Geometric and Analytic Problems in Several Complex Variables and Partial Differential Equations

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Proposal: DMS-9801258 Principal Investigators: Linda P. Rothschild, Salah M. Baouendi Abstract: The principal investigators will study the geometry of real submanifolds in complex space and the holomorphic mappings which send one such manifold into another. In particular, they will attempt to categorize those manifolds for which such mappings are determined by finitely many derivatives at a given point. They expect that this study will lead to discovery of new geometric, analytic, as well as algebraic invariants of these manifolds. A basic geometric and analytic problem is whether a holomorphic mapping defined on one side of a real hypersurface embedded in complex space extends to the other side of that hypersurface. More generally, one can consider smooth mappings between manifolds of higher codimension in complex space whose components satisfy the boundary Cauchy-Riemann equations. The principal investigators will study such mappings, in particular when the manifolds are given globally by polynomial equations, and will try to determine when such mappings extend holomorphically to the complex space. They will also study the relationship between algebraic and holomorphic equivalence of such manifolds. Complex numbers and functions of complex variables have been, since the 19th century, important tools in many fundamental problems in mathematics and its application to other areas of science and engineering. The solutions of some of these problems depend on a better understanding of these tools and their basic properties. For instance, the four dimensional space-time used in relativity theory can be considered as a two dimensional complex space. The quantitative and qualitative study of the transformations of geometric objects in complex spaces planned in this proposal may lead to new understanding and solutions of problems in control theory (e.g., motion of objects in space under certain physical constraints). Complex analysis also plays an important role in finding solutions to differenti al equations which model physical problems. Results of the research planned in this proposal could lead to the discovery of new properties of solutions of these equations and hence a better understanding of the related physical problems.