A spectral Bernstein theorem |
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Authors: | Pedro Freitas Isabel Salavessa |
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Institution: | 1.Department of Mathematics,Faculdade de Motricidade Humana (TU Lisbon), and Group of Mathematical Physics of the University of Lisbon,Lisboa,Portugal |
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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, +∞). This is the case when \({{\rm lim}_{\tilde{r}\to +\infty}\,\tilde{r}\kappa_i=0}\), where \({\tilde{r}}\) is the extrinsic distance to a point of M and κ i are the principal curvatures. (2) If the κ 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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