Alekseev Anton, Meinrenken Eckhard
Section de Mathématiques, Université de Genéve, Genéve, Suisse.
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4 Canada.
Commun Math Phys. 2024;405(10):229. doi: 10.1007/s00220-024-05075-7. Epub 2024 Sep 14.
A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincaré disk. Consider the infinite-dimensional Teichmüller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on . We prove that these Teichmüller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group of diffeomorphisms of the boundary. We give an explicit formula for the Hill potential on the boundary defining the moment map. Furthermore, using Fenchel-Nielsen parameters we prove a Wolpert formula for the symplectic form, leading to global Darboux coordinates on the Teichmüller space.
具有边界的曲面上的双曲0-度量是其内部的双曲度量，展现出庞加莱圆盘上标准度量的边界行为。考虑具有边界的定向曲面上双曲0-度量的无穷维泰希米勒空间，直至固定边界且同伦于恒等映射的微分同胚。我们证明这些空间具有自然的辛结构，仅取决于 上不变度量的选择。我们证明这些泰希米勒空间对于边界微分同胚群的万有覆盖的作用是哈密顿维拉索罗空间。我们给出了定义矩映射的边界上的希尔势的显式公式。此外，利用芬切尔 - 尼尔森参数，我们证明了辛形式的沃尔珀特公式，从而得到泰希米勒空间上的全局达布坐标。
