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

DOI:
https://doi.org/10.1090/S0002-9939-1991-1072337-1

MathSciNet review:
1072337

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Abstract: Let be a properly embedded, connected, complete surface in with non-zero constant mean curvature and with boundary a strictly convex plane curve . It is shown that if is contained in a vertical cylinder of , outside of some compact set of , and if is contained in a half-space of determined by , then inherits the symmetries of . In particular, is a Delaunay surface if is a circle. It is also shown that if has a finite number of vertical annular ends and the area of the flat disc bounded by is not "too small," then lies in a half-space.

**[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)****[2]**H. Hopf,*Lectures on differential geometry in the large*. Lecture Notes in Math., vol. 1000, 1983, Springer-Verlag, New York. MR**707850 (85b:53001)****[3]**N. Korevaar, R. Kusner, and B. Solomon,*The structure of complete embedded surfaces with constant mean curvature*, J. Differential Geom.**30**(1989), pp. 465-503. MR**1010168 (90g:53011)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1072337-1

Keywords:
Constant mean curvature,
Delaunay surface,
Alexandrov reflection principle

Article copyright:
© Copyright 1991
American Mathematical Society