Parameterization invariance and shape equations of elastic axisymmetric vesicles

TL;DR

This paper investigates how different parameterizations of axisymmetric vesicle shapes affect the associated Euler-Lagrange equations, concluding that for smooth, spherical vesicles they are equivalent, but differ when contours have discontinuities.

Contribution

It clarifies the impact of parameterization choices on the shape equations of vesicles, emphasizing the equivalence for smooth contours and differences with discontinuities.

Findings

Different parameterizations yield equivalent equations for smooth vesicles.

Discontinuities in contour derivatives lead to non-equivalent equations.

Elastic energy remains a global minimum for smooth contours.

Abstract

The issue of different parameterizations of the axisymmetric vesicle shape addressed by Hu Jian-Guo and Ou-Yang Zhong-Can [ Phys.Rev. E {\bf 47} (1993) 461 ] is reassesed, especially as it transpires through the corresponding Euler - Lagrange equations of the associated elastic energy functional. It is argued that for regular, smooth contours of vesicles with spherical topology, different parameterizations of the surface are equivalent and that the corresponding Euler - Lagrange equations are in essence the same. If, however, one allows for discontinuous (higher) derivatives of the contour line at the pole, the differently parameterized Euler - Lagrange equations cease to be equivalent and describe different physical problems. It nevertheless appears to be true that the elastic energy corresponding to smooth contours remains a global minimum.