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Metric and isoperimetric problems in symplectic geometry


Author: Claude Viterbo
Journal: J. Amer. Math. Soc. 13 (2000), 411-431
MSC (1991): Primary 53C15; Secondary 58F05, 49Q99, 58F22, 58E10
DOI: https://doi.org/10.1090/S0894-0347-00-00328-3
Published electronically: January 31, 2000
MathSciNet review: 1750956
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Abstract:

Our first result is a reduction inequality for the displacement energy. We apply it to establish some new results relating symplectic capacities and the volume of a Lagrangian submanifold in a number of different settings. In particular, we prove that a Lagrange submanifold always bounds a holomorphic disc of area less than $C_{n}\operatorname{vol}(L)^{2/n}$, where $C_{n}$ is some universal constant. We also explain how the Alexandroff-Bakelman-Pucci inequality is a special case of the above inequalities. Our inequality on displacement of reductions is also applied to yield a relation between length of billiard trajectories and volume of the domain. Two simple results concerning isoperimetric inequalities for convex domains and the closure of the symplectic group for the $W^{1/2,2}$ norm are included.


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Additional Information

Claude Viterbo
Affiliation: Département de Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay Cedex, France
Email: viterbo@dmi.ens.fr

DOI: https://doi.org/10.1090/S0894-0347-00-00328-3
Keywords: Symplectic geometry, Lagrangian submanifolds, minimal submanifolds, isoperimetric problems, billiards
Received by editor(s): March 3, 1998
Received by editor(s) in revised form: November 18, 1999
Published electronically: January 31, 2000
Additional Notes: The author was supported also by UMR 8628 du C.N.R.S. “Topologie et Dynamique" and Institut Universitaire de France.
Article copyright: © Copyright 2000 American Mathematical Society

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