Codimension Reduction of Minimal Submanifolds with Flat Normal Bundle in Spheres

Abstract

Let x:Mⁿ→𝕊^n+q be a minimal immersion of a complete connected manifold with flat normal bundle, where n≥3 and q≥2. We prove that if the squared norm of the second fundamental form S satisfies S<2n, then Mⁿ lies in 𝕊ⁿ⁺¹. The standard minimal products of three spheres show that the upper bound 2n is optimal. We also prove a codimension reduction theorem under the condition S+λ₂≤ 2n+2n/(3n−2), where λ₂ is the second largest eigenvalue of the Gram matrix of the shape operators, implying that Mⁿ lies in 𝕊ⁿ⁺¹.

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BibTeX RIS

BibTeX

@misc{li2026codimension,
  title = {{Codimension Reduction of Minimal Submanifolds with Flat Normal Bundle in Spheres}},
  author = {Juan Li and Zhiyuan Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07789},
  eprint = {2610.07789},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07789},
}

RIS

TY  - UNPB
TI  - Codimension Reduction of Minimal Submanifolds with Flat Normal Bundle in Spheres
AU  - Juan Li
AU  - Zhiyuan Xu
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07789
UR  - https://arxiv.org/abs/2610.07789
AB  - Let x:M^n→𝕊^n+q be a minimal immersion of a complete connected manifold with flat normal bundle, where n≥3 and q≥2. We prove that if the squared norm of the second fundamental form S satisfies S<2n, then M^n lies in 𝕊^n+1. The standard minimal products of three spheres show that the upper bound 2n is optimal. We also prove a codimension reduction theorem under the condition S+λ_2≤ 2n+2n/(3n−2), where λ_2 is the second largest eigenvalue of the Gram matrix of the shape operators, implying that M^n lies in 𝕊^n+1.
ER  -

Abstract

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