An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices II

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