A Spectral Bernstein Theorem

Details A Spectral Bernstein Theorem A Spectral Bernstein Theorem

2009-05-17

arxiv.org/abs/0905.2773v2

Mathematics

Differential Geometry

16 pages. v2. Final version: minor revisions, we add Proposition 3.2. Accepted for publication in the Annali di Matematica Pur

Scientific paper

10.1007/s10231-010-0139-0

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.

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