Fabian Rupp - A dynamic approach to the Canham problem

Fabian Rupp (Ulm University)
A dynamic approach to the Canham problem

Motivated by the Canham-Helfrich model for lipid bilayers, the minimization of the Willmore energy among surfaces of given topological type subject to the constraint of fixed isoperimetric ratio has been extensively studied throughout the last decade. In this talk, we consider a dynamical approach by introducing a non-local $L^2$-gradient flow for the Willmore energy, which preserves the isoperimetric ratio. For topological spheres with initial energy below an explicit threshold, we show global existence and convergence to a Helfrich immersion as $t\to\infty$. Our proof relies on a blow-up procedure and a constrained version of the Łojasiewicz--Simon gradient inequality.