Spectral invariants and length minimizing property of Hamiltonian paths

TL;DR

This paper establishes a criterion using spectral invariants for certain Hamiltonian paths to be length minimizing in their homotopy class, with applications to autonomous paths on symplectic manifolds.

Contribution

It introduces a new criterion based on spectral invariants for length minimizing Hamiltonian paths and applies it to autonomous paths with specific dynamical properties.

Findings

Autonomous Hamiltonian paths without contractible period-one orbits are length minimizing.

Spectral invariants can determine length minimizing properties of Hamiltonian paths.

The results extend previous work on spectral invariants and Hamiltonian dynamics.

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 (math.SG/0206092). As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in {\it its homotopy class} with fixed ends, when it has no contractible periodic orbits {\it 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.