  We examine the space of surfaces in $\RR^{3}$ 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
space $\Mk$ of all such surfaces with $k$ ends (where surfaces are identified
if they differ by an isometry of $\RR^{3}$) is locally a real analytic variety.
When the linearization of the quasilinear elliptic equation specifying mean
curvature equal to one has no $L^2-$nullspace we prove that $\Mk$ is locally
the quotient of a real analytic manifold of dimension $3k-6$ by a finite group
(i\.e\. a real analytic orbifold), for $k\geq 3$. 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 of $\Mk$ is independent of the topology of
the underlying punctured Riemann surface to which $\Sig$ is conformally
equivalent. These results also apply to hypersurfaces of $\HH^{n+1}$ with
nonzero constant mean curvature greater than that of a horosphere and whose
ends are cylindrically bounded.
