Gradient flow dynamics for cell membranes in the Canham-Helfrich model

TL;DR

This paper analyzes the gradient flow dynamics of cell membranes within the Canham-Helfrich model, proving global existence and convergence of solutions for specific geometries under certain energy conditions.

Contribution

It introduces new mathematical proofs for the global existence and convergence of solutions in the Canham-Helfrich model, utilizing a recently discovered multiplicity inequality.

Findings

Proved global existence of solutions for spheres and tori

Established convergence of solutions under energy thresholds

Applied multiplicity inequality to membrane dynamics

Abstract

The energetically most efficient way how a deformed red blood cell regains equilibrium is mathematically described by the gradient flow of the Canham-Helfrich functional, including a spontaneous curvature and the conservation of surface area and enclosed volume. Using a recently discovered multiplicity inequality, we prove global existence and convergence of smooth solutions for spheres and axisymmetric tori, provided the initial energy lies below explicit thresholds.