Carathéodory metric on some generalized Teichmüller spaces

We study the Carathéodory metric on some generalized Teichmüller spaces. Earle showed that the Carathéodory metric is complete on any Teichmüller space. Miyachi extended this result for Asymptotic Teichmüller spaces. We study the completeness of the Carathéodory metric on product Teichmüller spaces and on the Teichmüller space of a closed set in the Riemann sphere.