Diabatic limit, eta invariants and Cauchy–Riemann manifolds of dimension 3☆

Abstract

We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various η-invar-iants in CR geometry: on the one hand a renormalized η-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 η-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 Kähler-Einstein or an Einstein metric.1991 Mathematics Subject Classification. 32V05, 32V20, 53C20, 58J28. Key words and phrases. CR manifolds of dimension 3, pseudohermitian structures, eta invariants, contact complex.The second author is supported in part by a Young Researchers aci program of the French Ministry of Research.

On the Burns-Epstein invariants of spherical CR 3-manifolds

“…As an illustration, we present an example of the homology sphere Σ (2,3,11). Its fundamental group has the following presentation: (Σ(2, 3, 11)) = x1, x2, x3, h| h central, x By the computation in [14], we know that Σ(2, 3, 11) has five distinct irreducible representations into PU(2, 1).…”

Gravitational Chern–Simons and the adiabatic limit

“…As is noted in [3], the adiabatic limit has been known for some time, and has been studied in [4] and [5], for example.…”

“…(7). We borrow this terminology from [3], where one considers the adiabatic limit for the family of eta-invariants,…”

Analytic torsions on contact manifolds

“…However, we cannot prove equality of metrics in general, but only on particular contact manifolds called CR Seifert manifolds in [7]. These are CR manifolds (M, H, J) of dimension 3 admitting a transverse locally free circle action preserving the CR structure (H, J); see Definition 4.1.…”

“…independent of contact form) metric on det H * (E, d H ), where R and A are the Tanaka-Webster scalar curvature and torsion. Moreover spectrum when one takes the sub-Riemannian (or diabatic) limit ε 0 of calibrated metrics g ε = dθ(·, J·) + ε −1 θ 2 ; see [38,7]. Note that the classical Ray-Singer metric stays constant in this limit, being independent of the metric on M .…”