Remarks on Teichmüller Spaces of Circle Diffeomorphisms

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 Tα of the C1+α circle diffeomorphisms for each 0<α<1 and on the Teichmüller space T1 of the C1+Zygmund circle diffeomorphisms. Then we introduce characterizations of the elements of Tα 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.