A Hopf theorem for open surfaces in product spaces

  • Manfredo do Carmo
  • Isabel Fernández


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.


Line Field Product Space Open Surface Quadratic Differential Rotational Surface 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada I, ETS de InformíticaUniversidad de SevillaSevillaSpain
  2. 2.lnstituto de Matemática Pura e AplicadaRio de JaneiroBrasil

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