The goal of this project is to study geometric and analytic aspects of generic real submanifolds in complex manifolds (or, more generally, of manifolds with a CR-structure). Particular attention will be paid to the structure of mappings between such manifolds. Examples of generic manifolds include (smooth) boundaries of open subsets of complex Euclidian space. Important information about proper mappings between open sets can be gleaned from their restrictions to the boundary. Basic questions that will be investigated include the existence, uniqueness, and regularity of CR-mappings between given CR-manifolds, as well as geometric questions that arise in connection with this study. The objective is to gain a deeper understanding of mappings in CR-geometry, and their role in complex analysis and PDE. One part of the project will focus on these problems in the context of nontrivial CR-mappings of a Levi-nondegenerate hypersurface into another of higher dimension. The principal investigator expects that this research will lead to a better grasp of how the local CR-geometry of such hypersurfaces (which, in principle, is completely encoded in the Chern-Moser CR-curvature tensor) affects geometric properties of CR-mappings (e.g., various notions of nondegeneracy and rigidity). The principal investigator will also study geometric and analytic properties of CR-mappings between more general CR-manifolds. A particular analytic property of interest is that of finite jet determination. The equidimensional case is by now fairly well understood. The principal investigator intends to study the situation where the target manifold has a higher dimension than that of the source. This situation appears to be drastically different from the equidimensional one. The principal investigator anticipates that this study will involve the development of substantially new methods, which in turn will enhance our understanding of mappings 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 research, 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 principal investigator is hopeful that the investigations carried out in this project will enhance our understanding of the geometry of real submanifolds and partial differential equations in complex space, which will benefit research in adjacent areas of mathematics as well as in areas of theoretical physics. He expects the project to provide interesting research topics for graduate students. The seminar activity that results from the project should prove stimulating for both students and other researchers.
