  A classical result of Sampson and Schoen-Yau in 1978 states that every
diffeomorphism between compact hyperbolic Riemann surfaces is homotopic to an
harmonic diffeomorphism. As conjectured by Schoen in 1993 and partially proved
by Wan in 1992 and Tam-Wan in 1995, we prove in this article that this theorem
generalizes to the non compact case : every quasi symmetric homeomorphism of
the circle extends to a quasi isometric harmonic diffeomorphism of the
hyperbolic plane. This enables to parametrize the universal Teichmuller space
by bounded holomorphic quadratric differentials of the hyperbolic plane.
