A spectral Bernstein theorem

Abstract

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface M in ℝn+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, +∞). This is the case when limr̃→+∞ r̃Ki = 0, where r̃ is the extrinsic distance to a point of M and Ki are the principal curvatures. (2) If the Ki satisfy the decay conditions {pipe}Ki{pipe} ≤ 1/r̃ and strict inequality is achieved at some point y ε M, then there are no eigenvalues. We apply these results to minimal graphic and multigraphic hypersurfaces. © 2010 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.