The view from above

Albert Marden

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 languageEnglish (US)
Pages (from-to)383-394
Number of pages12
JournalPure and Applied Mathematics Quarterly
Volume7
Issue number2
DOIs
StatePublished - Apr 2011

Keywords

  • Floor-to-dome map
  • Quasiconformal
  • Sullivan theorem
  • Uniformly perfect

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