TY - JOUR

T1 - K-area, Hofer metric and geometry of conjugacy classes in Lie groups

AU - Entov, Michael

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0035600381&partnerID=8YFLogxK

U2 - 10.1007/s002220100161

DO - 10.1007/s002220100161

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AN - SCOPUS:0035600381

SN - 0020-9910

VL - 146

SP - 93

EP - 141

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -