A first-order formulation for axisymmetric Willmore surfaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917550911127552 |
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| author | Tu, Z. C. |
| author_facet | Tu, Z. C. |
| contents | We show that axisymmetric Willmore surfaces admit a first-order formulation obtained by combining two independent first integrals. If $ρ$ denotes the distance from the axis of revolution and $Ψ=\sinψ$, where $ψ$ is the tangent angle of the generating curve, then the profile satisfies \begin{equation*} \frac{\left[Ψ(ρΨ'-Ψ)^2+2(ρΨ'-Ψ)+2C_1ρ\right]^2}{1-Ψ^2} +\left[(ρΨ'-Ψ)^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 $ρ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01001 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A first-order formulation for axisymmetric Willmore surfaces Tu, Z. C. Mathematical Physics Soft Condensed Matter Analysis of PDEs We show that axisymmetric Willmore surfaces admit a first-order formulation obtained by combining two independent first integrals. If $ρ$ denotes the distance from the axis of revolution and $Ψ=\sinψ$, where $ψ$ is the tangent angle of the generating curve, then the profile satisfies \begin{equation*} \frac{\left[Ψ(ρΨ'-Ψ)^2+2(ρΨ'-Ψ)+2C_1ρ\right]^2}{1-Ψ^2} +\left[(ρΨ'-Ψ)^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 $ρ$. 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. |
| title | A first-order formulation for axisymmetric Willmore surfaces |
| topic | Mathematical Physics Soft Condensed Matter Analysis of PDEs |
| url | https://arxiv.org/abs/2606.01001 |