2011/12/17 by Vicent Gimeno, Vicente Palmer, Gimeno, Vicent +1
Mathematics · #53C20 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #math.DG #msc:53C20 #msc:53C42
paper · pdf · doi:10.48550/arxiv.1112.4042
20 pages
openalex publication_date 2011/12/17 · arxiv created 2012/04/19 · arxiv updated 2012/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a complete isometric immersion ϕ: Pm \longrightarrow Nn in an ambient Riemannian manifold Nn with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space Mnw, 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 \erren. 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).