Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-03T19:33:28.377Z Has data issue: false hasContentIssue false
  • English
  • Français

ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM

Published online by Cambridge University Press:  01 September 2009

G. PACELLI BESSA  and
M. SILVANA COSTA
G. PACELLI BESSA
Affiliation:
Universidade Federal do Ceara, Brazil e-mail: [email protected]
M. SILVANA COSTA
Affiliation:
Universidade Federal do Ceara, Brazil e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 3 , September 2009 , pp. 669 - 680
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Anderson, M., The compactification of a minimal subamnifold by the Gauss map, Unpublished preprint (Institute Des Hautes Etudes Scientifiques (IHES), Bures-sur-Yvette, France, 1985).Google Scholar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.Barta, J., Sur la vibration fundamental d'une membrane, C. R. Acad. Sci. 204 (1937), 472–473.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
7.Chern, S. S. and Osserman, R., Complete minimal surfaces in euclidean n-space, J. d'Anal. Math. 19 (1967), 15–34.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
9.Jorge, L. P. and Koutroufiotis, D., An estimative for the curvature of bounded submanifolds, Amer. J. Math. 103 (1981), 711–725.CrossRefGoogle Scholar
10.Jorge, L. P. and Meeks, W. III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), 203–221.CrossRefGoogle Scholar
11.Martín, F. and Morales, S., A complete bounded minimal cylinder in 3, Michigan Math. J. 47 (3) (2000), 499–514.CrossRefGoogle Scholar
12.Nadirashvili, N., Hadamard's and Calabi–Yau's conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996), 457–465.CrossRefGoogle Scholar
13.Filho, G. Oliveira, Compactification of minimal submanifolds of hyperbolic space, Comm. Anal. Geom. 1 (1) (1993), 1–29.CrossRefGoogle Scholar
14.Osserman, R., Global properties of minimal surfaces in 3 and n, Ann. Math. 80 (1964), 340–364.CrossRefGoogle Scholar
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.White, B., Complete surfaces of finite total curvature, J. Diff. Geom. 26 (1987), 315–326.Google Scholar