Mean curvature, volume and properness of isometric immersions
Abstract
We explore the relation among volume, curvature and properness of a $m$dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, 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 noncompact $2$Riemannian manifold $M$ with nonpositive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in $M$ . 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 arXiv:
 arXiv:1504.00055
 Bibcode:
 2015arXiv150400055G
 Keywords:

 Mathematics  Differential Geometry
 EPrint:
 17 pages, 1 figure