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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Metric and isoperimetric problems in symplectic geometry
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by Claude Viterbo PDF
J. Amer. Math. Soc. 13 (2000), 411-431 Request permission


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:
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 411-431
  • MSC (1991): Primary 53C15; Secondary 58F05, 49Q99, 58F22, 58E10
  • DOI:
  • MathSciNet review: 1750956