A Hopf theorem for open surfaces in product spaces

Abstract

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