{"id":"openai-math/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus","pdf":"https://github.com/openai/math/blob/main/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026","tags":["geometric-analysis","isoperimetric-problem","flat-tori","minimal-surfaces"],"score":0,"title":"The Isoperimetric Conjecture for the Cubic Flat Three-Torus","domain":"Differential geometry","family":"354","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"The isoperimetric profile of the cubic three-torus"}