A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces (Q1204884)

We construct a harmonic diffeomorphism from the Poincaré ball \(H^{n+1}\) to itself, whose boundary value is the identity on the sphere \(S^ n\), and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper half space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole \(\mathbb{R}_ +\), that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity to any point on the sphere.