Conformal metrics and Ricci tensors in the pseudo-Euclidean space (Q2701594)

Let \(M^n\), \(n\geq 3\), be a smooth manifold and \(T\) a smooth symmetric tensor field on \(M^n\) of order two. A natural question is whether there is a Riemannian metric \(g\) on \(M^n\) such that \(T\) is the Ricci curvature of \(g\); this question can be asked either locally or globally. This problem was studied by \textit{D.~DeTurck} [Invent. Math. 65, 179-207 (1981; Zbl 0489.53014); Ann. Math. Stud. 102, 525-537 (1982; Zbl 0478.53031)], with subsequent developments by other authors. NEWLINENEWLINENEWLINEHere the authors study the case that \(T=\sum_{i,j} \varepsilon_j c_{ij} dx_i \otimes dx_j\) is a constant tensor field in the pseudo-Euclidean space \((\mathbb{R}^n,g)\) where \(g_{ij}=\delta_{ij}\varepsilon_i\) with \(\varepsilon_i=\pm 1\) with at least one \(\varepsilon_i\) positive. They find necessary and sufficient conditions on \(c_{ij}\) for the existence of a metric \(\bar g\) such that \(\bar g=\varphi^{-2}g\) and \(\text{ Ric} \bar g=T\) for some smooth function \(\varphi\). The corresponding metrics are given explicitly, and most of them are globally defined. However, it is shown that no such \(\bar g\) can be complete. NEWLINENEWLINENEWLINEAs a consequence of these results, the authors find infinitely many explicit solutions \(\varphi\in C^\infty(\mathbb{R}^n)\) of the equation \(-\varphi \Delta_g\varphi + {n\over 2}\|\nabla_g \varphi \|^2 + \lambda \varphi^2 = 0\) for any constant \(\lambda\leq 0\) if \(g\) is Euclidean and any constant \(\lambda\in\mathbb{R}\) if \(g\) is pseudo-Euclidean. Furthermore, for certain unbounded functions \(\overline K\) defined on \(\mathbb{R}^n\), there are metrics \(\bar g\) conformal to the pseudo-Euclidean metric with scalar curvature \(\overline K\).