Effondrement, spectre et propriétés diophantiennes des flots riemanniens

Jammes, Pierre

Annales de l'Institut Fourier, Tome 60 (2010) no. 1, pp. 257-290.

On étudie le comportement des premières valeurs propres du laplacien agissant sur les formes différentielles lors d’un effondrement adiabatique d’un flot riemannien $ℱ$ sur une variété compacte $M$. Le nombre de petites valeurs propres peut alors se calculer en fonction de la cohomologie basique de $ℱ$, et on donne des critères spectraux pour l’annulation des classes d’Álvarez et d’Euler du flot. En outre, on définit un invariant de nature diophantienne du flot qui est lié au comportement asymptotique des petites valeurs propres. Un appendice est consacré aux propriétés arithmétiques des flots riemanniens.

We study the behavior of the first eigenvalues of the Hodge Laplacian acting on differential forms under adiabatic collapsing of a riemannian flow $ℱ$ on a closed manifold $M$. 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 Álvarez class and the Euler class of the flow. We also define a diophantine invariant of the flow which is related to the asymptotical behavior of the small eigenvalues. An appendix is devoted to arithmetic properties of riemannian flows.

Classification : 58J50, 58C40, 53C12
Mot clés : effondrements, formes différentielles, laplacien, petites valeurs propres, flots riemanniens, approximations diophantiennes
Keywords: Collapsing, differential forms, Laplacian, small eigenvalues, riemannian flows, diophantine approximations

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