Mean curvature, volume and properness of isometric immersions

TL;DR

This paper investigates how volume, curvature, and properness of isometric immersions are interconnected, establishing conditions under which properness correlates with finite volume and exploring curvature-topology relations in non-compact surfaces.

Contribution

It provides new criteria linking properness with volume finiteness under bounded mean curvature norms and relates curvature and topology of non-compact surfaces using focal point analysis.

Findings

Properness is equivalent to finite volume of extrinsic balls under certain curvature bounds.

Total absolute curvature influences the properness of surface immersions.

Focal point analysis relates curvature and topology of non-compact surfaces.

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 non-compact $2$-Riemannian manifold $M$ 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 $M$ . 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.