The principal investigator will analyze the problem of existence and regularity of holomorphic embeddings of certain CR manifolds of real hypersurface type. These are manifolds having the same internal structure as is induced on the boundary of a smooth, strongly pseudoconvex domain in complex Euclidean space. Key elements of the method to be applied are Henkin's solution operators for the tangential Cauchy-Riemann equations. For the local problem he will improve previous estimates and reduce, as much as possible, the considerable derivative loss in his existing arguments. The problem of global embeddings will be further analyzed. The study will lead to a deeper understanding of the concept of boundary of a complex manifold and of properties of certain over-determined systems of linear, first order, partial differential equations. An endless curve in the plane which wraps up on itself is an example of an immersion of the real line which is not an embedding. An embedding always manages to steer clear of itself. Holomorphic embeddings require that the function, which injects the curve or surface into the target manifold, be smooth in the complex analytic sense. Theorems from the study of CR or complex analytic manifolds have always seemed miraculous by the standards of traditional real analysis. The results to be completed by this project will be no different.
