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Surfaces of revolution with constant mean curvature in Lorentz-Minkowski space. (English) Zbl 0535.53002
In a classical paper [Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl. 6, 309-320 (1841)] Ch. Delaunay proved that the profile curve of a surface of revolution with nonzero constant mean curvature in Euclidean 3-space can be described as the locus of a focus, when a quadratic curve is rolled along the axis of revolution. In the present paper we study the problem in Lorentz- Minkowski 3-space by considering spacelike surfaces of revolution. In the case where the axis of revolution is spacelike or timelike we obtain results similar to Delaunay’s except that the nature of quadrics needs special attention. In the case where the axis is a null line we can determine the profile curves completely without giving a geometric interpretation.

MSC:
53A05 Surfaces in Euclidean and related spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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[1] CH. DELAUNAY, Sur la surface de revolution dont la courbure moyenne est constante, J. Math. Pures Appl. 6 (1841), 309-320.
[2] J. HANO AND K. NOMIZU, On isometric immersions of the hyperbolic plane into th Lorentz-Minkowski space and the Monge-Ampere equation of a certain type, Math. Ann.262 (1983), 245-253. · Zbl 0507.53042 · doi:10.1007/BF01455315 · eudml:163704
[3] W-Y. HSIANG AND W-C. Yu, A generalization of a theorem of Delaunay, J. Differentia Geometry 16 (1981), 161-177. · Zbl 0504.53044
[4] K. KENMOTSU, Surfaces of revolution with prescribed mean curvature, Thoku Math. J. 32 (1980), 147-153. · Zbl 0431.53005 · doi:10.2748/tmj/1178229688
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