p-Fundamental tone estimates of submanifolds with bounded mean curvature

Abstract

In this paper, we estimate the p-fundamental tone of submanifolds in a Cartan-Hadamard manifold. First, we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension-one -foliations of open subsets of Riemannian manifolds M and obtain lower bound estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of .