The isoperimetric conjecture for the cubic flat three-torus (balls, tubes, slabs)

Claims the full isoperimetric profile of $\mathbb R^3/\mathbb Z^3$, $I(V)=\min\{(36\pi)^{1/3}v^{2/3},2\sqrt{\pi v},2\}$, and a classification of all finite-perimeter minimizers: balls for $V\le4\pi/81$, tubes about shortest geodesics for $4\pi/81\le V\le1/\pi$, coordinate slabs for $1/\pi\le V\le1/2$, complements above $1/2$, with exactly the two adjacent types at each transition volume. The new step excludes three- and four-vertex nonstandard configurations at the two transition volumes via nested planar sections. It does NOT treat non-cubic flat tori, the cube with relative perimeter beyond the reflection correspondence, or higher dimensions.

Claims the full isoperimetric profile of $\mathbb R^3/\mathbb Z^3$, $I(V)=\min\{(36\pi)^{1/3}v^{2/3},2\sqrt{\pi v},2\}$, and a classification of all finite-perimeter minimizers: balls for $V\le4\pi/81$, tubes about shortest geodesics for $4\pi/81\le V\le1/\pi$, coordinate slabs for $1/\pi\le V\le1/2$, complements above $1/2$, with exactly the two adjacent types at each transition volume. The new step excludes three- and four-vertex nonstandard configurations at the two transition volumes via nested planar sections. It does NOT treat non-cubic flat tori, the cube with relative perimeter beyond the reflection correspondence, or higher dimensions.