2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/215679We 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}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 spaceDifferential GeometrySpectral Theory53C40; 58C40text