An improved second pinching theorem for minimal hypersurfaces with constant scalar curvature in spheres
Abstract
Let Mn be a closed minimal hypersurface of the unit sphere with constant squared norm S of the second fundamental form. For n≥4, we prove that S>n implies S>85n/59, improving the bound 10n/7 of Suh and Yang. No constancy assumption is imposed on the cubic trace f3. The proof combines a symmetrized Hessian estimate, an integral identity for a mixed moment, and a weighted estimate for the repeated-index components of the covariant derivative of the second fundamental form. An exact polynomial argument treats the entire excluded interval. We also prove an explicit positive lower bound for the average of the normalized quadratic defect P/S2, where P=f4−f32/S−S2/n. This second estimate holds in every dimension n≥2 and gives a quantitative obstruction to concentration near the locus of two principal curvatures.
Related
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- On the structure of stable constant anisotropic mean curvature hypersurfaces
BibTeX
@misc{chen2026improved,
title = {{An improved second pinching theorem for minimal hypersurfaces with constant scalar curvature in spheres}},
author = {Ruihan Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17586},
eprint = {2609.17586},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17586},
}
RIS
TY - UNPB
TI - An improved second pinching theorem for minimal hypersurfaces with constant scalar curvature in spheres
AU - Ruihan Chen
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.17586
UR - https://arxiv.org/abs/2609.17586
AB - Let M^n be a closed minimal hypersurface of the unit sphere with constant squared norm S of the second fundamental form. For n≥4, we prove that S>n implies S>85n/59, improving the bound 10n/7 of Suh and Yang. No constancy assumption is imposed on the cubic trace f_3. The proof combines a symmetrized Hessian estimate, an integral identity for a mixed moment, and a weighted estimate for the repeated-index components of the covariant derivative of the second fundamental form. An exact polynomial argument treats the entire excluded interval. We also prove an explicit positive lower bound for the average of the normalized quadratic defect P/S^2, where P=f_4−f_3^2/S−S^2/n. This second estimate holds in every dimension n≥2 and gives a quantitative obstruction to concentration near the locus of two principal curvatures.
ER -