Harmonic maps of the moduli space of compact Riemann surfaces
Based on Weil-Petersson metric, the moduli spaces derives from quotients of Teichmüller spaces regarded as V-manifolds have a natural Kähler structure with strictly negative sectional and Ricci curvatures. The author proves that for genus greater than one, the moduli spaces has strongly negative curvature in the sense of Siu. Applying Mazur's method to estimate the metric tensor for local parameter \(t_ j\to 0\), the author proves estimates for the Christoffel symbols and the curvature tensor. This in turns proves that the sectional, Ricci and scalar curvatures of \(M_ g\) are asymptotically bounded by -\(\Sigma\) log\(| t_ i|\). Together with the estimates of second order derivatives of the metric tensor, that the harmonic map on the completion of moduli space is holomorphic or antiholomorphic is proved. These will be used to prove a strong rigidity theorem.
scientific article; zbMATH DE number 3969063