On the spectrum of bounded immersions

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.