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

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.

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) |