The Enclosed Volume for Periodic Constant Mean Curvature Surfaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915743538348032 |
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| author | Heller, Lynn Heller, Sebastian Traizet, Martin |
| author_facet | Heller, Lynn Heller, Sebastian Traizet, Martin |
| contents | We establish a general formula for the enclosed volume of constant mean curvature (CMC) surfaces in Euclidean three space with translational periods forming a lattice. The formula relates the volume to the surface area, a Wess-Zumino-Witten-type term, and a newly defined curvature term of the associated family of flat connections, thereby extending the classical Minkowski formula for closed CMC surfaces. Interpreting the volume as a gauge-invariant quantity, we apply the result to a variety of examples and provide explicit computations. As an application, we construct a counterexample to the isoperimetric problem in $\mathbb{T}^2 \times \mathbb{R}$, disproving the conjecture that minimizers are restricted to spheres, cylinders, or pairs of planes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14935 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Enclosed Volume for Periodic Constant Mean Curvature Surfaces Heller, Lynn Heller, Sebastian Traizet, Martin Differential Geometry We establish a general formula for the enclosed volume of constant mean curvature (CMC) surfaces in Euclidean three space with translational periods forming a lattice. The formula relates the volume to the surface area, a Wess-Zumino-Witten-type term, and a newly defined curvature term of the associated family of flat connections, thereby extending the classical Minkowski formula for closed CMC surfaces. Interpreting the volume as a gauge-invariant quantity, we apply the result to a variety of examples and provide explicit computations. As an application, we construct a counterexample to the isoperimetric problem in $\mathbb{T}^2 \times \mathbb{R}$, disproving the conjecture that minimizers are restricted to spheres, cylinders, or pairs of planes. |
| title | The Enclosed Volume for Periodic Constant Mean Curvature Surfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2601.14935 |