Remarks on Teichmüller Spaces of Circle Diffeomorphisms

Abstract

In this paper, we first apply a gluing lemma developed in [1] to give an intrinsic and short proof of that the Teichmüller metric coincides with the Kobayashi metric on the Teichmüller space T_α of the C^1+α circle diffeomorphisms for each 0<α<1 and on the Teichmüller space T₁ of the C^1+Zygmund circle diffeomorphisms. Then we introduce characterizations of the elements of T_α and the elements of its tangent space at the base point in terms of conditions on two integral operators studied in [2] and [3] respectively. Some open questions are raised for T₁.