An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices II
Year: 2026
Type: preprint
Abstract: Let $M^n\subset S^{n+m}$ be a compact minimal submanifold of the unit sphere, and let $λ_1\geq\cdots\geqλ_m\geq0$ be the eigenvalues of its fundamental matrix. We prove that $\sum\limits_{α=1}^{\min\{n,m\}}λ_α+λ_2\leq n$ 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.
Source: arXiv (Cornell University)
Authors: Liu, Huimin, Yang, Ling
Institution: Fudan University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221104483