OpenAI research collection · 1 manuscript · Source checked 2026-10-19
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 4π/81 and 1/π, with exactly the adjacent two types minimizing at each transition.
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We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes 4π/81 and 1/π. The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.
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