Discontinuity of curvature on axisymmetric Willmore surfaces

TL;DR

This paper derives a new analytical solution for axisymmetric Willmore surfaces by using the tangential angle as a variable, revealing curvature discontinuities previously overlooked, and applicable to open vesicles with free edges.

Contribution

It introduces a novel approach to solving the Willmore equation with the tangential angle variable, leading to explicit solutions that show curvature discontinuities.

Findings

Curvature can be discontinuous on axisymmetric Willmore surfaces.

The solution satisfies boundary conditions for open vesicles with free edges.

The differential equation reduces to a Bernoulli form, simplifying analysis.

Abstract

The equilibrium shapes of vesicles are governed by the general shape equation which is derived from the minimization of the Helfrich elastic free energy and can be reduced to the Willmore equation in a special case. The general shape equation is a high order nonlinear partial differential equation and it is very difficult to find analytical solution even in axisymmetric case, which is reduced to a seconder ordinary differential equation. Traditional axisymmetric shape equation is with the turning radius as the variable. Here we study the shape equation with the tangential angle as the variable. In this case, the Willmore equation is reduced to the Bernoulli differential equation and the general solution is obtained conveniently. We find that the curvature in this solution is discontinuous in some cases, which was ignored by previous researchers. This solution can satisfy the boundary conditions for open vesicle with free edges.