The geometric Cauchy problem for the membrane shape equation

TL;DR

This paper investigates the geometric Cauchy problem for membrane surfaces governed by the shape equation of vesicles, employing advanced differential geometry methods to analyze equilibrium configurations of lipid bilayer vesicles.

Contribution

It introduces a novel geometric approach using moving frames and exterior differential systems to solve the membrane shape equation's Cauchy problem.

Findings

Established existence and uniqueness conditions for solutions

Derived explicit geometric characterizations of vesicle shapes

Provided a framework for analyzing membrane equilibrium configurations

Abstract

We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surface area. Our approach uses the method of moving frames and techniques from the theory of exterior differential systems.