On the Chen flow


In this talk we give survey what is currently known for Chen’s flow, and discuss some very recent results. Chen’s flow is the biharmonic heat flow for immersions, where the velocity is given by the rough Laplacian of the mean curvature vector. This operator is known as Chen’s biharmonic operator and the solutions to the elliptic problem are called biharmonic submanifolds. The flow itself is very similar to the mean curvature flow (this is essentially the content of Chen’s conjecture), however proving this requires quite different strategies compared to the mean curvature flow. We focus on results available in low dimensions – curves, surfaces, and 4-manifolds. We provide characterisations of finite-time singularities and global analysis. The case of curves is particularly challenging. Here we identify a new shrinker (the Lemniscate of Bernoulli) and use some new observations to push through the analysis. Some numerics is also presented. The work reported on in the talk is in collaboration with Yann Bernard, Matthew Cooper, Philip Schrader and Glen Wheeler.