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.
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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.
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Carmo, M.d., Fernández, I. (2012). A Hopf theorem for open surfaces in product spaces. In: Tenenblat, K. (eds) Manfredo P. do Carmo – Selected Papers. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25588-5_33
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DOI: https://doi.org/10.1007/978-3-642-25588-5_33
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