The geometric Cauchy problem for the membrane shape equation

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.

v2: Examples included; references added; minor clarifications made. 20 pages, 8 figures. Version to appear on J. Phys. A

References in corpus (5)

• Geometric theory on the elasticity of bio-membranes
• Cylindrical equilibrium shapes of fluid membranes
• mKdV Surfaces
• Hamilton's equations for a fluid membrane: axial symmetry
• A class of overdetermined systems defined by tableaux: involutiveness and Cauchy problem

Cited by in corpus (3)

• On a stack of surfaces obtained from the $\mathbb{C}{P}^{N-1}$ sigma models
• On the Björling problem for Willmore surfaces
• Critical Robertson-Walker universes