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The Carathéodory metric on Teichmüller space of genus two surface
Let $\Tei_{g,n}$ be the Teichmüller space of Riemann surfaces of genus $g$ with $n$ marked points. It is conjectured that the Teichmüller and Carathéodory metrics coincide on a Teichmüller disk exactly when every zero of the associated holomorphic quadratic differential (away from marked points) has
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