Some explicit real analytic trivializations of the Teichmüller curves

Abstract

ABSTRACT. The barycentric extension of homeomorphisms from the unit circle to the closed unit disk produces real analytic trivializations of the Teichmüller curves.1. Introduction.In part because of their universal properties (see [5, 6, and 9]) the Teichmüller curves have played a central role in the deformation theory of Riemann surfaces. Since Teichmüller spaces are contractible, the Teichmüller curves are topologically trivial fibrations. At times (see for example [12]), smooth trivializations are useful. Our purpose here is to present explicit real analytic trivializations. These trivializations, which we obtain from the barycentric extension introduced by Douady and the author in [3], have invariance properties that may be useful. We present our basic construction in the next section. The final section describes the resulting trivializations.We assume some familiarity with the Ahlfors-Bers theory of Teichmüller spaces. Facts for which no reference is given can be found in the recent books [8, 10, or