Rigidity of minimal submanifolds in hyperbolic space

TL;DR

This paper establishes that complete minimal submanifolds in hyperbolic space with small total scalar curvature are topologically simple, having only one end, and do not admit nontrivial $L^2$ harmonic 1-forms, revealing rigidity properties.

Contribution

It proves new rigidity results for minimal submanifolds in hyperbolic space based on curvature conditions, linking geometric and topological properties.

Findings

Minimal submanifolds with small total scalar curvature have only one end.

Such submanifolds admit no nontrivial $L^2$ harmonic 1-forms.

The results connect curvature bounds with topological and harmonic form properties.

Abstract

We prove that if an $n$-dimensional complete minimal submanifold $M$ in hyperbolic space has sufficiently small total scalar curvature then $M$ has only one end. We also prove that for such $M$ there exist no nontrivial $L^2$ harmonic 1-forms on $M$.