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.

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