$L^2$ harmonic 1-forms on minimal submanifolds in hyperbolic space

TL;DR

This paper establishes conditions under which no nontrivial $L^2$ harmonic 1-forms exist on certain minimal submanifolds in hyperbolic space, linking stability, eigenvalues, and harmonic forms.

Contribution

It proves the nonexistence of $L^2$ harmonic 1-forms under specific eigenvalue bounds and provides criteria for super stability of minimal submanifolds in hyperbolic space.

Findings

Nonexistence of $L^2$ harmonic 1-forms under eigenvalue bounds

Conditions for super stability of minimal submanifolds

Relation between eigenvalues and harmonic form properties

Abstract

In this paper, we prove the nonexistence of $L^2$ harmonic 1-forms on a complete super stable minimal submanifold $M$ in hyperbolic space under the assumption that the first eigenvalue $\lambda_1 (M)$ for the Laplace operator on $M$ is bounded below by $(2n-1)(n-1)$. Moreover, we provide sufficient conditions for minimal submanifolds in hyperbolic space to be super stable.