Diffuse Interfaces and Topology: A Phase-Field Model for Willmore's Energy

Motivated from a biological model, we consider the problem of minimising Willmore's curvature energy in the class of connected surfaces with prescribed area embedded into a bounded domain. From a computational point of view, it may be favourable to approximate the curvature energy by a phase field model. Diffuse Interfaces, however, can easily separate into multiple components along a gradient flow evolution. This is overcome using a topological penalty term in the functional involving a geodesic distance function. I will present a proof of Gamma-convergence to the sharp interface limit in two and three dimensions and numerical evidence of the effectiveness of our method in two dimensions. Time permitting, I will also demonstrate that it is not possible to get a sharper control of the topology of a surface (i.e. genus) even in the sharp interface limit.