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Some structure theorems for complete constant mean curvature surfaces with boundary a convex curve

Authors: Ricardo Sa Earp and Harold Rosenberg
Journal: Proc. Amer. Math. Soc. 113 (1991), 1045-1053
MSC: Primary 53A10; Secondary 49Q05, 53C45
MathSciNet review: 1072337
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Abstract: Let $ M$ be a properly embedded, connected, complete surface in $ {\mathbb{R}^3}$ with non-zero constant mean curvature and with boundary a strictly convex plane curve $ C$. It is shown that if $ M$ is contained in a vertical cylinder of $ \mathbb{R}_ + ^3$, outside of some compact set of $ {\mathbb{R}^3}$, and if $ M$ is contained in a half-space of $ {\mathbb{R}^3}$ determined by $ C$, then $ M$ inherits the symmetries of $ C$. In particular, $ M$ is a Delaunay surface if $ C$ is a circle. It is also shown that if $ M$ has a finite number of vertical annular ends and the area of the flat disc $ D$ bounded by $ C$ is not "too small," then $ M$ lies in a half-space.

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  • [1] F. Brito, R. Earp, W. Meeks, and H. Rosenberg, Structure theorems for constant mean curvature surfaces bounded by a planar curve, Indiana Math. J. 40 (1991), 333-343. MR 1101235 (93e:53009)
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Keywords: Constant mean curvature, Delaunay surface, Alexandrov reflection principle
Article copyright: © Copyright 1991 American Mathematical Society

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