A Hopf theorem for open surfaces in product spaces

Hopf’s theorem has been recently extended to compact genus zero surfaces with constant mean curvature H in a product space $$ \mathcal{M}^2_k \, X \, \mathbb{R}\,where\,\mathcal{M}^2_k $$ is a surface with constant Gaussian curvature $$ k \,\neq\, 0 \, {\rm{[AbRo]}}$$ . It also has been observed that, rather than H = const., it suffices to assume that the differential dH of His appropriately bounded [AdCT]. Here, we consider the case of simply-connected open surfaces with boundary in $$ \mathcal{M}^2_k \, X \, \mathbb{R}\,{\rm{such \, that}} \,dH $$ is appropriately bounded and certain conditions on the boundary are satisfied, and show that such surfaces can all be described.