Axially Symmetric Helfrich Spheres

TL;DR

The paper investigates closed axially symmetric Helfrich surfaces by reducing the Euler–Lagrange equation to a second-order reduced membrane equation (RME) $H+c_o=-\frac{\nu_3}{z}$ and studying its $\mathcal{C}^3$-regular solutions. It proves that such RME solutions are real analytic Helfrich surfaces intersecting the plane $\{z=0\}$ orthogonally in geodesic circles, and that a global rescaling condition $\int_\Sigma (H+c_o)\,d\Sigma=0$ is necessary (and, when nonzero, sufficient alongside RME) for axisymmetric closed genus-zero spheres to be Helfrich critical points. The work analyzes axially symmetric generating curves via an ODE system, shows how to glue two disc-type surfaces along $\{z=0\}$ to obtain closed genus-zero surfaces, and constructs multiple non-round examples with evidence of a broader infinite discrete family, including a conjecture that all such spheres are symmetric about $\{z=0\}$. These results advance understanding of Helfrich morphology by linking reduced equations, regularity, symmetry, and global energy balance, and by providing a pathway to a complete classification of 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.

Figures (7)

• Figure 1: Six curves $\gamma(s)=(r(s),z(s))$ constructed from solutions of \ref{['system1']}-\ref{['system3']} with initial conditions \ref{['conditions']} and different initial heights $z_0\neq 0$. For all the curves we have fixed $c_o=1$. The curves on the top row are unduloid-type curves, the first curve on the bottom row is an ovaloid-type curve, and the last curve is a nodoid-type curve. The curve in the center of the bottom row is the horizontal straight line at height $z=-1/c_o$ (cf., Remark \ref{['line']}). (Observe that the representations of the top and bottom rows have different scale.)
• Figure 2: Axially symmetric closed Helfrich surfaces of genus zero with non-constant mean curvature.
• Figure 3: Left: The graph of $\varphi"(\ell)$ as a function of the initial height $z_0>0$. Center: The graph of $r_*=r(\ell)$ as a function of the initial height $z_0>0$. The green horizontal line represents the radius of the critical cylinder, ie., $r=1/(2c_o)$. Right: A piece of the planar curve $z_0>0\longmapsto (\varphi"(\ell),r_*)$. The green point corresponds to the values of the critical cylinder, ie., $(0,1/(2c_o))$. For the three cases we have fixed $c_o=1$.
• Figure 4: Three circular biconcave discoids, that is, axially symmetric topological spheres with non-constant mean curvature solution of \ref{['EL']} for $r>0$. The green points are the 'poles', where the surfaces are only of class $\mathcal{C}^1$. Due to this lack of regularity, these discoids are not Helfrich surfaces.
• Figure :

Theorems & Definitions (51)

• Remark 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Theorem 2.6
• Remark 2.7
• Lemma 2.8
• proof