2009/01/09 by Isabel M. C. Salavessa, Salavessa, Isabel M. C.
Mathematics · #53C40 #58C40 #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP) #math.DG #math.SP #msc:53C40 #msc:58C40
paper · pdf · doi:10.48550/arxiv.0901.1246
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
We prove, under a certain boundedness condition at infinity on the (X\top, X\bot)-component of the second fundamental form, the vanishing of the essential spectrum of a complete minimal X-bounded and X-properly immersed submanifold on a Riemannian manifold endowed with a strongly convex vector field 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 \citeBM