Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces

Let M = Mg,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 [15]. Let P = Pg,k = Rg,k × R+ 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 Rg,k. Then the parabolic classifying map, Φ : M → P , which assigns to a CMC surface its induced conformal structure and asymptotic necksizes, is a proper, real analytic map. It follows that Φ 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 Φ 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 (compare [7, 8] for the case k = 3).