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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On hypersurfaces of hyperbolic space infinitesimally supported by horospheres


Author: Robert J. Currier
Journal: Trans. Amer. Math. Soc. 313 (1989), 419-431
MSC: Primary 53C40; Secondary 57R30, 58F17
MathSciNet review: 935532
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Abstract: This paper is concerned with complete, smooth immersed hypersurfaces of hyperbolic space that are infinitesimally supported by horospheres. This latter condition may be restated as requiring that all eigenvalues of the second fundamental form, with respect to a particular unit normal field, be at least one. The following alternative must hold: either there is a point where all the eigenvalues of the second fundamental form are strictly greater than one, in which case the hypersurface is compact, imbedded and diffeomorphic to a sphere; or, the second fundamental form at every point has $ 1$ as an eigenvalue, in which case the hypersurface is isometric to Euclidean space and is imbedded in hyperbolic space as a horosphere.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0935532-0
PII: S 0002-9947(1989)0935532-0
Article copyright: © Copyright 1989 American Mathematical Society