On the spectrum of bounded immersions

TL;DR

This paper explores how the spectral properties of bounded minimal surfaces relate to the geometric measure of their limit sets, providing conditions for discreteness of spectrum and examples illustrating the sharpness of these conditions.

Contribution

It establishes a link between the discreteness of the spectrum of bounded minimal immersions and the Hausdorff measure of their limit sets, answering a question posed by Yau.

Findings

Discrete spectrum occurs when the limit set has zero generalized Hausdorff measure.

Counterexamples demonstrate the sharpness of the spectral criteria.

The results apply to minimal disks and solutions of Plateau's problem.

Abstract

In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $\varphi \colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\lim\varphi$. In particular, we prove that if $\varphi \colon \!M^2 \ra D \subseteq \R^3$ is a minimal immersion into an open, bounded, strictly convex subset $D$ with $C^2$-boundary, then $M$ has discrete spectrum provided that $\haus_\Psi(\lim\varphi \cap D)=0$, where $\haus_\Psi$ 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 $\R^3$, 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.