This is the contents of a course that the author has given at the C.I.M.E. Summer School held in Martina Franca, June 30–July 6, 2002. The author describes some recent results from the theory of extremal discs for CR manifolds. In particular he introduces the notions of extremal discs, stationary discs, defective discs, and supporting and positive lifts for smooth generic CR manifolds. He proves some relations between these concepts and considers the special case of stationary discs for quadrics. After that, he gives some general existence theorems for stationary discs, investigates the geometry of the lifts and contemplates the notion of defective manifolds, conjecturing that no quadrics are defective. Finally he applies the extremal discs to the question of the regularity of CR mappings, using the idea that a CR mapping preserves the lifts of the extremal discs.
|Name||Itogi Nauki i Tekhniki|
|Publisher||Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow|