Harmonic functions on Riemann surfaces

Summary

A real function on a Riemann surface is harmonic when it is harmonic in every holomorphic coordinate chart. This makes sense because holomorphic changes of coordinates preserve harmonicity, even though the surface carries no preferred Laplacian. Such functions are the starting point of the potential theory of surfaces and of the classification of open surfaces. They also give the classical existence proofs for holomorphic functions and differentials.