Math. Ann. 314, 529–553 (1999) Mathematische Annalen c  Springer-Verlag 1999 Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes Sebasti´ an Montiel ⋆ Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Granada, E-18071 Granada, Spain (e-mail: smontiel@goliat.ugr.es) Received: 17 May 1997 / in final form: 28 January 1998 Mathematics Subject Classification (1991): 53A10, 53C42, 53C50, 35Q75, 83C99 1. Introduction Maximal and nonzero constant mean curvature spacelike hypersurfaces in Lorentzian manifolds are objects worthy of a big amount of interest, from both physical and mathematical points of view (cfr. [B, CFM,MT,St]). The main questions on this type of spacelike hypersurfaces are existence and uniqueness and both two started to be considered in the simplest space- times: Lorentzian space forms such as Minkowski and de Sitter spaces, or others being close, in a geometrical sense, to them (see [BS, C, CY, Ge, Tr]). In this paper we are interested in the problem of uniqueness for this type of hypersurfaces, when the ambient spacetime has a big number of constant mean curvature hypersurfaces. In fact, we ask If a spacetime is foliated by means of constant mean curvature space- like hypersurfaces, are the leaves of that foliation the only spacelike hypersurfaces with constant mean curvature? With this generality, it is too difficult to answer this question. For that we reduce the possible ambient spacetimes to a case where the foliation has umbilical leaves and their unit normals integrate into geodesic: we shall consider spacetimes M n+1 which are equipped with a closed conformal timelike vector field X . Then, the distribution on M n+1 orthogonal to X ⋆ Research partially supported by a DGICYT grant PB97–0785