Abstract

The Canham-Helfrich energy is widely used to describe the elastic properties of biological membranes at the sub-cellular level. The membranes are modeled as surfaces in the 3-dimensional space and their shape minimize the energy which penalizes the curvatures of the surface. The Canham-Helfrich energy can be generalized to the multiphase case in order to model also heterogeneous biological membranes, for instance in the presence of proteins on the membrane. In this seminar first of all I will review some tools of differential geometry and I will introduce the Canham-Helfrich functional. A first step is to look at rotational symmetric shapes, both in one phase and in multiphase: this has been done in 2013 by Choksi, Morandotti and Veneroni. If no symmetry of the minimizers is assumed, the problem requires other tools. I will briefly motivate the necessity of a weaker notion of a surface considering some examples coming from the soap film theory, which is easier to understand. Finally, I will briefly discuss existence of single and multiphase minimizers under area and enclosed volume constraints and regularity of minimizers. This is a joint work with K. Brazda and U. Stefanelli both at the University of Vienna.