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

The main result is Theorem 1. Let u: 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() is the interior of the ideal polygon with vertices given by {1,e iα ,ω,ωe iα ,,ω m ,ω m e iα } in the unit disc model of 2 , where ω=e 2πi/(m+1) ,

α=α m (ν)=2tan -1 sin(π/(m+1)) cos(π/(m+1))+e 2ν ,

and ν=π|b|/(2(m+1)) is the common length of the finite edges of the -tree associated to the quadratic differential.

Reviewer: A. Neagu (Iaşi)
53C43Differential geometric aspects of harmonic maps