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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Sur les metriques admettant
les plans comme surfaces minimales


Author: M. Bekkar
Journal: Proc. Amer. Math. Soc. 124 (1996), 3077-3083
MSC (1991): Primary 49Q05, 53A10
DOI: https://doi.org/10.1090/S0002-9939-96-03530-7
MathSciNet review: 1346962
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Abstract: We establish the system of partial differential equations satisfied by the riemannian metrics on open subsets of ${\mathbb {R}}^{3}$ which admit planes as minimal surfaces. This is a nonlinear system of 10 partial differential equations, with the euclidian metric as a particular solution. In a previous work, we solved this system for axially symmetrical metrics. In this paper we linearize the system at the euclidian metric and solve the linear system. We obtain a 20-dimensional space of solutions.


References [Enhancements On Off] (What's this?)

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  • [B2] ------, Métriques riemanniennes qui admettent le plan comme surface minimale, Thèse de Doctorat, Université de Haute Alsace, Mulhouse, Septembre 1993.
  • [B3] Mohamed Bekkar, Sur une caractérisation des métriques de Heisenberg, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 11, 1017–1019 (French, with English and French summaries). MR 1281883
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    Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409

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Additional Information

M. Bekkar
Affiliation: Université de Haute Alsace, 4 rue des Frères Lumière, F 68093 Mulhouse cedex, France et Université d’Oran Es-Sénia, Institut de Mathématiques, Oran, Algérie
Email: M.Bekkar{@}univ-mulhouse.fr

DOI: https://doi.org/10.1090/S0002-9939-96-03530-7
Keywords: Minimal surfaces
Received by editor(s): March 15, 1995
Communicated by: Christopher Croke
Article copyright: © Copyright 1996 American Mathematical Society