Three-dimensional homogeneous Lorentzian Yamabe solitons
A Yamabe soliton \((M,g)\) is a (pseudo-)Riemannian manifold \(M\) on which there exists a Yamabe vector field \(X\) whose Lie derivative satisfies \[ L_X g = (\tau-\lambda)g, \] where \(\tau\) is the scalar curvature and \(\lambda\) a real constant. If \(X=\nabla f\) for some smooth function \(f\), then \(M\) is called a gradient Yamabe soliton. A Yamabe soliton is called non-trivial if \(X\) is not a Killing vector field. The authors study 3-dimensional \(M\) with a non-flat Lorentzian homogeneous metric \(g\). Their first main result is that such an \(M\) is a non-trivial Yamabe soliton if and only if \(M\) is locally conformally flat and its Ricci operator is 2-step nilpotent. Such an \(M\) belongs to the class of space-times called homogeneous plane waves, and an earlier result due to Blau and O'Loughlin states that there exist two possible types of homogeneous plane waves. The authors find that the Lorentzian non-trivial Yamabe solitons in dimension three are precisely of this kind. Finally, the authors study left-invariant Yamabe vector fields on Lorentzian Lie groups of dimension three. If the soliton is to be non-trivial, there is only one 3-parameter family of such Lie groups admitting a left-invariant Yamabe vector field. In particular, it follows from the authors' results that there exist more Yamabe solitons than just the left-invariant ones in dimension three.
scientific article; zbMATH DE number 6134640
scientific article