Compactness makes the continuous function $|f|$ attain a maximum. Apply the preceding argument in a chart at a maximizing point to make $f$ constant on a neighborhood. The permitted <identity theorem> and connectedness then make it constant on all of $S$. Hence \b[every <holomorphic function> from a compact connected <Riemann surface> to $\mathbb C$ is constant].

Solution

Past exam of the geometry course of the University of Oxford / 2007 / B / 8 / iv / Solution

 Solution (source code)