Mathematics – Differential Geometry
Scientific paper
1994-08-19
Mathematics
Differential Geometry
15 pages, amsTeX
Scientific paper
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.
Kusner Rob
Mazzeo Rafe
Pollack Daniel
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