THE MODULI SPACE OF COMPLETE EMBEDDED CONSTANT MEAN-CURVATURE SURFACES

We examine the space of finite topology surfaces in R(3) which are com plete, properly embedded and have nonzero constant mean curvature. The se surfaces are noncompact provided we exclude the case of the round s phere. We prove that the space M(k) of all such surfaces with k ends ( where surfaces are identified if they differ by an isometry of R(3)) i s locally a real analytic variety. When the linearization of the quasi linear elliptic equation specifying mean curvature equal to one has no L(2)-nullspace, we prove that M(k) is locally the quotient of a real analytic manifold of dimension 3k-6 by a finite group (i.e. a real ana lytic orbifold), for k greater than or equal to 3. This finite group i s the isotropy subgroup of the surface in the group of Euclidean motio ns. It is of interest to note that the dimension of M(k) is independen t of the genus of the underlying punctured Riemann surface to which Si gma is conformally equivalent. These results also apply to hypersurfac es of H-n+1 with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.