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

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Abstract

Given a closed symplectic manifold (M, ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M, ω) by means of the Hofer metric on Ham (M, ω). We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold (M, ω) 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 S2.

Original languageEnglish
Pages (from-to)93-141
Number of pages49
JournalInventiones Mathematicae
Volume146
Issue number1
DOIs
StatePublished - 2001

ASJC Scopus subject areas

  • General Mathematics

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