Given a map from a closed surface to an arbitrary closed Riemannian manifold, there is a gradient flow of the harmonic map energy that would like to flow it to be a branched minimal immersion. This is impossible in general, so the flow develops singularities. In recent work we described what sort of singularities can happen asymptotically at infinite time. In forthcoming work we plan to describe what can happen at finite time. When combined, the work gives a description of how the arbitrary map decomposes automatically into a collection of minimal immersions. I’m planning to make this accessible to a general audience of geometers and/or analysts.
Joint work particularly with Melanie Rupflin.