Symplectic Capacities in Manifolds
Künzle, Alfred
Banach Center Publications, Tome 38 (1997), p. 77-87 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:208681 
@article{bwmeta1.element.bwnjournal-article-bcpv39z1p77bwm,
     author = {K\"unzle, Alfred},
     title = {Symplectic Capacities in Manifolds},
     journal = {Banach Center Publications},
     volume = {38},
     year = {1997},
     pages = {77-87},
     zbl = {0949.37030},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv39z1p77bwm}
}

Künzle, Alfred. Symplectic Capacities in Manifolds. Banach Center Publications, Tome 38 (1997) pp. 77-87. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv39z1p77bwm/

[000] [A84] J. P. Aubin, L'analyse non linéaire et ses motivations économiques, Masson, Paris, 1984. | Zbl 0551.90001

[001] [CE80] F. H. Clarke, I. Ekeland, Hamiltonian trajectories with prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), 103-116. | Zbl 0403.70016

[002] [E90] I. Ekeland, Convexity methods in Hamiltonian mechanics, Springer, Berlin-Heidelberg, 1990.

[003] [EH89] I. Ekeland, H. Hofer, Symplectic Topology and Hamiltonian Dynamics I, Math. Z. 200 (1989), 355-378. See also C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 37-40. | Zbl 0641.53035

[004] [EH90] I. Ekeland, H. Hofer, Symplectic Topology and Hamiltonian Dynamics II, Math. Z. 203 (1990), 553-567. | Zbl 0729.53039

[005] [G85] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. | Zbl 0592.53025

[006] [H91] M. Herman, Exemples de flots Hamiltoniens dont aucune perturbation en topologie C∞ n’a d’orbites périodiques sur un ouvert de surfaces d’énergies, C. R. Acad. Sci. Sér. I Math. 312 (1991), 989-994. | Zbl 0759.58016

[007] [HZ87] H. W. Hofer, E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math. 90 (1987), 1-9. | Zbl 0631.58022

[008] [HZ90] H. W. Hofer, E. Zehnder, A new capacity for symplectic manifolds, in: Analysis et cetera, Academic Press, 1990, 405-429.

[009] [HZ94] H. Hofer, E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, Basel-Boston-Berlin, 1994. | Zbl 0805.58003

[010] [K90] A. F. Künzle, Une capacité symplectique pour les ensembles convexes et quelques applications, Ph.D. thesis, Université Paris IX Dauphine, June 1990.

[011] [K91] A. F. Künzle, The least characteristic action as symplectic capacity, preprint, Forschungsinstitut für Mathematik, ETH Zürich, May 1991.

[012] [K93] A. F. Künzle, Singular Hamiltonian systems and Symplectic Capacities, in: Singularities and Differential Equations, Banach Center Publ. 33 (1996), 171-187. | Zbl 0855.58027

[013] [R70] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton N.J., 1970.

[014] [Si93] K. F. Siburg, Symplectic capacities in two dimensions, Manuscripta Math. 78 (1993), 149-163. | Zbl 0791.58043

[015] [Si90] J. C. Sikorav, Systèmes Hamiltoniens et topologie symplectique, Lecture Notes, Dipartimento di Matematica dell'Università di Pisa, August 1990.

[016] [V87] C. Viterbo, A proof of the Weinstein conjecture in ℝ2n, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 337-357.