Abstract:

Varifolds are Radon measures that generalize the notion of differentiable submanifolds of Euclidean spaces. In this talk, I will first give a gentle introduction to this branch of geometric measure theory, focusing on the class of curvature varifolds with orientation and boundary. In the second part, I will apply these concepts to describe configurations of biomembranes, for example the biconcave shapes of human red blood cells, which can be modeled as surfaces of minimal Canham-Helfrich energy. This quadratic curvature functional can be seen as a modification of the Willmore energy, including different bending rigidities and a spontaneous curvature parameter. I will present results on multiphase membranes with sharp phase-interfaces and on the gradient flow evolution of biomembranes, based on joint work with Martin Kruzik, Luca Lussardi, and Ulisse Stefanelli.