Axially symmetric membranes with polar tethers

TL;DR

This paper derives a simplified equilibrium equation for axially symmetric membranes with polar tethers, revealing the role of external forces and presenting exact solutions illustrating shape transitions.

Contribution

It introduces a lower-order differential equation governing equilibrium shapes of symmetric membranes, highlighting the influence of external axial forces and providing explicit solutions.

Findings

Equilibrium shapes satisfy a second-order differential equation.

Curvature singularities occur at poles when a parameter c is non-zero.

Exact solutions demonstrate a transition from discocyte to stomatocyte shapes.

Abstract

Axially symmetric equilibrium configurations of the conformally invariant Willmore energy are shown to satisfy an equation that is two orders lower in derivatives of the embedding functions than the equilibrium shape equation, not one as would be expected on the basis of axial symmetry. Modulo a translation along the axis, this equation involves a single free parameter c.If c\ne 0, a geometry with spherical topology will possess curvature singularities at its poles. The physical origin of the singularity is identified by examining the Noether charge associated with the translational invariance of the energy; it is consistent with an external axial force acting at the poles. A one-parameter family of exact solutions displaying a discocyte to stomatocyte transition is described.