I ) Introduction 1 to CR geometry and subelliptic harmonic maps . II ) Boundary values of Bergman - harmonic maps

We give an elementary introduction to CR and pseudohermitian geometry, starting from H. Lewy’s legacy (cf. [20]) i.e. tangential Cauchy-Riemann equations on the boundary of the Siegel domain. In this context we describe fundamental objects, such as contact structures, Levi forms, the Tanaka-Webster connection and the Fefferman metric (cf. e.g. [4]). Also naturally arising Hörmander systems of vector fields, the associated sublaplacians, and J. Jost and C-J. Xu’s subelliptic harmonic maps (cf. [16]), a first geometric interpretation of which is given in terms of Lorentzian geometry (cf. [2]). A second, more specialized talk scheduled for the afternoon of the same day is devoted to the discussion of boundary values of Bergman-harmonic maps. There we start from A. Korányi & H.M. Reimann’s crucial observation (cf. [17]) that, as a consequence of Fefferman’s asymptotic expansion formula (cf. [9]) for the Bergman kernel of a smoothly bounded strictly pseudoconvex domain Ω ⊂ C K(ζ, z) = CΩ|∇φ(z)| · detLφ(z) ·Ψ(ζ, z)−(n+1) + E(ζ, z), |E(ζ, z)| ≤ C ′ Ω |Ψ(ζ, z)| −(n+1)+1/2 · |log |Ψ(ζ, z)|| , the Kählerian geometry of the interior of Ω may be effectively related to the contact geometry of its boundary ∂Ω. Then we make use of the Graham-Lee connection (cf.[13]) to derive the compatibility equations on ∂Ω satisfied by the boundary values of a Bergman-harmonic map Φ : Ω→ S which is C∞ up the boundary. We are led to a geometric interpretation (cf. [5]) of Jost & Xu’s subelliptic harmonic maps from an open set U ⊂ R2n−1 carrying a given Hörmander system of vector fields. Talks given at The State University of New Jersey Rutgers Campus at Camden, Department of Mathematical Sciences, within A Day for Complex Analysis and Geometry, May 3, 2013. Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, Campus Macchia Romana, 85100 Potenza, Italy, e-mail [email protected] 1