Asymptotically extrinsic tamed submanifolds

TL;DR

This paper investigates the asymptotic structure of open submanifolds immersed in real space forms with non-positive curvature, using decay properties of the second fundamental form to describe their behavior at infinity.

Contribution

It introduces a new approach to analyze the ends of tamed submanifolds via decay of the second fundamental form and estimates their number based on extrinsic volume growth.

Findings

Description of the asymptotic structure at infinity of tamed submanifolds.

Lower bounds on the number of ends based on extrinsic volume growth.

Extension of previous structural results to a broader class of submanifolds.

Abstract

We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $\kappa \leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in the papers Internat. Math. Res. Notices 1994, no. 9, authored by R. E. Greene, P. Petersen and S. Zhou and Math. Ann. 2001, 321 (4), authored by A. Petrunin and W. Tuschmann. We shall obtain too an estimation from below of the number of its ends in terms of the volume growth of a special class of extrinsic domains, the extrinsic balls.