Classifying complex geodesics for the Carath\'eodory metric on low-dimensional Teichm\"uller spaces

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