Isoperimetric Inequalities on Slabs with applications to Cubes and Gaussian Slabs

We study isoperimetric inequalities on "slabs", namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension-one base. As our two main applications, we consider the case when the base is the flat torus R² / 2 Z² and the standard Gaussian measure in Rⁿ⁻¹. The isoperimetric conjecture on the three-dimensional cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. Our analysis confirms the isoperimetric conjecture on the three-dimensional cube with side lengths (β,1,1) in a new range of relatives volumes v ∈ [0,1/2]. In particular, we confirm the conjecture for the standard cube (β=1) for all v ≤ 0.120582, when β ≤ 0.919431 for the entire range where spheres are conjectured to be minimizing, and also for all v ∈ [0,1/2] (1/π - β/4,1/π + β/4). When β ≤ 0.919431 we reduce the validity of the full conjecture to establishing that the half-plane \ x ∈ [0,β] × [0,1]² \; ; \; x₃ ≤ 1/π \ is an isoperimetric minimizer. We also show that the analogous conjecture on a high-dimensional cube [0,1]ⁿ is false for n ≥ 10. In the case of a slab with a Gaussian base of width $T>0$, we identify a phase transition when T = √2 π and when T = π. In particular, while products of half-planes with $[0,T]$ are always minimizing when T ≤ √2 π, when T > π they are never minimizing, being beaten by Gaussian unduloids. In the range T ∈ (√2 π,π], a potential trichotomy occurs.