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 $\phi$, that are close to a half sphere with small radius, sliding on the boundary $S$ of a domain $\Omega$ 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 and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. If time allows we conclude by investigating the convergence of the flow.