= Solution
The identity theorem says that if two holomorphic functions on a connected Riemann surface agree on a set having an accumulation point, then they agree everywhere.
Apply local coordinates to their difference $F$. The power-series proof in the plane shows that near any point, either $F$ vanishes identically or its zeros are isolated: if the first nonzero Taylor coefficient has order $m$, then $F(z)=(z-z_0)^mG(z)$ with $G(z_0)\ne0$. An accumulation point of the zero set therefore has a neighbourhood on which $F=0$. The set of points having such a neighbourhood is nonempty and open. It is also closed: near a limit point, a coordinate neighbourhood contains an open set on which $F=0$, and the planar identity theorem makes $F$ vanish throughout that coordinate neighbourhood. Connectedness now makes this set the whole surface.
For a nonempty connected open set $U\subset\mathbb R^2$, a function $h:U\to\mathbb R$ is harmonic when $h\in C^2(U)$ and
$$
\Delta h=h_{xx}+h_{yy}=0.
$$
On a sufficiently small disc, the one-form
$$
-h_y\,dx+h_x\,dy
$$
is closed and hence equals $dk$ for some $k$. The Cauchy--Riemann equations then make $h+ik$ holomorphic. Holomorphic functions are smooth, so $h\in C^\infty(U)$.
A real-valued function $H$ on a Riemann surface is harmonic when, in every holomorphic chart $z$, its coordinate expression is harmonic. This definition is chart-independent. If $w=w(z)$ is a holomorphic transition map, the chain rule and Cauchy--Riemann equations give
$$
\Delta_z(H\circ w)=|w'(z)|^2(\Delta_w H)\circ w.
$$
\b[Thus vanishing of the Laplacian is preserved by every change of holomorphic coordinate.]
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