Harmonic mappings and moduli spaces of Riemann surfaces

Abstract

Harmonic maps constitute a powerful tool for the analysis of moduli and Teichmuller spaces of compact Riemann surfaces. These moduli and Teichmuller spaces parametrize the different conformal structures a given compact differentiable surface F of some genus p can carry, together with a topological marking in the Teichmuller case. That is, each point in the moduli space Mp or the Teichmuller space Tp corresponds to some (marked) Riemann surface of genus p, that is, to some conformal structure on F . Mp then is a quotient of Tp by the mapping class group Γp, the group of homotopy classes of oriented diffeomorphisms of F . Since some nontrivial elements of Γp have fixed points, Mp acquires some quotient singularities. However, a suitable finite covering of Mp is free from singularities, that is, a manifold. Therefore, for many aspects, the singularities of Mp can be ignored, and this will sometimes simplify our discussion. – In this review, we shall confine ourselves mostly to the case p ≥ 2 which is the most difficult and most interesting case. Harmonic maps come from Riemannian geometry. They are defined as maps h : M → N between Riemannian manifolds that minimize a certain variational integral, called the energy. Thus, they depend on the Riemannian metrics of M and N . The theory works best when the metric of N has nonpositive sectional curvature. Harmonic maps can then be applied to moduli or Teichmuller spaces in two different ways. On one hand, one can look at harmonic maps to, from, or between Riemann surfaces and study how the harmonic maps or quantities associated to them, like their energy, depend on the underlying conformal structures. This is facilitated by the fact that the harmonic maps in question are unique in their homotopy classes, as we are looking at the case of genus p ≥ 2 which implies that the Riemann surfaces can be equipped with a hyperbolic metric, that is, one with constant negative curvature. In that manner,