Flat tori with large Laplacian eigenvalues in dimensions up to eight

TL;DR

This paper investigates the maximization of Laplacian eigenvalues on flat tori in dimensions up to eight, revealing sequences of tori with eigenvalues significantly exceeding asymptotic expectations and exploring their geometric properties.

Contribution

It introduces a computational approach to identify flat tori with large Laplacian eigenvalues, extending understanding of spectral optimization in higher dimensions.

Findings

Constructed sequences of flat tori with large eigenvalues in dimensions up to eight.

Eigenvalues exceed Weyl's law predictions by factors of 1.54 to 2.01.

Derived and verified stationarity conditions for these tori.

Abstract

We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $\lambda_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For larger $k$, this is equivalent to the NP-hard problem of finding the $d$-dimensional (dual) lattice with longest $k$-th shortest lattice vector. As a result of extensive computations, for $d \leq 8$, we obtain a sequence of flat tori, $T_{k,d}$, each of volume one, such that the $k$-th Laplacian eigenvalue of $T_{k,d}$ is very large; for each (finite) $k$ the $k$-th eigenvalue exceeds the value in (the $k\to \infty$ asymptotic) Weyl's law by a factor between 1.54 and 2.01, depending on the dimension. Stationarity conditions are derived and numerically verified for $T_{k,d}$ and we describe the degeneration of the tori as $k \to \infty$.