Spectral invariants and length minimizing property of Hamiltonian paths

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In this paper we provide a criterion for the quasi-autonomous Hamiltonian path (``Hofer's geodesic'') on arbitrary closed symplectic manifolds (M,ω) to be length minimizing in its homotopy class in terms of the spectral invariants ρ(G;1) that the author has recently constructed (math.SG/0206092). 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, when it has no contractible periodic orbits of period one, has a maximum and a minimum point which are generically under-twisted and all of its critical points are nondegenerate in the Floer theoretic sense. This is a sequel to the papers math.SG/0104243 and math.SG/0206092.