Axially Symmetric Helfrich Spheres

TL;DR

This paper characterizes axially symmetric Helfrich spheres, showing that solutions to a specific reduced membrane equation correspond to these spheres and revealing their geometric symmetry and discrete family structure.

Contribution

It establishes a converse relationship between solutions of the reduced membrane equation and Helfrich spheres under axial symmetry and rescaling conditions.

Findings

Solutions are either round or satisfy the reduced membrane equation.

Axially symmetric solutions satisfying a rescaling condition are Helfrich spheres.

These surfaces exhibit symmetry with respect to a plane and belong to an infinite discrete family.

Abstract

Smooth axially symmetric Helfrich topological spheres are either round or else they must satisfy a second order equation known as the reduced membrane equation [17]. In this paper, we show that, conversely, axially symmetric closed genus zero solutions of the reduced membrane equation which, in addition, satisfy a rescaling condition are axially symmetric Helfrich spheres. We also exploit this characterization to geometrically describe these surfaces and present convincing evidence that they are symmetric with respect to a suitable plane orthogonal to the axis of rotation and that they belong to a particular infinite discrete family of surfaces.