Axially Symmetric Helfrich Spheres

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.

💡 Research Summary

The paper addresses the geometric classification of closed, embedded, genus‑zero surfaces (topological spheres) that are critical points of the Helfrich energy, a model for lipid bilayer membranes. The Helfrich functional is given by
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