The Moduli Space of Complete Embedded Constant Mean Curvature Surfaces

TL;DR

This paper studies the moduli space of complete, properly embedded constant mean curvature surfaces in three-dimensional space, revealing its local structure as a real analytic variety and describing its dimension and symmetry properties.

Contribution

It characterizes the local structure of the moduli space of such surfaces, showing it is a real analytic variety and, under certain conditions, a quotient of a real analytic manifold by a finite group.

Findings

The moduli space is locally a real analytic variety.

When the linearized operator has no $L^2$-nullspace, the space is a real analytic orbifold of dimension $3k-6$.

The dimension of the moduli space is independent of the surface topology.

Abstract

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.