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 $\varphi \colon M^m \ra N^n$andtheHausdorffdimensionofitslimitset$\lim\varphi$. In particular, we prove that if $\phi \text{\hspace{0.17em}⁣}:\text{\hspace{0.17em}⁣}{M}^2\text{ra}D\subseteq {\mathbb{R}}^3$ is a minimal immersion into an open, bounded, strictly convex subset $D$ with $C^2$-boundary, then $M$ has discrete spectrum provided that ${\text{haus}}_{\mathrm{\Psi }}(\lim \phi \cap D)=0$, where ${\text{haus}}_{\mathrm{\Psi }}$ is the generalized Hausdorff measure of order $\mathrm{\Psi }(t)=t^2\mathrm{\mid }\log t\mathrm{\mid }$. 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.