Length minimizing Hamiltonian paths for symplectically aspherical manifolds

Kerman, Ely; Lalonde, François

Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1503-1526.

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 $P,Q$ globaux fixés qui sont “sous-tendus" et n’a aucune orbite contractile de période $1$ et d’action hors de l’intervalle $\left[𝒜\right(Q),𝒜(P\left)\right]$. 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 $P,Q$ and no contractible periodic orbits with period one and action outside the interval $\left[𝒜\right(Q),𝒜(P\left)\right]$. 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.

Classification : 37J05, 53D35, 53D40, 58B20
Keywords: Hofer's geometry, Hamiltonian diffeomorphism, Floer homology, length minimizing paths, coisotropic submanifolds
Mot clés : géométrie de Hofer, difféomorphismes hamiltoniens, homologie de Floer, difféotopies minimisantes, sous-variétés co-isotropes

[BP] M. Bialy; L. Polterovich Geodesics of Hofer's metric on the group of Hamiltonian diffeomorphisms, Duke Math. J., Volume 76 (1994), pp. 273-292 | MR | Zbl

[En] M. Entov K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math, Volume 146 (2001), pp. 93-141 | DOI | MR | Zbl

[Fl1] A. Floer Morse Theory for Lagrangian intersections, J. Diff. Geom., Volume 28 (1988), pp. 513-547 | MR | Zbl

[Fl2] A. Floer Witten's complex and infinite dimensional Morse Theory, J. Diff. Geom., Volume 30 (1989), pp. 202-221 | Zbl

[Fl3] A. Floer Symplectic fixed points and holomorphic spheres, Commun. Math. Phys, Volume 120 (1989), pp. 575-611 | DOI | MR | Zbl

[Ho1] H. Hofer On the topological properties of symplectic maps, Proc. Royal Soc. Edinburgh, Volume 115 (1990), pp. 25-38 | DOI | MR | Zbl

[Ho2] H. Hofer Estimates for the energy of a symplectic map, Comment. Math. Helv, Volume 68 (1993), pp. 48-72 | DOI | MR | Zbl

[HS] H. Hofer; D. Salamon; (H. Hofer, C. Taubes Floer homology and Novikov rings, The Floer memorial volume (Progress in Mathematics), Volume 133 (1995), pp. 483-524 | Zbl

[KL2] E. Kerman; F. Lalonde Floer homology and length minimizing Hamiltonian paths (in preparation)

[La] F. Lalonde Floer and Quantum homologies in fibrations over surfaces (in preparation)

[LM1] F. Lalonde; D. McDuff Hofer's $L^{\infty }$-geometry: Energy and stability of Hamiltonian flows, part I, Invent. Math, Volume 122 (1995), pp. 1-33 | DOI | MR | Zbl

[LM2] F. Lalonde; D. McDuff Hofer's $L^{\infty }$-geometry: Energy and stability of Hamiltonian flows, part II, Invent. Math, Volume 122 (1995), pp. 35-69 | DOI | MR | Zbl

[LP] F. Lalonde; L. Polterovich Symplectic diffeomorphisms as isometries of Hofer's norm, Topology, Volume 3 (1997), pp. 711-727 | DOI | MR | Zbl

[Mc] D. McDuff Geometric variants of the Hofer norm (2002) (Preprint) | MR | Zbl

[McSl] D. McDuff; J. Slimowitz Hofer-Zehnder capacity and length minimizing Hamiltonian paths, Geom. Topol, Volume 5 (2001), pp. 799-830 | DOI | MR | Zbl

[Oh1] Y.-G. Oh Symplectic topology as the geometry of action functional, Journ. Diff. Geom, Volume 46 (1997), pp. 499-577 | MR | Zbl

[Oh2] Y.-G. Oh Symplectic topology as the geometry of action functional II, Comm. Anal. Geom, Volume 7 (1999), pp. 1-55 | MR | Zbl

[Oh3] Y.-G. Oh Chain level Floer theory and Hofer's geometry of the Hamiltonian Diffeomorphism group (2001) (preprint) | Zbl

[Po] L. Polterovich The geometry of the group of symplectomorphisms, Birkhäuser, 2001 | MR

[Sc] M. Schwarz On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math, Volume 193 (2000), pp. 419-461 | DOI | MR | Zbl

[Se] P. Seidel ${\pi }_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal, Volume 7 (1997), pp. 1046-1095 | DOI | MR | Zbl

[Si] K.F. Siburg New minimal geodesics in the group of symplectic diffeomorphisms, Calc. Var, Volume 3 (1995), pp. 299-309 | DOI | MR | Zbl

[Us] I. Ustilovsky Conjugate points on geodesics of Hofer's metric, Diff. Geometry Appl, Volume 6 (1996), pp. 327-342 | DOI | MR | Zbl

[Vi] C. Viterbo Symplectic topology as the geometry of generating functions, Math. Annalen, Volume 292 (1992), pp. 685-710 | DOI | MR | Zbl