Chow, Ben (ed.) et al., Elliptic and parabolic methods in geometry. Proceedings of a workshop, Minneapolis, MN, USA, May 23–27, 1994. Wellesley, MA: A K Peters. 57-66 (1996).
It is well-known that conformal and quasiconformal maps as well as harmonic maps are closely related to the deformation theory of Riemann surfaces. However, the mapping behavior of harmonic maps is more subtle than that of conformal and quasiconformal maps. An interesting general question is: How can one characterize the images of harmonic diffeomorphisms from the complex plane onto the Poincaré disk ? M. Wolf [Topology 30, No. 4, 517-540 (1991; Zbl 0747.58027)] and Y. Minsky [J. Differ. Geom. 35, No. 1, 151-217 (1992; Zbl 0763.53042)] have studied the geometric behavior of harmonic maps between compact hyperbolic surfaces. H. I. Choi and A. Treibergs [J. Differ. Geom. 32, No. 3, 775-817 (1990; Zbl 0717.53038)] have constructed examples of harmonic diffeomorphisms of into regions of which are ideal convex polygons.
In this paper, the author describes the mapping geometry of harmonic maps from a surface into , using the geometry of the associated holomorphic quadratic differential. The following theorem is proved: If is an orientation-preserving harmonic map whose associated quadratic differential is a polynomial, then is an ideal convex polygon.
The author also studies the conformal module of ring type domains on and which are related by a harmonic map. A nonexistence result of harmonic diffeomorphism from onto under a certain energy growth condition is shown.