On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces

TL;DR

This paper proves that the spectrum of certain complete minimal surfaces immersed in a ball is discrete when their mean curvature is small, answering a question posed by Yau.

Contribution

It establishes conditions under which the spectrum of minimal surfaces immersed in a ball is discrete, extending understanding of spectral properties of minimal submanifolds.

Findings

Spectrum of minimal surfaces in a ball is discrete.

Discreteness holds when the mean curvature vector norm is sufficiently small.

Provides a positive answer to Yau's question on spectral discreteness.

Abstract

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 $\mathbb{R}^{3}$ is discrete. This gives a positive answer to a question of Yau.