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Images of harmonic maps with symmetry. (English) Zbl 1087.53059

The main result is Theorem 1. Let \(u: \mathbb{C}\to\mathbb{H}^2\) 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 \(u(\mathbb{C})\) 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 \(\mathbb{H}^2\), 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 \(\mathbb{R}\)-tree associated to the quadratic differential.
Reviewer: A. Neagu (Iaşi)

MSC:

53C43 Differential geometric aspects of harmonic maps
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References:

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