  In this paper we consider the length minimizing properties of Hamiltonian
paths generated by quasi-autonomous Hamiltonians on symplectically aspherical
manifolds. Motivated by the work of L. Polterovich and M. Schwarz, we study the
role of the fixed global extrema in the Floer complex of the generating
Hamiltonian. Our main result determines a natural condition on a fixed global
maximum of a Hamiltonian 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 periodic orbits with period one and action greater than
the fixed extrema. 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 recent theorem of D. McDuff which states that
quasi-autonomous Hamiltonians generate length minimizing paths over short time
intervals.
