Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

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



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
Published electronically: January 31, 2000
MathSciNet review: 1750956
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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.

Journals Transfer

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

  • [BG] V. Benci and F. Giannoni.
    Periodic bounce trajectories with a low number of bounce points.
    Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6:73-93, 1989. MR 90i:58161
  • [Bollo] B. Bollobas.
    Linear analysis. An introductory course.
    Cambridge Mathematical Textbooks.
    Cambridge University Press, 1990. MR 92a:46001
  • [Ca] X. Cabré.
    On the Alexandroff-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations.
    Commun. Pure Appl. Math. 48:539-570, 1995. MR 96c:35023
  • [CC] L. Caffarelli and X. Cabré.
    Fully nonlinear elliptic equations.
    American Math. Soc. Colloquium publications, vol. 43,
    Amer. Math. Soc. Providence, R.I., U.S.A. MR 96h:35046
  • [Chek] Y. Chekanov.
    Lagrangian intersections, symplectic energy and areas of holomorphic curves.
    Duke Math. J. 95:213-226, 1998. MR 99k:58034
  • [Chen] B.Y. Chen.
    Geometry of submanifolds and applications.
    Science University of Tokyo, 1980. MR 82m:53051
  • [Cr] C. Croke.
    Area and the length of the shortest closed geodesic.
    J. Differential Geometry 27:1-21, 1988. MR 89a:53050
  • [EH1] I. Ekeland and H. Hofer.
    Symplectic topology and Hamiltonian dynamics.
    Math. Z. 200:355-378, 1989. MR 90a:58046
  • [EH2] I. Ekeland and H. Hofer.
    Symplectic topology and Hamiltonian dynamics II.
    Math. Z. 203:553-567, 1990. MR 91e:58053
  • [Fer] E. Ferrand.
    Thèse de Doctorat.
    Ecole Polytechnique, Palaiseau, 1997
  • [G] M. Gromov.
    Pseudo Holomorphic Curves in symplectic Manifolds.
    Inventiones Math. 82:307-347, 1985. MR 87j:53053
  • [H1] D. Hermann.
    Thèse de Doctorat.
    Université de Paris-Sud. Orsay, 1997.
  • [H2] D. Hermann.
    Personnal Communication (in french).
  • [H3] D. Hermann.
    Non-equivalence of symplectic capacities for open sets with restricted contact type boundary.
    Preprint 1998.
  • [Ho] H. Hofer.
    On the topological properties of symplectic maps.
    Proc. R. Soc. Edinb., Sect. A 115:25-38, 1990. MR 91h:58042
  • [J] F. John.
    Extremum problems with inequalities as subsidiary conditions.
    Courant Anniversary volume, Interscience New-York 1948.
    (see also) Collected papers, Ed. by Jürgen Moser, Birkhäuser, 1985. MR 10:719b
  • [LalS] F. Lalonde and J.C. Sikorav.
    Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents.
    Comment. Math. Helvetici 66:18-33, 1991. MR 92f:58060
  • [LauS] F. Laudenbach and J. C. Sikorav.
    Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibré cotangent.
    Inventiones Math. 82:349-357, 1985. MR 87c:58042
  • [O1] Y.G. Oh.
    Second variation and stabilities of minimal Lagrangian submanifolds.
    Inventiones Math. 101:501-519, 1990. MR 91f:58022
  • [O2] Y.G. Oh.
    Volume minimization of Lagrangian submanifolds under Hamiltonian deformations.
    Math. Z. 212:175-192, 1993. MR 94a:58040
  • [Pol1] L. Polterovich.
    Symplectic displacement energy for Lagrangian submanifolds.
    Ergodic Theory Dyn. Syst. 13:357-367, 1993. MR 94h:58081
  • [Pol1] L. Polterovich.
    The surgery of Lagrange submanifolds.
    Geom. Funct. Anal. 1:198-210, 1991. MR 93D:57062
  • [Rez1] A. Reznikov.
    Affine symplectic geometry.
    Israel J. Math. 80:207-224, 1992. MR 95a:53113
  • [Rez2] A. Reznikov.
    Characteristic Classes in Symplectic Topology.
  • [San] L. A. Santalo.
    Integral geometry and geometric probability.
    Encyclopedia of Mathematics and Its Applications. Vol. 1, Addison-Wesley Publishing Company (1976). MR 55:6340
  • [Th] D. Théret.
    A complete proof of Viterbo's uniqueness theorem on generating functions.
    Topology And Its Applications 96-3:249-266, 1999. CMP 2000:01
  • [V1] C. Viterbo.
    Symplectic topology as the geometry of generating functions.
    Math. Annalen 292:685-710, 1992. MR 93b:58058
  • [V2] C. Viterbo.
    Capacités symplectiques et applications.
    Séminaire Bourbaki, Juin 89, Exposé 714.
    Astérisque, 177-178, 1990. MR 91m:58069
  • [V3] C. Viterbo.
    A new obstruction to embedding Lagrangian tori.
    Invent. Math. 100:301-320, 1990. MR 91d:58085

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 53C15, 58F05, 49Q99, 58F22, 58E10

Retrieve articles in all journals with MSC (1991): 53C15, 58F05, 49Q99, 58F22, 58E10

Additional Information

Claude Viterbo
Affiliation: Département de Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay Cedex, France

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

American Mathematical Society