The Spectrum of the Martin-Morales-Nadirashvili Minimal Surfaces Is Discrete

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of ℝ³ is discrete. This gives a positive answer to a question of Yau (Asian J. Math. 4:235–278, 2000).