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

Abstract

Take a riemanniann nilmanifold, lift its metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the laplacian for the dirichlet problem on them. We describe the asymptotic behaviour of the spectrum when the radius of these balls goes to infinity. Furthermore we show that the first macroscopic eigenvalue is bounded from above, by an uniform constant for the three dimensional heisenberg group, and by a constant depending on the Albanese's torus for the other nilmanifolds. We also show that the Heisenberg groups belong to a family of nilmanifolds, where the equality characterizes some pseudo left invariant metrics.

The length of harmonic forms on a compact Riemannian manifold

“…The macroscopic spectrum of a nilmanifold is given by the asymptotic behaviour of the eigenvalues of a Laplace-Beltrami operator acting on the function on the metric balls of the universal covering of a nilmanifold, as the radius of the balls goes to infinity. In [Ver02] the second author showed that the first eigenvalue of this macroscopic spectrum satisfies an inequality, whose equality case is attained by the nilmanifolds having all harmonic 1-forms of constant norm. This shows that the nilmanifolds with left-invariant metrics are not the only ones satisfying the equality case, as in the torus case.…”