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Shearing coordinates and convexity of length functions on Teichmüller space. (English) Zbl 1286.30032

Fenchel-Nielsen coordinates are an important way to parametrize Teichmüller space, which are determined by the lengths and the twists of the components of pants decompositions. However note that there is no canonical way to determine twist parameters. Let \(X\) be a complete, finite area, hyperbolic surface of genus \(g\) with \(n\) cusps, and let \(\mathcal{P}\) be a pants decomposition of \(X\). In the paper under review, the authors show that there are Fenchel-Nielsen coordinates associated to \(\mathcal{P}\) such that the length function for any essential curve is convex. Furthermore it is strictly convex if the curve intersects all component of \(\mathcal{P}\). It is derived from the result that the length function is convex with respect to the shearing coodinates associated to a maximal lamination with finitely many leaves in \(X\).

MSC:

30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57N16 Geometric structures on manifolds of high or arbitrary dimension
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