The moduli space of Riemann surfaces is Kähler hyperbolic. (English) Zbl 0988.32012

McMullen, Curtis T.

Ann. Math. (2) 151, No. 1, 327-357 (2000).

Let \({\mathcal M}_{g,n}\) be the moduli space of \(n\)-pointed Riemann surfaces of genus \(g\). The Teichmüller space \({\mathcal T}_{g,n}\) is the universal cover of \({\mathcal M}_{g,n}\). \({\mathcal T}_{g,n}\) has two natural metrics: the Teichmüller metric and the Weil-Petersson metric (defined using the hyperbolic metric on the surface).
The Teichmüller metric is complete but it is not Riemannian when \(\dim {\mathcal T}_{g,n}>1\). On the other hand, the Weil-Petersson metric is Kähler but it is not complete.
The main result of the paper says that the Teichmüller metric on the moduli space \({\mathcal M}_{g,n}\) is comparable to a Kähler metric which is Kähler hyperbolic in the sense of M. Gromov [see J. Differ. Geom. 33, No. 1, 263-292 (1991; Zbl 0719.53042)].
This theorem has several nice applications, e.g., the following complex isoperimetric inequality. There exists a linear bound for the volume of any even-dimensional compact submanifold of \({\mathcal T}_{g,n}\) in terms of the volume of the boundary. Let \({\mathcal M}_{g,n}\) be the moduli space of \(n\)-pointed Riemann surfaces of genus \(g\). The Teichmüller space \({\mathcal T}_{g,n}\) is the universal cover of \({\mathcal M}_{g,n}\). \({\mathcal T}_{g,n}\) has two natural metrics: the Teichmüller metric and the Weil-Petersson metric (defined using the hyperbolic metric on the surface).

Reviewer: Adrian Langer (Coventry)

MSC:

32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H10	Families, moduli of curves (algebraic)
14H55	Riemann surfaces; Weierstrass points; gap sequences
32Q15	Kähler manifolds

Keywords:

Teichmüller space; moduli space of Riemann surfaces; hyperbolic manifolds; Bers embedding

Citations:

Zbl 0719.53042