The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Abstract

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g,c) consisting of a left‐invariant Riemannian metric g and a positive constant c such that Ric(g)=cT, where Ric(g) is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that Ric(g)=cT is solvable for some left‐invariant Riemannian metric g.

The prescribed Ricci curvature problem for naturally reductive metrics on compact Lie groups

“…The first such work, due to Hamilton, appeared in [Ham84], where he showed that for a left-invariant metric T on SU(2) there exists a left-invariant metric g, unique up to scaling, such that (1.2) is satisfied with c > 0. Buttsworth extended this result in [But19] to all signatures of T and all unimodular Lie groups of dimension 3. In this case, the equation may fail to have a solution.…”

The Prescribed Ricci Curvature Problem for Homogeneous Metrics

“…The question of solvability of (2.2) was settled by Hamilton and the first-named author in [34,13]. Specifically, the following result holds.…”

“…In [13], the first-named author settled the question of the solvability of equation (2.2) for left-invariant Riemannian metrics on unimodular Lie groups in 3 dimensions. We summarise the results in Theorem 4.4 below.…”

On the variational properties of the prescribed Ricci curvature functional

“…The case where M is a homogeneous space G/H has been studied extensively; see the survey [11] and the more recent references [12,25,26,4,3]. In some situations, the equation can be solved explicitly, as shown, e.g., in [31,9]. Assume that T is positive-definite.…”