The isoperimetric profile of the cubic three-torus

Proves the isoperimetric conjecture for the cubic flat three-torus: balls, tubes, slabs and their complements.

1 paper September 24, 2026 Explained by AI

Picture a cube whose opposite faces are glued, so leaving one face brings you back through the opposite one. The question is how to enclose a given volume inside it with the least surface area. The paper proves the long-standing guess: small volumes use a ball, middle volumes a tube around a straight loop, and larger ones a flat slab. Past half the volume, the answers are the complements of these. A Lean proof covers the full result.

First step

2004

Hauswirth, Pérez, Romon and Ros, who state the ball–tube–slab prediction explicitly

Significance

Solid

It settles a concrete, well-known periodic isoperimetric problem and classifies every minimizer, including the transition cases.

Trust

Formal proof listed

Field

Differential geometry

1 paper

The explanation

Start with a unit cube whose opposite faces are glued. A bug leaving through the top comes back through the bottom. This space is the cubic flat three-torus. Now enclose one tenth of its volume with as little surface area as you can. A ball has surface area (36π)^(1/3) × 0.1^(2/3), about 1.042. A solid tube around a straight loop that closes up through two faces has area 2√(0.1π), about 1.121. A slab between two parallel faces always has area 2. At volume 0.1 the ball wins.

At volume one quarter, the ball needs about 1.919 and the tube about 1.772. The tube now wins. At larger volumes the slab wins, since its area stays at 2. The guess was that the best shape always follows this order. It appears in work of Hauswirth, Pérez, Romon and RosSource 1. Some pieces were known. Balls win for small enough volumes, by Morgan and JohnsonSource 2. Slabs are the only winners near half the volume, by Acerbi, Fusco and MoriniSource 3. But those results did not reach the switching volumesSource 4. Milman reduced the full guess to checking just two volumesSource 5.

The paper proves the full guessSource 6. The least area at volume V is the smallest of the three numbers aboveSource 6. Above one half, use 1 − V in place of V. The ball wins up to volume 4π/81, about 0.155. There the ball has radius 1/3, the tube radius 2/9, and both have area 4π/9, about 1.396. The tube wins up to volume 1/π, about 0.318. There tube and slab both have area 2. Then the slab wins up to one half. Past one half, the winners are the complementsSource 7. At each switch both shapes win and nothing else doesSource 8.

The remaining difficulty was to rule out a surface with handles, of higher genus, that might do betterSource 9. The new ingredient slices a possible rival by parallel planes near the tube–slab switchSource 10. Nesting of those slices forces extra length in some of them, which adds up to too much area. Explicit arithmetic bounds finish the last one-variable estimateSource 11.

How sure is this? The Lean scope note says the formalization determines the least-area profile and all minimizers in the unit cubic flat three-torus. It includes every equality case at the volumes 4π/81 and 1/π. No specialist review has been published.

What it doesn't show. This does not treat flat tori of other shapes, such as non-cubic boxes, or higher dimensions.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Solid. Lean covers the full profile and the classification of minimizers, including the transition cases. No specialist review is published.

Words used

Flat three-torus: A cube with opposite faces glued, so that moving out through one face brings you back through the opposite one.
Isoperimetric problem: The question of how to enclose a fixed volume with the least boundary area.
Isoperimetric profile: The least boundary area needed to enclose each volume.
Minimizer: A region that achieves the least boundary area for its volume.
Shortest closed geodesic: A shortest straight loop in the torus, here a line that runs across the cube parallel to one edge.
Slab: The region between two parallel flat faces of the cube.
Genus: The number of handles of a surface; a sphere has genus 0 and a doughnut surface genus 1.

The original paper

The Isoperimetric Conjecture for the Cubic Flat Three-Torus (PDF, opens in a new tab)
September 24, 2026

Lean notes

All quoted lines (11)

The ball–cylinder–slab prediction appears explicitly in the periodic isoperimetric problem of Hauswirth, P\'erez, Romon, and Ros and in Ros's account of the cubic torus and cube.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
Morgan and Johnson's sharp small-volume comparison implies that balls minimize on a compact flat manifold for sufficiently small volumes.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
Acerbi, Fusco, and Morini proved that coordinate slabs are the only minimizers near half volume in the cubic three-torus, up to translations and coordinate permutations.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
These small-volume and near-half-volume neighborhoods were obtained without effective bounds reaching the proposed transitions.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
He also made explicit the reduction of the profile conjecture to $4\pi/81$ and $1/\pi$ , using the method of Hauswirth, P\'erez, Romon, and Ros.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes $4\pi/81$ and $1/\pi$ .
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
For $V>1/2$ , the minimizing regions are the complements of those at volume $1-V$ .
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
For $0<V\le1/2$ , every minimizing region is, up to an ambient isometry and a null set, one of the following: $\begin{array}{c|c} \text{volume range}&\text{region}\\ \hline 0<V\le 4\pi/81 & \text{a ball of radius }(3V/(4\pi))^{1/3}\\[2pt] 4\pi/81\le V\le1/\pi &\text{a solid circular tube of radius }\sqrt{V/\pi}\\ &\text{about a shortest closed geodesic}\\[2pt] 1/\pi\le V\le1/2 &\text{a slab of width }V\text{ between coordinate tori}. \end{array}$ Both listed types occur at each common endpoint of adjacent ranges, and there are no further equality cases.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
The remaining geometric difficulty is to exclude connected boundaries of higher genus.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
The main new ingredient is a quantitative use of nested planar sections at the cylinder–slab transition.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text
The other configurations are excluded by scalar inequalities, and explicit outward arithmetic bounds complete the remaining one-variable estimate.
The Isoperimetric Conjecture for the Cubic Flat Three-Torus Back to text

Other sources

openai/math, Lean scope note for family 354 The formalization determines the isoperimetric profile and all minimizers in the unit cubic flat three-torus.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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