Classifying Complex Geodesics for the Carathéodory Metric on Low-Dimensional Teichmüller Spaces

It was recently shown that the Caratheodory and Teichmuller metrics on the Teichmuller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmuller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichmuller disks on which the two metrics agree, and we conjecture that the Caratheodory and Teichmuller metrics agree on a Teichmuller disk if and only if the Teichmuller 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 Teichmuller spaces of the five-times punctured sphere and the twice-punctured torus. We also extend the result that the Caratheodory and Teichmuller metrics are different to the case of compact surfaces with punctures.