Global Invariants for CR Geometry and Isolated Singularities

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Abstract Award: DMS-0503868 Principal Investigator: Stephen S.T. Yau This research proposal contains projects in five areas: (1) Fundamental problem in CR geometry, (2) Explicit computation of CR automorphism groups, (3) Simultaneous embedding problem for a CR family of compact CR manifolds, (4) Deformation of CR manifolds and deformation of isolated singularities, (5) Holomorphic De Rham cohomology and complex Plateau problem. Yau introduces a new Bergman function for any strongly pseudoconvex complex manifold, which is invariant under biholomorphic maps. He intends to use this Bergman function to study the CR equivalent problem of strongly pseudoconvex CR manifolds lying on the same variety with isolated normal singularities. He developed a new technique to define a continuous numerical invariant on strongly pseudoconvex CR manifolds lying on the same variety. He shows that his invariant varies continuously in R when the CR structure of strongly pseudoconvex CR manifold changes in the variety. Yau observes that his new Bergman functions put a lot of restriction on biholomorphic maps between strongly pseudoconvex CR manifold, from which the automorphism groups of the CR manifolds can be determined explicitly. He illustrates how this works in a concrete example. The proposed work in (3) builds on his joint work with Xiaojun Huang and Hing Sun Luk on simultaneous embedding of CR manifolds. It will be done in collaboration with Xiaojun Huang. They intend to use their prior work on further extensions to study the most subtle problem on simultaneous embedding of 3-dimensional CR manifolds. Yau plans to use the techniques developed in (1) and (3) above to understand the relation between the Kuranishi family of CR manifolds and versal deformation of isolated normal singularities in a more concrete manner. Finally the work in (5) is part of Yau's ongoing project on holomorphic De Rham cohomology and complex Plateau problem. He has shown that the vanishing of holomorphic De Rham cohomology of the CR manifold X gives a lot of restriction on the singularities of the variety which X bounds. He plans to use his result to solve the clasical complex Plateau problem for 3-dimensional strongly pseudoconvex CR manifolds in 3-dimensional complex Euclidean space. This research proposal has an important theme of unifying different fields such as CR Geometry, Complex Analysis, Singularities Theory and Algebraic Geometry together. The philosophy and technique of the proposal will also be useful in biotechnology. With the completion of human genomic sequence, it is an urgent task to accurate identify protein coding regions (exons) from genomic sequence. Similar to Yau's proposal, one can try to find numerical characters of exons which introns do not have. In this way, one may be able to identify all possible human proteins. The next important task is to predict the properties of these newly discovered proteins. Using the similar technique developed in Yau's proposal, one may able to represent each protein as a point in certain n-dimensional space. Proteins which are closed to each other in this space should have similar properties. In this way, one can predict some properties of newly found proteins. Biologists can do experiment to verify these properties.