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

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.