On the spectrum of bounded immersions

Abstract

In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $φ\colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\lim \phi$. In particular, we prove that if $φ\colon \!M^2 \ra D \subseteq \R^3$ is a minimal immersion into an open, bounded, strictly convex subset $D$ with $C^2$-boundary, then $M$ has discrete spectrum provided that $\haus_Ψ(\limφ\cap D)=0$, where $\haus_Ψ$ is the generalized Hausdorff measure of order $\Psi (t)=t^2|\log t|$. Our theorem applies to a number of examples recently constructed by various authors in the light of N. Nadirashvili's discovery of complete, bounded minimal disks in ${\mathbb{R}}^3$, as well as to solutions of Plateau's problems, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. Suitable counter-examples show the sharpness of our results: in particular, we develop a simple criterion for the existence of essential spectrum which is suited for the techniques developed after Jorge-Xavier and Nadirashvili's examples.

24 pages. Submitted for publication

References in corpus (4)

• Density theorems for complete minimal surfaces in R^3
• Discreteness of the spectrum of the Laplace-Beltrami operator
• Complete proper minimal surfaces in convex bodies of $R^3$
• Compact complete minimal immersions in R^3

Cited by in corpus (3)

• Density and spectrum of minimal submanifolds in space forms
• On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature
• Spectrum of the Laplacian on radial graphs