# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $ℂ$ onto the Poincaré disk $ℍ$? 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 $ℂ$ into regions of $ℍ$ which are ideal convex polygons.

In this paper, the author describes the mapping geometry of harmonic maps from a surface into $ℍ$, using the geometry of the associated holomorphic quadratic differential. The following theorem is proved: If $u:ℂ\to ℍ$ is an orientation-preserving harmonic map whose associated quadratic differential is a polynomial, then $\overline{u\left(ℂ\right)}$ is an ideal convex polygon.

The author also studies the conformal module of ring type domains on $ℂ$ and $ℍ$ which are related by a harmonic map. A nonexistence result of harmonic diffeomorphism from $ℂ$ onto $ℍ$ under a certain energy growth condition is shown.

##### MSC:
 58E20 Harmonic maps between infinite-dimensional spaces 30F30 Differentials on Riemann surfaces