Some structure theorems for Weingarten surfaces

Let $M\subset\mathbb{R}^3$ be a properly embedded, connected, complete surface with boundary a convex planar curve $C$, satisfying an elliptic equation $H=f(H^2-K)$, where $H$ and $K$ are the mean and the Gauss curvature respectively - which we will refer to as Weingarten equation. When $M$ is contained in one of the two halfspaces determined by $C$, we give sufficient conditions for $M$ to inherit the symmetries of $C$. In particular, when $M$ is vertically cylindrically bounded, we get that $M$ is rotational if $C$ is a circle. In the case in which the Weingarten equation is linear, we give a sufficient condition for such a surface to be contained in a halfspace. Both results are generalizations of results of Rosenberg and Sa Earp, for constant mean curvature surfaces, to the Weingarten setting. In particular, our results also recover and generalize the constant mean curvature case.