---
tag: MD99PYBCE6
kind: concept
name: Harmonic map heat flow
msc: ["58E20", "53E99", 35K55]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_map
---

## Summary

The harmonic map heat flow deforms a map between [[Riemannian manifolds|7CBT4SM83Y]] in the direction of its [[tension field|W8F8S8DX5X]].
It is the gradient flow of the [[energy|E6HDE1DFXD]], so the energy decreases along it, and its stationary solutions are the [[harmonic maps|KZJYG9DS3M]].
James Eells and Joseph Sampson introduced it to find harmonic maps in a given homotopy class.
For compact manifolds and a target of nonpositive sectional curvature, they proved that it exists for all time.
It then converges to a harmonic map along a sequence of times.
In general the flow can develop singularities, and its analysis led to the study of bubbling.

## Explanations

### L1 human

Start with any map and let it relax, the way a hot metal plate relaxes to an even temperature.
At each moment every point moves in the direction that lowers the stretching energy fastest.
For maps into the real line, this is exactly the heat equation, and the map tends to a harmonic function.
For a closed curve wrapped around a cylinder, the flow straightens the curve into a circle of the cylinder, a closed geodesic, which cannot be shortened further.