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 convergence result for initial data below an energy threshold which depends on the prescribed boundary conditions. Moreover, we show optimality by constructing singular examples for the Willmore flow above this energy threshold. Finally, a Li-Yau inequality for open curves in H 2 is proved.