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

TL;DR

This paper establishes conditions under which complete submanifolds in certain Riemannian manifolds have finite topology and relates volume growth to the number of ends, extending classical inequalities and providing Bernstein type results.

Contribution

It generalizes volume growth and topology relations for submanifolds in radially symmetric spaces, extending classical inequalities and Bernstein theorems.

Findings

Derived conditions ensuring properness and finite topology of submanifolds.

Established inequalities linking volume growth and number of ends in symmetric spaces.

Extended Bernstein type results for minimal submanifolds in Euclidean and Hyperbolic spaces.

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 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).