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An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices II
Abstract
Let Mn⊂ Sn+m be a compact minimal submanifold of the unit sphere, and let λ1≥⋯≥λm≥0 be the eigenvalues of its fundamental matrix. We prove that ∑α=1min{n,m}λα+λ2≤ n forces M to be a totally geodesic subsphere, a generalized Clifford torus, or a Veronese manifold. We first show that equality everywhere in the pinching condition forces the second fundamental form to be parallel. Next, by computing the eigenvalues of the fundamental matrices of irreducible symmetric R-spaces, we obtain a complete classification of the equality cases among minimal submanifolds with parallel second fundamental form.
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BibTeX
@misc{liu2026optimal,
title = {{An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices II}},
author = {Huimin Liu and Ling Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11485},
eprint = {2610.11485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11485},
}
RIS
TY - UNPB
TI - An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices II
AU - Huimin Liu
AU - Ling Yang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11485
UR - https://arxiv.org/abs/2610.11485
AB - Let M^n⊂ S^n+m be a compact minimal submanifold of the unit sphere, and let λ_1≥⋯≥λ_m≥0 be the eigenvalues of its fundamental matrix. We prove that ∑_α=1^min{n,m}λ_α+λ_2≤ n forces M to be a totally geodesic subsphere, a generalized Clifford torus, or a Veronese manifold. We first show that equality everywhere in the pinching condition forces the second fundamental form to be parallel. Next, by computing the eigenvalues of the fundamental matrices of irreducible symmetric R-spaces, we obtain a complete classification of the equality cases among minimal submanifolds with parallel second fundamental form.
ER -