Hopf Differentials and the Images of Harmonic Maps

In [Hz], Heinz proved that there is no harmonic diffeomorphism from the unit disk D onto the complex plane C. The result was generalized by Schoen [S] and he proved that there is no harmonic diffeomorphism from the unit disk onto a complete surface of nonnegative curvature. Unlike conformal or quasi-conformal maps between Riemann surfaces, the inverse of a harmonic map is not harmonic in general. Hence it is an interesting question whether there is any harmonic diffeomorphism from C onto D equipped with the Poincaré metric. In fact a general form of this question was formulated by Schoen [S] as follows: Is it true that Riemann surfaces which are related by a harmonic diffeomorphism are necessarily quasi-conformally related? Let us first recall some facts on harmonic maps between surfaces. Let Σ1 and Σ2 be two Riemann surfaces with conformal metrics ρ 2(z)|dz|2 and σ2(h)|dh|2 respectively. The harmonic map equation for maps from Σ1 into Σ2 can be written as hzz + 2(log σ)hhzhz = 0. Define ||∂h|| = ρ−1σ|hz|, and ||∂h|| = ρ−1σ|hz|. Hence ||∂h|| and ||∂h|| are the norms of the (1, 0)-part and (0, 1)-part of dh. The energy density of h is given by e(h) = ||∂h||2+||∂h||2, and the Jacobian of h is given by J(h) = ||∂h||2−||∂h||2. The Hopf differential of h is defined as φdz = σ(h)hzhzdz , which is the (2, 0)-part of h∗ ( σ2(h)|dh|2 )