Asymptotically Extrinsic Tamed Submanifolds

We study, from the extrinsic point of view, the structure at infinity of open submanifolds, \(\varphi :M^m \hookrightarrow \mathbb {M}^{n}(\kappa )\) isometrically immersed in the real space forms of constant sectional curvature \(\kappa \le 0\). We shall use the decay of the second fundamental form of the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in Greene et al. (Int Math Res Not 1994:364–377, 1994) and Petrunin and Tuschmann (Math Ann 321:775–788, 2001) and 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.