On the spectrum of bounded immersions

Abstract

In this article, we investigate some of the relations between the spectrum of a non-compact, extrinsically bounded submanifold $\varphi : M^m \to N^n$ and the Hausdorff dimension of its limit set $\lim \varphi$. In particular, we prove that if $\varphi : M^2 \to \mathbb{R}^3$ is a (complete) minimal surface immersed into an open, bounded, strictly convex subset $\Omega$ with $C^3$-boundary, then $M$ has discrete spectrum, provided that $\mathcal{H}_{\Psi} (\lim \varphi \cap \Omega) = 0$, where $\mathcal{H}_{\Psi}$ is the Hausdorff measure of order $\Psi(t) = t^2 \lvert \log t \rvert$. Our main theorem, Thm. 2.4, applies to a number of examples recently constructed, by many authors, in the light of Nadirashvili’s discovery of complete bounded minimal disks in $\mathbb{R}^3$, as well as to solutions of Plateau’s problems for non-rectifiable Jordan curves, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. On the other hand, we present a simple criterion, called the ball property, whose fulfilment guarantees the existence of elements in the essential spectrum. As an application, we show that some of the examples of Jorge-Xavier and Rosenberg-Toubiana of complete minimal surfaces between two planes have essential spectrum $\sigma_{\mathrm{ess}}(-\Delta) = [0, \infty)$.