A Spectral Bernstein Theorem

Abstract

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface $M$ in ${\mathbb{R}}^{n+1}$. (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 $|{\kappa }_i|\le 1/\tilde{r}$, 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.

16 pages. v2. Final version: minor revisions, we add Proposition 3.2. Accepted for publication in the Annali di Matematica Pura ed Applicata, on the 05/03/2010

References in corpus (3)

• Topologically non-trivial quantum layers
• On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces
• Curvature integral estimates for complete hypersurfaces