Work will be done on mathematical problems arising in the field of several complex variables and certain partial differential equations which occur naturally in this context. In all, five areas will be treated. The first concerns a reflection principle analogous to the classical one. One is given a holomorphic mapping defined on one side of a surface within the space of several complex variables and would like to extend the map across the surface to a full neighborhood. Recent results in two complex dimensions completely resolve the question for non- Levi-flat hypersurfaces. This condition is not sufficient in higher dimensions and efforts will be made to find the additional conditions necessary on the surface to ensure the reflection principle. A second line of investigation concerns properties of holomorphic mappings between hypersurfaces which are related by means of a holomorphic map from one into the other. Two main questions to be addressed concern conditions in which the map is essentially finite and the determination of conditions under which one can decide whether one given hypersurface can be holomorphically mapped into another (in some nonsingular fashion). Work will also be done on holomorphic extensions of functions defined on generic manifolds or, equivalently: can one identify restrictions of holomorphic functions along sector of the boundary of a domain, for example, when the domain is a wedge and the boundary is the edge of that wedge? It is known that under certain restrictive conditions CR-mappings between pseudoconvex smooth hypersurfaces which are diffeomorphisms are actually infinitely differentiable. Work is continuing in an effort to obtain smoothness information on such mappings under the weakest possible conditions, both on the manifolds and the mappings. Considerable progress has been made during the past year in this regard. The final theme of this research concerns a boundary analogue to the well-known result that a holomorphic function which vanishes to infinite order at an interior point of a domain vanishes throughout the connected component of the point. This need not be the case at a boundary point, although the same phenomenon obtains at boundary points for a large class of functions. It will be one objective of this project to determine precise conditions where this property of "unique continuation" can occur.
