K-area, Hofer metric and geometry of conjugacy classes in Lie groups
K-area, Hofer metric and geometry of conjugacy classes in Lie groups
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李群中共轭类的 K 面积、Hofer 度量和几何
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发表时间：
2000
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通讯作者：
Michael Entov
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作者：
Michael Entov
给定一个闭辛流形（M，ω），我们借助于Ham（M，ω）上的霍费尔度量，在Ham（M，ω）的泛覆盖中引入一个与共轭类元组相关联的量。我们利用伪全纯曲线来估计辛流形（M，ω）的Floer上同调上的乘法结构，并将其表示为相关Hamilton流的周期轨道的作用量.作为推论，我们得到了酉矩阵乘积特征值的Agnihotri-Belkale-Woodward不等式的一种新的证明方法。作为另一个推论，我们得到了由某些自治Hamilton算子生成的Hamilton流的测地线性质（关于霍费尔度量）的一个新的证明。我们的主要技术工具是K-面积定义的哈密顿纤维在一个表面上的精神L。Polterovich关于S2上Hamilton纤维化的工作。
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.