It was proved by the authors
that given a quasiconformal harmonic diffeomorphism F on ℍ2, there is a
neighborhood 𝒩 of the class F represented by F in the universal Teichmüller space
such that if H∈𝒩, then the boundary map of H can be extended to a
quasiconformal harmonic diffeomorphism on ℍ2, i.e. the class H can be
represented by a quasiconformal harmonic diffeomorphism. More precisely, it was
proved that if F is a quasiconformal harmonic diffeomorphism on ℍ2, and if
G is a quasiconformal map on ℍ2 such that the dilatation of G is small
enough, then there exists quasiconformal harmonic diffeormophisms with the
same boundary data with F ∘ G and G ∘ F. The purposes of this paper is
to study the higher dimensional generalization to this result and related
problems.
