Harmonic maps of the moduli space of compact Riemann surfaces
type: Item
instance of: scholarly article
title: Harmonic maps of the moduli space of compact Riemann surfaces
zbMATH Open document ID: 0601.32025
DOI: 10.1007/BF01458616
author: Georg Schumacher
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164161
review text: 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.
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 58C05
Mathematics Subject Classification ID: 53A45
zbMATH DE Number: 3969063
zbMATH Keywords: Siu's curvature condition
zbMATH Keywords: Riemann surface with nodes
zbMATH Keywords: Weil-Petersson metric
zbMATH Keywords: harmonic map
zbMATH Keywords: completion of moduli space
zbMATH Keywords: strong rigidity
MaRDI profile type: Publication
cites work: Curvature properties of Teichmüller's space
cites work: Q4046382
cites work: Theta functions on Riemann surfaces
cites work: The extension of the Weil-Petersson metric to the boundary of Teichmüller space
cites work: The complex-analyticity of harmonic maps and the strong rigidity of compact Kaehler manifolds
description: scientific article; zbMATH DE number 3969063