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]**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**, https://doi.org/10.1512/iumj.1991.40.40017**[2]**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****[3]**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**

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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