Rotation hypersurfaces in spaces of constant curvature

do Carmo, M.; Dajczer, M.

doi:10.1090/S0002-9947-1983-0694383-X

Rotation hypersurfaces in spaces of constant curvature
HTML articles powered by AMS MathViewer

by M. do Carmo and M. Dajczer

Trans. Amer. Math. Soc. 277 (1983), 685-709

DOI: https://doi.org/10.1090/S0002-9947-1983-0694383-X

PDF | Request permission

Abstract:

Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the belicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space $H^3$; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in $H^3$.

References

J. L. Barbosa and M. do Carmo, Helicoids, catenoids, and minimal hypersurfaces of $\textbf {R}^{n}$ invariant by an $\ell$-parameter group of motions, An. Acad. Brasil. Ciênc. 53 (1981), no. 3, 403–408. MR 663233
David E. Blair, On a generalization of the catenoid, Canadian J. Math. 27 (1975), 231–236. MR 380637, DOI 10.4153/CJM-1975-028-8
M. do Carmo and C. K. Peng, Stable complete minimal hypersurfaces, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Sci. Press Beijing, Beijing, 1982, pp. 1349–1358. MR 714373
Élie Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. 17 (1938), no. 1, 177–191 (French). MR 1553310, DOI 10.1007/BF02410700

Sur les surfaces de revolution dont la courbure moyenne est constante

6

Notes on isoparametric hypersurfaces

H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
Hiroshi Mori, Minimal surfaces of revolution in $H^{3}$ and their global stability, Indiana Univ. Math. J. 30 (1981), no. 5, 787–794. MR 625602, DOI 10.1512/iumj.1981.30.30057
Seiki Nishikawa and Yoshiaki Maeda, Conformally flat hypersurfaces in a conformally flat Riemannian manifold, Tohoku Math. J. (2) 26 (1974), 159–168. MR 338967, DOI 10.2748/tmj/1178241240
Tominosuke Ôtsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970), 145–173. MR 264565, DOI 10.2307/2373502
Patrick J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J. (2) 21 (1969), 363–388. MR 253243, DOI 10.2748/tmj/1178242949