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Exit radii of submanifolds from cylindrical domains in warped product manifolds

Published online by Cambridge University Press:  03 June 2015

Hironori Kumura
Hironori Kumura*
Affiliation:
Department of Mathematics, Shizuoka University, Shizuoka 422-8529, Japan, (smhkumu@ipc.shizuoka.ac.jp)
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Abstract

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 559 - 569
Copyright
Copyright © Royal Society of Edinburgh 2015 

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References

1 Alías, L. J., Bessa, G. P., Montenegro, J. F. and Piccione, P.. Curvature estimates for submanifolds in warped products. Preprint, 2010. (Available at http://arxiv.org/abs/1009.3467.)Google Scholar
2 Collin, P. and Rosenberg, H.. Notes sur la démonstration de Nadirashvili des conjectures de Hadamard et Calabi–Yau. Bull. Sci. Math. 123 (1999), 563–575.Google Scholar
3 Jorge, L. and Koutroufiotis, D.. An estimate for the curvature of bounded submanifolds. Am. J. Math. 103 (1981), 711–725.Google Scholar
4 Jorge, L. and Xavier, F.. An inequality between the exterior diameter and the mean curvature of bounded immersion. Math. Z. 178 (1981), 77–82.Google Scholar
5 Kasue, A.. Applications of Laplacian and Hessian comparison theorems. In Geometry of Geodesics and Related Topics: Proc. Symp., University of Tokyo, November 29-December 3, 1982. Advanced Studies in Pure Mathematics, vol. 3, pp. 333–386 (Elsevier, 1984).Google Scholar
6 Kasue, A.. Estimates for solutions of Poisson equations and their application to submanifolds. Lecture Notes in Mathematics, vol. 1090, pp. 1–14 (Springer, 1984).Google Scholar
7 Martín, F. and Morales, S.. A complete bounded minimal cylinder in ℝ3 . Michigan Math. J. 47 (2000), 499–514.Google Scholar
8 Nadirashvili, N. S.. Hadamard’s and Calabi–Yau conjectures on negatively curved and minimal surfaces. Invent. Math. 126 (1996), 457–465.Google Scholar
9 Omori, H.. Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn 19 (1967), 205–214.Google Scholar
10 Zhou, D.. Laplace inequalities with geometric applications. Arch. Math. 67 (1996), 50–58.Google Scholar