For years the author, R. Bartnik, and J. Urbas have been considering the problem of finding a closed hypersurface of prescribed curvature in a complete -dimensional manifold . That is, if is a connected open subset of , is a positive function of class , and is a smooth, symmetric function defined in an open cone , then the problem is to find a hypersurface such that , where means that is evaluated at the vector whose components are the principal curvatures of . The author studied the case when is a Riemannian manifold and is the mean curvature. In [J. Differ. Geom. 43, 612–641 (1996; Zbl 0861.53058) and Math. Z. 224, 167–194 (1997; Zbl 0871.53045)] he proved the existence of closed strictly convex Weingarten hypersurfaces of a Riemannian manifold, provided there exist appropriate barrier hypersurfaces.
In this paper, the author obtains similar results for closed strictly convex spacelike hypersurfaces in a globally hyperbolic Lorentzian manifold. The class of curvature functions permitted in the Lorentzian case is somewhat smaller than in the Riemannian case. This is because in the Lorentzian case the Gauss equations give rise to a term in the equation for the second fundamental form with the opposite sign from that in the Riemannian case. The author proves his result by studying the corresponding curvature flow problem and proving convergence to a stationary solution with the aid of suitable a priori estimates.