CR extension for tube-like CR manifolds of CR dimension 1

Abstract

Although previous research on CR extension has emphasized the concept of wedge extendability, wedges do not have some of the properties we expect of the regions described by a general theory. In particular, the regions we describe should be of roughly the same size and shape as the full regions of extendability, they should vary smoothly as one varies the base point or the size of the open neighborhood on the manifold, and they should satisfy a natural containment condition. We illustrate through an example the failure of wedges to satisfy these conditions. We then develop an alternative description of the sort outlined above for the class of tube-like CR submanifolds of $\mathbb{C}^n$ of CR dimension 1.