Papers › Flat tori with large Laplacian eigenvalues in dimensions up to eight
Flat tori with large Laplacian eigenvalues in dimensions up to eight
Chiu-Yen Kao, Braxton Osting, Jackson C. Turner
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
We consider the optimization problem of maximizing the k-th Laplacian eigenvalue, λₖ, 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 ≤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→∞ 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 →∞.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections