Symplectic Capacities in Manifolds

Alfred Künzle

Banach Center Publications (1997)

  • Volume: 39, Issue: 1, page 77-87
  • ISSN: 0137-6934

Abstract

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Symplectic capacities coinciding on convex sets in the standard symplectic vector space are extended to any subsets of symplectic manifolds. It is shown that, using embeddings of non-smooth convex sets and a product formula, calculations of some capacities become very simple. Moreover, it is proved that there exist such capacities which are distinct and that there are star-shaped domains diffeomorphic to the ball but not symplectomorphic to any convex set.

How to cite

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Künzle, Alfred. "Symplectic Capacities in Manifolds." Banach Center Publications 39.1 (1997): 77-87. <http://eudml.org/doc/208681>.

@article{Künzle1997,
abstract = {Symplectic capacities coinciding on convex sets in the standard symplectic vector space are extended to any subsets of symplectic manifolds. It is shown that, using embeddings of non-smooth convex sets and a product formula, calculations of some capacities become very simple. Moreover, it is proved that there exist such capacities which are distinct and that there are star-shaped domains diffeomorphic to the ball but not symplectomorphic to any convex set.},
author = {Künzle, Alfred},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {77-87},
title = {Symplectic Capacities in Manifolds},
url = {http://eudml.org/doc/208681},
volume = {39},
year = {1997},
}

TY - JOUR
AU - Künzle, Alfred
TI - Symplectic Capacities in Manifolds
JO - Banach Center Publications
PY - 1997
VL - 39
IS - 1
SP - 77
EP - 87
AB - Symplectic capacities coinciding on convex sets in the standard symplectic vector space are extended to any subsets of symplectic manifolds. It is shown that, using embeddings of non-smooth convex sets and a product formula, calculations of some capacities become very simple. Moreover, it is proved that there exist such capacities which are distinct and that there are star-shaped domains diffeomorphic to the ball but not symplectomorphic to any convex set.
LA - eng
UR - http://eudml.org/doc/208681
ER -

References

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  9. [HZ90] H. W. Hofer, E. Zehnder, A new capacity for symplectic manifolds, in: Analysis et cetera, Academic Press, 1990, 405-429. 
  10. [HZ94] H. Hofer, E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, Basel-Boston-Berlin, 1994. Zbl0805.58003
  11. [K90] A. F. Künzle, Une capacité symplectique pour les ensembles convexes et quelques applications, Ph.D. thesis, Université Paris IX Dauphine, June 1990. 
  12. [K91] A. F. Künzle, The least characteristic action as symplectic capacity, preprint, Forschungsinstitut für Mathematik, ETH Zürich, May 1991. 
  13. [K93] A. F. Künzle, Singular Hamiltonian systems and Symplectic Capacities, in: Singularities and Differential Equations, Banach Center Publ. 33 (1996), 171-187. Zbl0855.58027
  14. [R70] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton N.J., 1970. 
  15. [Si93] K. F. Siburg, Symplectic capacities in two dimensions, Manuscripta Math. 78 (1993), 149-163. Zbl0791.58043
  16. [Si90] J. C. Sikorav, Systèmes Hamiltoniens et topologie symplectique, Lecture Notes, Dipartimento di Matematica dell'Università di Pisa, August 1990. 
  17. [V87] C. Viterbo, A proof of the Weinstein conjecture in ℝ 2 n , Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 337-357. 

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