Volume Growth, Number of Ends, and the Topology of a Complete Submanifold

• Published in: The Journal of geometric analysis 2014-07, Vol.24 (3), p.1346-1367
• Main Authors: Gimeno, Vicent, Palmer, Vicente
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Original Research
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-012-9376-3

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  , 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 ℝ n . 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 ).