A singular example for the Willmore flow

Abstract

In this article it will be proven that there are surfaces that become singular under the Willmore flow. For this purpose, we investigate the Willmore flow of radially symmetric immersions of the sphere using a blow-up construction due to E. Kuwert and R. Schätzle. It will be shown that in this situation the blow-up limit is a surface of revolution as well and is either a round sphere or consists of planes and catenoids. Furthermore, we give an estimate for the number of these planes and catenoids in terms of the Willmore energy of the initial surface. This will enable us to show that there are immersions of the sphere with a Willmore energy arbitrarily close to 8 π that do not converge to a Willmore immersion under the Willmore flow. Either a small quantum of the curvature concentrates or the diameter of the surface does not stay bounded under the Willmore flow.