On the convergence of the Willmore flow with Dirichlet boundary conditions

Abstract

Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a global existence and convergence result for solutions of the Willmore flow with initial data below an explicit, sharp energy threshold. Strikingly, this threshold depends on the prescribed boundary conditions — it can even be made to be 0. We show sharpness for some critical boundary data by constructing surfaces above this energy threshold so that the corresponding Willmore flow develops a singularity. Finally, a Li–Yau inequality for open curves in H2 is proved.