Four-dimensional homogeneous critical metrics for quadratic curvature functionals (Q6631311)

This paper provides a complete classification of Riemannian homogeneous manifolds which are critical for some quadratic curvature functional in dimension four.\N\NThe space of scalar quadratic curvature invariants of a Riemannian manifold \((M,g)\) is generated by \(\{\tau^2, \Delta \tau, ||\rho||^2, ||R||^2\}\), where \(\Delta \tau\) denotes the Laplacian of the scalar curvature \(\tau\), \(R\) is the curvature tensor and \(\rho\) stands for the Ricci tensor.\N\NDimension four is special and, for critical metrics, this is manifested by the Gauss-Bonnet theorem, which states that \(\int_M (||R||^2-4||\rho||^2+\tau^2) \, \mathrm{dvol}_g = 8\pi^2\chi(M)\). This implies that every quadratic curvature functional is equivalent to \N\[\N\mathcal{S}: g\mapsto \int_M \tau^2\, \mathrm{dvol}_g, \quad \mathcal{F}_t: g\mapsto \int_M (||\rho||^2+t||\tau||^2)\, \mathrm{dvol}_g.\N\]\NRemark that the functional \(g\mapsto \int_M ||R||^2 \, \mathrm{dvol}_g\) is equivalent to \(\mathcal{F}_{-1/4}\).\N\NAn important point is that it follows, from the Euler-Lagrange equations for these functionals (restricted to metrics of volume one), that, in dimensions 3 and 4, Einstein metrics are critical for all quadratic curvature functionals. Furthermore, in the homogeneous setting, the scalar curvature in constant which implies a simplication of the Euler-Lagrange equations.\N\NIt is well known that any simply connected homogeneous manifold is either symmetric or it is isometric to a Lie group with a left-invariant metric.\N\NThe analysis begins with symmetric manifolds and the authors show that \(\mathbb{S}^2\times \mathbb{H}^2\) is the only non-Einstein homogeneous example (up to homothety) which is critical for all quadratic curvature funcionals. In the solvable Lie group case, each family is considered separately to obtain a description of all metrics which are critical for some quadratic curvature functional. Remarkably, there are many quadratic curvature functionals in dimension four which do not have any homogeneous critical metric and this is in contrast with the situation in dimension 3. Special attention is given to critical metrics with zero energy. It is shown that any homogeneous Ricci soliton is a critical metric with zero energy for some quadratic curvature functional.