Isoperimetric Inequalities on Slabs with applications to Cubes and Gaussian Slabs

Abstract

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 $\mathbb{R}^2 / 2 \mathbb{Z}^2$ and the standard Gaussian measure in $\mathbb{R}^{n-1}$. 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. This has only been established for relative volumes close to $0$, $1/2$ and $1$ by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three-dimensional cube with side lengths $(\beta,1,1)$ in a new range of relatives volumes $\bar v \in [0,1/2]$. In particular, we confirm the conjecture for the standard cube ($\beta=1$) for all $\bar v \leq 0.120582$, when $\beta \leq 0.919431$ for the entire range where spheres are conjectured to be minimizing, and also for all $\bar v \in [0,1/2] \setminus (\frac{1}{\pi} - \frac{\beta}{4},\frac{1}{\pi} + \frac{\beta}{4})$. When $\beta \leq 0.919431$ we reduce the validity of the full conjecture to establishing that the half-plane $\{ x \in [0,\beta] \times [0,1]^2 \; ; \; x_3 \leq \frac{1}{\pi} \}$ is an isoperimetric minimizer. We also show that the analogous conjecture on a high-dimensional cube $[0,1]^n$ is false for $n \geq 10$. In the case of a slab with a Gaussian base of width $T>0$, we identify a phase transition when $T = \sqrt{2 \pi}$ and when $T = \pi$. In particular, while products of half-planes with $[0,T]$ are always minimizing when $T \leq \sqrt{2 \pi}$, when $T > \pi$ they are never minimizing, being beaten by Gaussian unduloids. In the range $T \in (\sqrt{2 \pi},\pi]$, a potential trichotomy occurs.