Spectral invariants and 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 (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.

The hypothesis in Theorem II is replaced by a more restricted condition of ``nondegeneracy in the Floer theoretic sense''

References in corpus (1)

• Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group

Cited by in corpus (5)

• Spectral invariants, analysis of the Floer moduli spaces and geometry of the Hamiltonian diffeomorphism group
• Length minimizing property, Conley-Zehnder index and $C^1$-perturbations of Hamiltonian functions
• Hofer's geometry and Floer theory under the quantum limit
• Floer homology in symplectic geometry and in mirror symmetry
• Spectral invariants in Lagrangian Floer theory