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

Markovic, Vladimir (2018) Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases. Duke Mathematical Journal, 167 (3). pp. 497-535. ISSN 0012-7094. doi:10.1215/00127094-2017-0041.

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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 T_g, the 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 φ. First, we construct a holomorphic embedding ℰ:H^k → Tg,n corresponding to φ. The criterion says that the two metrics agree on this disk if and only if a certain function Φ: ℰ (H^k) → H can be extended to a holomorphic function Φ : T_(g,n) → H. We then show by explicit computation that this is not the case for quadratic differentials arising from L-shaped pillowcases.

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Additional Information:© 2018 Duke University Press. Received: 19 September 2016. Revised: 31 July 2017. First available in Project Euclid: 24 January 2018. We made several remarks comparing different aspects of our work with that of McMullen [18] and Knese [14]. Although we actually do not use any of the results from [18] and [14], these papers played an influential role in the writing of this article. We thank the referees for very useful suggestions which improved this article. Also, we thank F. Gardiner for sending us comments and the paper [10]. S. Antonakoudis has also reported making significant progress toward understanding Carathéodory’s metric on Teichmüller spaces. Markovic’s work was partially supported by Simons Investigator Award 409745 from the Simons Foundation, by the Institute for Advanced Study, by National Science Foundation grant DMS-1201463, and by the Isaac Newton Institute in Cambridge.
Funding AgencyGrant Number
Simons Foundation409745
Institute for Advanced StudyUNSPECIFIED
Isaac Newton InstituteUNSPECIFIED
Subject Keywords:Carathéodory metric; Teichmüller space; L-shaped pillowcases
Issue or Number:3
Classification Code:Primary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]
Record Number:CaltechAUTHORS:20180315-080429098
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Official Citation:Markovic, Vladimir. Carathéodory’s metrics on Teichmüller spaces and L -shaped pillowcases. Duke Math. J. 167 (2018), no. 3, 497-535. doi:10.1215/00127094-2017-0041.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85324
Deposited By: Ruth Sustaita
Deposited On:16 Mar 2018 15:22
Last Modified:15 Nov 2021 20:27

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