Mathematical Sciences: The Leviforms and CR Extension

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This project will continue work on problems of partial differential equations and geometric questions centering on the theory of several complex variables and complex geometry. Work will be done on restrictions of holomorphic functions to submanifolds. These are functions which always satisfy certain equations known as the tangential Cauchy Riemann equations. The investigations to be carried out on this project involve the converse question: What local geometric conditions on a submanifold of complex Euclidean space guarantee that each function satisfying the tangential Cauchy Riemann equations is locally the restriction of holomorphic function? Methods of complex geometry and analysis will be applied in studying this fundamental question. The geometric conditions are described by curvature conditions as measured by the Levi forms and sector property conditions on the submanifold. The analytic disc technique and the FBI transform will be used to construct the extension of a given CR function. Work will also be done in applying the technique of analytic discs to prove Hardy Littlewood maximal function type estimates on domains in complex Euclidean space whose boundaries possess some weak form of convexity. Results of this research are expected to have application to the theory of partial differential equations and to differential geometry.