---
tag: PK2KNMWNW3
kind: concept
name: Harmonic functions on Riemannian manifolds
msc: [31C12, 58J05]
sources:
  - url: https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator
---

## Summary

A harmonic function on a [[Riemannian manifold|7CBT4SM83Y]] is a function killed by the [[Laplace–Beltrami operator|F4NWZNTRMD]] of the metric.
This extends the classical [[harmonic functions|0DEB0JMN0W]] of Euclidean space to curved spaces.
Their behaviour depends on the geometry: on a connected closed manifold they are constant.
On a complete manifold with nonnegative Ricci curvature, the positive ones are constant.
They link analysis with geometry, through gradient estimates, Liouville theorems and the study of the manifold at infinity.

## Explanations

### L1 human

On a curved surface, one can still ask for functions whose value at each point is the average of nearby values, in a sense suited to the curvature.
The Laplace–Beltrami operator makes this precise, and harmonic functions are the functions it sends to $0$.
On a closed surface such as a sphere, only constant functions are harmonic, because a nonconstant function would have a maximum where it cannot be an average.
On the hyperbolic plane there are many bounded harmonic functions, which reflects the large size of the space at infinity.