Lower-semicontinuity for the Helfrich problem

TL;DR

This paper establishes a lower-semicontinuity result for the Helfrich energy minimization problem in the class of closed surfaces with fixed topological and geometric constraints, using varifold compactness and partial regularity techniques.

Contribution

It proves a lower-semicontinuity estimate for Helfrich energy minimizers, overcoming previous counterexamples, by analyzing partial regularity and adjusting geometric quantities via diffeomorphisms.

Findings

Lower-semicontinuity of Helfrich energy established

Partial regularity of the limit surface demonstrated

Method to compare Helfrich energy locally to biharmonic graphs

Abstract

We minimise the Canham-Helfrich energy in the class of closed immersions with prescribed genus, surface area and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false by a counter example by Gro{\ss}e-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of $\mathbb{R}^3$