Geodesic length functions and the Nielsen problem. (English) Zbl 0616.53039

Wolpert, Scott A.

J. Differ. Geom. 25, 275-296 (1987).

In view of the author’s result establishing the noncompleteness of the Weil-Petersson metric for the Teichmüller space \(T_{g,n}\), the geodesical convexity of \(T_{g,n}\) was an open problem. The present paper is devoted to this question. Roughly speaking, the author shows that the geodesic structure in the Weil-Petersson geometry is quite similar to the one of a complete Riemannian metric of negative curvature.
The basic idea in the author’s approach is to substitute for completeness the use of the geodesic length functions introduced by Fricke-Klein. The paper contains a comprehensive exposition of Teichmüller theory and the Weil-Petersson geometry by means of harmonic Beltrami differentials. Regarding the geometric properties of \(T_{g,n}\) the author proves that every pair of points is joined by a unique geodesic, the exponential map is a homeomorphism and the Weil-Petersson distance is measured along geodesics. As applications he gives new proofs that \(T_{g,n}\) is Stein and of the Nielsen realization problem.

Reviewer: D.Motreanu

MSC:

53C22	Geodesics in global differential geometry
30F30	Differentials on Riemann surfaces
30F35	Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)

Keywords:

geodesical convexity; Weil-Petersson geometry; geodesic length functions; Teichmüller theory; harmonic Beltrami differentials; exponential map; Nielsen realization problem