Effondrement, spectre et propri\'et\'es diophantiennes des flots riemanniens

TL;DR

This paper investigates how the eigenvalues of the Hodge Laplacian behave under collapsing of riemannian flows, linking spectral properties to cohomology, diophantine invariants, and arithmetic aspects.

Contribution

It introduces a diophantine invariant for riemannian flows and establishes spectral criteria relating eigenvalues to topological and arithmetic properties.

Findings

Number of small eigenvalues relates to basic cohomology.

Spectral criteria for vanishing of the Alvarez and Euler classes.

Definition of a diophantine invariant connected to eigenvalue behavior.

Abstract

Let F be a riemannian flow on a closed manifold M. We study the behavior of the first eigenvalues of the Hodge Laplacian acting on differential forms under adiabatic collapsing of the flow. We show that the number of small eigenvalues is related to the basic cohomology of F, and give spectral criteria for the vanishing of the \'Alvarez class and the Euler class of F. We also define a diophantine invariant of the flow wich is related to the asymptotical behavior of the small eigenvalues. An appendix is devoted to arithmetic properties of riemannian flows.