Abstract

It is known that there is no nonconstant bounded harmonic map from the Euclidean space Rⁿ to the hyperbolic space H^m. 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 H^m 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 assumptions, 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 coeficients of φ and the vertices of the image of the harmonic map. The second purpose of this paper is to investigate this problem. We will explicitly demonstrate some families of polynomial holomorphic quadratic differentials, such that the harmonic maps from R² into H² with Hopf differentials in the same family will have the same image. In proving this, we first study the asymptotic behaviors of harmonic maps from R² into H² with polynomial Hopf differentials φdz². The result may have independent interest.

Authors