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

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.