HARMONIC EXTENSIONS OF QUASI-CONFORMAL MAPS TO HYPERBOLIC SPACE

We show that the set of quasiconformal (quasisymmetric, if n = 2) maps h : Sn-1 --> Sn-1 which admit a quasi-isometric harmonic extension H : H-n --> H-n is open in the set of quasiconformal (quasisymmetric, re sp.) self-maps of Sn-1. The proof involves first deforming a harmonic map by a quasi-isometry, and then using that deformed map to set harmo nic map Dirichlet problems on a compact exhaustion of H-n. The solutio ns to these Dirichlet problems then converge to a harmonic map of boun ded energy density which is at finite distance from the original defor med map.