The macroscopic sound of tori

Abstract

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.

Variational convergence over metric spaces

“…This also has some applications to a homogenization problem, [31], and convergence of Dirichlet forms, [16,17,28], including finitedimensional approximation problems, [19,18]. The results of such studies could possibly extend to the nonlinear case.…”

The macroscopic spectrum of nilmanifolds with an emphasis on the Heisenberg groups

“…In the light of theorem 11, the good sense is the compact convergence of the resolvent like in our paper on the macroscopical sound of tori [Ver02]. The proof is quite similar, but needs some adaptation to the geometry of nilmanifolds.…”

“…The theorem and definition of the previous section is useful, for the goal we would like to achieve thanks to the following one (whose proof can be found in [Ver02] for example), which links the convergence of the resolvents R α ζ associated to a family of functionals A α , with their spectra.…”

“…In the particular case of tori D. Burago and S. Ivanov [BI95] showed that the asymptotic volume is bounded from below by the constant arising from the flat cases, which depends only on the dimension, and furthermore that the equality caracterises flat tori. In [Ver02] there is an alternate proof using the macroscopic spectra in the 2-dimensional case.…”

“…Thus we are naturally lead to explore the equality case (see [Ver02] for the case of tori). Feeding with new assumptions on the nilmanifold we get for example the following theorem (see also section 5), introducing a family of metric which are not far from being left invariant (see definition 18).…”