Mathematical Sciences: Applications of the d-bar Neumann Problem to the Study of Weakly Pseudoconvex Domains

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The central theme of this work is geometric problems in several complex variables. Three questions will be addressed in the current project, the first being the Kuranishi problem. Here one is interested in real manifolds which have structure similar to boundaries of domains in complex space. At issue is whether or not smooth pseudoconvex C-R manifolds admit embeddings. The analytic side of this question is one of solving a complex partial differential equation with mixed boundary conditions. If the research proceeds as predicted, the embedding result should be expressible in terms of the first two eigenvectors of the Levi form for the manifold (they must be positive). Another line of work of a similar nature concerns the fundamental first order d-bar-Neumann problem on pseudoconvex domains. One is interested in finding solutions of the inhomogeneous d-bar equation on a domain which are orthogonal to the space of holomorphic functions on that domain. The question of global regularity of the solutions depends not only on the geometry of the domain but also on the metric (which determines the volume element.). Work will be done in establishing conditions for global regularity on smoothly bounded pseudoconvex domains in terms of curvature inequalities which must be satisfied by the metric. The third element of this research concerns automorphisms or self-maps of bounded domains. Recent work of Bell and Pinchuk has essentially classified all pseudoconvex domains in two- dimensional complex space with noncompact automorphism group (the domains must also be of finite type). A number of related questions remain open which will be analyzed during this undertaking.