On the area preserving Willmore flow of small bubbles sliding on a domain's boundary

Abstract:

We consider the area preserving Willmore evolution of surfaces ϕ, that are close to a half sphere with small radius, sliding on the boundary S of a domain Ω while meeting it orthogonally. We prove that the flow exists for all times and keeps a `half spherish' shape. Additionally we investigate the asymptotic behaviour of the flow. If time allows we conclude by investigating the  convergence of the flow.