Let \(N\) be a globally hyperbolic \((n+1)\)-dimensional Lorentzian manifold having a compact Cauchy hypersurface \(\mathcal S_ 0\). For a connected open subset \(\Omega\) of \(N\), let \(f\) and \(F\) be functions such that \(f\in C^{2,\alpha}(\overline\Omega)\) and \(F\) is smooth and symmetric in an open cone \(\Gamma\subset \mathbb R^ n\). Many authors considered the problem of existence of a hypersurface \(M\in \Omega\) such that \(F_{\,| M} = f(x)\), where \(F_{\,| M}\) means that \(F\) is evaluated at the vector \((\kappa_i(x))\) whose components are the principal curvatures of \(M\).
The case when \(F\) is the mean curvature \(H_ 1\) was studied by the author [Commun. Math. Phys. 89, No. 4, 523–553 (1983; Zbl 0519.53056), Math. Z. 235, 83–97 (2000; Zbl 1017.53053)] and R. Bartnik [Commun. Math. Phys. 94, 155–175 (1984; Zbl 0548.53054), Commun. Math. Phys. 117, No. 4, 615–624 (1988; Zbl 0647.53044)]. In his previous paper [Indiana Univ. Math. J. 49, No. 3, 1125–1153 (2000; Zbl 1034.53064)], the author proved the existence of closed, space-like hypersurfaces having a prescribed curvature function \(F_{| M}=f(x)\) in a globally hyperbolic Lorentzian manifold \(N^{n,1}\) with a compact Cauchy hypersurface \({\mathcal S}_0\).
In this paper, the author considers the case when \(F\) is the scalar curvature operator \(H_ 2\). A hypersurface \(M\) with \(F=H_ 2\) prescribed at each point is not in general a hypersurface of prescribed scalar curvature, because the scalar curvature of a hypersurface \(M\) depends also on \(\overline R_{\alpha\beta}\nu^\alpha\nu^\beta\), where \(\overline R_{\alpha\beta}\) is the Ricci curvature of the ambient manifold \(N\) and \(\nu\) is the past-directed normal vector field to \(M\). Hence, \(F\) for \(M\) must satisfy \(F_{\,| M} = f(x,\nu)\).
\(M\) is said to be admissible if the vector of principal curvatures \((\kappa_ i)\) with respect to the past directed normal of \(M\) belongs to the cone \(\{\lambda\in \mathbb R^n;\;\sum\lambda_i>0,\;\sum_{i<j}\lambda_i\lambda_j>0 \}\). A hypersurface \(M_ 1\) is said to be a lower barrier for the pair \((F, f)\) if \(F_{| \Sigma}\leq f(x,\nu)\) for all points \(\Sigma\subset M_ 1\) where \(M_ 1\) is admissible. A hypersurface \(M_ 2\) is an upper barrier for \((F, f)\) if \(F_{\,| M_ 2}\geq f(x,\nu)\) and \(M_ 2\) is admissible.
Let \(\Omega\) be a precompact, connected, open subset of \(N\), that is bounded by two achronal, connected, space-like hypersurfaces \(M_ 1\) and \(M_ 2\) of class \(C^{4,\alpha}\), where \(M_ 1\) is supposed to lie in the past of \(M_ 2\). The main result of the paper states that if \(M_ 1\) and \(M_ 2\) are lower and upper barriers for \((F,f)\), where \(F = H_ 2\), then, the problem \(F_{| M}=f(x,\nu)\) has an admissible solution \(M\subset \overline \Omega\) of class \(C^{4,\alpha}\) that can be written as a graph over \(\mathcal S_ 0\) provided there exists a strictly convex function \(\chi\in C^ 2(\Omega)\).