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The Moduli Space of Complete Embedded Constant Mean Curvature Surfaces
dg-ga · Submitted 19 Aug 1994
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We examine the space of surfaces in $\RR^{3}$ which are complete, properly embedded and have nonzero constant mean curvature. | background 0.61 | 0.62 | 0.03 | asserted 0.88 |
| s1 | These surfaces are noncompact provided we exclude the case of the round sphere. | background 0.88 | 0.24 | 0.03 | asserted 0.94 |
| s2 | 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. | result 0.92 | 0.95 | 0.55 | asserted 1.00 |
| s3 | 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 $k\geq 3$. | result 0.99 | 0.94 | 0.57 | asserted 1.00 |
| s4 | This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. | result 0.65 | 0.72 | 0.04 | asserted 1.00 |
| s5 | 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. | interpretation 0.65 | 0.71 | 0.06 | asserted 0.96 |
| s6 | 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. | result 0.78 | 0.70 | 0.06 | asserted 0.98 |
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Original 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.
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