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Title: Carathéodory's metrics on Teichmüller spaces and L-shaped pillowcases. Duke Math. J. 167 (2018), no. 3, 497–535.
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 Teichmüller space of a closed surface of genus g≥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.
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