Isoperimetric Inequalities on Slabs with applications to Cubes and Gaussian Slabs

TL;DR

This work develops a general two-step framework to study sharp isoperimetric inequalities on slabs: (i) reduce the slab problem to a two-dimensional model slab built from the base isoperimetric profile, and (ii) analyze the resulting model via symmetrization, CMC equations, and stability/ODI tools. For the base being a flat torus and for Gaussian bases, the authors obtain detailed isoperimetric characterizations, including new results for the 3D cube: spheres about corners are minimizers for a broad range of volumes, cylinders about the short edge emerge as minimizers in an intermediate band, and flat planes minimize beyond the Gaussian boundary, with a precise dependence on the short edge length. In high dimensions, the conjectured profile on the cube fails for all dimensions with \(n \ge 10\), consistent with known instability phenomena in large dimensions. In Gaussian slabs, a phase transition at \(T = \sqrt{2\pi}\) and another at \(T = \pi\) delineate regimes where horizontal half-planes, Gaussian unduloids, or vertical half-planes minimize, revealing a rich trichotomy in the intermediate regime. Overall, the paper provides a robust variational and geometric toolkit for isoperimetry on slabs, yielding both sharp positive results and a clear counterexample in high dimensions, with implications for cubes and Gaussian slabs alike.

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

Figures (9)

• Figure 1: Cylinders in $\mathbf{Q}^3(\beta)$ are conjectured to be minimizers for (weighted) volumes $\bar{v}$ between the yellow and green plots. We are able to show they are minimizers between the yellow and blue plots.
• Figure 2: Plot of conjectured isoperimetric profile $\mathcal{I}_m$ of the three-dimensional cube $\mathbf{Q}^3(\beta)$ with side lengths $(\beta,1,1)$. The regimes when the interior of an eighth sphere about a corner and a quarter cylinder about the short edge are minimizing are highlighted in red and green, respectively. The true isoperimetric profile of the base $\mathbf{Q}^2$ (on $[0,1/2]$) is obtained by replacing the red curve by the dashed-green one.
• Figure 3: Various one-sided generalized unduloids $\{f(t)\}$ with different initial values $f(0) = s_1 = \Phi_{\mathbf{T}^2}^{-1}(v_1)$. For convenience, the $s$-axis is parametrized on $(0,1+\frac{2}{\pi})$ instead of $(-\frac{1}{2} - \frac{1}{\pi},\frac{1}{2} + \frac{1}{\pi})$. Note that when $s_1 \leq \frac{2}{\pi}$ (equivalently, $v_1 \leq \frac{1}{\pi}$), $\Sigma$ is a quarter circle in the model slab, corresponding to an eighth sphere in $\mathbf{Q}^3(\beta)$. When $s_1 > \frac{2}{\pi}$ the one-sided unduloids are no longer nested (and some require time $> \beta$ to reach the bottom).
• Figure 4: Plot of $F(\xi)$, lowerbounding $\beta V(\xi)$.
• Figure 5: By direct verification, $k \mapsto \mathcal{I}_m^{(k)}(v_s(10))$ is strictly decreasing on the sequence of integers $\{1,\ldots,10\}$, although not on the interval $[9,10]$. $k \mapsto \mathcal{I}_m^{(k)}(v_s(11))$ is strictly decreasing on the sequence of integers $\{1,\ldots,11\}$ (and on the entire $[1,11]$ as well).

Theorems & Definitions (100)

• Conjecture 1.1: Isoperimetric conjecture on $3$-dimensional cube
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Conjecture 1.5: Isoperimetric conjecture on $n$-dimensional cube
• Theorem 1.6
• Lemma 1.7
• proof
• Theorem 1.8
• Definition 2.1: Weighted Riemannian Manifold