PSEUDOHERMITIAN GEOMETRY, SUBELLIPTIC THEORY AND SPACE-TIME PHYSICS 1

This is a collection of three lectures, at a graduate school level, devoted to the main problems in CR and pseudohermitian geometry and focusing on the impact of subelliptic theory, as a tool similar to elliptic theory in Riemannian geometry, and on the interrelations between pseudohermitian geometry and general relativity theory. The classical themes in CR geometry are the CR embedding problem and the CR extension problem, to which one may add the relationship between hyperbolic and subelliptic theories, as stemming from the existence of Fefferman’s metric, a Lorentzian metric associated to a positively ori- ented contact form on a strictly pseudoconvex CR manifold. Topics to be dealt with lie at the cross road of three main mathematical disciplines i.e. complex analysis in several complex variables, partial differential equations, and differential geometry.