Global minimizers for the doubly-constrained Helfrich energy: the axisymmetric case

TL;DR

This paper proves the existence of global minimizers for the Helfrich energy functional in the axisymmetric case, contributing to the mathematical understanding of biomembrane shape optimization.

Contribution

It establishes the existence of global minimizers for the Canham-Helfrich functional under volume and area constraints in axisymmetric geometries.

Findings

Existence of global minimizers proven for axisymmetric surfaces.

Results applicable to shape analysis of biomembranes.

Advances understanding of curvature-dependent energy minimization.

Abstract

Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volume and having fixed surface area. By restricting attention to axisymmetric surfaces, we prove the existence of global minimizers.