The view from above

Abstract

We will consider plane regions Ω ⊂ C, Ω ≠ C. This article is an exposition of the theory, initiated by Sullivan and Thurston, governing the geometric relationship of Ω to a particular surface in upper half of euclidean 3-space that will be explicitly constructed below. It lies over Ω as the dome in a domed stadium lies over the playing field. The fact of the geometric relationship opens a new direction for viewing Ω itself. It is astonishing that in the case Ω is simply connected, there is a universal relationship independent of the particular shape of Ω. There are important applications to complex analysis and to the study of hyperbolic 3-manifolds.