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IMMERSION OF MANIFOLDS WITH UNBOUNDED IMAGE AND A MODIFIED MAXIMUM PRINCIPLE OF YAU

Part of: Global differential geometry

Published online by Cambridge University Press:  01 October 2008

ALBERT BORBÉLY
ALBERT BORBÉLY*
Affiliation:
Faculty of Science, Department of Mathematics and Computer Science, Kuwait University, PO Box 5969, Safat 13060, Kuwait (email: borbely.albert@gmail.com)
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Abstract

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Let N be a complete Riemannian manifold isometrically immersed into a Hadamard manifold M. We show that the immersion cannot be bounded if the mean curvature of the immersed manifold is small compared with the curvature of M and the Laplacian of the distance function on N grows at most linearly. The latter condition is satisfied if the Ricci curvature of N does not approach too fast. The main tool in the proof is a modification of Yau’s maximum principle.

Keywords

immersionmean curvaturemaximum principle

MSC classification

Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 2 , October 2008 , pp. 285 - 291
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Nadirashvili, N., ‘Hadamard’s and Calabi-Yau’s conjectures on negatively curved minimal surfaces’, Invent. Math. 126 (1996), 457–465.CrossRefGoogle Scholar
[2]Peterson, P., Riemannian Geometry (Springer, New York, 1998).CrossRefGoogle Scholar
[3]Yau, S.-T., ‘Harmonic functions on complete Riemannian manifolds’, Comm. Pure Appl. Math. 28 (1975), 201–228.CrossRefGoogle Scholar