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.