Spectral invariants and the length minimizing property of Hamiltonian paths

Abstract

In this paper we provide a criterion for the quasi-autonomous Hamiltonian path ("Hofer's geodesic") on arbitrary closed symplectic manifolds (M, omega) to be length minimizing in its homotopy class in terms of the spectral invariants rho(G; 1) that the author has recently constructed. As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in its homotopy class with fixed ends, as long as it has no contractible periodic orbits of period one and it has a maximum and a minimum that are generically under-twisted, and all of its critical points are non-degenerate in the Floer theoretic sense.