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...

Let $(M,J,\omega )$ be a manifold with an almost complex structure $J$ tamed by a symplectic form $\omega$. We suppose that $M$ has the complex dimension two, is Levi-convex and with bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of $M$ can be foliated by the boundaries of pseudoholomorphic discs.

We consider a compact almost complex manifold $(M,J,\omega )$ with smooth Levi convex boundary $\partial M$ and a symplectic tame form $\omega$. Suppose that $S^2$ is a real two-sphere, containing complex elliptic and hyperbolic points and generically embedded into $\partial M$. We prove a result on filling $S^2$ by holomorphic discs.