Take a torus with a
Riemannian metric. Lift the metric on its universal cover. You get a distance
which in turn yields balls. On these balls you can look at the Laplacian.
Focus on the spectrum for the Dirichlet or Neumann problem. We describe
the asymptotic behaviour of the eigenvalues as the radius of the balls goes
to infinity, and characterise the flat tori using the tools of homogenisation
our conclusion being that “Macroscopically, one can hear the shape of a
flat torus". We also show how in the two dimensional case we can recover
earlier results by D. Burago, S. Ivanov and I. Babenko on the asymptotic
volume.
