K-area, Hofer metric and geometry of conjugacy classes in Lie groups

Abstract

Given a closed symplectic manifold $(M,\omega )$ we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group $Ham(M,\omega )$ by means of the Hofer metric on $Ham(M,\omega )$. We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold $(M,\omega )$ to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of unitary matrices. As another corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by certain autonomous Hamiltonians. Our main technical tool is K-area defined for Hamiltonian fibrations over a surface with boundary in the spirit of L.Polterovich's work on Hamiltonian fibrations over $S^2$.

arXiv commentCorrected final version, accepted for publication in Inventiones Mathematicae

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