Mean curvature, volume and properness of isometric immersions
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Title
Mean curvature, volume and properness of isometric immersionsDate
2017Publisher
American Mathematical SocietyISSN
1088-6850; 0002-9947Bibliographic citation
GIMENO, Vicent; PALMER, Vicente. Mean curvature, volume and properness of isometric immersions. Transactions of the American Mathematical Society, 2017, vol. 369, no 6, p. 4347-4366Type
info:eu-repo/semantics/articlePublisher version
http://www.ams.org/journals/tran/2017-369-06/S0002-9947-2017-06892-6/Subject
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 ... [+]
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
≤
p
≤∞
, 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. [-]
Is part of
Transactions of the American Mathematical Society, 2017, vol. 369, núm. 6Rights
© 2017 American Mathematical Society.
First published in Proceedings of the American Mathematical Society in volume 369, number 6, June 2017, published by the American Mathematical Society.
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