On hypersurfaces of hyperbolic space infinitesimally supported by horospheres

Currier, Robert J.

doi:10.1090/S0002-9947-1989-0935532-0

On hypersurfaces of hyperbolic space infinitesimally supported by horospheres
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by Robert J. Currier PDF

Trans. Amer. Math. Soc. 313 (1989), 419-431 Request permission

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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