Rigidity and stability estimates for minimal submanifolds in the hyperbolic space

TL;DR

This paper derives conditions on the second fundamental form of complete minimal submanifolds in hyperbolic space to determine when they are totally geodesic, and provides sharp eigenvalue bounds for certain surfaces.

Contribution

It establishes new rigidity conditions based on the second fundamental form and eigenvalue estimates for minimal submanifolds in hyperbolic space.

Findings

Conditions for total geodesicity based on second fundamental form

Sharp upper bounds for eigenvalues of super stability operator

Rigidity results for minimal submanifolds in hyperbolic space

Abstract

In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold $M^n$ in the hyperbolic space $\mathbb{H}^{n+m}$ in order to show that $M^n$ is totally geodesic. We also obtain sharp upper bounds estimates for the first eigenvalue of the super stability operator in the case of $M$ is a surface in $\mathbb{H}^{4}$.