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

Abstract

We prove, under a certain boundedness condition at infinity on the $({\bar{X}}^{\top },{\bar{X}}^{\perp })$-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}

v.2 13 pages. We improve some theorems, correct some misprints and add a new theorem (theorem 5) in section 2 that shows complete bounded minimal immersed submanifolds have unbounded second fundamental form, for suitable curvature conditions of the ambient space