Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces
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 c...
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| Format: | Journal article |
| Language: | English |
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Springer
2020
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| _version_ | 1826256754593234944 |
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| author | Gekhtman, D Markovic, V |
| author_facet | Gekhtman, D Markovic, V |
| author_sort | Gekhtman, D |
| collection | OXFORD |
| description | 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.
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| first_indexed | 2024-03-06T18:07:16Z |
| format | Journal article |
| id | oxford-uuid:01dadee3-9ddb-4662-93af-6997a8c2416b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T18:07:16Z |
| publishDate | 2020 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:01dadee3-9ddb-4662-93af-6997a8c2416b2022-03-26T08:37:13ZClassifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spacesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:01dadee3-9ddb-4662-93af-6997a8c2416bEnglishSymplectic ElementsSpringer2020Gekhtman, DMarkovic, VIt 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. |
| spellingShingle | Gekhtman, D Markovic, V Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces |
| title | Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces |
| title_full | Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces |
| title_fullStr | Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces |
| title_full_unstemmed | Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces |
| title_short | Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces |
| title_sort | classifying complex geodesics for the caratheodory metric on low dimensional teichmuller spaces |
| work_keys_str_mv | AT gekhtmand classifyingcomplexgeodesicsforthecaratheodorymetriconlowdimensionalteichmullerspaces AT markovicv classifyingcomplexgeodesicsforthecaratheodorymetriconlowdimensionalteichmullerspaces |