Abstract
Given a complete isometric immersion φ:P m⟶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 guarantee that the immersion is proper and that P has finite topology in line with the results reported in Bessa et al. (Commun. Anal. Geom. 15(4):725–732, 2007) and Bessa and Costa (Glasg. Math. J. 51:669–680, 2009). When the ambient manifold is a radially symmetric space, an inequality is shown between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends, which generalizes the classical inequality stated in Anderson (Preprint IHES, 1984) for complete and minimal submanifolds in ℝn. As a corollary we obtain the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in hyperbolic space, together with Bernstein-type results for such submanifolds in Euclidean and hyperbolic spaces, in the manner of the work Kasue and Sugahara (Osaka J. Math. 24:679–704, 1987).
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The authors wish to thank the referee for his/her useful suggestions.
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Communicated by Marco Abate.
Work partially supported by the Caixa Castelló Foundation, and DGI grant MTM2010-21206-C02-02.
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Gimeno, V., Palmer, V. Volume Growth, Number of Ends, and the Topology of a Complete Submanifold. J Geom Anal 24, 1346–1367 (2014). https://doi.org/10.1007/s12220-012-9376-3
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DOI: https://doi.org/10.1007/s12220-012-9376-3
Keywords
- Volume growth
- Minimal submanifold
- End
- Hessian-Index comparison theory
- Extrinsic distance
- Total extrinsic curvature
- Second fundamental form
- Gap theorem
- Bernstein-type theorem