On gradient Ricci solitons conformal to a pseudo-Euclidean space
The article under review is devoted to particular gradient Ricci solitons. By definition, a semi-Riemannian manifold \((M^n, g)\), \(n\geq 3\), is called a gradient Ricci soliton, if there exists a smooth function \(f:M^n\to \mathbb R\), called a potential function, such that \[ \mathrm{Ric}_g + \mathrm{Hess}_g (f) = \rho g, \] where \(\mathrm{Ric}_g\) is the Ricci tenzor, \(\mathrm{Hess}_g(f)\) is the Hessian of \(f\) with respect to \(g\), and \(\rho\in\mathbb R\) is a constant. The main attention is paid to gradient Ricci solitons, which are steady (\(\rho=0\)) and conformally flat. In the Riemannian case, gradient Ricci solitons which are complete, steady and locally conformally flat, are classified in [\textit{H.-D. Cao} and \textit{Q. Chen}, Trans. Am. Math. Soc. 364, No. 5, 2377--2391 (2012; Zbl 1245.53038)]. In the Lorentzian case the situation is more complicated, non-trivial examples are discussed in [\textit{W. Batat} et al., Bull. Lond. Math. Soc. 43, No. 6, 1219--1227 (2011; Zbl 1266.53049)], [\textit{M. Brozos-Vaźquez} et al, J. Geom. Anal. 23, No. 3, 1196--1212 (2013; Zbl 1285.53059)], [\textit{K. Onda}, Geom. Dedicata 147, 313--322 (2010; Zbl 1203.53044)]. The present article aims at providing an explicit description of steady, conformally flat gradient Ricci solitons, which are invariant under the action of an \((n-1)\)-dimensional translation group. Namely, let \((\mathbb R^n, g)\) be a pseudo-Euclidean space endowed with Cartesian coordinates \((x_1, \dots, x_n)\) such that \(g=\sum\varepsilon_j dx_j^2\), where \(\varepsilon_j=\pm 1\), \(1\leq j\leq n\), with at least one \(\varepsilon_j=1\). Let \(\xi = \sum\alpha_j x_j\) be a basic invariant for an \((n-1)\)-dimensional translation group. The problem is to find smooth functions \(\varphi(\xi)\) and \(f(\xi)\) such that the metric \(\tilde g = g / \varphi^2\) satisfies the equation \[ \mathrm{Ric}_{\tilde g} + \mathrm{Hess}_{\tilde g} (f) = \rho {\tilde g} \] for some constant \(\rho\), which is vanishes in the steady case. First, the authors derive a corresponding system of ordinary differential equations for \(\varphi\) and \(f\). Theorem A. Let \((\mathbb R^n, g)\) be a pseudo-Euclidean space, \(n\geq 3\), with Cartesian coordinates \((x_1, \dots, x_n)\) and \(g=\sum\varepsilon_j dx_j^2\). Consider smooth functions \(\varphi(\xi)\) and \(f(\xi)\), where \(\xi = \sum\alpha_j x_j\), \(\alpha_j\in \mathbb R\), and \(\sum \varepsilon_j\alpha_j^2 = \varepsilon_{j_0}\) or \(\sum \varepsilon_j\alpha_j^2 = 0\). Then \((\mathbb R^n, \tilde g)\) with \(\tilde g = g / \varphi^2\) is a gradient Ricci soliton with \(f\) as a potential function if, and only if, \(\varphi\) and \(f\) satisfy \noindent (i) \[ \varphi^{\prime\prime} = \varphi^{\prime} f^{\prime} + (n-1) \frac{(\varphi^{\prime})^2 }{\varphi} + \varepsilon_{j_0}\frac{\rho}{\varphi}, \] \[ f^{\prime\prime} = - n \frac{\varphi^{\prime}}{\varphi}f^{\prime} - (n-1)(n-2)\left( \frac{\varphi^{\prime}}{\varphi}\right)^2 - \varepsilon_{j_0}(n-2)\frac{\rho}{\varphi^2}, \] whenever \(\sum \varepsilon_j\alpha_j^2 = \varepsilon_{j_0}\), and \noindent (ii) \[ (n-2)\varphi^{\prime\prime} + \varphi f^{\prime\prime} + 2 f^{\prime}\varphi^{\prime}=0, \quad \rho=0, \] whenever \(\sum \varepsilon_j\alpha_j^2 = 0\). Next, the steady case \(\rho=0\) is considered. Solutions of the system (i) are described by the following statements. Theorem B1. Let \((\mathbb R^n, g)\) be a pseudo-Euclidean space, \(n\geq 3\), with Cartesian coordinates \((x_1, \dots, x_n)\) and \(g=\sum\varepsilon_j dx_j^2\). Consider functions \(\varphi(\xi)\) and \(f(\xi)\), where \(\xi = \sum\alpha_j x_j\), \(\alpha_j\in \mathbb R\), \(\sum \varepsilon_j\alpha_j^2 = \pm 1\), given by \[ \varphi_\pm (\xi) = \gamma\left( \pm \frac{1}{\sqrt{n-1}}\xi +b \right)^{\frac{1}{n-2}(-1\pm\sqrt{n-1})}, \quad \gamma>0, b\in\mathbb R, \] \[ f_\pm(\xi) = \mp \sqrt{n-1} \ln \left( \pm\frac{1}{\sqrt{n-1}}\xi +b \right). \] Then \((\mathbb R^n, \tilde g_\pm)\) with \(\tilde g_\pm = g / \varphi_\pm^2\) are steady gradient Ricci solitons with \(f_\pm\) as potential functions. These solutions are defined on the half space \(\sum\alpha_j x_j>\mp b\sqrt{n-1}\). In the Riemannian case, i.e., if \(\varepsilon_j=1\) for all \(j\), the metric \(\tilde g\) is not complete. Theorem B2. Let \((\mathbb R^n, g)\) be a pseudo-Euclidean space, \(n\geq 3\), with Cartesian coordinates \((x_1, \dots, x_n)\) and \(g=\sum\varepsilon_j dx_j^2\). Consider functions \(v(\xi)\) and \(h(\xi)\), where \(\xi = \sum\alpha_j x_j\), \(\alpha_j\in \mathbb R\), \(\sum \varepsilon_j\alpha_j^2 = \pm 1\), given by \[ \frac{v^\prime}{v} = \frac{\gamma}{\sqrt{a}} (av+c)^{\frac{1}{2}-\frac{1}{2\sqrt{n-1}}}(av+d)^{\frac{1}{2}+\frac{1}{2\sqrt{n-1}}}, \quad \gamma>0, \] \[ h = \gamma\sqrt{a} (av+c)^{-\frac{1}{2}-\frac{1}{2\sqrt{n-1}}}(av+d)^{-\frac{1}{2}+\frac{1}{2\sqrt{n-1}}}, \] where \[ a=(n-1)(n-2), \quad b=2(n-1), \quad c=n-1-\sqrt{n-1}, \quad d=n-1+\sqrt{n-1}. \] Let \(\varphi(\xi)\) and \(f(\xi)\) be functions obtained by integrating \[ \frac{\varphi^\prime}{\varphi} = h \, v, \quad f^\prime = h. \] Then \((\mathbb R^n, \tilde g)\) with \(\tilde g = g / \varphi^2\) is a steady gradient Ricci soliton with \(f\) as a potential function. In the Riemannian case the metric \(\tilde g\) is not complete. Theorems B1 and B2 represent two kinds of solutions for (i), and it turns out that there are no other solutions. Theorem C. Let \((\mathbb R^n, g)\) be a pseudo-Euclidean space, \(n\geq 3\), with Cartesian coordinates \((x_1, \dots, x_n)\) and \(g=\sum\varepsilon_j dx_j^2\). Let \((\mathbb R^n, \tilde g)\) with \(\tilde g = g / \varphi^2\) be a steady gradient Ricci soliton with \(f\) as a potential function. Then \(\varphi\) and \(f\) are invariant under an \((n-1)\)-dimensional translation group whose basic invariant is \(\xi = \sum\alpha_j x_j\), \(\alpha_j\in \mathbb R\), with \(\sum \varepsilon_j\alpha_j^2 = \pm 1\) if, and only if, \(\varphi\) and \(f\) are given as in Theorems B1 and B2. On the other hand, solutions of (ii) are described by the following statement. Theorem D. Let \((\mathbb R^n, g)\) be a pseudo-Euclidean space, \(n\geq 3\), with Cartesian coordinates \((x_1, \dots, x_n)\) and \(g=\sum\varepsilon_j dx_j^2\). Let \(\varphi(\xi)\) be an arbitrary nonvanishing smooth function depending on \(\xi = \sum\alpha_j x_j\), \(\alpha_j\in \mathbb R\), \(\sum \varepsilon_j\alpha_j^2 = 0\). Then the function \(f(\xi)\) given by \[ f (\xi) = - (n-2) \int \frac{1}{\varphi^2}\left( \int\varphi\varphi^{\prime\prime} d\xi\right) d\xi \] satisfies (ii), and \((\mathbb R^n, \tilde g)\) with \(\tilde g = g / \varphi^2\) is a steady gradient Ricci soliton with \(f\) as a potential function. Any such metric \(\tilde g\) has scalar curvature \(\tilde K\equiv 0\). It would be interesting to provide a similar analysis for the case of \(\rho\not = 0\), which would result in a description of particular classes of shrinking/expanding gradient Ricci solitons.
scientific article; zbMATH DE number 6308421