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On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space. (English) Zbl 1129.30030

Papadopoulos, Athanase (ed.), Handbook of Teichmüller theory. Volume I. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-029-6/hbk). IRMA Lectures in Mathematics and Theoretical Physics 11, 111-204 (2007).
In this relatively long paper, the authors consider connected orientable Riemann surfaces of genus \(g\) at least 2. The Teichmüller space of such a Riemann surface carries several metrics. In this paper two of these metrics, namely the Teichmüller’s metric and the Thurston’s asymmetric metric, which are Finsler metrics, have been studied. On these metric spaces, the respective norms have been defined. In the rest of this paper, some questions regarding these metrics are discussed to determine the differences and similarities.
For the entire collection see [Zbl 1113.30038].

MSC:

30F60 Teichmüller theory for Riemann surfaces
57M50 General geometric structures on low-dimensional manifolds
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Keywords:

moduli space
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