Length minimizing Hamiltonian paths for symplectically aspherical manifolds

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 Polterovioh 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 [script A sign(Q), script A sign(P)]. 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.