Kobayashi’s and Teichmüller’s Metrics and Bers Complex Manifold Structure on Circle Diffeomorphisms

Abstract

Given a modulus of continuity ω, we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk. We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms. Using these distortion properties, we give the Bers complex manifold structure on the Teichmuller space TC1+H as the union of TC1+α over all 0 < α ⪯ 1, which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure. Furthermore, we prove that with the Bers complex manifold structure on TC1+H, Kobayashi’s metric and Teichmuller’s metric coincide.