Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms

We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $\Sigma$ of curvature $-1$ and infinite area (thus of non parabolic conformal type). The vertical projection of this graph yields a harmonic diffeomorphism from $S$ onto $\Sigma$. The proof uses the theory of divergence lines to construct minimal graphs.

We also generalize a theorem of R.~Schoen. Let $g_1$ and $g_2$ be two complete metrics on a orientable surface $S$ with compact boundary and suppose $$\int_{S_r^2}K_{g_2}^-d\sigma_{g_2}\le C\ln(2+r)$$ for some $C>0$ and all $r>0$. If there is a harmonic diffeomorphism from $(S,g_1)$ to $(S,g_2)$, then $(S,g_1)$ is parabolic.