Nous étudions dans cette note les chemins de difféomorphismes engendrés par des hamiltoniens quasi-autonomes sur des variétés symplectiquement asphériques. Motivés par le travail de Polterovich et Schwarz , nous examinons le rôle des extrema globaux et fixes au cours du temps dans le complexe de Floer de l’hamiltonien. Notre principal résultat donne une condition suffisante naturelle pour que l’isotopie hamiltonienne minimise la partie positive de la norme de Hofer. On en déduit qu’un hamiltonien quasi- autonome engendre une isotopie minimisant la norme de Hofer s’il a des extrema globaux fixés qui sont “sous-tendus" et n’a aucune orbite contractile de période et d’action hors de l’intervalle . Ceci nous permet de construire de nouveaux exemples d’hamiltoniens autonomes qui induisent des flots minimisant pour tous les temps. Ces constructions sont basées sur la géométrie des variétés co-isotropes. On donne enfin une nouvelle preuve du fait que tout Hamiltonien quasi-autonome engendre une isotopie minimisante sur un intervalle suffisamment petit.
In this note we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of Polterovich and Schwarz, we study the role, in the Floer complex of the generating Hamiltonian, of the global extrema which remain fixed as the time varies. Our main result determines a natural condition which implies that the corresponding path minimizes the positive Hofer length. We use this to prove that a quasi-autonomous Hamiltonian generates a length minimizing path if it has under-twisted fixed global extrema and no contractible periodic orbits with period one and action outside the interval . This, in turn, allows us to produce new examples of autonomous Hamiltonian flows which are length minimizing for all times. These constructions are based on the geometry of coisotropic submanifolds. Finally, we give a new proof of the fact that quasi-autonomous Hamiltonians generate length minimizing paths over short time intervals.
@article{AIF_2003__53_5_1503_0,
author = {Kerman, Ely and Lalonde, Fran\c cois},
title = {Length minimizing Hamiltonian paths for symplectically aspherical manifolds},
journal = {Annales de l'Institut Fourier},
volume = {53},
year = {2003},
pages = {1503-1526},
doi = {10.5802/aif.1986},
mrnumber = {2032941},
zbl = {02014684},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2003__53_5_1503_0}
}
Kerman, Ely; Lalonde, François. Length minimizing Hamiltonian paths for symplectically aspherical manifolds. Annales de l'Institut Fourier, Tome 53 (2003) pp. 1503-1526. doi : 10.5802/aif.1986. http://gdmltest.u-ga.fr/item/AIF_2003__53_5_1503_0/
[BP] Geodesics of Hofer's metric on the group of Hamiltonian diffeomorphisms, Duke Math. J., Tome 76 (1994), pp. 273-292 | MR 1301192 | Zbl 0819.58006
[En] K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math, Tome 146 (2001), pp. 93-141 | Article | MR 1859019 | Zbl 1039.53099
[Fl1] Morse Theory for Lagrangian intersections, J. Diff. Geom., Tome 28 (1988), pp. 513-547 | MR 965228 | Zbl 0674.57027
[Fl2] Witten's complex and infinite dimensional Morse Theory, J. Diff. Geom., Tome 30 (1989), pp. 202-221 | Zbl 0678.58012
[Fl3] Symplectic fixed points and holomorphic spheres, Commun. Math. Phys, Tome 120 (1989), pp. 575-611 | Article | MR 987770 | Zbl 0755.58022
[Ho1] On the topological properties of symplectic maps, Proc. Royal Soc. Edinburgh, Tome 115 (1990), pp. 25-38 | Article | MR 1059642 | Zbl 0713.58004
[Ho2] Estimates for the energy of a symplectic map, Comment. Math. Helv, Tome 68 (1993), pp. 48-72 | Article | MR 1201201 | Zbl 0787.58017
[HS] Floer homology and Novikov rings, The Floer memorial volume, Birkhäuser (Progress in Mathematics) Tome 133 (1995), pp. 483-524 | Zbl 0842.58029
[KL2] Floer homology and length minimizing Hamiltonian paths (in preparation)
[La] Floer and Quantum homologies in fibrations over surfaces (in preparation)
[LM1] Hofer's -geometry: Energy and stability of Hamiltonian flows, part I, Invent. Math, Tome 122 (1995), pp. 1-33 | Article | MR 1354953 | Zbl 0844.58020
[LM2] Hofer's -geometry: Energy and stability of Hamiltonian flows, part II, Invent. Math, Tome 122 (1995), pp. 35-69 | Article | MR 1354953 | Zbl 0844.58020
[LP] Symplectic diffeomorphisms as isometries of Hofer's norm, Topology, Tome 3 (1997), pp. 711-727 | Article | MR 1422431 | Zbl 0874.53028
[Mc] Geometric variants of the Hofer norm (2002) (Preprint) | MR 1959582 | Zbl 1037.37033
[McSl] Hofer-Zehnder capacity and length minimizing Hamiltonian paths, Geom. Topol, Tome 5 (2001), pp. 799-830 | Article | MR 1871405 | Zbl 1002.57056
[Oh1] Symplectic topology as the geometry of action functional, Journ. Diff. Geom, Tome 46 (1997), pp. 499-577 | MR 1484890 | Zbl 0926.53031
[Oh2] Symplectic topology as the geometry of action functional II, Comm. Anal. Geom, Tome 7 (1999), pp. 1-55 | MR 1674121 | Zbl 0966.53055
[Oh3] Chain level Floer theory and Hofer's geometry of the Hamiltonian Diffeomorphism group (2001) (preprint) | Zbl 1038.53084
[Po] The geometry of the group of symplectomorphisms, Birkhäuser (2001) | MR 1826128
[Sc] On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math, Tome 193 (2000), pp. 419-461 | Article | MR 1755825 | Zbl 1023.57020
[Se] of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal, Tome 7 (1997), pp. 1046-1095 | Article | MR 1487754 | Zbl 0928.53042
[Si] New minimal geodesics in the group of symplectic diffeomorphisms, Calc. Var, Tome 3 (1995), pp. 299-309 | Article | MR 1385290 | Zbl 0828.58028
[Us] Conjugate points on geodesics of Hofer's metric, Diff. Geometry Appl, Tome 6 (1996), pp. 327-342 | Article | MR 1422339 | Zbl 0864.58004
[Vi] Symplectic topology as the geometry of generating functions, Math. Annalen, Tome 292 (1992), pp. 685-710 | Article | MR 1157321 | Zbl 0735.58019