On the spectrum of bounded immersions

In this article, we investigate some of the relations between the spectrum of a non-compact, extrinsically bounded submanifold φ: Mm → Nn and the Hausdorff dimension of its limit set lim φ. In particular, we prove that if φ:M2 → R3 is a (complete) minimal surface immersed into an open, bounded, strictly convex subset ω with C3-boundary, then M has discrete spectrum, provided that Hφ(limφ ∩ ω) = 0, where Hφ is the Hausdorff measure of order Φ(t) = t2\ logt|. 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 R3, 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 [64], [65]. 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 [36] and Rosenberg-Loubiana [60] of complete minimal surfaces between two planes have essential spectrum σess(-δ) = [0, ∞).