Eigenvalue Estimates for submanifolds of $N \times \mathbb{R}$ with locally bounded mean curvature

TL;DR

This paper derives lower bounds for the fundamental tone of submanifolds with bounded mean curvature in product manifolds, providing sharp estimates for minimal immersions and insights into minimal surfaces' spectral properties.

Contribution

It introduces new lower bounds for the fundamental tone of submanifolds in product spaces with bounded mean curvature, including sharp estimates for minimal cases.

Findings

Lower bounds for the fundamental tone in $N imes R$

Sharp estimates for minimal immersions

Positive fundamental tone for cylindrically bounded minimal surfaces

Abstract

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N \times \mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} \leq \kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone.