Teichmüller space exhibits several properties characteristic of hyperbolic spaces, such as: Some geodesics behave like they do in hyperbolic space. Random walks on Teichmüller space converge almost surely to a point on the Thurston boundary. Some of these features can be explained by the study of maps from Teichmüller space to the curve complex, which is known to be hyperbolic. Complex geometry The Bers embedding gives a complex structure as an open subset of Metrics coming from the complex structure Since Teichmüller space is a complex manifold it carries a Carathéodory metric. Teichmüller space is Kobayashi hyperbolic and its Kobayashi metric coincides with the Teichmüller metric. This latter result is used in Royden's proof that the mapping class group is the full group of isometries for the Teichmüller metric. The Bers embedding realises Teichmüller space as a domain of holomorphy and hence it also carries a Bergman metric. Kähler metrics on Teichmüller space The Weil–Petersson metric is Kähler but it is not complete. Cheng and Yau showed that there is a unique complete Kähler–Einstein metric on Teichmüller space. It has constant negative scalar curvature. Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by that is Kähler-hyperbolic. Equivalence of metrics With the exception of the incomplete Weil–Petersson metric, all metrics on Teichmüller space introduced here are quasi-isometric to each other. See also Moduli of algebraic curves p-adic Teichmüller theory Inter-universal Teichmüller theory Teichmüller modular form References Sources Further reading The last volume contains translations of several of Teichmüller's papers. Riemann surfaces Moduli theory Differential geometry Geometric group theory