A Hopf theorem for open surfaces in product spaces

Manfredo do Carmo () and Isabel Fernández ()
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Manfredo do Carmo: Universidad de Sevilla, Departamento de Matemática Aplicada I, ETS de Informítica
Isabel Fernández: Universidad de Sevilla, Departamento de Matemática Aplicada I, ETS de Informítica

A chapter in Manfredo P. do Carmo – Selected Papers, 2012, pp 457-469 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-25588-5_33

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DOI: 10.1007/978-3-642-25588-5_33

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