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

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.

Fixed typos, incorporated referee's suggestions. Accepted for publication in Journal d'Analyse Mathématique