×

On the surfaces of revolution with constant mean curvature in the hyperbolic space. (Sur les surfaces de révolution à courbure moyenne constante dans l’espace hyperbolique.) (French) Zbl 0921.53029

The meridians of a Euclidean surface of revolution with constant mean curvature have a nice kinematic generation. This paper shows that in hyperbolic three-space there exists a hyperbolic kinematics counterpart. The author studies plane curves \(\gamma _{e,p}\) which share some focal properties with the Euclidean conic sections. The path of its focus under the plane hyperbolic rolling of \(\gamma\) on a geodesic \(a\) shall be denoted by \(c\). Hyperbolic rotation of \(c\) around the axis \(a\) gives a hyperbolic surface of revolution with meridian \(c\). This surface then has constant (hyperbolic) mean curvature.
Reviewer: O.Röschel (Graz)

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A35 Non-Euclidean differential geometry
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Ballmann ( W. ) .- On spaces of nonpositive curvature , Birkhäuser , 1995 . MR 1377265 · Zbl 0834.53003
[2] Buser ( P. ) . - Geometry and spectra of compact Riemann surfaces , Birkhäuser , Boston , 1992 . MR 1183224 | Zbl 0770.53001 · Zbl 0770.53001
[3] Delaunay ( C. ) .- Sur la surface de révolution dont la courbure moyenne est constante , J. Math. Pure et Appl. 16 ( 1841 ), pp. 309 - 320 . Article
[4] Gomes ( J.M. ) . - Spherical surfaces with constant mean curvature in hyperbolic space , Bol. Soc. Bras. Mat. 18 , n^\circ 2 ( 1987 ), pp. 49 - 73 . MR 1018445 | Zbl 0752.53034 · Zbl 0752.53034
[5] Hsiang ( W.Y. ) .- On generalization of theorems of A. D. Alexandrov and C. Delaunay on hypersurfaces of constant mean curvature , Duke Math. J. 49 , n^\circ 3 ( 1982 ), pp. 485 - 496 . Article | MR 672494 | Zbl 0496.53006 · Zbl 0496.53006
[6] Hsiang ( W. ) and Yu ( W.C. ) .- A generalization of a theorem of Delaunay , J. Diff. Geom. 16 ( 1981 ), pp. 161 - 177 . MR 638783 | Zbl 0504.53044 · Zbl 0504.53044
[7] Kobayashi ( S. ) et Nomizu ( K. ) .- Foundations of differential geometry I , Interscience , New-York , 1963 . MR 152974 | Zbl 0119.37502 · Zbl 0119.37502
[8] Rosenberg ( H. ) et Sá Earp ( R. ) .- The geometry of properly embedded special surfaces in R3; e.g. surfaces satisfying aH + bK = 1, where a and b are positive , Duke Math. J. 73 , n^\circ 2 ( 1994 ), pp. 291 - 306 . Article | MR 1262209 | Zbl 0802.53002 · Zbl 0802.53002
[9] Sterling ( I. ) .- A Generalization of a theorem of Delaunay to rotational W-hypersurfaces of \sigma l-type in Hn+1 and Sn+1 , Pacific J. Math. 127 , n^\circ 1 ( 1987 ), pp. 187 - 197 . Article | MR 876025 | Zbl 0579.53041 · Zbl 0579.53041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.