Mathematical Sciences: Analytic Discs in Several Complex Variables

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This award supports mathematical research on problems concerned primarily with the extension of functions and mappings defined on boundaries of domains in several complex variables. The boundaries are generalizations of surfaces, known as (real) manifolds. The idea of extending a function from the manifold to a holomorphic function on the domain is actually a boundary value problem in partial differential equations. Functions which can be extended must satisfy certain (Cauchy-Riemann) conditions on the manifold. It has recently been shown that all CR functions on a generic manifold extend over a wedge in a neighborhood of some point of the manifold. Work will now be done adapting newly developed techniques to establish the location of the wedge. Another research direction will be that of determining geometric conditions for extending CR functions over a neighborhood of the manifold. Analyticity or wedge-extendibility of a CR function on a CR manifold propagates along complex curves in the manifold. A natural consideration is also the propagation of CR extendibility. Work will be done in showing that CR extendibility is transported in parallel with respect to some connection on M. A basic tool in all the work is that of using analytic discs to localize the analysis. Work on mappings between CR manifolds will focus on continuous maps between analytic manifolds to determine the extent to which continuity actually implies that the mappings are holomorphic. Some initial smoothness assumptions will be made coupled with the use of the reflection principle on attached analytic discs.