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Math-2659 · mathematics · Geometry

Isoperimetric profile of the cubic flat three-torus

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

On $\mathbb R^3/\mathbb Z^3$, let $V\in(0,1)$ be the prescribed volume and put $v=\min(V,1-V)$. Is the least perimeter \[\min\left\{(36\pi)^{1/3}v^{2/3},\;2\sqrt{\pi v},\;2\right\}?\] Are all minimizers, modulo isometry and null sets, balls for $v\leq4\pi/81$, round tubes along shortest geodesics for $4\pi/81\leq v\leq1/\pi$, coordinate slabs for $v\geq1/\pi$, and their complements for $V>1/2$, with exactly both adjacent types at transition volumes?

Short English statement

On $\mathbb R^3/\mathbb Z^3$, let $V\in(0,1)$ be the prescribed volume and put $v=\min(V,1-V)$. Is the least perimeter \[\min\left\{(36\pi)^{1/3}v^{2/3},\;2\sqrt{\pi v},\;2\right\}?\] Are all minimizers, modulo isometry and null sets, balls for $v\leq4\pi/81$, round tubes along shortest geodesics for $4\pi/81\leq v\leq1/\pi$, coordinate slabs for $v\geq1/\pi$, and their complements for $V>1/2$, with exactly both adjacent types at transition volumes?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2659-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2659

Cubic three-torus isoperimetry: The manuscript claims a complete isoperimetric profile for the unit cubic flat three-torus. Balls, tubes around shortest geodesics, coordinate slabs, and complements are the only minimizers, including the transition equality cases. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Isoperimetric profile of the cubic flat three-torus

OpenAI's claim

Cubic three-torus isoperimetry: The manuscript claims a complete isoperimetric profile for the unit cubic flat three-torus. Balls, tubes around shortest geodesics, coordinate slabs, and complements are the only minimizers, including the transition equality cases.

Mathematical statement of the claimed result

Mathematical claim 1.
On the unit cubic flat torus $\mathbb R^3/\mathbb Z^3$, for $0<V<1$ set $v=\min(V,1-V)$. The claimed finite-perimeter isoperimetric profile is \[I(V)=\min\{(36\pi)^{1/3}v^{2/3},\ 2\sqrt{\pi v},\ 2\}.\] For $V\le1/2$ every minimizer, up to isometry and a null set, is a radius-$(3V/(4\pi))^{1/3}$ ball when $V\le4\pi/81$, a radius-$\sqrt{V/\pi}$ tube about a shortest closed geodesic when $4\pi/81\le V\le1/\pi$, or a coordinate-torus slab of width $V$ when $1/\pi\le V\le1/2$. Adjacent types both minimize at common endpoints; no other equality types occur. Larger-volume minimizers are complements.

Outline of the argument

Cubic three-torus isoperimetry: A reduction leaves nonstandard competitors to be excluded at two critical volumes. Geometric pattern classification and slice inequalities eliminate the three- and four-vertex possibilities except one parameterized case. Nested-area estimates reduce that case to positivity of an explicitly checked scalar expression, and the two-volume reduction recovers every volume and equality case.

Formal verification

Family 354 describes Lean coverage for the profile and equality-case classification in the unit cubic three-torus.

Cubic three-torus isoperimetry: The manuscript links CubicTorus.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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