Volume growth, number of ends and the topology of complete submanifolds
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comunitat-uji-handle2:10234/43662
comunitat-uji-handle3:10234/43643
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Title
Volume growth, number of ends and the topology of complete submanifoldsDate
2012Publisher
arXIVAnother identifiers
arXiv:1112.4042Type
info:eu-repo/semantics/articlePublisher version
http://link.springer.com/article/10.1007%2Fs12220-012-9376-3Version
info:eu-repo/semantics/submittedVersionSubject
Abstract
Given a complete isometric immersion $\phi: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional ... [+]
Given a complete isometric immersion $\phi: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space $M^n_w$, we determine a set of conditions on the extrinsic curvatures of $P$ that guarantees that the immersion is proper and that $P$ has finite topology, in the line of the paper "On Submanifolds With Tamed Second Fundamental Form", (Glasgow Mathematical Journal, 51, 2009), authored by G. Pacelli Bessa and M. Silvana Costa. When the ambient manifold is a radially symmetric space, it is shown an inequality between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends which generalizes the classical inequality stated in Anderson's paper "The compactification of a minimal submanifold by the Gauss Map", (Preprint IEHS, 1984), for complete and minimal submanifolds in $\erre^n$. We obtain as a corollary the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in the Hyperbolic space together with Bernstein type results for such submanifolds in Euclidean and Hyperbolic spaces, in the vein of the work due to A. Kasue and K. Sugahara "Gap theorems for certain submanifolds of Euclidean spaces and hyperbolic space forms", (Osaka J. Math. 24,1987). [-]
Description
eprint de ArXIV. Pendent de publicar a Journal of Geometric Analysis, 2012
Is part of
The Journal of Geometric Analysis, July 2014, Volume 24, Issue 3, pp 1346-1367Rights
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info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
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- INIT_Articles [747]