Hypersurfaces of Prescribed Scalar Curvature in Lorentzian Manifolds

The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers. 0. Introduction Consider the problem of finding a closed hypersurface of prescribed curvature F in a globally hyperbolic (n+1)-dimensional Lorentzian manifold N having a compact Cauchy hypersurface S0. To be more precise, let Ω be a connected open subset of N, f ∈ C2,α(Ω̄), F a smooth, symmetric function defined in an open cone Γ ⊂ Rn, then we look for a space-like hypersurface M ⊂ Ω such that (0.1) F|M = f(x) ∀x ∈ M, where F|M means that F is evaluated at the vector (κi(x)) the components of which are the principal curvatures of M . The prescribed function f should satisfy natural structural conditions, e. g. if Γ is the positive cone and the hypersurface M is supposed to be convex, then f should be positive, but no further, merely technical, conditions should be imposed. In [1, 2, 8, 14] the case F = H, the mean curvature, has been treated, and in [15] we solved the problem for curvature functions F of class K∗ that includes the Gaussian curvature, see [15, Section 1] for the definition, but excludes the symmetric polynomials Hk for 1 < k < n. Among these, H2, that corresponds to the scalar curvature operator, is of special interest. Received by the editors September 30, 2001. 2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05.