Harmonic Rough Isometries into Hadamard Space

§0 Introduction. In a paper of Wan [W], he proved that for each holomorphic quadratic differential defined on the Euclidean 2-disk, D, there exists a harmonic diffeomorphism from the real hyperbolic plane H R into itself, such that its Hopf differential is the given holomorphic quadratic differential after representing H R by the Poincaré model. In addition, he also proved that this correspondence is one-to-one from the set of holomorphic quadratic differentials that are bounded with respect to the hyperbolic metric to the set of quasi-conformal harmonic diffeomorphisms from H R onto itself. It is known in the literature that a quasi-conformal diffeomorphism of H R, when viewed as a quasi-conformal diffeomorphism of D , induces a quasisymmetric homeomorphism from S, the boundary of D, onto itself. Conversely, every quasi-symmetric homeomorphism of S can be extended to a quasi-conformal diffeomorphism of H R by identifying the geometric boundary of H 2 R with S . Inspired by all these, R. Schoen [S] posed the following conjecture: