  Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant
mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid
motions, endowed with the real analytic structure described in [kmp]. Let $P =
P_{g,k} = r_{g,k} \times R_+^k$ be the space of parabolic structures over
Riemann surfaces of genus g with k (marked) punctures, the real analytic
structure coming from the 3g-3+k local complex analytic coordinates on the
Riemann moduli space r_{g,k}. Then the parabolic classifying map, Phi: M --> P,
which assigns to a CMC surface its induced conformal structure and asymptotic
necksizes, is a proper, real analytic map. It follows that Phi is closed and in
particular has closed image. For genus g=0, this can be used to show that every
conformal type of multiply punctured Riemann sphere occurs as a CMC surface,
and -- under a nondegeneracy hypothesis -- that Phi has a well defined (mod 2)
degree. This degree vanishes, so generically an even number of CMC surfaces
realize any given conformal structure and asymptotic necksizes.
