Sur les metriques admettant les plans comme surfaces minimales

Bekkar, M.

doi:10.1090/S0002-9939-96-03530-7

Sur les metriques admettant les plans comme surfaces minimales
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by M. Bekkar

Proc. Amer. Math. Soc. 124 (1996), 3077-3083

DOI: https://doi.org/10.1090/S0002-9939-96-03530-7

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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

M. Bekkar, Exemples de surfaces minimales dans l’espace de Heisenberg, Rend. Sem. Fac. Sci. Univ. Cagliari 61 (1991), no. 2, 123–130 (French, with English and French summaries). MR 1193456
—, Métriques riemanniennes qui admettent le plan comme surface minimale, Thèse de Doctorat, Université de Haute Alsace, Mulhouse, Septembre 1993.
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
R. L. Bryant, On metrics in 3-space for which the planes are minimal, Preprint, Duke University, September 1994.
Robert Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR 0256278