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 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 coefficients 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.
