Differential geometry and analysis on CR manifolds / Sorin Dragomir, Giuseppe Tomassini

In this monographs the authors present some properties and results related to the differential geometry and analysis on CR manifolds. In the first part, they use as a central tool the canonical Tanaka-Webster connection defined in the case of the nondegenerate CR manifolds of hypersurface type, in order to study the pseudo-Hermitian geometry of CR manifolds. Next they consider the Fefferman metric, studying some CR invariants, the wave operators, the curvature, the Monge Ampere equations etc. The authors study the Yamabe problem, obtaining embedding results, regularity results, existence and uniqueness of extremals as well as some open problems. Then they consider the theory of pseudoharmonic maps obtaining geometric interpretations, some variational approach and some results related to the Hörmander systems. In the second part of the book, the authors study the pseudo-Einsteinian structures, for which the pseudo-Hermitian Ricci tensor of the Tanaka-Webster connection is proportional to the Lévi form. They present the achievements in the field together with the Lee conjecture. Next they deal with pseudo-Hermitian immersions, quasiconformal mappings and Yang-Mills fields on CR manifolds. At the end, the authors study some aspects from the spectral geometry: commutation formulas, Bochner type formulas, integral identities, a Greenleaf theorem, the Folland-Stein operator, some results of Zhong Jiaqing and Yang Hongcang. (Zentralblatt)