Flowing to Free Boundary Minimal Surfaces

In this talk, I will discuss an approach to free boundary minimal surfaces which comes out of recent work by Struwe on a non-local energy, called the half-energy. I will introduce the gradient flow of this functional and its theory in the already studied case of disc type domains, covering existence, uniqueness, regularity and singularity analysis and highlighting the striking parallels with the theory of the classical harmonic map flow. Then I will go on to present new work, joint with Melanie Rupflin and Michael Struwe, which extends this theory to all compact surfaces with boundary. This relies upon combining the above ideas with those of the Teichmüller harmonic map flow introduced by Rupflin and Topping.