Caratheodory metrics on Teichmuller spaces

Abstract

Let $S$ be an arbitrary Riemann surface whose Teichmüller space $T(S)$ has dimension at least two. A long standing problem is to determine whether the Carathéodory metric $d_C$ agrees with the Teichmüller metric $d_T$ on $T(S)$. It was shown that $d_C\ne d_T$ when $S$ is a closed surface of genus at least two. In this paper we study the general case, and prove that $d_C\ne d_T$ on $T(S)$ except possibly on the following seven Teichmüller spaces: $T_{0,0}^1$, $T_{0,1}^1$, $T_{0,0}^2$, $T_{0,2}^1$, $T_{0,1}^2$, $T_{0,0}^3$, and $T_{0,1}^3$.