Spectral and Isoperimetric Bounds on Flat Tori

Abstract

We record several elementary relations between spectral and isoperimetric parameters of a flat torus $\mathbb{T}_Λ= \mathbb{R}^n/Λ$ and the covariance structure of a fundamental domain $K$ for the lattice $Λ\subset \mathbb{R}^n$. For every measurable fundamental domain $K$ and nonzero vector $ξ$ in the dual lattice $Λ^*$, we observe the sharp directional variance estimate \[ \left\langle \operatorname{Cov}_K ξ,ξ\right\rangle \geq \frac{1}{12}. \] This yields a lower bound on the torus spectral gap $λ_{\mathrm{SG}}(\mathbb{T}_Λ)$ (equivalently, the length of the shortest nonzero dual vector $λ_1(Λ^*)$) in terms of the maximal covariance of $K$: \[ λ_{\mathrm{SG}}(\mathbb{T}_Λ) = 4π^2 λ_1(Λ^*)^2 \geq \frac{π^2}{3\left\|\operatorname{Cov}_K\right\|_{\mathrm{op}}}. \] Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant $D_{\mathrm{Che}}(\mathbb{T}_Λ)$ using an old argument of Hadwiger. In particular, when the Voronoi cell $K_Λ$ of a lattice with $\det Λ= 1$ is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that \[ D_{\mathrm{Che}}(\mathbb{T}_Λ),\quad λ_{\mathrm{SG}}(\mathbb{T}_Λ),\quad λ_1(Λ^*) \geq c > 0, \] where $c > 0$ is a universal constant independent of dimension $n$; this may be thought of as a positive resolution of the Kannan--Lovász--Simonovits conjecture for all flat tori. While there are lattices $Λ$ and corresponding Voronoi cells $K = K_Λ$ for which no dimension-independent converse inequality to the spectral-gap bound above can hold, we show that under a certain sectional tiling hypothesis, this inequality is in fact an equivalence (up to numerical constants).

Disclosure

“ting point in this work was Hadwiger’s theorem, which implies an equivalence between the spectral and isoperimetric properties of KΛ and TΛ under a certain tiling condition on KΛ , yielding a version of Theorem 1.5; when prompted to check, ChatGPT found that this equivalence cannot hold for general lattices. Finally, ChatGPT provided a first draft of this note, which was checked and polished by the author. 2 A universal covariance lower bound We begin with the elementary Haar-m”

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2026-08-23T05:13:00.569474Z

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