The non-parabolicity of infinite volume ends

TL;DR

This paper proves that for certain complete noncompact manifolds immersed in Hadamard spaces with finite mean curvature norm, each end is either finite in volume or non-parabolic, revealing geometric constraints on their structure.

Contribution

It establishes a new dichotomy for the ends of manifolds with finite mean curvature in Hadamard spaces, linking volume finiteness and parabolicity.

Findings

Ends are either finite volume or non-parabolic.

Finite mean curvature norm implies geometric restrictions.

Results extend understanding of manifold ends in nonpositive curvature.

Abstract

Let $M^m$, with $m\geq 3$, 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 $2\leq p\leq m$. We prove that each end of $M$ must either have finite volume or be non-parabolic.