Geodesic
ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM
Glasgow mathematical journal, Tome 51 (2009) no. 3, pp. 669-680

Voir la notice de l'article provenant de la source Cambridge University Press

Based on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725–732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 is proper (compact if N is compact). In addition, if N is Hadamard, then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below. 
BESSA, G. PACELLI; COSTA, M. SILVANA. ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM. Glasgow mathematical journal, Tome 51 (2009) no. 3, pp. 669-680. doi: 10.1017/S0017089509990085
@article{10_1017_S0017089509990085,
     author = {BESSA, G. PACELLI and COSTA, M. SILVANA},
     title = {ON {SUBMANIFOLDS} {WITH} {TAMED} {SECOND} {FUNDAMENTAL} {FORM}},
     journal = {Glasgow mathematical journal},
     pages = {669--680},
     year = {2009},
     volume = {51},
     number = {3},
     doi = {10.1017/S0017089509990085},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089509990085/}
}
TY  - JOUR
AU  - BESSA, G. PACELLI
AU  - COSTA, M. SILVANA
TI  - ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM
JO  - Glasgow mathematical journal
PY  - 2009
SP  - 669
EP  - 680
VL  - 51
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089509990085/
DO  - 10.1017/S0017089509990085
ID  - 10_1017_S0017089509990085
ER  - 
%0 Journal Article
%A BESSA, G. PACELLI
%A COSTA, M. SILVANA
%T ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM
%J Glasgow mathematical journal
%D 2009
%P 669-680
%V 51
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089509990085/
%R 10.1017/S0017089509990085
%F 10_1017_S0017089509990085

[1] 1.Anderson, M., The compactification of a minimal subamnifold by the Gauss map, Unpublished preprint (Institute Des Hautes Etudes Scientifiques (IHES), Bures-sur-Yvette, France, 1985). Google Scholar

[2] 2.Barta, J., Sur la vibration fundamental d'une membrane, C. R. Acad. Sci. 204 (1937), 472–473. Google Scholar

[3] 3.Bessa, G. P. and Fabio Montenegro, J., An extension of Barta's theorem and geometric applications, Ann. Global Anal. Geom. 31 (4) (2007), 345–362. Google Scholar | DOI

[4] 4.Bessa, G. P., Jorge, L. P. and Fabio Montenegro, J., Complete submanifolds of n with finite topology, Comm. Anal. Geom. 15 (4) (2007), 725–732. Google Scholar | DOI

[5] 5.Bessa, G. P. and Costa, M. Silvana, Eigenvalue estimates for submanifolds with locally bounded mean curvature in N ×, Proc. Amer. Math. Soc. 137 (2009), 1093–1102. Google Scholar | DOI

[6] 6.Cheeger, J., Critical points of distance functions and applications to geometry, in Geometric topology: Recent developments, Lecture Notes in Mathematics, vol. 1504 (Morel, J.-M., Takens, F. and Teissier, B., Editors) (Springer, Berlin, 1991), 1–38. Google Scholar | DOI

[7] 7.Chern, S. S. and Osserman, R., Complete minimal surfaces in euclidean n-space, J. d'Anal. Math. 19 (1967), 15–34. Google Scholar | DOI

[8] 8.Dubrovin, B. A., Fomenko, A. T. and Novikov, S. P., Modern geometry – methods and applications. Part II. The geometry and topology of manifolds (Axler, S. and Ribet, K. A., Editors) (Graduate Texts in Mathematics, vol. 104, Springer-Verlag, New York, Heidelberg, Tokyo, 1985, ISBN 0-387-96162-3). Google Scholar | DOI

[9] 9.Jorge, L. P. and Koutroufiotis, D., An estimative for the curvature of bounded submanifolds, Amer. J. Math. 103 (1981), 711–725. Google Scholar | DOI

[10] 10.Jorge, L. P. and Meeks, W. III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), 203–221. Google Scholar | DOI

[11] 11.Martín, F. and Morales, S., A complete bounded minimal cylinder in 3, Michigan Math. J. 47 (3) (2000), 499–514. Google Scholar | DOI

[12] 12.Nadirashvili, N., Hadamard's and Calabi–Yau's conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996), 457–465. Google Scholar | DOI

[13] 13.Filho, G. Oliveira, Compactification of minimal submanifolds of hyperbolic space, Comm. Anal. Geom. 1 (1) (1993), 1–29. Google Scholar | DOI

[14] 14.Osserman, R., Global properties of minimal surfaces in 3 and n, Ann. Math. 80 (1964), 340–364. Google Scholar | DOI

[15] 15.Schoen, R. and Yau, S., Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, vol. 1 (Yau, S. T., Editor) (International Press, Somerville, MA, 1994). Google Scholar

[16] 16.White, B., Complete surfaces of finite total curvature, J. Diff. Geom. 26 (1987), 315–326. Google Scholar

Cité par Sources :