Spectral and Isoperimetric Bounds on Flat Tori

Abstract

We record several elementary relations between spectral and isoperimetric parameters of a flat torus $\mathbb{T}_\Lambda = \mathbb{R}^n/\Lambda$ and the covariance structure of a fundamental domain $K$ for the lattice $\Lambda \subset \mathbb{R}^n$. For every measurable fundamental domain $K$ and nonzero vector $\xi$ in the dual lattice $\Lambda^*$, we observe the sharp directional variance estimate \[ \left\langle \operatorname{Cov}_K \xi,\xi\right\rangle \geq \frac{1}{12}. \] This yields a lower bound on the torus spectral gap $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)$ (equivalently, the length of the shortest nonzero dual vector $\lambda_1(\Lambda^*)$) in terms of the maximal covariance of $K$: \[ \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda) = 4\pi^2 \lambda_1(\Lambda^*)^2 \geq \frac{\pi^2}{3\left\|\operatorname{Cov}_K\right\|_{\mathrm{op}}}. \] Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant $D_{\mathrm{Che}}(\mathbb{T}_\Lambda)$ using an old argument of Hadwiger. In particular, when the Voronoi cell $K_\Lambda$ of a lattice with $\det \Lambda = 1$ is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that \[ D_{\mathrm{Che}}(\mathbb{T}_\Lambda),\quad \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda),\quad \lambda_1(\Lambda^*) \geq c > 0, \] where $c > 0$ is a universal constant independent of dimension $n$; this may be thought of as a positive resolution of the Kannan--Lov\'asz--Simonovits conjecture for all flat tori. While there are lattices $\Lambda$ and corresponding Voronoi cells $K = K_\Lambda$ 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).