Some structure theorems for Weingarten surfaces

Abstract Let be a properly embedded, connected, complete surface with boundary a convex planar curve , satisfying an elliptic equation , where and are the mean and the Gauss curvature, respectively—which we will refer to as Weingarten equation . When is contained in one of the two halfspaces determined by , we give sufficient conditions for to inherit the symmetries of . In particular, when is vertically cylindrically bounded, we show that is rotational if 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, to the Weingarten setting, of results of Rosenberg and Sa Earp for constant mean curvature surfaces. In particular, our results also recover and generalize the constant mean curvature case.