Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $W'$.

References [Enhancements On Off] (What's this?)

  • 1. R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, second ed., Springer-Verlag, New York, 1988. MR 89f:58001
  • 2. Adrian Butscher, Regularizing a singular special Lagrangian variety, Submitted February 2001.
  • 3. Reese Harvey and H. Blaine Lawson, Jr., Calibrated geometries, Acta Math. 148 (1982), 47-157. MR 85i:53058
  • 4. Mark Haskins, Constructing special Lagrangian cones, math.DG/0005164.
  • 5. Nigel J. Hitchin, The moduli space of special Lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, 503-515, Dedicated to Ennio De Giorgi. MR 2000c:32075
  • 6. Dominic Joyce, Constructing special Lagrangian $m$-folds in $\mathbf{C}^m$ by evolving quadrics, math.DG/0008154. Math. Ann. 320 (2001), 757-797.
  • 7. -, Evolution equations for special Lagrangian 3-folds in $\mathbf{C}^3$, math.DG/0010036. Ann. Global Anal. and Geom. 20 (2001), 345-403.
  • 8. -, Lectures on Calabi-Yau and special Lagrangian geometry, math.DG/0108088.
  • 9. -, Ruled special Lagrangian 3-folds in $\mathbf{C}^3$, math.DG/0012060. Proc. London Math. Soc. 85 (2002), 233-256.
  • 10. -, Singularities of special Lagrangian fibrations and the SYZ-conjecture, math.DG/0011179.
  • 11. -, Special Lagrangian 3-folds and integrable systems, math.DG/0101249.
  • 12. -, Special Lagrangian $m$-folds in $\mathbf{C}^m$ with symmetries, math.DG/0008021.
  • 13. Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, Second ed., The Clarendon Press Oxford University Press, New York, 1998. MR 2000g:53098
  • 14. Robert C. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), no. 4, 705-747. MR 99j:53083
  • 15. Sema Salur, Deformations of special Lagrangian submanifolds, Commun. Contemp. Math. 2 (2000), no. 3, 365-372. MR 2002g:53094
  • 16. R. Schoen and J. Wolfson, Minimizing volume among Lagrangian submanifolds, Differential Equations: La Pietra 1996 (Shatah Giaquinta and Varadhan, eds.), Proc. of Symp. in Pure Math., vol. 65, 1999, pp. 181-199. MR 99k:53130
  • 17. Richard Schoen, Lecture Notes in Geometric PDEs on Manifolds, Course given in the Spring of 1998 at Stanford University.
  • 18. Günter Schwarz, Hodge Decomposition--A Method for Solving Boundary Value Problems, Springer-Verlag, Berlin, 1995. MR 96k:58222

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58J05

Retrieve articles in all journals with MSC (2000): 58J05

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

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

American Mathematical Society