Gradient flows for the harmonic map energy

Professor Peter Topping (Warwick). Part of the Analysis and Applications Seminar Series.

The standard $L^2$ gradient flow for the harmonic map energy is the celebrated harmonic map flow, which is the original geometric flow introduced by Eells and Sampson in 1964. After giving a brief survey of the two-dimensional theory for this flow, we will take a look at the alternative gradient flow for this energy that was introduced jointly with Melanie Rupflin, which evolves both a map and the metric on its domain simultaneously. Now the gradient flow wants to converge to a minimal immersion rather than a harmonic map.

A theory for this flow has been developed over recent years in order to understand its singularity formation and its asymptotics. We will take a look at some of the main points of this theory, including how the flow decomposes a map into minimal immersions. The main focus of the talk will be the recent work joint with Kohout and Rupflin that addresses the question of whether the flow enjoys unique limits or not. We will see when it does and when it does not.