The Carath\'eodory metric on Teichm\"uller space of genus two surface

TL;DR

This paper confirms a conjecture relating the Carathéodory and Teichmüller metrics on genus two Teichmüller space, showing they agree under specific conditions on quadratic differentials.

Contribution

It proves the conjecture for genus two surfaces, extending previous results and clarifying the metric equivalence conditions.

Findings

Confirmed the conjecture for genus two surface Teichmüller space

Established the equivalence of metrics under even order zeros of quadratic differentials

Extended the understanding of metric behavior in Teichmüller spaces

Abstract

Let $\Tei_{g,n}$ be the Teichm\"uller space of Riemann surfaces of genus $g$ with $n$ punctures. It is conjectured that the Teichm\"uller and Carath\'{e}odory metrics agree on a Teichm\"{u}ller disk if and only if all the zeros of the corresponding holomorphic quadratic differential are of even order. The conjecture was proved by Gekhtman and Markovic for $\Tei_{0,5}\cong \Tei_{1,2}$. We confirm the conjecture for $\Tei_{2,0}\cong\Tei_{0,6}$.