Let $X$ be a compact Riemann surface and $u: X \longrightarrow \mathbb{R}$ be a smooth real-valued function.
If $u(\operatorname{div}(f))=0$ for every meromorphic function $f$ on $X$, does this imply that $u$ is constant?
Here let $D=\sum n_i P_i$ be a divisor on $X$, $u(D)$ is defined to be $u(D) := \sum n_i u(P_i)$.
My guess is that $u(\operatorname{div}(f))=0$ for every meromorphic $f$ would force $\Delta u=0$, since $\Delta \log|f| \sim \delta_{\operatorname{div}(f)}$ (here by "$\sim$" I mean up to some $2\pi i$ thing).