Weekly Seminar (ONLINE via ZOOM)
Canham-Helfrich energy and geometric measure theory
Sascha Eichmann (University of Tübingen (Germany))
Thu, 06 Aug 2020 • 12:15-13:30h • 12:15 ONLINE coffee, 12:30 talk (access data will be published)

Abstract

Helfrich (1973) and Canham (1970) introduced the following geometric curvature energy to model the shape of human red blood cells. The idea is that the two dimensional boundary layer $\Sigma\subset\mathbb{R}^3$ of such a cell minimises

$$\int_\Sigma |H-H_0|^2 dA$$

under suitable constraints on e.g. the enclosed volume and surface area. Here $H$ is the scalar mean curvature of $\Sigma$ and $H_0\in\mathbb{R}$ is a parameter called the spontaneous curvature, which represents an asymmetry in the boundary layer of the cell. This induces a prefered curvature of the cell. If this asymmetry is not desired, i.e. $H_0=0$, this Canham-Helfrich energy becomes a variant of the famous Willmore energy.

To show existence of such a minimiser, we will implement the direct method of the calculus of variations. Compactness for a minimising sequence under varifold convergence can be easily obtained. Unfortunately lower-semicontinuity of the Helfrich energy under this varifold convergence is in general not correct by a counterexample of Große-Brauckmann (1993). Nevertheless we can actually show a lower-semicontinuity estimate for the minimising sequence itself.

Throughout the talk we will highlight the main tools used from geometric measure theory. We explain these in some detail and how they are applied to our problem.

In the last part of the talk we will discuss some directions for future research in this area, i.e. some open problems and some modifications to the Canham-Helfrich energy itself.