The volume-preserving Willmore flow

We consider a closed surface in \(\mathbb{R}^3\) evolving by the volume-preserving Willmore flow and prove a lower bound for the existence time of smooth solutions. For spherical initial surfaces with Willmore energy below \(8\pi\) we show long time existence and convergence to a round sphere by performing a suitable blow-up and by proving a constrained Lojasiewicz-Simon inequality.