The non-parabolicity of infinite volume ends

Abstract

Let $M^m$, with $m\ge 3$, be an $m$-dimensional complete noncompact manifold isometrically immersed in a Hadamard manifold $\bar{M}$. Assume that the mean curvature vector has finite $L^p$-norm, for some $2\le p\le m$. We prove that each end of $M$ must either have finite volume or be non-parabolic.

Minor corrections in Theorem B. To appear in Proc. Amer. Math. Soc