Prescribed diagonal Schouten tensor in locally conformally flat manifolds

Abstract

We consider the pseudo-euclidean space \({(\mathbb{R}^n, g)}\) , with n ≥ 3 and \({g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}\) and tensors of the form \({T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}\) . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric \({\bar{g}}\) , conformal to g, so that \({A_{\bar g}=T}\) , where \({A_{\bar g}}\) is the Schouten Tensor of the metric \({\bar g}\) . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric \({\bar g}\) is complete on \({\mathbb{R}^n}\) . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics \({\bar g}\) , conformal to g, such that \({\sigma_2 ({\bar g })}\) or \({\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}\) is equal to a given function. We prove that for some functions, f 1 and f 2, there exist complete metrics \({\bar{g} = g/{\varphi^2}}\) , such that \({\sigma_2 ({\bar g }) = f_1}\) or \({\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}\) .