Mathematical Sciences: Geometry and Analysis of 3-Dimensional CR-Structures

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ABSTRACT Proposal: DMS-962304 PI: Epstein CR-geometry is the natural odd dimensional analogue of complex geometry. From work of Kohn, Boutet de Monvel, and Harvey and Lawson it is known that any compact strictly pseudoconvex CR-manifold of dimension 5 or greater can be realized as the boundary of a compact normal Stein space. A CR-manifold with such a realization is called embeddable. It has been known since the 1960s that the situation in 3-dimensions is quite different: the generic perturbation of an embeddable CR-structure is not embeddable. Our goal is to describe the set of embeddable CR-structures on a compact three manifold as a subset of the set of all CR-structures. For the case of deformations of the unit 3-sphere the small embeddable perturbations are essentially an infinite dimensional and infinite codimensional analytic submanifold of the set of all CR-structures. The research outlined in this proposal is directed towards extending such results to more general classes of 3-manifolds. In earlier work the investigator introduced a stratification of the set of embeddable CR-structures. Locally the stratification is defined by formally analytic relations. A major thrust of the proposed work is to analyze these equations using methods arising from the Nash-Moser implicit function theorem. It is hoped that it can be shown that the strata have a transverse analytic structure. General position arguments and a generalization of a result of Kiremidjian being sought by the investigator and G. Henkin would then show that the stratification has only finitely many distinct strata. This would imply that the set of embeddable structures is a closed subset in a reasonable topology. Using extremal determinants analogous to those used by Osgood, Phillips and Sarnak it is hoped to define a natural exhaustion function for the space of embeddable structures. The importance of mathematics in physics, economics, chemistry, engineering, etc. stems from the fact that EQUATIONS describe the rela tionships among the variables that arise in these fields. Making predictions is thereby reduced to solving the equations. One therefore needs criteria to determine whether the equations have solutions, and algorithms for solving them. In practice this can only be done approximately so it is important to have an estimate for the size of he errors one is making. In this proposal we consider a FAMILY of related equations that arise naturally in analytic geometry. These equations often do not have solutions. Even though the equations in the family are closely related, the property of solvability can change very wildly as one moves through the family. Our principal aim is to understand this instability and give criteria that describe the well behaved members of the family. There are similarities between the issues that arise in this analysis and problems encountered in image reconstruction techniques used, e.g., in CAT scans. It is hoped that a better understanding of the case at hand will provide insight into other cases where similar instabilities arise.