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We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-type trace formulae.Comment: 40 page

Abstract. Let A be a self-adjoint operator acting over a space X endowed with a partition. We give lower bounds on the energy of a mixed state ρ from its distribution in the partition and the spectral density of A. These bounds improve with the refinement of the partition, and generalize inequalities by Li-Yau and Lieb-Thirring for the Laplacian in R n . They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of ρ, as seen from X, and some spectral entropy, with respect to its energy distribution. On R n , this yields lower bounds on the sum of the entropy of the densities of ρ and its Fourier transform. A general log-Sobolev inequality is also shown. It holds on mixed states, without Markovian or positivity assumption on A.

One knows that the large time heat decay exponent on a nilpotent group is given by half the growing rate of the volume of its large balls. This work deals with the similar problem of trying to interpret geometrically the heat decay on (one) forms. We will show how it is (partially) related to the depth of the relations required to define the group. The tools used apply in general on Carnot-Carathéodory manifolds.

We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay F (λ) = χ A (]0, λ]) 1,∞ , without assuming e −tA is sub-Markovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum-mechanical sense. As an illustration, one can relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of the p,2 -cohomology for some p ≥ 2.

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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.