A first-order formulation for axisymmetric Willmore surfaces

We show that axisymmetric Willmore surfaces admit a first-order formulation obtained by combining two independent first integrals. If $\rho$ denotes the distance from the axis of revolution and $\Psi=\sin\psi$, where $\psi$ is the tangent angle of the generating curve, then the profile satisfies \begin{equation*} \left[\frac{\Psi(\rho\Psi'-\Psi)^2+2(\rho\Psi'-\Psi)+2C_1\rho}{\sqrt{1-\Psi^2}}\right]^2 +\left[(\rho\Psi'-\Psi)^2-2\right]^2=C_2, \end{equation*} where $C_1$ and $C_2$ are constants of integration and the prime denotes differentiation with respect to $\rho$. This equation reduces the axisymmetric Willmore equation to a first-order ordinary differential equation and provides a convenient classification scheme for Willmore surfaces of revolution. The sphere and the Clifford torus are discussed as elementary checks of the formulation.