On Chern's Conjecture for Minimal Submanifolds with Flat Normal Bundle in Spheres

Let $M^n$ $(n⩾3)$ be a closed minimal submanifold in the unit sphere ${\mathbb{S}}^{n+m}$ $(m⩾2)$ with flat normal bundle, and let $S$ denote the squared norm of its second fundamental form. We prove an explicit second-gap rigidity theorem for $S$. More precisely, if $S$ is constant and $$0⩽S⩽n+\delta ,$$ where $\delta$ is an explicit constant satisfying $\delta ⩾\frac{n}{87}$, then either $S\equiv 0$ and $M$ is a totally geodesic sphere, or $S\equiv n$ and $M$ is a Clifford torus contained in a totally geodesic ${\mathbb{S}}^{n+1}\subset {\mathbb{S}}^{n+m}$. %We observe that the flat-normal-bundle assumption is necessary here. The flat-normal-bundle condition is essential in the general higher-codimensional setting: without it, the corresponding rigidity statement already fails in dimension two. This theorem provides positive evidence for Chern’s conjecture in higher codimension.