ABSTRACT

We describe the mapping geometry of harmonic maps from a surface into the hyperbolic plane, using the geometry of the associated holomorphic quadratic differential. In particular we describe a theorem that an orientation preserving harmonic map from the complex plane to the hyperbolic plane with a polynomial growth quadratic differential has to map onto an ideal convex polygon in the hyperbolic plane. We also study the conformal module of ring type domains in the complex plane and the hyperbolic plane which are related by a harmonic map. We prove a nonexistence result of harmonic diffeomorphism from the complex plane onto the hyperbolic plane under certain energy growth condition.