Harmonic maps, length, and energy in Teichmüller space

Abstract

The limiting behavior of high-energy harmonic maps between closed hyperbolic surfaces is analyzed. In general a measured foliation on the domain is shown to be mapped very nearly (exponentially in the energy) to its geodesic representative in the range. This foliation is in fact the horizontal foliation h of the Hopf differential of the harmonic map. h is also characterized as nearly maximizing (up to an additive constant) the ratio of squared hyperbolic length in the range to extremal length in the domain, among all simple closed curves in the domain. The same ratio gives the energy of the map up to an additive constant. This can be viewed as an analogy to other canonical maps between surfaces, for which different optimization problems are characterized by corresponding length-ratio maximizations.