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Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces

Gekhtman, Dmitri and Markovic, Vladimir (2020) Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces. Journal d'Analyse Mathématique, 140 (2). pp. 669-694. ISSN 0021-7670. https://resolver.caltech.edu/CaltechAUTHORS:20200514-134833618

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Abstract

It was recently shown that the Carathéodory and Teichmüller metrics on the Teichmüller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmüller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichmüller disks on which the two metrics agree, and we conjecture that the Carathéodory and Teichmüller metrics agree on a Teichmüller disk if and only if the Teichmüller disk is generated by a differential with no odd-order zeros. Using dynamical results of Minsky, Smillie, and Weiss, we show that it suffices to consider disks generated by Jenkins-Strebel differentials. We then prove a complex-analytic criterion characterizing Jenkins-Strebel differentials which generate disks on which the metrics coincide. Finally, we use this criterion to prove the conjecture for the Teichmüller spaces of the five-times punctured sphere and the twice-punctured torus. We also extend the result that the Carathéodory and Teichmüller metrics are different to the case of compact surfaces with punctures.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s11854-020-0102-yDOIArticle
https://rdcu.be/b4aD2PublisherFree ReadCube access
https://arxiv.org/abs/1711.04722arXivDiscussion Paper
Additional Information:© 2020 Hebrew University Magnes Press. Received 03 December 2017; Revised 22 July 2018; Published 20 April 2020. We would like to thank Alex Wright for helpful discussions.
Issue or Number:2
Record Number:CaltechAUTHORS:20200514-134833618
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200514-134833618
Official Citation:Gekhtman, D., Markovic, V. Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces. JAMA 140, 669–694 (2020). https://doi.org/10.1007/s11854-020-0102-y
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:103205
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:14 May 2020 21:01
Last Modified:14 May 2020 21:01

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