Structure of Mappings in Several Complex Variables and Cauchy-Riemann Geometry

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The study of invariant geometries is an important part of mathematics. Different types of invariant geometries arise in different areas of mathematics, e.g., in several complex variables (SCV), partial differential equations (PDE), and algebraic, complex, and differential geometry. In this research project, the principal investigator Peter Ebenfelt will investigate a particular geometry that arises in the study of SCV, and which has also deep connections with contemporary topics in mathematical physics such as quantum field theory and string theory, as well as applications in, e.g., systems engineering and control theory. Tools that are needed for this study come from a variety of different areas in mathematics, such as complex analysis, PDE, and differential geometry, and the techniques and tools developed in this study influence these areas as well. Ebenfelt expects that the project will also provide interesting research topics for graduate students and postdocs. The seminar activity that results from the project should be stimulating for both students and other researchers. The goal of this mathematics research project by Peter Ebenfelt is to study geometric, analytic, and algebraic aspects of generic real submanifolds in complex varieties (more generally, of Cauchy-Riemann (CR) structures) and their mappings. Ebenfelt will consider questions regarding existence, uniqueness, and regularity of CR maps, as well as related questions that arise in connection with this study. He will consider the context of CR maps of a Levi nondegenerate hypersurface into a nondegenerate hyperquadric of higher dimension. This study will enhance our understanding of the CR submanifold structure of the hyperquadrics, which constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The proposed work should also provide insight into how the local CR geometry of such hypersurfaces (in principle completely encoded in the CR curvature tensor) affects various properties, such as notions of nondegeneracy and rigidity, of their CR maps. Ebenfelt will also continue his study of CR maps between generic submanifolds of infinite type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. He will, in particular, focus on understanding the recently discovered phenomenon that for infinite type hypersurfaces the biholomorphic, formal, and smooth CR equivalence classifications are different. He will also study a conjecture regarding finite jet determination of local automorphisms. These investigations are expected to shed new light on the nature of CR maps between infinite type manifolds, and lead to a better understanding of the Pfaffian systems arising in this context. Ebenfelt will also continue his work on normal forms for infinite type hypersurfaces in complex 2-space. Finally, Ebenfelt will study CR invariants on unit circle bundles over Kähler manifolds, specifically Cartan's umbilical tensor in the 3-dimensional case and those arising in the expansions of the Bergman and Szegö kernels. He intends to develop new methods for detecting umbilical points on general 3-dimensional CR manifolds, and resolve an open problem regarding existence of umbilical points on compact CR manifolds embedded in complex 2-space.