Conformal metrics and Ricci tensors in the pseudo-Euclidean space

Abstract:

We consider constant symmetric tensors $T$ on $R^n$, $n\geq 3$, and we study the problem of finding metrics $\bar {g}$ conformal to the pseudo-Euclidean metric $g$ such that $\mbox {Ric} \bar {g} = T$. We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics $\bar {g}$. As a consequence of these results we get solutions globally defined on $R^n$ for the equation $- \varphi \Delta _g \varphi +n ||\nabla _g \varphi ||^2/2 + \lambda \varphi ^2 = 0.$ Moreover, we show that for certain unbounded functions $\overline {K}$ defined on $R^n$, there are metrics conformal to the pseudo-Euclidean metric with scalar curvature $\overline {K}$.