Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces
- Creators
- Gekhtman, Dmitri
- Markovic, Vladimir
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.
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.Attached Files
Accepted Version - 1711.04722.pdf
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Additional details
- Eprint ID
- 103205
- Resolver ID
- CaltechAUTHORS:20200514-134833618
- Created
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2020-05-14Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field