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 HΨ(lim ϕ ∩ Ω) = 0, where HΨ is the Hausdorff measure of order Ψ(t) = t2 | log t|. 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. 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 σess(−Δ) = [0, ∞).