University of Warwick
Gradient flows for the harmonic map energy
In 1964, Eells and Sampson introduced the harmonic map flow, thus starting the field of geometric flows. Their idea was essentially to consider the \(L^2\) gradient flow of the harmonic map energy, i.e. the Dirichlet energy, but instead of considering real-valued functions (which would give the heat equation) they considered more general maps, for example taking values into a sphere or a more general Riemannian manifold.
The critical domain dimension for this nonlinear PDE is two. In 1985 Struwe initiated an analysis of the blow-up behaviour of the flow in this dimension, and described it in terms of the notion of bubbling in the spirit of Sacks-Uhlenbeck.
After giving a brief overview of this story I will describe a different gradient flow for the harmonic map energy that I introduced with Melanie Rupflin, coinciding with the harmonic map flow on \(S^2\) and a flow introduced on the torus \(T^2\) by Ding-Li-Liu. In our flow we allow not only the map to evolve, but also the domain metric. This changes the purpose of the flow from finding harmonic maps to finding minimal surfaces, as I will explain. From a PDE perspective it introduces a selection of new blow-up behaviours that are reminiscent of bubbling and yet throw up completely new phenomena and possibilities. I will survey some of what has been found over the past few years, including hopefully our most recent work joint with Rupflin and Kohout, and mention some open problems.
I will not be assuming great geometry prerequisites. I will, however, explain some of the elegant classical geometry that opens up this subject to those coming from a nonlinear PDE background.