Some structure theorems for complete constant mean curvature surfaces with boundary a convex curve
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- by Ricardo Sa Earp and Harold Rosenberg PDF
- Proc. Amer. Math. Soc. 113 (1991), 1045-1053 Request permission
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.References
- Ricardo Earp, Fabiano Brito, William H. Meeks III, and Harold Rosenberg, Structure theorems for constant mean curvature surfaces bounded by a planar curve, Indiana Univ. Math. J. 40 (1991), no. 1, 333–343. MR 1101235, DOI 10.1512/iumj.1991.40.40017
- Heinz Hopf, Differential geometry in the large, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983. Notes taken by Peter Lax and John Gray; With a preface by S. S. Chern. MR 707850, DOI 10.1007/978-3-662-21563-0
- Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465–503. MR 1010168
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 1045-1053
- MSC: Primary 53A10; Secondary 49Q05, 53C45
- DOI: https://doi.org/10.1090/S0002-9939-1991-1072337-1
- MathSciNet review: 1072337