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

Abstract

Let M M be a properly embedded, connected, complete surface in R 3 {\mathbb {R}^3} with non-zero constant mean curvature and with boundary a strictly convex plane curve C C . It is shown that if M M is contained in a vertical cylinder of R + 3 \mathbb {R}_ + ^3 , outside of some compact set of R 3 {\mathbb {R}^3} , and if M M is contained in a half-space of R 3 {\mathbb {R}^3} determined by C C , then M M inherits the symmetries of C C . In particular, M M is a Delaunay surface if C C is a circle. It is also shown that if M M has a finite number of vertical annular ends and the area of the flat disc D D bounded by C C is not "too small," then M M lies in a half-space.