Geometry of Real Submanifolds in Complex Space and CR Structures

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DMS 0401215PI: Peter EbenfeltUC San DiegoGeometry of real submanifolds in complex space . . .Abstract:The principal investigator will study geometric, analytic, and algebraic aspects of generic real submanifolds in complex manifolds (or, more generally, of manifolds with a CR structure) and their mappings. He will investigate the existence, uniqueness, and regularity of CR mappings between given CR manifolds, as well as geometric questions that arise in connection with this study.More specifically, the proposer will focus on these problems in the context of studying CR embeddings of a strictly pseudoconvex hypersurface into another of higher dimension. In contrast to the case in which the manifolds are of the same dimension, little is known here and there are a number of unresolved but very basic questions. The PI will also study geometric properties ofCR mappings between generic submanifolds of higher codimension, as well as continuing his study of CR mappings between real hypersurfaces of infinite type (in the sense of Kohn and Bloom--Graham) by investigating closer the prolongation of the system defining CR mappings to a singular Pfaffian system on the jet bundle. Another problem that the PI plans to study is that of normal forms for real hypersurfaces in complex manifolds.The study of real submanifolds in complex manifolds is central to the theory of several complex variables, and has close connections to other areas of mathematics, such as partial differential equations, differential geometry, and algebraic geometry, as well as to contemporary topics in mathematical physics. A real submanifold of a complex manifold inherits, from its ambient manifold, a partial complex structure, called a CR (for Cauchy-Riemann) structure, which in general is much more rigid than the complex structure of the ambient manifold. Much of the research in this project is motivated by the desire to classify such CR structures up to equivalences which leave the ambient manifold invariant. This is one of the most fundamental questions in this field.