HARMONIC MAPS FROM Rn TO Hm WITH SYMMETRY

It is known that there is no nonconstant bounded harmonic map from the Euclidean space Rn to the hyperbolic space Hm. This is a particular case of a result of S.-Y. Cheng. However, there are many polynomial growth harmonic maps from R2 to H2 by the results of Z. Han, L.-F. Tam, A. Treibergs and T. Wan. One of the purposes of this paper is to construct harmonic maps from Rn to Hm by prescribing boundary data at infinity. The boundary data is assumed to satisfy some symmetric properties. On the other hand, it was proved by Han-Tam-Treibergs-Wan that under some reasonable assump-tions, the image of a harmonic diffeomorphism from R2 into H2 is an ideal polygon with n + 2 vertices on the geometric boundary of H2 if and only if its Hopf differential is of the form φdz2 where φ is a polynomial of degree n. It is unclear whether one can find explicit relation between the coefficient