UNIQUENESS AND REGULARITY OF PROPER HARMONIC MAPS-II

This is a continuation of [L-T2] of the study of harmonic maps between hyperbolic spaces. In this paper, we prove a uniqueness theorem on quasi-conformal harmonic diffeomorphisms of H-2. We also construct many harmonic maps on H(m) with the same general boundary data on the geometric boundary of H(m). We improve the regularity and existence results in [L-T2] so that in some sense they are optimal.