Harmonic functions on Riemann surfaces

Remarks

It suffices to check the condition for the charts of one atlas. Indeed, transition maps are holomorphic, and harmonicity is preserved by holomorphic maps. Equivalently, $u$ is harmonic if each point of $U$ has a neighbourhood on which $u = \operatorname{Re} f$ for a holomorphic function $f$. Example: on $X = \C \setminus \set{0}$ with the chart $z \mapsto z$, the function $u(z) = \log\abs{z}$ is harmonic, since $\Delta \log\abs{z} = 0$ there. Non-example: $u(z) = \abs{z}^2$ on $X = \C$ is not harmonic, since in the chart $z \mapsto z$ its Laplacian is $4$. On the Riemann sphere, or on any compact Riemann surface, the maximum principle forces each harmonic function on the whole surface to be constant. The Laplacian of $u$ depends on the chart, but the condition $\Delta(u \circ \phi^{-1}) = 0$ does not; in coordinate-free form it reads $\partial \bar\partial u = 0$. Subharmonic and superharmonic functions on $X$ are defined in the same way, chart by chart.