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On Teichmüller spaces of surfaces with boundary. (English) Zbl 1138.32006

In the paper under review, the author first characterizes hyperbolic metrics on a triangulated surface with boundary using a variational principle. He then applies this characterization to provide a new parametrization of the Teichmüller space of a hyperbolic surface with nonempty boundary so that the lengths of the boundary components are fixed. In this new parametrization, the Teichmüller space is an open convex polytope. When a sequence of hyperbolic metrics on a surface with boundary converges to a cusped metric, the coordinate introduced in the paper converges to the simplicial coordinate introduced by R. C. Penner [Commun. Anal. Geom. 12, 793–820 (2004; Zbl 1072.32008)].

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M50 General geometric structures on low-dimensional manifolds
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)

Citations:

Zbl 1072.32008
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