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 super-Riemann 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 a certain cohomology group is trivial, namely the first Cech cohomology
group of the body Riemann surface with coefficients in the sheaf consisting of
the reciprocal of a line bundle tensor the holomorphic vector fields over the
body. In particular, this gives a general criteria for when a DeWitt N=1
superanalytic super-Riemann surface is N=1 superanalytically equivalent to a
ringed-space (1,1)-supermanifold, as studied in the algebro-geometric setting.
This general classification result implies 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 N=2 superconformal
DeWitt super-Riemann surfaces with simply connected body are classified up to
N=2 superconformal equivalence by conformal equivalence classes of holomorphic
line bundles over the underlying body Riemann surface. In addition, N=2
superconformal DeWitt super-Riemann surfaces with compact genus-one body and
transition functions which correspond to the trivial cocycle in the first Cech
cohomology group of the body Riemann surface with coefficients in the
reciprocal of a line bundle tensor the sheaf of holomorphic vector fields over
the body are classified up to N=2 superconformal equivalence by holomorphic
line bundles over the torus modulo conformal equivalence. The corresponding
results for the uniformization of N=1 superanalytic DeWitt super-Riemann
surfaces of genus zero or one are presented.