A spectral Bernstein theorem

作者：Freitas Pedro; Salavessa Isabel^*

来源：Annali di Matematica Pura ed Applicata, 2011, 190(1): 77-90.

DOI：10.1007/s10231-010-0139-0

• Spectrum
• Minimal submanifolds

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, +infinity). This is the case when lim((r) over tilde ->+infinity) (r) over tilde kappa(i) = 0, where (r) over tilde 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 vertical bar kappa(i)vertical bar <= 1/(r) over tilde and strict inequality is achieved at some point y is an element of M, then there are no eigenvalues. We apply these results to minimal graphic and multigraphic hypersurfaces.

• 出版日期2011-1