Axially symmetric Helfrich spheres
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.
Fields of Study & Subjects
Nonlinear Waves and Solitons
Control and Stability of Dynamical Systems
Geometric Analysis and Curvature Flows
Publication details
- Journal / source
- Communications on Pure & Applied Analysis
- Publisher
- American Institute of Mathematical Sciences (AIMS)
- Volume
- 29
- Issue
- 0
- Pages
- 163–192
- Publication date
- 1 January 2026