Parametrizations of Teichmüller space and its Thurston boundary.(English)Zbl 1044.32005

Hildebrandt, Stefan (ed.) et al., Geometric analysis and nonlinear partial differential equations. Berlin: Springer (ISBN 3-540-44051-8/hbk). 81-88 (2003).
Let $$T_g$$ $$(g\geq 2)$$ denote the Teichmüller space of hyperbolic metrics on a closed surface of genus $$g$$, and let us denote its boundary by $$\partial T_g$$. The question dealt with here is to find the minimum number of curves needed to obtain an imbedding of $$\partial T_g$$ into the real projective space.
The number $$9g$$-$$9$$ is well known to do this. P. Schmutz [Comment. Math. Helv. 68, 278–288 (1993; Zbl 0790.30036)] showed that for every $$g\geq 2$$, the Teichmüller space $$T_g$$ can be parametrized by $$6g$$-$$5$$ length functions of simple closed geodesics. Here the author gives another proof of this result by simplifying the proof of Schmutz.
For the entire collection see [Zbl 1007.00073].

MSC:

 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30F60 Teichmüller theory for Riemann surfaces

Zbl 0790.30036