Abstract
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.
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Acknowledgements
Vicent Gimeno: Work partially supported by the Research Program of University Jaume I Project UJI-B2016-07, and DGI -MINECO Grant (FEDER) MTM2013-48371-C2-2-P. Vicente Palmer: Work partially supported by the Research Program of University Jaume I Project UJI-B2016-07, DGI -MINECO Grant (FEDER) MTM2013-48371-C2-2-P, and Generalitat Valenciana Grant PrometeoII/2014/064. G. Pacelli Bessa: Work partially supported by CNPq- Brazil grant # 301581/2013-4.
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Bessa, G.P., Gimeno, V. & Palmer, V. Asymptotically Extrinsic Tamed Submanifolds. J Geom Anal 28, 448–472 (2018). https://doi.org/10.1007/s12220-017-9828-x
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DOI: https://doi.org/10.1007/s12220-017-9828-x