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Remarks on the geometric behavior of harmonic maps between surfaces. (English) Zbl 0871.58025
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 $\bbfC$ onto the Poincaré disk $\bbfH$? {\it M. Wolf} [Topology 30, No. 4, 517-540 (1991; Zbl 0747.58027)] and {\it 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. {\it H. I. Choi} and {\it A. Treibergs} [J. Differ. Geom. 32, No. 3, 775-817 (1990; Zbl 0717.53038)] have constructed examples of harmonic diffeomorphisms of $\bbfC$ into regions of $\bbfH$ which are ideal convex polygons. In this paper, the author describes the mapping geometry of harmonic maps from a surface into $\bbfH$, using the geometry of the associated holomorphic quadratic differential. The following theorem is proved: If $u:\bbfC\to\bbfH$ is an orientation-preserving harmonic map whose associated quadratic differential is a polynomial, then $\overline{u(\bbfC)}$ is an ideal convex polygon. The author also studies the conformal module of ring type domains on $\bbfC$ and $\bbfH$ which are related by a harmonic map. A nonexistence result of harmonic diffeomorphism from $\bbfC$ onto $\bbfH$ under a certain energy growth condition is shown. For the entire collection see [Zbl 0853.00042].

58E20Harmonic maps between infinite-dimensional spaces
30F30Differentials on Riemann surfaces