P-Harmonic functions and connectedness at infinity of complete submanifolds in a Riemannian manifold

Abstract

In this paper, we study the connectedness at infinity of complete submanifolds by using the theory of p-harmonic function. For lower-dimensional cases, we prove that if M is a complete orientable noncompact hypersurface in \(\mathbb {R}^{n+1}\) and if \(\delta \)-stability inequality holds on M, then M has only one p-nonparabolic end. It is also proved that if \(M^n\) is a complete noncompact submanifold in \({\mathbb {R}}^{n+k}\) with sufficiently small \(L^n\) norm of the traceless second fundamental form, then M has only one p-nonparabolic end. Moreover, we obtain a lower bound of the fundamental tone of the p Laplace operator on complete submanifolds in a Riemannian manifold.