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An application of Moser iteration to complete minimal submanifolds in a sphere

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Leung-Fu Cheung  and
Pui-Fai Leung
Leung-Fu Cheung
Affiliation:
Department of Applied MathematicsThe Hong Kong Polytechnic University, Hung Hom Kowloon, Hongkong e-mail: malfcheu@hkpu07.polyu.edu.hk
Pui-Fai Leung
Affiliation:
Department of MathematicsNational University of SingaporeLower Kent Ridge RoadSingapore 119260, Singapore e-mail: matfredl@nus.edu.sg
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Abstract

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We apply the Moser iteration method to obtain a pointwise bound on the norm of the second fundamental form from a bound on its Ln norm for a complete minimal submanifold in a sphere. As an application we show that a complete minimal submanifold in a sphere with finite total curvature and Ricci curvature bounded away from -∞ must be compact. This complements similar results of Osserman and Oliveira in the case the ambient space is the Euclidean and the hyperbolic space respectively.

MSC classification

Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 2 , April 2004 , pp. 151 - 166
Copyright
Copyright © Australian Mathematical Society 2004

References

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