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Images of harmonic maps with symmetry. (English) Zbl 1087.53059
The main result is Theorem 1. Let $u: \bbfC\to\bbfH^2$ be the unique (up to equivalence) complete orientation preserving harmonic embedding associated to a quadratic differential equivalent to $[z^{2m}- (a+ ib) z^{m-1}]\,dz^2$. Then, up to isometry, the image $u(\bbfC)$ is the interior of the ideal polygon with vertices given by $\{1, e^{i\alpha}, \omega, \omega e^{i\alpha},\dots, \omega^m, \omega^m e^{i\alpha}\}$ in the unit disc model of $\bbfH^2$, where $\omega= e^{2\pi i/(m+1)}$, $$\alpha= \alpha_m(\nu)= 2\tan^{-1} \Biggl({\sin(\pi/(m+ 1))\over\cos(\pi/(m+ 1))+ e^{2\nu}}\Biggr),$$ and $\nu= \pi|b|/(2(m+ 1))$ is the common length of the finite edges of the $\bbfR$-tree associated to the quadratic differential.
Reviewer: A. Neagu (Iaşi)

53C43Differential geometric aspects of harmonic maps
Full Text: DOI Euclid
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