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

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Real surfaces or submanifolds in multidimensional complex spaces exhibit a rich local as well as global geometry. The interplay between real and complex geometry and analysis is a fundamental ingredient in studying these objects. Techniques from differential and algebraic geometry, as well as real and complex analysis and partial differential equations, will be used in the proposed work. The principal investigators plan to determine when two such manifolds are equivalent under invertible holomorphic transformations. They will look for new constructive criteria to build such mappings and to determine their convergence. They also plan to classify mappings from one surface in a multidimensional complex space into another embedded in a complex space of higher dimension. Of particular interest are those hypersurfaces defined by quadratic equations and admitting large symmetry groups. The latter will serve as models to discover more general phenomena and formulate basic properties of the manifolds under consideration. In particular, they will focus on determining when it is possible to reduce the biholomorphic equivalence problem to solving systems of polynomial equations with complex coefficients. They also plan to determine when a formal mapping sending a real submanifold into another is necessarily convergent. In addition, they will attempt to categorize those submanifolds for which such mappings are determined by finitely many derivatives at a given point. They expect that this study will lead to the discovery of new geometric, analytic, and algebraic invariants of these submanifolds. The principal investigators will initiate new studies of overdetermined systems of nonlinear partial differential equations of first and higher order. They will study the properties of solutions of such systems in order to classify all possible submanifolds with prescribed Cauchy data. Several problems discussed here have attracted the attention of many mathematicians and physicists since the beginning of the twentieth century, starting with the work of Henri Poincare and Elie Cartan. A number of fundamental problems remain unsolved to the present time. The study of the geometry of real manifolds in complex spaces is central to the field of several complex variables and to other areas of science, including geometry, mathematical physics, and engineering. Progress on the problems proposed by the principal investigators will likely have impact on these areas as well.