Volume gap for minimal submanifolds in spheres

TL;DR

This paper establishes volume inequalities for minimal submanifolds in spheres, characterizes equality cases, and applies results to verify Yau's conjecture and improve classical volume gap theorems.

Contribution

It introduces new volume gap inequalities for minimal submanifolds and hypersurfaces, providing characterizations and applications to longstanding conjectures.

Findings

Proves a volume gap inequality for minimal submanifolds in spheres.

Verifies the non-embedded case of Yau's conjecture for certain volume bounds.

Improves classical volume gap results for minimal hypersurfaces with constant scalar curvature.

Abstract

For a closed minimal submanifold $f:M^n\looparrowright \mathbb{S}^{N}$ in the unit sphere $(n<N)$, we prove $${\rm Vol}(M^n) \geq\frac{n+1}{n+2}\int_{M}\left( 1+\varphi_{p}^2\right) \geq m{\rm Vol}(\mathbb{S}^{n}),$$ where $\varphi_{p}(x):=\langle f(x),p\rangle$ is the height function in direction $p\in f(M)$, $m$ denotes the multiplicity of $p\in f(M)$ and ${\rm Vol}$ 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.