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 $\mathcal{M}^2_k \, X \, \mathbb{R}\,where\,\mathcal{M}^2_k$ 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.

2000 Mathematics Subject Classification: 53C42; 53C40.

First author supported by CNPq and FAPERJ.

Second author research partially supported by MEC-FEDER grant number MTM2004-00 160.