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PreprintCC BY-NC-ND 4.0arXiv

The Second Gap and Rigidity in Chern's Conjecture with Constant Cubic Trace

  • Jianquan Ge
  • Huixin Tan
  • Wenjiao Yan
  • Yunheng Zhang
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Abstract

Let Mn⊂𝕊n+1(1) be a closed minimal hypersurface with constant S=|h|2 and constant f3=tr h3, where h is the shape operator. We prove that S>n implies S≥2n, and that the equality images are precisely the minimal Cartan isoparametric hypersurfaces with three principal curvatures, which occur only in dimensions n=3, 6, 12, 24.

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Cite

BibTeXRIS
BibTeX
@misc{ge2026second,
  title = {{The Second Gap and Rigidity in Chern's Conjecture with Constant Cubic Trace}},
  author = {Jianquan Ge and Huixin Tan and Wenjiao Yan and Yunheng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27661},
  eprint = {2609.27661},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27661},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The Second Gap and Rigidity in Chern's Conjecture with Constant Cubic Trace
AU  - Jianquan Ge
AU  - Huixin Tan
AU  - Wenjiao Yan
AU  - Yunheng Zhang
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27661
UR  - https://arxiv.org/abs/2609.27661
AB  - Let M^n⊂𝕊^n+1(1) be a closed minimal hypersurface with constant S=|h|^2 and constant f_3=tr h^3, where h is the shape operator. We prove that S>n implies S≥2n, and that the equality images are precisely the minimal Cartan isoparametric hypersurfaces with three principal curvatures, which occur only in dimensions n=3, 6, 12, 24.
ER  -