Abstract/Details

## The numerical computation of the Douady-Earle extension and Teichmueller mappings

Ye, Taiping.   University of Connecticut ProQuest Dissertations Publishing,  1992. 9326225.

### Abstract (summary)

When we have an automorphism of the unit circle S$\sp1,$ naturally we should ask whether we can extend the map to the unit disk. The answer, as we already know, is Yes. The classical extension methods while elegant, do not preserve as many natural properties as the Douady-Earle extension does. Among all the automorphisms we have some which are the boundary maps of lifts of Teichmuller maps between two finite type Riemann surfaces (with the least quasiconformal coefficient among all the quasiconformal homeomorphisms). Since the Douady-Earle extension is quasiconformal provided that the automorphism of S$\sp1$ allows a quasiconformal extension, the question we should ask is whether the Douady-Earle extension of the boundary map of the Teichmuller map and the Teichmuller map itself are the same.

Between compact Riemann surfaces with genus greater than 1, every Teichmuller map has singularities, while the Douady-Earle extension is a diffeomorphism (without any singularity), so for those automorphisms the Teichmuller map and the Douady-Earle extension of its boundary map are different.

Here we consider one of the best cases of a Teichmuller map between genus 1 Riemann surfaces. Specifically, we look at maps between the punctured square torus and the punctured diamond torus. Using error estimates and delicate computer computation, we prove that even in this very nice case the two maps are not the same. At the same time we discuss various numerical techniques, provide the major computer programs used in this study and a user's guide to these programs.

### Indexing (details)

Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
The numerical computation of the Douady-Earle extension and Teichmueller mappings
Author
Ye, Taiping
Number of pages
117
Degree date
1992
School code
0056
Source
DAI-B 54/04, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-208-76068-0
University/institution
University of Connecticut
University location
United States -- Connecticut
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9326225
ProQuest document ID
303988498
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
https://www.proquest.com/docview/303988498/abstract