On Uniformization of N=2 Superconformal and N=1 Superanalytic Dewitt Super-riemann Surfaces

We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt superRiemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition functions containing no odd functions of the even variable if and only if the first Čech cohomology group of the body Riemann surface with coefficients in the sheaf of holomorphic vector fields over the body is trivial. As a consequence, we give a constructive proof that there is a countably infinite family of N=2 superconformal equivalence classes of N=2 superconformal DeWitt super-Riemann surfaces with genus-zero compact body, and that N=2 superconformal DeWitt super-Riemann surfaces with simply connected body are classified up to N=2 superconformal equivalence by holomorphic line bundles over the underlying body Riemann surface, up to conformal equivalence. In addition, we prove that N=2 superconformal DeWitt super-Riemann surfaces with compact genus-one body and transition functions which correspond to the trivial cocycle in the first Čech cohomology group of the body Riemann surface with coefficients in the sheaf of holomorphic vector fields over the body are classified up to N=2 superconformal equivalence by theta functions associated to the underlying torus up to type modulo the trivial theta functions, or in other words, by holomorphic line bundles over the torus modulo conformal equivalence. We also give the corresponding results for the uniformization of N=1 superanalytic DeWitt super-Riemann surfaces of genus zero or one.