Rotational Hypersurfaces with Constant Gauss-Kronecker Curvature

Abstract

The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in ℝ ⁿ. They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such hypersurfaces. In particular, they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume, which can be seen as the higher-dimensional generalization of the pseudo-sphere.

Keywords

Differential geometry / Gauss-Kronecker curvature / Ordinary differential equation

References

[1] Breiner C, Kapouleas N. Complete constant mean curvature hypersurfaces in Euclidean spaces of dimension four or higher. Am. J. Math., 2021, 143(4): 1161-1259

[2] Buenoa A, Galvezb J A, Mirac P. Rotational hypersurfaces of prescribed mean curvature. J. Differ. Equations, 2020, 268(5): 2394-2413

[3] do Carmo M P. Differential Geometry of Curves and Surfaces, 2016, New York: Dover., Publ. Revised and Updated Second Edition

[4] do Carmo M P, Dajczer M. Rotational hypersurfaces in spaces of constant curvature. Trans. Amer. Math. Soc., 1983, 277: 685-709

[5] Delaunay C. Sur la surface de revolution dont la courbure moyenne est constant. Journal de Mathematiques Pures et Appliquees, 1841, 6: 309-320

[6] Hsiang W. Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces. I. J. Diff. Geom., 1982, 17(2): 337-356

[7] Hsiang W, Yu W. A generalization of a theorem of Delaunay. J. Diff. Geom., 1981, 16: 161-177

[8] Leite L. Rotational hypersurfaces of space forms with constant scalar curvature. Manuscripta Math., 1990, 67: 285-304

[9] Montiel S, Ros A. Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures. Pitman Monogr. Surveys Pure Appl. Math., 1991, 52: 279-296

[10] Palmas O. Complete rotational hypersurfaces with H _k constant in space forms. Bull. Braz. Math. Soc., 1999, 30(2): 139-161

[11] Ros A. Compact hypersurfaces with higher order mean curvatures. Rev. Mat. Iberoamericana, 1987, 3(3–4): 447-453

[12] Rosenberg H, Spruck J. On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space. J. Diff. Geom., 1994, 40: 379-409

[13] Wang Z. A prescribed Gauss-Kronecker curvature problem on the product of two unit spheres. International Mathematics Research Notices, 2010, 2010(23): 4399-4433