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 ${\mathbb{H}}^2$ is proved.

Revised version with some additional details, 41 pages

References in corpus (2)

• The elastic flow of curves in the hyperbolic plane
• On the area-preserving Willmore flow of small bubbles sliding on a domain's boundary

Cited by in corpus (1)

• Singularities of the hyperbolic elastic flow: Convergence, quantization and blow-ups