Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are 4\pi/81 and 1/\pi, with exactly the adjacent two types minimizing at each transition. (Lean)
Manuscript
The Isoperimetric Conjecture for the Cubic Flat Three-Torus
September 24, 2026 · source · cite as \cite{OAI:The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026}
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 minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.
Result 354 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md