Mean curvature, volume and properness of isometric immersions

Abstract

We explore the relation among volume, curvature and properness of an m m -dimensional isometric immersion in a Riemannian manifold. We show that, when the L p L^p -norm of the mean curvature vector is bounded for some m ≤ p ≤ ∞ m \leq p\leq \infty , and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact 2 2 -Riemannian manifold M M with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in M M . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.