Talk by Fabian Rupp
On May 2nd, 2022, Dr. Fabian Rupp (Universität Wien) gave a talk about "Li-Yau inequalities for the Helfrich functional" as part of the research seminar Analysis of the FernUniversität in Hagen.
In order to study the shape of red blood cells, Canham and Helfrich suggested a variational model involving a curvature dependent energy functional. In the simplest case, this reduces to the Willmore functional, a well-studied quantity in conformal geometry. An essential tool in the study of the Willmore functional is an inequality of Li and Yau which implies that any immersed surface with energy strictly below 8π must not intersect itself. In this talk, we will prove an extension of this inequality to the Helfrich energy. One crucial consequence is that smoothly embedded minimizers in the Canham-Helfrich model exist if the energy is not too large.
This is joint work with Christian Scharrer (Bonn).