Stable minimal hypersurfaces in the hyperbolic space

TL;DR

This paper establishes bounds on the first eigenvalue of the Laplacian for stable minimal hypersurfaces in hyperbolic space, providing conditions for stability and describing specific stable minimal surfaces like catenoids and helicoids.

Contribution

It introduces an upper bound for the first eigenvalue on stable minimal hypersurfaces in hyperbolic space and characterizes the stability of certain classical minimal surfaces.

Findings

Upper bound for the first eigenvalue of the Laplacian

Conditions for stability of minimal hypersurfaces

Existence of stable higher-dimensional catenoids

Abstract

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.