Let $\Sigma_1$ and $\Sigma_2$ be Riemann surfaces with nonnegative and negative curvature respectively. Show that any harmonic map $f:\Sigma_1 \rightarrow \Sigma_2$ is a constant.
Thanks to the comment of Anthony Carapetis, it follows from Bochner identity and Riemann-Hurwitz that $f$ must be a constant.