On the spectrum of $\bar{X}$-bounded minimal submanifolds

TL;DR

This paper proves that under certain boundedness conditions, the essential spectrum of specific minimal submanifolds in Riemannian manifolds is zero, extending previous results using an advanced version of Barta's theorem.

Contribution

It extends existing spectral results for minimal submanifolds by establishing the vanishing of the essential spectrum under new boundedness conditions.

Findings

Essential spectrum of certain minimal submanifolds vanishes.

Results apply to submanifolds in manifolds with convex vector fields.

Generalizes previous spectral theorems for minimal surfaces.

Abstract

We prove, under a certain boundedness condition at infinity on the $(\bar{X}^{\top}, \bar{X}^{\bot})$-component of the second fundamental form, the vanishing of the essential spectrum of a complete minimal $\bar{X}$-bounded and $\bar{X}$-properly immersed submanifold on a Riemannian manifold endowed with a strongly convex vector field $\bar{X}$. The same conclusion also holds for any complete minimal $h$-bounded and $h$-properly immersed submanifold that lies in a open set of a Riemannian manifold $\oM$ supporting a nonnegative strictly convex function $h$. This extends a recent result of Bessa, Jorge and Montenegro on the spectrum of Martin-Morales minimal surfaces. Our proof uses as main tool an extension of Barta's theorem given in \cite{BM}