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

Author: 

Gekhtman, D
Markovic, V

Publication Date: 

20 April 2020

Journal: 

Journal d'Analyse Mathematique

Last Updated: 

2021-04-20T10:22:36.98+01:00

Issue: 

2

Volume: 

140

DOI: 

10.1007/s11854-020-0102-y

page: 

669-694

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.

Symplectic id: 

1109984

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article