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PreprintarXiv licencearXiv

On the Bryant conjecture for minimal hypersurfaces in S4

  • Chao Qian
  • Ya Tao
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Abstract

We prove that a local minimal hypersurface M3 in the unit sphere S4 with constant scalar curvature and three distinct principal curvatures at every point must be locally congruent to the minimal Cartan isoparametric hypersurface in S4, which provides a positive answer to the Bryant conjecture \cite{ChangLocal}. The key ingredient is to prove that, for each fixed S0>3, the set 𝒒S0 of fourth-order jets of normalized local graph solutions satisfying |A|2≑ S0 is compact, where A denotes the shape operator.

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Cite

BibTeXRIS
BibTeX
@misc{qian2026bryant,
  title = {{On the Bryant conjecture for minimal hypersurfaces in $S^4$}},
  author = {Chao Qian and Ya Tao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00886},
  eprint = {2610.00886},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00886},
}
RIS
TY  - UNPB
TI  - On the Bryant conjecture for minimal hypersurfaces in S^4
AU  - Chao Qian
AU  - Ya Tao
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00886
UR  - https://arxiv.org/abs/2610.00886
AB  - We prove that a local minimal hypersurface M^3 in the unit sphere S^4 with constant scalar curvature and three distinct principal curvatures at every point must be locally congruent to the minimal Cartan isoparametric hypersurface in S^4, which provides a positive answer to the Bryant conjecture \cite{ChangLocal}. The key ingredient is to prove that, for each fixed S_0>3, the set 𝒒_S_0 of fourth-order jets of normalized local graph solutions satisfying |A|^2≑ S_0 is compact, where A denotes the shape operator.
ER  -