Hk, (k=2,⋯, g−1) are constant and there exist enough multiple principal curvatures, then the CMC hypersurface is an isoparametric hypersurface. When g=4, if H2 and H3 are constant, then the CMC hypersurface is an isoparametric hypersurface.

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Cite

BibTeXRIS
BibTeX
@misc{cheng2026local,
  title = {{Local rigidity of constant mean curvature hypersurfaces in space forms (II)}},
  author = {Xinxin Cheng and Yayun Chen and Tongzhu Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13949},
  eprint = {2609.13949},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13949},
}
RIS
TY  - UNPB
TI  - Local rigidity of constant mean curvature hypersurfaces in space forms (II)
AU  - Xinxin Cheng
AU  - Yayun Chen
AU  - Tongzhu Li
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13949
UR  - https://arxiv.org/abs/2609.13949
AB  - This is the second article of a sequence of research on the local rigidity of constant mean curvature (CMC) hypersurfaces in space forms. In the previous one, we studied the local rigidity of CMC hypersurfaces whose the number of the distinct principal curvatures satisfies g≤ 3. In this paper, we study the local rigidity of CMC hypersurfaces with g≥ 4. When g>4, we prove that if the k-order mean curvatures H_k, (k=2,⋯, g−1) are constant and there exist enough multiple principal curvatures, then the CMC hypersurface is an isoparametric hypersurface. When g=4, if H_2 and H_3 are constant, then the CMC hypersurface is an isoparametric hypersurface.
ER  -