Geometry of Real Submanifolds in Complex Space and CR Structures

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Abstract Eberfelt In this project, the principal investigator (PI) studies geometric, analytic, and algebraic aspects of real submanifolds in complex manifolds and, more generally, of manifolds with a CR structure. More specifically, he focuses on questions that are related to the local classification problem, which asks for a local description of the CR structure on a manifold near a distinguished point up to equivalence. For instance, for a real submanifold in complex space, one would like to know which other real submanifolds are equivalent to it by a local biholomorphic transformation. The PI studies this problem extrinsically by trying to find normal forms in classes of real submanifolds, and intrinsically as an equivalence problem for systems of differential equations. An important part of the classification problem is to understand the group of transformations preserving the structure or, more generally, the set of mappings between two given structures. The PI of this project studies the local stability group of a real submanifold in complex space, i.e. the group of local biholomorphisms preserving the real submanifold and a given distinguished point on it. He investigates under what conditions this group can be embedded as a subgroup of the jet group of a predetermined order, and seeks to describe the subgroups that arise in this way in more detail. He looks for conditions that imply coercivity results such as e.g. convergence of all formal mappings between real-analytic submanifolds, or real-analyticity of all smooth CR mappings. He also investigates closer the prolongation of the system defining CR mappings to a Pfaffian system, and explores its applications. The theory of several complex variables is a rapidly developing subject in mathematics which has applications in contemporary mathematical physics (e.g quantum field theory and string theory) as well as in engineering (e.g. control theory). The study of the geometry of real submanifolds in complex spaces, such as e.g. smooth boundaries of domains, is central to this theory, and is also related to other areas of mathematics such as partial differential equations and differential geometry. In this project, we investigate questions regarding the geometry of real submanifolds in complex space and their mappings that arise in the classification problem of such up to equivalences that preserve the complex structure of the ambient space.