Harmonic map heat flow

Summary

The harmonic map heat flow deforms a map between Riemannian manifolds in the direction of its tension field. It is the gradient flow of the energy, so the energy decreases along it, and its stationary solutions are the harmonic maps. 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.