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The Bernstein problem for complete Lagrangian stationary surfaces

Author: Chikako Mese
Journal: Proc. Amer. Math. Soc. 129 (2001), 573-580
MSC (1991): Primary 58E12; Secondary 53C15
Published electronically: July 27, 2000
MathSciNet review: 1707155
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In this paper, we investigate the global geometric behavior of lagrangian stationary surfaces which are lagrangian surfaces whose area is critical with respect to lagrangian variations. We find that if a complete oriented immersed lagrangian surface has quadratic area growth, one end and finite topological type, then it is minimal and hence holomorphic. The key to the proof is the mean curvature estimate of Schoen and Wolfson combined with the observation that a complete immersed surface of quadratic area growth, finite topology and $L^2$ mean curvature has finite total absolute curvature.

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Additional Information

Chikako Mese
Affiliation: Department of Mathematics DRB155, University of Southern California, 1042 West 36th Place, Los Angeles, California 90089
Address at time of publication: Box 5657, Department of Mathematics, Connecticut College, 270 Mohegan Ave., New London, Connecticut 06320

Received by editor(s): April 12, 1999
Published electronically: July 27, 2000
Additional Notes: The author would like to thank Professor Richard Schoen for introducing her to this problem and Professor Paul Yang for his interest in this work. Additionally, she thanks Professor Francis Bonahon for many useful conversations.
Communicated by: Bennett Chow
Article copyright: © Copyright 2000 American Mathematical Society