Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with vanishing Gauss-Kronecker curvature

Author(s): T. Hasanis; A. Savas-Halilaj; T. Vlachos
Journal: Trans. Amer. Math. Soc. 359 (2007), 2799-2818.
MSC (2000): Primary 53C40; Secondary 53C42, 53C50
Posted: January 26, 2007
MathSciNet review: 2286057
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, a nowhere vanishing second fundamental form and a scalar curvature bounded from below.

References:

1.: L.J. Alias and B. Palmer, Curvature properties of zero mean curvature surfaces in four-dimensional Lorentzian space forms, Math. Proc. Cambridge Philos. Soc. 124 $\left( 1998\right)$ , 315-327. MR 1631131 (99f:53061)
2.: S. C. de Almeida and F.G.B. Brito, Minimal hypersurfaces of $\mathbb{S}^{4}$ with constant Gauss-Kronecker curvature, Math. Z. 195 $\left( 1987\right)$ , 99-107. MR 0888131 (88i:53095)
3.: R.L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 $\left( 1969\right)$ , 1-49. MR 0251664 (40:4891)
4.: M. do Carmo and M. Dajczer, Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277 $\left( 1983\right)$ , 685-709. MR 0694383 (85b:53055)
5.: S.Y. Cheng and S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 $\left( 1975\right)$ , 333-354. MR 0385749 (52:6608)
6.: M. Dajczer and D. Gromoll, Gauss parametrizations and rigidity aspects of submanifolds, J. Differential Geom. 22 $\left( 1985\right)$ , 1-12. MR 0826420 (87g:53088a)
7.: T. Hasanis, A. Savas-Halilaj and T. Vlachos, Minimal hypersurfaces with zero Gauss-Kronecker curvature, Illinois J. Math. 49 $\left( 2005\right)$ , 523-529. MR 2164350 (2006e:53107)
8.: T. Hasanis, A. Savas-Halilaj and T. Vlachos, Complete minimal hypersurfaces of $\mathbb{S}^{4}$ with zero Gauss-Kronecker curvature, Math. Proc. Cambridge Philos. Soc., to appear.
9.: H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 $\left( 1967\right)$ , 205-214. MR 0215259 (35:6101)
10.: B. O' Neill, Semi-Riemannian Geometry; With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc. New York, $\left( 1983\right)$ . MR 0719023 (85f:53002)
11.: R. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc.22 $\left( 1957\right)$ . MR 0121424 (22:12162)
12.: J. Ramanathan, Minimal hypersurfaces in $\mathbb{S}^{4}$ with vanishing Gauss-Kronecker curvature, Math. Z. 205 $\left( 1990\right)$ , 645-658. MR 1082881 (91m:53048)
13.: S.T. Yau, Harmonic functions on complete Riemannian manifolds,Comm. Pure Appl. Math. 28 $\left( 1975\right)$ , 201-228. MR 0431040 (55:4042)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C40, 53C42, 53C50

Retrieve articles in all Journals with MSC (2000): 53C40, 53C42, 53C50

Additional Information:

T. Hasanis
Affiliation: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Email: thasanis@cc.uoi.gr

A. Savas-Halilaj
Affiliation: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Email: me00499@cc.uoi.gr

T. Vlachos
Affiliation: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Email: tvlachos@cc.uoi.gr

DOI: 10.1090/S0002-9947-07-04231-6
PII: S 0002-9947(07)04231-6
Keywords: Hyperbolic space, minimal hypersurface, second fundamental form, Gauss-Kronecker curvature, stationary surface.
Received by editor(s): April 27, 2005
Posted: January 26, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|