Volume gap for minimal submanifolds in spheres

Abstract

For a closed minimal submanifold $f:M^n\looparrowright {\mathbb{S}}^N$ in the unit sphere $(n<N)$, we prove $$\mathrm{Vol}(M^n)\ge \frac{n+1}{n+2}{\int }_M\left(1+\ddot{I}{†}_p^2\right)\ge m\mathrm{Vol}({\mathbb{S}}^n),$$ where $\ddot{I}{†}_p(x):=⟨f(x),p⟩$ is the height function in direction $p\in f(M)$, $m$ denotes the multiplicity of $p\in f(M)$ and $\mathrm{Vol}$ denotes the Riemannian volume functional, and each equality holds if and only if $M$ is totally geodesic. As an application, if the volume of $M^n$ is less than or equal to the volume of any $n$-dimensional minimal Clifford torus, then $M^n$ must be embedded, verifying the non-embedded case of Yau's conjecture. In addition, we also get volume gaps for minimal hypersurfaces with constant scalar curvature, improving Cheng-Li-Yau's classical volume gap in this case. Some other volume gaps and related pinching rigidities are also obtained.

Accepted by The Journal of Geometric Analysis on July 17, 2026