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arXiv · 2601.14935

The Enclosed Volume for Periodic Constant Mean Curvature Surfaces

Lynn Heller·Sebastian Heller·Martin Traizet
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Abstract

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.

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Lynn Heller, Sebastian Heller, Martin Traizet. 2026-01-21. The Enclosed Volume for Periodic Constant Mean Curvature Surfaces. https://arxiv.org/abs/2601.14935

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