The Enclosed Volume for Periodic Constant Mean Curvature Surfaces

TL;DR

The paper develops a gauge-theoretic framework to compute the enclosed volume of periodic constant mean curvature surfaces in $\mathbb{R}^3$, recasting volume as a gauge-invariant quantity tied to the associated family of flat connections $\nabla^\lambda$ and a curvature term $K$. It derives a generalized Minkowski formula: $K = -\frac{i}{2\pi}\mathcal{A}(f) + \frac{3i}{2\pi}\mathcal{V}(f) - \frac{3i}{2\pi}\mathcal{V}_\Sigma(\rho)$, with $\mathcal{V}(f)$, $\mathcal{A}(f)$, and the WZW-type term $\mathcal{V}_\Sigma(\rho)$ capturing global-period data; when the lattice rank is $\le 2$, the $\mathcal{V}_\Sigma(\rho)$ term vanishes and the classical Minkowski relation is recovered. The authors develop two computation schemes for $K$: spectral data for equivariant surfaces and residue formulas from DPW potentials, enabling explicit volume calculations from integrable-systems data. They apply the theory to Lawson-type doubly periodic CMC surfaces in $\mathbb{T}^2\times\mathbb{R}$, constructing families via DPW and obtaining area/volume formulas; these are used to analyze isoperimetric profiles and produce a counterexample to the isoperimetric conjecture, showing minimizers need not be spheres, cylinders, or planes. The work provides a coherent link between differential geometry, gauge theory, and integrable systems, with concrete computational tools for volumes of periodic CMC surfaces and implications for geometric optimization in flat 3-manifolds.

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.

Figures (5)

• Figure 1: The conjugate cousin of the Lawson minimal surface $\xi_{2,2}$ in $\mathbb{S}^3$ (left) is a doubly periodic CMC surface in $\mathbb{R}^3$ with hexagonal period lattice and genus 2 in the quotient. The conjugate cousin of $\xi_{3,3}$ shown on the right is a doubly periodic CMC surface with square period lattice and genus 2 in the quotient. The surfaces correspond to the Lawson-type surface $\mathfrak L_{3,\tfrac{\pi}{4}}$ and $\mathfrak L_{4,\tfrac{\pi}{4}}$ in our notation. Images by the third author using a C++ implementation of the DPW method and the numerical data of Section \ref{['sec:iso']}.
• Figure 2: The conjugate cousin of the Lawson minimal surface $\xi_{5,5}$ in $\mathbb{S}^3$ is a doubly periodic CMC surface with hexagonal period lattice and genus 3 in the quotient. The surface corresponds to the Lawson-type surface $\mathfrak L_{6,\tfrac{\pi}{4}}$ in our notation. Image by the third author.
• Figure 3: Comparison of isoperimetric profiles in $\mathbb{T}^2\times\mathbb{R}$ for the equilateral torus. The figure plots the area as a function of enclosed volume for the (rescaled) Lawson-type family $\mathfrak L_{3,\varphi}$ (blue curves) and for the conjectured isoperimetric minimizers -- spheres, cylinders, and pairs of horizontal planes (red curves). Near the critical volume $\tfrac{3}{4\pi}$, corresponding to the transition from cylinders to planes, the Lawson-type family attains smaller area than the conjectured competitors, providing a counterexample to the isoperimetric conjecture of HPRR.
• Figure 4: Two views of the Lawson-type CMC surface $\mathfrak L_{3,\varphi}$ at $\varphi=1.454838491$. (a) Periodic extension of the surface in $\mathbb{R}^3$. (b) The same surface restricted to a fundamental domain of the equilateral lattice together with a finite height in the $\mathbb{R}$-direction. Images by the third author.
• Figure 5: Red: isoperimetric profile $\mathcal{V}\mapsto\mathcal{A}$ of the conjectured minimizers (sphere, cylinder, and pair of planes) in $\mathbb{T}^2\times\mathbb{R}$, where $\mathbb{T}^2$ is the square torus. Blue: isoperimetric profile of the $\mathfrak L_{4,\varphi}$ family for $\varphi\in[\tfrac{\pi}{6},\tfrac{9\pi}{20}]$.

Theorems & Definitions (51)

• Proposition 1
• proof
• Remark 1
• Remark 2: Recovering the immersion: the Sym--Bobenko formula
• Remark 3: Area formula
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 4