15 February 2018 Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases
Vladimir Markovic
Duke Math. J. 167(3): 497-535 (15 February 2018). DOI: 10.1215/00127094-2017-0041


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 Tg, the Teichmüller space of a closed surface of genus g2. 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 φ. First, we construct a holomorphic embedding E:HkTg,n corresponding to φ. The criterion says that the two metrics agree on this disk if and only if a certain function Φ:E(Hk)H can be extended to a holomorphic function Φ:Tg,nH. We then show by explicit computation that this is not the case for quadratic differentials arising from L-shaped pillowcases.


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Vladimir Markovic. "Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases." Duke Math. J. 167 (3) 497 - 535, 15 February 2018. https://doi.org/10.1215/00127094-2017-0041


Received: 19 September 2016; Revised: 31 July 2017; Published: 15 February 2018
First available in Project Euclid: 24 January 2018

zbMATH: 06848178
MathSciNet: MR3761105
Digital Object Identifier: 10.1215/00127094-2017-0041

Primary: 20H10

Keywords: Carathéodory metric , L-shaped pillowcases , Teichmüller space

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 3 • 15 February 2018
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