The Enclosed Volume for Periodic Constant Mean Curvature Surfaces

TL;DR

This paper derives a general volume formula for periodic constant mean curvature surfaces in Euclidean space, linking geometric quantities and providing insights into isoperimetric problems in specific manifolds.

Contribution

It introduces a novel volume formula for periodic CMC surfaces that extends classical results and applies gauge theory concepts to geometric analysis.

Findings

Derived a volume formula relating surface area, a Wess-Zumino-Witten term, and curvature.

Applied the formula to compute volumes of examples and counterexamples.

Constructed a counterexample to the isoperimetric problem in a torus-product space.

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.