On the Weil-Petersson convex geometry of Teichmüller space. (English. Japanese original) Zbl 1393.30035
Sugaku Expo. 30, No. 2, 159-186 (2017); translation from Sūgaku 65, No. 2, 174-198 (2013).
Summary: The family of conformal structures that can be realized on a given topological surface constitute the Teichmüller space of the surface. It is known to be simply connected and finite dimensional when the surface is of finite type. We will equip the Teichmüller space with a particular Riemannian metric, the Weil-Petersson metric, and investigate the convex geometry induced from the resulting distance function. In particular, we illustrate that the Weil-Petersson geometry offers a synthetic view of the so-called augmented Teichmüller space, through the theory of the CAT(0) space. Additionally, we will consider the Weil-Petersson metric defined on the universal Teichmüller space, where each finite-dimensional Teichmüller space is isometrically embedded. In conclusion, we formulate a question concerning the synthetic Weil-Petersson geometry of the universal Teichmüller space.
MSC:
| 30F60 | Teichmüller theory for Riemann surfaces |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 30-03 | History of functions of a complex variable |
| 32-03 | History of several complex variables and analytic spaces |
| 01A60 | History of mathematics in the 20th century |
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