Diabatic Limit, Eta Invariants and Cauchy-Riemann Manifolds of Dimension 3

Olivier Biquard (1) , Marc Herzlich (2) , Michel Rumin (3)

1 IRMA - Institut de Recherche Mathématique Avancée
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various eta-invariants in CR geometry: on the one hand a renormalized eta-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding the size of the Reeb field; on the other hand the eta-invariant of the middle degree operator of the contact complex. We then provide explicit computations for a class of examples: transverse circle invariant CR structures on Seifert manifolds. Applications are given to the problem of filling a CR manifold by a complex hyperbolic manifold, and more generally by a Kahler-Einstein or an Einstein metric.

Domaines: Géométrie différentielle [math.DG]

Olivier Biquard, Marc Herzlich, Michel Rumin. Diabatic Limit, Eta Invariants and Cauchy-Riemann Manifolds of Dimension 3. Annales Scientifiques de l'École Normale Supérieure, 2007, 40 (4), pp.589-631. ⟨10.1016/j.ansens.2007.06.001⟩. ⟨hal-00005356⟩