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

arXiv0901.1246MaRDI QIDQ6212242

Isabel Salavessa

Publication date: 9 January 2009

Abstract: We prove, under a certain boundedness condition at infinity on the $$-component of the second fundamental form, the vanishing of the essential spectrum of a complete minimal $$-bounded and $$-properly immersed submanifold on a Riemannian manifold endowed with a strongly convex vector field $$. 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}

Mathematics Subject Classification ID

Global submanifolds (53C40) Spectral theory; eigenvalue problems on manifolds (58C40)

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