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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Deformations of minimal Lagrangian submanifolds with boundary


Author: Adrian Butscher
Journal: Proc. Amer. Math. Soc. 131 (2003), 1953-1964
MSC (2000): Primary 58J05
Published electronically: October 24, 2002
MathSciNet review: 1955286
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Abstract: Let $L$ be a special Lagrangian submanifold of a compact Calabi-Yau manifold $M$ with boundary lying on the symplectic, codimension 2 submanifold $W$. It is shown how deformations of $L$ which keep the boundary of $L$ confined to $W$can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian submanifolds near $L$ with boundary on $W$ is found to be finite dimensional and is parametrized over the space of harmonic 1-forms of $L$ satisfying Neumann boundary conditions. The second is that if $W'$ is a symplectic, codimension 2 submanifold sufficiently near $W$, then, under suitable conditions, there exists a minimal Lagrangian submanifold $L'$ near $L$ with boundary on $W'$.


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Additional Information

Adrian Butscher
Affiliation: Max Planck Institute for Gravitational Physics, am Muehlenberg 1, 14476 Golm Brandenburg, Germany
Address at time of publication: Department of Mathematics, University of Toronto at Scarborough, Scarborough, Ontario, Canada M1C 1A4
Email: butscher@aei-potsdam.mpg.de, butscher@utsc.utoronto.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06800-4
PII: S 0002-9939(02)06800-4
Received by editor(s): October 11, 2001
Received by editor(s) in revised form: January 24, 2002
Published electronically: October 24, 2002
Communicated by: Bennett Chow
Article copyright: © Copyright 2002 American Mathematical Society