Eigenvalue estimates for hypersurfaces in $H^m \times R$ and applications

TL;DR

This paper establishes eigenvalue bounds for minimal hypersurfaces in hyperbolic space times a line, with applications to minimal surface stability, index, and volume growth in specific dimensions.

Contribution

It provides new lower bounds for the Laplacian spectrum on minimal hypersurfaces in $H^m imes R$ and applies these to analyze stability and geometric properties of minimal surfaces.

Findings

Finite index for complete minimal surfaces with finite total extrinsic curvature in dimension 2.

Upper bounds on the spectrum's infimum for stable minimal surfaces in $H^3$ and $H^2 imes R$.

Volume growth estimates for minimal hypersurfaces.

Abstract

In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into $H^m \times R$. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature has finite index. On the other hand, for stable, minimal surfaces in $H^3$ or in $H^2 \times \R$, we give an upper bound on the infimum of the spectrum of the Laplacian and on the volume growth.