The View from Above

We will consider plane regions Omega subset of, Omega not equal C. This articleis an exposition of the theory, initiated by Sullivan and Thurston, governing the geometric relationship of Omega to a particular surface in upper half of euclidean 3-space that will be explicitly constructed below. It lies over Omega as the dome in a domed stadium lies over the playing field. The fact of the geometric relationship opens a new direction for viewing-itself. It is astonishing that in the case Omega is simply connected, there is a universal relationship in dependent of the particular shape of Omega. There are important applications to complex analysis and to the study of hyperbolic 3-manifolds.