---
tag: P214QJ0S3B
kind: concept
name: Harmonic functions on Riemann surfaces
msc: [30F15, 31C05]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_surface
---

## 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.

## Explanations

### L1 human

A Riemann surface looks locally like a piece of the complex plane, but the local pictures are glued by holomorphic maps.
A function is harmonic on the surface if it looks harmonic in each local picture.
For example, on the punctured plane $\C \setminus \set{0}$ the function $\log\abs{z}$ is harmonic.
On a compact Riemann surface such as the Riemann sphere, every harmonic function is constant, because it attains a maximum.