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.
Original language | English (US) |
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Pages (from-to) | 383-394 |
Number of pages | 12 |
Journal | Pure and Applied Mathematics Quarterly |
Volume | 7 |
Issue number | 2 |
DOIs | |
State | Published - Apr 2011 |
Keywords
- Floor-to-dome map
- Quasiconformal
- Sullivan theorem
- Uniformly perfect