Mean curvature, volume and properness of isometric immersions

Gimeno, Vicent; Palmer, Vicente

doi:10.1090/tran/6892

Mean curvature, volume and properness of isometric immersions
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by Vicent Gimeno and Vicente Palmer PDF

Trans. Amer. Math. Soc. 369 (2017), 4347-4366 Request permission

Abstract:

We explore the relation among volume, curvature and properness of an $m$-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.

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