On Yau rigidity theorem for minimal submanifolds in spheres

TL;DR

This paper extends Yau's rigidity theorem for minimal submanifolds in spheres, establishing conditions under which such submanifolds are classified as totally geodesic, product of spheres, or Veronese surface, using the DDVV inequality.

Contribution

It proves a new rigidity result for minimal submanifolds in spheres with a curvature bound, extending to submanifolds with parallel mean curvature in space forms.

Findings

Classifies minimal submanifolds under curvature conditions

Identifies specific geometric models: geodesic spheres, product of spheres, Veronese surface

Extends rigidity to submanifolds with parallel mean curvature

Abstract

In this note, we investigate the well-known Yau rigidity theorem for minimal submanifolds in spheres. Using the parameter method of Yau and the DDVV inequality verified by Lu, Ge and Tang, we prove that if $M$ is an $n$-dimensional oriented compact minimal submanifold in the unit sphere $S^{n+p}(1)$, and if $K_{M}\geq\frac{sgn(p-1)p}{2(p+1)},$ then $M$ is either a totally geodesic sphere, the standard immersion of the product of two spheres, or the Veronese surface in $S^4(1)$. Here $sgn(\cdot)$ is the standard sign function. We also extend the rigidity theorem above to the case where $M$ is a compact submanifold with parallel mean curvature in a space form.