Partial differential equations on semi-Riemannian manifolds (Q1970943)

The author studies the existence and nonexistence of Lorentzian warped product metrics with prescribed scalar curvature functions on Lorentzian warped product manifolds. Let \((N, g)\) be a closed (compact without boundary) Riemannian manifold of dimension \(n\) \((n \geq 3)\). For \(a > 0\), consider the product manifold \(M:= [a, \infty) \times_f N\) with the Lorentzian warped metric \(h = - dt^2 + f^2(t) g,\) where \(f\) is a positive function defined on \([a, \infty)\). Denoting the scalar curvatures of \(M\) and \(N\) by \(R(t, x)\) and \(R_g\), respectively, \(R(t,x)\) satisfies a nonlinear second order ODE with respect to the variable \(t\). From a result due to Kazdan and Warner, any compact manifold of dimension \(\geq 3\) can be classified into three categories related with prescribed scalar curvature. The author proves various results about the existence and the nonexistence of the warping functions in each category. There are similar results for Riemannian warped product manifolds due to \textit{M. C. Leung} [Commun. Partial Differ. Equations 20, 367-417 (1995; Zbl 0833.53038)].