A Spectral Bernstein Theorem

TL;DR

This paper investigates the spectral properties of the Laplace operator on minimal hypersurfaces in Euclidean space, establishing conditions under which the spectrum is purely essential and identifying when eigenvalues are absent.

Contribution

It provides new spectral theorems for minimal hypersurfaces under volume growth and curvature decay conditions, extending understanding of their Laplacian spectrum.

Findings

Essential spectrum is [0, +∞) under certain geometric conditions.

No eigenvalues exist if principal curvatures decay faster than 1/tilde{r}.

Results apply to minimal graphs and multigraphs.

Abstract

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface $M$ in $\R^{n+1}$. (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that $M$ has only essential spectrum consisting of the half line $[0, +\infty)$. This is the case when $\lim_{\tilde{r}\to +\infty}\tilde{r}\kappa_i=0$, where $\tilde{r}$ is the extrinsic distance to a point of $M$ and $\kappa_i$ are the principal curvatures. (2) If the $\kappa_i$ satisfy the decay conditions $|\kappa_i|\leq 1/\tilde{r}$, and strict inequality is achieved at some point $y\in M$, then there are no eigenvalues. We apply these results to minimal graphic and multigraphic hypersurfaces.