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

We relate a recently introduced non-local invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various η-invariants: 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 transverse circle invariant CR structures on Seifert manifolds. This yields obstructions to filling a CR manifold by complex hyperbolic, Kähler-Einstein, or Einstein manifolds.