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 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. Ebenfelt will consider 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 themselves constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The work should also provide insight into how the local CR geometry of such hypersurfaces (in principle completely encoded in the CR curvature tensor) affects properties, such as e.g. various notions of nondegeneracy and rigidity, of their CR maps. There are highly interesting and nontrivial problems even in the special case where both the source and target manifolds are hyperquadrics (e.g. the Gap Conjecture). Ebenfelt will continue his study of CR maps between generic submanifolds of infinite (commutator) type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. The PI will, in particular, focus on a conjecture in this context regarding finite jet determination of local automorphisms. The study is 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 continue his work on normal forms for infinite type hypersurfaces in complex 2-space.; results in this direction will yield results on the finite jet determination problem and, possibly, on the problem of convergence of formal mappings in this context. Ebenfelt will also study CR maps between more general CR manifolds: one topic of interest is that of transversality, and he will try to improve recent transversality results (joint with with Duong and Baouendi--Rothschild) for CR maps into a higher dimensional space. Current conditions for transversality in this context involve eigenvalues of the Levi form. Ebenfelt believes that "deeper" invariants of the CR structures are actually involved, as in the equidimensional case. This study will likely require development of substantially new methods, which in turn will benefit the theory of CR maps into higher dimensional spaces.
The study of real submanifolds in complex manifolds is central to complex analysis and to other areas of mathematics and physics. In this mathematics research project, tools from a wide range of areas such as real and complex analysis, partial differential equations, and algebraic geometry are used and further developed. The investigations carried out in this project will benefit research in adjacent areas of mathematics as well as in areas of theoretical physics. The methods and techniques developed will be useful in other areas of mathematics, and likely also in physics (e.g., string theory) and engineering (e.g., control theory; systems engineering). Ebenfelt expects that the project will 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.
