Spectral invariants and length minimizing property of Hamiltonian paths

Details Spectral invariants and length minimizing property of Hamiltonian paths Spectral invariants and length minimizing property of Hamiltonian paths

2002-12-24

arxiv.org/abs/math/0212337v5

Asian J. Math. 9 (2005), 1--18

Mathematics

Symplectic Geometry

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

Scientific paper

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.

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