Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases

Abstract

One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each ${\mathcal{T}}_g$, the Teichmüller space of a closed surface of genus $g\ge 2$. The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential $\phi$. First, we construct a holomorphic embedding $\mathcal{E}:{\mathbb{H}}^k\to {\mathcal{T}}_{g,n}$ corresponding to $\phi$. The criterion says that the two metrics agree on this disk if and only if a certain function $\mathbf{\Phi }:\mathcal{E}\left({\mathbb{H}}^k\right)\to \mathbb{H}$ can be extended to a holomorphic function $\mathbf{\Phi }:{\mathcal{T}}_{g,n}\to \mathbb{H}$. We then show by explicit computation that this is not the case for quadratic differentials arising from $L$-shaped pillowcases.