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 R^n-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 (, 1, 1) in a new range of relatives volumes v 0, 1/2. In particular, we confirm the conjecture for the standard cube (=1) for all v 0. 120582, when 0. 919431 for the entire range where spheres are conjectured to be minimizing, and also for all v 0, 1/2 (1 - 4, 1 + 4). When 0. 919431 we reduce the validity of the full conjecture to establishing that the half-plane \ x 0, 0, 1² \; ; \; x₃ 1{ \} is an isoperimetric minimizer. We also show that the analogous conjecture on a high-dimensional cube 0, 1ⁿ 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 and when T =. In particular, while products of half-planes with 0, T are always minimizing when T 2, when T > they are never minimizing, being beaten by Gaussian unduloids. In the range T (2, ], a potential trichotomy occurs.