Three's company in six dimensions: irreducible, isospectral, non-isometric flat tori

TL;DR

This paper presents the first known example of three six-dimensional flat tori that are non-isometric yet share the same Laplace spectrum, extending the understanding of spectral geometry in low dimensions.

Contribution

It provides the first explicit example of a triplet of mutually non-isometric flat tori in six dimensions with identical Laplace spectra.

Findings

First known triplet of non-isometric flat tori in six dimensions with identical spectra

Extends spectral geometry understanding to new low-dimensional cases

Demonstrates existence of spectral isospectrality beyond previously known dimensions

Abstract

In 1964, John Milnor, using a construction of two lattices by Witt, produced the first example of two flat tori that are not globally isometric and whose Laplacians for exterior forms have the same sequence of eigenvalues. The aforementioned flat tori are sixteen-dimensional. One is reducible while the second is irreducible. In the ensuing years, pairs of non-isometric flat tori that share a common Laplace spectrum have been shown to exist in dimensions four and higher. In dimensions three and lower, Alexander Schiemann proved in 1994 that any flat tori that are isospectral are in fact isometric, so four is the lowest dimension in which such pairs exist. Using a four-dimensional such pair, one can easily construct an eight-dimensional such triplet. However, triplets of mutually non-isometric flat tori that share a common Laplace spectrum in dimensions 4, 5, 6, and 7 have eluded researchers - until now. We present here the first example.