The conformal theory of Alexandrov embedded constant mean curvature surfaces in $R^3$
The conformal theory of Alexandrov embedded constant mean curvature surfaces in $R^3$
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$R^3$ 中嵌入常平均曲率曲面的 Alexandrov 共形理论
DOI：
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发表时间：
2001
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通讯作者：
D. Pollack
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文献类型：
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作者：
R. Mazzeo;F. Pacard;D. Pollack
我们首先证明了一个一般的胶合定理，创建新的非退化常数平均曲率曲面附加半Delaunay表面与任何非退化CMC表面的任意点的小颈尺寸。证明使用柯西数据匹配的方法从\cite{MP}，cf。也可以引用{MPP}。在本文的第二部分中，我们开发的后果，这一结果和（至少部分）特征的图像相关联的地图，每个完整的，亚历山德罗夫嵌入CMC表面与有限的拓扑结构，其相关的共形结构，这是一个紧凑的黎曼曲面与有限数量的穿刺。特别是，我们表明，这个“健忘”的地图是满射时，属是零。这特别证明了CMC模空间具有复杂的拓扑结构。后者的结果是密切相关的最近的工作Kusner \cite{Ku}。
We first prove a general gluing theorem which creates new nondegenerate constant mean curvature surfaces by attaching half Delaunay surfaces with small necksize to arbitrary points of any nondegenerate CMC surface. The proof uses the method of Cauchy data matching from \cite{MP}, cf. also \cite{MPP}. In the second part of this paper, we develop the consequences of this result and (at least partially) characterize the image of the map which associates to each complete, Alexandrov-embedded CMC surface with finite topology its associated conformal structure, which is a compact Riemann surface with a finite number of punctures. In particular, we show that this `forgetful' map is surjective when the genus is zero. This proves in particular that the CMC moduli space has a complicated topological structure. These latter results are closely related to recent work of Kusner \cite{Ku}.