A topological sphere theorem for pinched Legendrian submanifolds

Abstract

Let Mⁿ be a closed simply connected Legendrian submanifold of 𝕊²ⁿ⁺¹, where n≥5, and let B and H=trB denote its second fundamental form and mean-curvature vector. We prove that M is homeomorphic to 𝕊ⁿ if |B|²≤ ((n−1)(n+2))/n+(n²+3n−2)/2n²|H|²−((n−1)(n−2))/2n²|H|√(|H|²+4n). The equality can be attained by the Calabi products, so simple connectivity cannot be omitted. For n≥4, we also prove a sphere theorem under an explicit bound and the weaker assumption b₁(M;ℝ)=0. The proof combines phase-rotated Lawson--Simons variations with a systolic inequality for finite two-complexes. We also give local constructions showing that the pinching condition allows negative isotropic curvature at a point.

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BibTeX RIS

BibTeX

@misc{cui2026topological,
  title = {{A topological sphere theorem for pinched Legendrian submanifolds}},
  author = {Qing Cui and Linlin Sun},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18293},
  eprint = {2610.18293},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18293},
}

RIS

TY  - UNPB
TI  - A topological sphere theorem for pinched Legendrian submanifolds
AU  - Qing Cui
AU  - Linlin Sun
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18293
UR  - https://arxiv.org/abs/2610.18293
AB  - Let M^n be a closed simply connected Legendrian submanifold of 𝕊^2n+1, where n≥5, and let B and H=trB denote its second fundamental form and mean-curvature vector. We prove that M is homeomorphic to 𝕊^n if |B|^2≤ ((n−1)(n+2))/n+(n^2+3n−2)/2n^2|H|^2−((n−1)(n−2))/2n^2|H|√(|H|^2+4n). The equality can be attained by the Calabi products, so simple connectivity cannot be omitted. For n≥4, we also prove a sphere theorem under an explicit bound and the weaker assumption b_1(M;ℝ)=0. The proof combines phase-rotated Lawson--Simons variations with a systolic inequality for finite two-complexes. We also give local constructions showing that the pinching condition allows negative isotropic curvature at a point.
ER  -

Abstract

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