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Mean curvature, volume and properness of isometric immersions


Authors: Vicent Gimeno and Vicente Palmer
Journal: Trans. Amer. Math. Soc. 369 (2017), 4347-4366
MSC (2010): Primary 53C20, 53C40; Secondary 53C42
DOI: https://doi.org/10.1090/tran/6892
Published electronically: February 8, 2017
MathSciNet review: 3624412
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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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Additional Information

Vicent Gimeno
Affiliation: Departament de Matemàtiques- IMAC, Universitat Jaume I, Castelló, Spain
Email: gimenov@uji.es

Vicente Palmer
Affiliation: Departament de Matemàtiques- INIT, Universitat Jaume I, Castellon, Spain
Email: palmer@mat.uji.es

DOI: https://doi.org/10.1090/tran/6892
Keywords: Focal distance, geodesic ray, total curvature, properness, Calabi's conjecture
Received by editor(s): October 21, 2015
Published electronically: February 8, 2017
Additional Notes: The first author’s work was partially supported by the Research Program of University Jaume I Project P1-1B2012-18, and DGI -MINECO grant (FEDER) MTM2013-48371-C2-2-P
The second author’s work was partially supported by the Research Program of University Jaume I Project P1-1B2012-18, DGI -MINECO grant (FEDER) MTM2013-48371-C2-2-P, and Generalitat Valenciana Grant PrometeoII/2014/064
Article copyright: © Copyright 2017 American Mathematical Society