Mathematical Sciences: Analytic and Geometric Function Theory

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The primary areas of investigation in this project are studies of potential theory in several complex variables and complex differential geometry related to manifolds in the space of several complex variables. Work in potential theory concerns the analysis of plurisubharmonic functions arising as solutions of a partial differential equation given by the complex Monge-Ampere operator. The plurisubharmonic functions are the basic building blocks for potential theory. They arise as solutions of Monge-Ampere equations. However, there are many unanswered questions regarding the nature of the Monge-Ampere operator which impede development of a plurisubharmonic theory. They include determining whether or not the plurisubharmonic functions form the natural domains of Monge-Ampere operator and establishing the boundedness of solutions of the inhomogeneous Monge-Ampere equation when the forcing term is a Borel measure. Work will also be done studying the analytic and geometric properties of complex manifolds and their boundaries, specifically Cauchy-Riemann manifolds. Certain renormalized characteristic classes, finite on strictly pseudoconvex domains, have been identified. They have proved effective in showing that the Chern form for Kahler-Einstein surfaces is a nonnegative (2,2) form. From this follows the remarkable fact that the integral of this Chern form can be broken into a part only dependent on the boundary topology. Work is now proceeding to determine the set of all renormalized characteristic classes, that is, global integrals of Chern-Weyl polynomials of curvature tensors of Kahler-Einstein metrics which are finite.