On metrics and super-Riemann surfaces (Q1107182)

The author considers a super-Riemann surface M with compact body \(\epsilon\) (M) and shows that for any given metric \(\mu_ 0\) on \(\epsilon\) (M) there is a metric \(\mu\) in M whose body is \(\mu_ 0\) (theorem 1) and that if the genus \((\epsilon (M))>1\) then M admits a unique metric whose lift to the universal cover \(\tilde M\) is superconformally equivalent to the standard metric on the super-half- plane (Baranov-Shvarts metric). This result explains a rather surprising result that the Teichmüller space for arbitrary super-Riemann surface with compact body (obtained by the author in a previous paper) coincides with the one obtained by Crane and Rabin, although they imposed a strong metrizability assumption.