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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.\n  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 $r>0$. 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