A Hopf theorem for open surfaces in product spaces

Chapter

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.

Keywords

Line Field Product Space Open Surface Quadratic Differential Rotational Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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