Some Recent Results in Cr Geometry

CR geometry originated from the study of real hypersurfaces in complex manifolds. In their foundational work Chern and Moser [CM] developed local invariants for CR manifolds. Shortly afterward Tanaka [T] and Webster [W2] introduced a canonical connection associated to a given pseudohermitian structure (i.e. a contact form), so pseudohermitian geometry was born. Today CR geometry or pseudohermitian geometry has become an independent subject with fascinating connections with complex analysis, Riemannian and sub-Riemannian geometry and other areas of mathematics. We will only consider oriented strictly pseudoconvex CR manifolds with a chosen pseudohermitian structure. With the induced Tanaka-Webster connection and the adapted metric one may try to develop a whole theory parallel to Riemannian geometry. There are many different directions. On the geometric side one can study the induced Carnot-Caratheodory distance, its geodesics, the Hausdorff measures etc. We will not say anything on these fascinating topics except by giving a few references. On the Carnot-Caratheodory distance in the more general subRiemannian setting one can read Gromov’s long paper [Gr] which contains a wealth of fascinating ideas. There have been a lot works on the isoperimetric problem in the Heisenberg space, cf. the book [CDPT]. Partly motivated by this problem, Paul Yang and his collaborators have developed a theory of p-mean curvature for surfaces in 3-D pseudohermitian manifolds. We refer to his survey [Y] and references therein. On the more analytic side, there are also many natural problems. The CR Yamabe problem initiated by Jerison and Lee [JL1] has been quite well understood and a recent reference is the book [DT]. We will not discuss it here. It is also natural to study the fundamental operators, the sub-Laplacian and the Kohn Laplacian on functions, and their spectrum on pseudohermitian manifolds. One would hope that this study will be as fruitful as the study of the spectrum of the Laplacian in Riemannian geometry. These operators are not elliptic and therefore their analysis involves new analytic challenges. Another major new complication is that the Tanaka-Webster connection has nontrivial torsion. In this paper we discuss some recent results in this direction. In Section 2 we give a quick summary of the basics in CR geometry. We take the opportunity to discuss a basic classification result which seems missing from the literature. In Section 3 we discuss some estimates on the eigenvalues of the fundamental operators. In Section 4 we discuss some Obata-type results in CR geometry and address the rigidity question in the sharp eigenvalue estimate. The last Section, in which we discuss a problem on the CR structure on circle bundles over