The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ℝ3 which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceMk of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ℝ3) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL2-nullspace, we prove thatMk is locally the quotient of a real analytic manifold of dimension 3k−6 by a finite group (i.e. a real analytic orbifold), fork ≥ 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofMk is independent of the genus of the underlying punctured Riemann surface to which Σ is conformally equivalent. These results also apply to hypersurfaces of Hn+1 with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.