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

TL;DR

This paper classifies when the Carathéodory and Teichmüller metrics agree on Teichmüller disks, proving the conjecture for certain punctured surfaces and extending the difference result to punctured compact surfaces.

Contribution

It provides a complex-analytic criterion for Jenkins-Strebel differentials and confirms the conjecture for specific low-dimensional Teichmüller spaces.

Findings

Metrics agree on disks generated by differentials with no odd-order zeros

Criterion characterizes Jenkins-Strebel differentials where metrics coincide

Metrics differ on compact surfaces with punctures

Abstract

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.