The Problem of CR Extension

Document Type

Presentation Abstract

Presentation Date

2-10-2005

Abstract

We begin by considering the following question: Given an open set V in ℂn , is it possible to find a larger open set U containing V such that every holomorphic function on V extends to a holomorphic function on U? If , n=1 this is never possible. But in several variables this can sometimes be done, and characterizing those sets for which extension is possible is a difficult problem. Perhaps even more surprising is the fact that, under suitable hypotheses on a hypersurface in ℂn, there may exist a common open set in ℂn to which every sufficiently smooth solution to a certain system of partial differential equations extends holomorphically. Furthermore, such an extension phenomenon may even be observed for sets of higher codimension if they retain some of the complex structure of the ambient space. These sets are the CR manifolds. The associated partial differential equations are referred to as the Cauchy-Riemann equations, and their solutions are the CR functions. The problem of CR extension, then, is to understand under what conditions there exists a common open set in ℂn to which every CR function on a CR manifold extends holomorphically, and, when CR extension is possible, to describe this set.

In this talk, I will discuss the properties we expect of the regions for CR extension. I will describe work on a model class of manifolds that illustrates the limitations of earlier descriptions of CR extension and develops an alternative meeting the proposed criteria.

Additional Details

Thursday, 10 February 2005
4:10 p.m. in Skaggs 117

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