Caratheodory metrics on Teichmuller spaces

TL;DR

This paper investigates whether the Carathéodory metric coincides with the Teichmüller metric on Teichmüller spaces of Riemann surfaces, showing they differ in most cases except for seven specific spaces.

Contribution

It generalizes previous results by proving the inequality between the metrics for most Teichmüller spaces, identifying only seven exceptions where they may coincide.

Findings

d_C ≠ d_T on most Teichmüller spaces

Seven specific Teichmüller spaces are potential exceptions

Advances understanding of metric equivalence in Teichmüller theory

Abstract

Let $S$ be an arbitrary Riemann surface whose Teichm\"uller space $T(S)$ has dimension at least two. A long standing problem is to determine whether the Carath\'eodory metric $d_C$ agrees with the Teichm\"uller 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 Teichm\"uller spaces: $T^1_{0,0}$, $T^1_{0,1}$, $T^2_{0,0}$, $T^1_{0,2}$, $T^2_{0,1}$, $T^3_{0,0}$, and $T^3_{0,1}$.