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

• 成果类型: Article
• 署名作者: Entov, M
• 署名单位: Weizmann Institute of Science; Tel Aviv University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s002220100161
• 发表日期: 2001
• 页码: 93-141
• 关键词: mathematical-theory; morse-theory; conjecture; HOMOLOGY; energy

摘要

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-holomorpbic 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.