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

One of the most important results in Teichm¨uller theory is the
theorem of Royden which says that the Teichm¨uller and Kobayashi metrics
agree on the Teichm¨uller space of a given closed Riemann surface. The problem that remained open is whether the Carath`eodory metric agrees with the
Teichm¨uller metric as well. In this paper we prove that these two metrics
disagree on each Tg, the Teichm¨uller space of a closed surface of genus g ≥ 2.
The main step is to establish a criterion to decide when the Teichm¨uller and
Carath´eodory metrics agree on the Teichm¨uller disc corresponding to a rational Jenkins-Strebel differential ϕ. First we construct a holomorphic embedding
E : Hk → Tg,n corresponding to ϕ. The criterion says that the two metrics
agree on this disc if and only if a certain function Φ : E(Hk) → H can be
extended to a holomorphic function Φ : Tg,n → H. We then show by explicit
computation that this is not the case for quadratic differentials arising from
L-shaped pillowcases.