Homogeneous and curvature homogeneous Lorentzian critical metrics

Proceedings of the Royal Society of Edinburgh, page 1 of 25 DOI:10.1017/prm.2022.44 Homogeneous and curvature homogeneous Lorentzian critical metrics M. Brozos-V´azquez CITMAga, 15782 Santiago de Compostela, Spain and Universidade da Coru˜na, Campus Industrial de Ferrol, Department of Mathematics, 15403 Ferrol, Spain (miguel.brozos.v[email protected]) S. Caeiro-Oliveira and E. Garc´ıa-R´ıo CITMAga, 15782 Santiago de Compostela, Spain and Faculty of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain ([email protected],[email protected]) (Received 23 December 2021; accepted 11 June 2022) We determine all three-dimensional homogeneous and 1-curvature homogeneous Lorentzian metrics which are critical for a quadratic curvature functional. As a result, we show that any quadratic curvature functional admits different non-Einstein homogeneous critical metrics and that there exist homogeneous metrics which are critical for all quadratic curvature functionals without being Einstein. Keywords: Quadratic curvature functional; Lorentzian; critical metric; curvature homogeneous; homogeneous; Ricci soliton 2020 Mathematics subject classification: Primary: 53B30 Secondary: 53C50, 53C24 1. Introduction Einstein metrics are central in geometry and physics. Being critical for the Einstein–Hilbert functional EH:g→EH(g)=Mτgdvolgsubject to a volume constraint, they provide optimal metrics in the sense that the scalar curvature τis more evenly distributed about the manifold. Since the space of scalar curvature invariants of order one is generated by the scalar curvature, in the search of optimal metrics on a given manifold, it is natural to consider other functionals defined by integrating polynomial curvature invariants of higher order. The space of scalar curvature invariants of order two has dimension at most four and it is generated by {τ2,ρ2,R2,Δτ}. In dimension three, the curvature tensor Ris totally determined by the Ricci tensor ρand R2=2ρ2−1 2τ2. Thus, the space of quadratic curvature functionals in dimension three is generated by S:g→ M τ2 gdvolgand T:g→ Mρg2dvolg. ©The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. 1 https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 2M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo Hence, every quadratic curvature functional can be expressed as a multiple of Sor Ft=T+tS, for some t∈R. There are many special cases of the functionals Ftthat can be found in the literature. We cite just a few examples. The functional defined by the L2-norm of the curvature tensor corresponds to F−1/4.Fort=−1 3one has the functional defined by the norm of the trace-free Ricci tensor ρ0=ρ−1 3τg. The norm of the Schouten tensor S=ρ−1 4τg is given by S2=ρ2−5 16 τ2, thus defining the functional F−5/16. The functional corresponding to t=−3 8is equivalent to the functional σ2:g→ Mσ2(Sg)dvol gdefined by the second symmetric elementary function of the eigenvalues of the Schouten tensor (see [15]). Some of these functionals also play a role in relativistic physics. The functional F−3/8appears in three-dimensional massive gravity, which is a correction to Einstein’s theory of gravity based on the equivalent functional EH− 1 m2F−3/8, where mis the relative mass parameter [3]. Other critical gravity theories complement the Einstein–Hilbert functional with a conformally invariant term (as is the case of conformal gravity) or with a term that extends topological invariants which depend on the dimension (as in the Lovelock theory). The Branson Q-curvature gives rise to higher-curvature theories of gravity whose action is given by a series of dimensionally extended conformal invariants. The Q-curvature of a three-dimensional manifold is defined by (see [4]) Q=−1 4Δτ−2ρ2+23 32 τ2. Hence the functional defined by the total Q-curvature is equivalent to Ftfor t=−23 64 and the corresponding theories are based on the functional EH− 1 m2F−23/64 (see [11] and references therein). Euler–Lagrange equations characterizing critical metrics for quadratic curvature functionals subject to a volume constraint were given in [2]. For the functional S and dimension three the equations read 2∇2τ−2 3Δτg−2τρ−1 3τg=0.(1.1) Critical metrics for the functional Tin dimension three are those satisfying −Δρ+∇2τ−2R[ρ]−1 3ρ2g=0,(1.2) where R[ρ] denotes the action of the curvature tensor on the Ricci tensor. From (1.1) and (1.2), the Euler–Lagrange equations for the functionals Fthave the expression −Δρ+(1+2t)∇2τ−2 3tΔτg−2(R[ρ]−1 3ρ2g)−2tτ(ρ−1 3τg)=0.(1.3) Although the functionals Sand Twere initially introduced in the compact setting, one extends them to non-compact manifolds as long as the integral exists. Alternatively one works on compact subsets K⊂Mand the investigation focusses on the restriction of the metric to those subsets, considering variations with constant volume vanishing at the boundary. Variations of metrics with compact support and constant volume result in the Euler–Lagrange equations (1.1) and (1.2), which can be analysed without the compactness assumption [15]. It follows directly from equations (1.1), (1.2) and (1.3) that, if a metric is critical for two different quadratic curvature functionals, then it is critical for all quadratic https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 3 curvature functionals. Einstein metrics are critical for all quadratic curvature functionals in dimension three, but there are also non-Einstein pp-waves which are S and Ft-critical for all t∈R(see [6]). The purpose of this work is to investigate three-dimensional Lorentzian critical metrics for quadratic curvature functionals with a high degree of symmetry. Hence we focus on the homogeneous and the 1-curvature homogeneous settings. Thus, we classify homogeneous metrics which are critical for quadratic curvature functionals defined on the whole space of metrics (not necessarily homogeneous) restricted to constant volume over compact subsets of M.In§2we consider the case of 1-curvature homogeneous spaces and show that they are Ft-critical if and only if t=−1 2(in which case the underlying structure is a Brinkmann wave and the energy of the functional vanishes), or t⩾−5 2(in which case there is a single Ricci curvature which is a double root of the minimal polynomial of the Ricci operator). Furthermore the energy of the functional is negative, zero or positive depending on the value of t. We collect some basic facts about three-dimensional homogeneous Lorentzian 3manifolds in §3and, following the work [9], reduce the analysis to the context of left-invariant metrics on Lie groups. Three-dimensional Lie groups naturally split into the unimodular and the non-unimodular ones. In §4we describe all the critical metrics on Lorentzian 3-dimensional Lie groups. In the unimodular case we obtain that the possible unimodular Lie groups admitting left-invariant Ft-critical metrics are the following (modulo isomorphism): •The Heisenberg group, which only admits non-Einstein left invariant Ft-critical metrics for the value t=−3. •The Poincar´egroupE(1,1), which admits non-Einstein Ft-critical metrics for all t∈R. •The Euclidean group E(2), which only admits non-Einstein left-invariant Ftcritical metrics for the value t=−1. •The Lie group SL(2,R), which admits non-Einstein Ft-critical metrics for all t∈R. •The Lie group SU(2), which admits non-Einstein Ft-critical metrics for t∈ (−3,−1 2). While unimodular Lorentzian Lie groups were fully described by Rahmani in [20], the description of the non-unimodular ones given in [12] is not complete. This leads to new situations in the non-unimodular case that were not considered previously in the literature. On a non-unimodular Lie group, let Adenote the matrix associated to the characteristic endomorphism of the unimodular kernel (see §3.2.2). The analysis in§4shows that the Lorentzian signature framework is much richer than the Riemannian one, where Lie groups with Anormalized by tr A= 2 admit critical metrics only if det A⩽1(see[5]). Assuming the normalization tr A= 2, results for non-unimodular Lie groups can be summarized as follows: https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 4M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo •Lie groups with det A= 0 admit critical metrics so that the restriction of the metric to the unimodular kernel is Lorentzian (theorem 4.12-(2)), Riemannian (theorem 4.16-(1)) or degenerate (theorem 4.20-(1)). •Lie groups with det A⩽1 admit critical metrics whose unimodular kernel is Lorentzian (theorem 4.12-(1)), Riemannian (theorem 4.16-(2), with sectional curvature K= 1 if det A= 1), and degenerate (theorem 4.20-(2), with sectional curvature K= 0 if det A= 1). •For any value of det Athere exist non-unimodular Lie groups that admit critical metrics such that the unimodular kernel is Lorentzian (theorem 4.12-(3)). Finally, §5is devoted to analyse some special families of Ft-critical metrics in more detail. Finally a relation is shown between algebraic Ricci solitons and critical metrics with zero energy. 2. Three-dimensional curvature homogeneous Lorentzian spaces A Lorentzian manifold (M, g) is said to be k-curvature homogeneous if for any pair of points p, q ∈Mthere exists a linear isometry Φpq :TpM→TqMsatisfying Φ∗ pq∇iRq=∇iRpfor all 0 ⩽i⩽k. Clearly any locally homogeneous Lorentzian manifold is k-curvature homogeneous for all k⩾0. However, the converse does not hold in general. A three-dimensional Lorentzian manifold is 0-curvature homogeneous if the Ricci operator has constant eigenvalues and the corresponding Jordan normal form does not change from point to point. In dimension three, 2-curvature homogeneity guarantees local homogeneity (see [14]), but there are exactly two classes of 1-curvature homogeneous Lorentzian manifolds which are not locally homogeneous. Attending to the Jordan normal form of the Ricci operator, these classes are described as follows [7]. (A) Diagonalizable Ricci operator. Let {u1,u 2,u 3}be a pseudo-orthonormal local frame field satisfying u1,u 1=u2,u 3= 1. The brackets given by [u1,u 2]=−κu 2,[u1,u 3]=−2u2+κu 3,[u2,u 3]=−2κu 1+√2Φ u2, (2.1) where κ∈Rand Φ is a function satisfying u1(Φ) = κΦandu2(Φ) = −√2 2b, b∈R, define 1-curvature homogeneous manifolds with Ricci tensor of the form ρ=−2κ2u1⊗u1+2bu 2⊗u3. (B) Non-diagonalizable Ricci operator. Let {u1,u 2,u 3}be a pseudo-orthonormal local frame field satisfying u1,u 1=u2,u 3= 1. The brackets [u1,u 2]=(α−β)u2,[u1,u 3]=−Ψu2−(α+β)u3,[u2,u 3]=0,(2.2) where α, β ∈R,α=β,β= 0 and Ψ is a function satisfying u1(Ψ) = 2ε− 2(α+β)Ψ and u2(Ψ) = 0, ε2= 1, define 1-curvature homogeneous manifolds https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 5 with Ricci tensor given by ρ=−2β2u1⊗u1−4β2u2⊗u3−2εu3⊗u3. Any k-curvature homogeneous manifold has constant scalar curvature, so equation (1.1) reduces to τ(ρ−1 3τg) = 0. Therefore, for any k⩾0, a k-curvature homogeneous metric is S-critical if and only if it is Einstein or its scalar curvature vanishes. Equation (1.3) also simplifies if τis constant and a k-curvature homogeneous metric is Ft-critical if and only if it satisfies Δρ+2(R[ρ]−1 3ρ2g)+2tτ(ρ−1 3τg)=0.(2.3) We classify Ft-critical metrics which are 1-curvature homogeneous but not homogeneous as follows. Theorem 2.1. Let (M, g)be a 1-curvature homogeneous Lorentzian manifold which is not locally homogeneous. Then gis Ft-critical for some t∈Rif and only if it satisfies one of the following assertions: (1) (M, g)belongs to the class (A) above with κ=0and b=0. In this case, g is Ft-critical for t=−1 2. (2) (M, g)belongs to the class (B) above. In this case, gis Ft-critical for t= α2+αβ−β2 3β2⩾−5 12 . Proof. Following [7], since (M, g) is a non-homogeneous 1-curvature homogeneous manifold, it either belongs to class (A) or class (B). Assume first that (M, g) belongs to class (A) and let {u1,u 2,u 3}be a pseudo-orthonormal local frame satisfying u1,u 1=u2,u 3= 1 with Lie brackets given by (2.1). A straightforward calculation shows that R[ρ]=−2bκ2u1⊗u1+2(b2+bκ2+2κ4)u2⊗u3, Δρ=−4κ(b+2κ2)κu 1⊗u1−κu 2⊗u3+2u3⊗u3. Hence, from equation (2.3), Δρ(u3,u 3)=−8κ(b+2κ2)=0.Sob=−2κ2or κ=0. In the former case the metric is Einstein, so it is locally homogeneous and must be excluded. If κ= 0, then equation (2.3) reduces to 4 3b2(1 + 2t)(u1⊗u1−u2⊗u3)= 0. Hence b=0ort=−1 2.Ifb= 0 the manifold is Einstein, so we conclude t=−1 2. This corresponds to assertion (1). Now we assume that (M, g) belongs to class (B). A straightforward calculation shows that R[ρ]=2β22β2u1⊗u1+4β2u2⊗u3+εu 3⊗u3, Δρ=−4(2α2+2αβ −β2)εu 3⊗u3. Hence, equation (2.3) reduces to 8{α2+αβ −(1 + 3t)β2}εu 3⊗u3= 0. Since ε, β = 0, we conclude that t=α2+αβ−β2 3β2, which corresponds to assertion (2).  https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 6M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo Remark 2.2. Manifolds in theorem 2.1-(1) admit a parallel null line field L= span{u2}, so they are Brinkmann waves, but not pp-waves (see §3.3). Since, moreover, they have non-vanishing constant scalar curvature, they are a particular family of the metrics described in theorem 5.1 of [6]. Remark 2.3. The Ricci operator of manifolds in theorem 2.1-(1) takes the form Ric = diag[0,b,b]. Note that, since these metrics are Ft-critical for the value t= −1 2, the functional has zero energy: ρ2−1 2τ2=0. On the other hand, the Ricci operator of manifolds in theorem 2.1-(2) has a single eigenvalue λ=−2β2, which is a double root of the minimal polynomial. This family of manifolds provides Ft-critical metrics for all t⩾−5 12 . Moreover, the energy of the Ftfunctionals is given by ρ2+tτ2= (12 + 36t)β4, so it is positive if t>−1 3, negative if −5 12 ⩽t<−1 3, and zero if t=−1 3. Remark 2.4. A three-dimensional Lorentzian manifold is locally conformally flat if and only if the Schouten tensor is Codazzi (i.e., ∇XSYZ =∇YSXZ) or, equivalently, if the Cotton tensor vanishes. A straightforward calculation shows that a three-dimensional non-homogeneous 1-curvature homogeneous Lorentzian manifold is locally conformally flat if and only if it corresponds to class (B) with β=−2α, in which case the metric is Ft-critical for t=−5 12 . An immediate calculation from the expression of tin theorem 2.1-(2) shows that a Lorentzian three-dimensional 1-curvature homogeneous (non-homogeneous) manifold is locally conformally flat if andonlyifitisFtcritical for t=−5 12 . Remark 2.5. Any locally conformally flat three-dimensional 0-curvature homogeneous Lorentzian manifold that is not 1-curvature homogeneous admits a local pseudo-orthonormal frame {u1,u 2,u 3}with u1,u 1=u2,u 3= 1 so that (see [10]) [u1,u 2]=0,[u1,u 3]=−Φu1−Ξu2+Ψu3,[u2,u 3]=−3Ψu1−Υu2, where Φ,Ψ,Ξ,Υ are smooth functions satisfying u1(Ψ) = 2Ψ2+1 4(κ+ε),u 2(Ψ) = 0, u1(Ξ) = u3(Φ) + ΦΥ −Φ2−ΨΞ + 2ε, u2(Ξ) = u3(Ψ) −3ΦΨ, u1(Υ) = u2(Ξ) + ΨΥ,u 2(Υ) = 1 2(κ+ε)−5Ψ2, where ε2= 1. A straightforward calculation shows that the Ricci operator has a single eigenvalue κ+ε, which is a double root of the minimal polynomial. Thus, it is given by ρ=(κ+ε){u1⊗u1+u2⊗u3+u3⊗u2}−2εu3⊗u3. Moreover R[ρ]=(κ+ε)2{u1⊗u1+u2⊗u3+u3⊗u2}−ε(κ+ε)u3⊗u3, Δρ=−3ε(κ+ε)u3⊗u3, and thus equation (2.3) reduces to ε(κ+ε)(12t+ 5) = 0. Hence, either κ+ε=0 and (M, g) is critical for all quadratic curvature functionals or, otherwise, it is https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 7 Ft-critical for t=−5 12 . Note that there exist locally conformally flat homogeneous metrics whose Ricci operator has complex eigenvalues which are not critical for any quadratic curvature functional Ft. This analysis is carried out in §5.2. Remark 2.6. A Lorentzian manifold is said to be semi-symmetric if the curvature tensor satisfies R(X, Y )·R= 0 for all vector fields, where R(X, Y ) acts as a derivation on R. Equivalently, the curvature tensor at each point coincides with that of a symmetric space (but possibly changing from point to point). Hence, if a nonhomogeneous 0-curvature homogeneous Lorentzian manifold is semi-symmetric, then the Ricci operator is two-step nilpotent, as in Cahen–Wallach symmetric spaces, or Ric = diag[0,λ,λ], as in direct products R×N(c) with Na surface of constant Gauss curvature. As a consequence, the only semi-symmetric 1-curvature homogeneous Lorentzian manifolds which are not locally homogeneous correspond to class (A) with κ=0andb= 0. This shows that a Lorentzian three-dimensional 1-curvature homogeneous (non-homogeneous) manifold is semi-symmetric if and only if it is Ft-critical for t=−1 2. Note that there exist F−1/2-critical homogeneous metrics whose Ricci operator has complex eigenvalues and thus they are not semi-symmetric (see §5.3). 3. Three-dimensional homogeneous Lorentzian spaces The set of three-dimensional homogeneous Lorentzian manifolds splits into two categories as follows. Theorem 3.1 [9]. Let (M, g)be a three-dimensional connected,simply connected, complete homogeneous Lorentzian manifold. Then,either (M, g)is symmetric, or it is isometric to a three-dimensional Lie group equipped with a left-invariant Lorentzian metric. The previous result is also true at the local level, therefore a locally homogeneous Lorentzian 3-manifold is either locally symmetric or locally isometric to a Lie group with a left-invariant Lorentzian metric. 3.1. Symmetric manifolds Indecomposable but not irreducible Lorentzian symmetric spaces are locally isometric to Cahen–Wallach symmetric spaces and, hence, they are a particular family of plane waves [8]. Otherwise, three-dimensional symmetric Lorentzian manifolds are of constant sectional curvature or (locally) a product of the form R×N(c), where N(c) is a Riemannian or a Lorentzian surface of constant Gauss curvature. Since manifolds of constant sectional curvature are Einstein, they are critical for all quadratic curvature functionals. It has been shown in [6] that Cahen–Wallach symmetric spaces are also critical for all quadratic curvature functionals, whereas products of the form R×N(c) are critical for t=−1 2. 3.2. Lie groups with left invariant metric We work at the Lie algebra level. Let gbe a three-dimensional Lie algebra endowed with a non-degenerate scalar product ·,· :g×g→R.Let×:g×g→g https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 8M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo Table 1. Unimodular Lie algebras Type Brackets Conditions Ia e1,e 1=e2,e 2=−e3,e 3=1 [e1,e 2]=−λ3e3,[e1,e 3]=−λ2e2,[e2,e 3]=λ1e1 Ib e1,e 1=e2,e 2=−e3,e 3=1 [e1,e 2]=−βe2−αe3,[e1,e 3]=−αe2+βe3,[e2,e 3]=λe1β=0 II u1,u 2=u3,u 3=1 [u1,u 2]=λ2u3,[u1,u 3]=−λ1u1−εu2,[u2,u 3]=λ1u2ε=±1 III u1,u 2=u3,u 3=1 [u1,u 2]=u1+λu3,[u1,u 3]=−λu1,[u2,u 3]=λu2+u3 Table 2. Non-unimodular Lie algebras Type Brackets Conditions IV.1 −e1,e 1=e2,e 2=e3,e 3=1 [e1,e 2]=0,[e1,e 3]=αe1+βe2,[e2,e 3]=γe1+δe2α+δ=0 IV.2 e1,e 1=e2,e 2=−e3,e 3=1 [e1,e 2]=0,[e1,e 3]=αe1+βe2,[e2,e 3]=γe1+δe2α+δ=0 IV.3 u1,u 1=u2,u 3=1 [u1,u 2]=0,[u1,u 3]=αu1+βu2,[u2,u 3]=γu1+δu2α+δ=0 be the cross product satisfying ei×ej,e k= det(ei,e j,e k) for all orthonormal bases {e1,e 2,e 3}. The endomorphism Ldetermined by L(ei×ej)=[ei,e j], 1 ⩽ i, j ⩽3, is referred to as the structure operator of g. 3.2.1. Unimodular Lie groups Three-dimensional unimodular Lie groups are characterized by the self-adjointness of the structure operator L[19,20]. Attending to the Jordan normal form of L, Rahmani showed in [20] that unimodular Lorentzian Lie algebras (g,,) are equivalent to one of the four types given in table I, where {e1,e 2,e 3}and {u1,u 2,u 3}are orthonormal and pseudo-orthonormal bases as specified in each case. 3.2.2. Non-unimodular Lie groups The Lie algebra of a non-unimodular Lie group is a semi-direct product g=uR, where u={x∈g;trad(x)=0}denotes the unimodular kernel, which is an abelian ideal of gcontaining the commutator ideal [g,g](see[19]). These semi-direct products are determined by the endomorphism −ad(e3), which does not depend on the choice of e3/∈u. Thus, for a basis {e1,e 2}of u, the endomorphism −ad(e3) is given by −ad(e3)(e1)=αe1+βe2and −ad(e3)(e2)=γe1+δe2.LetAdenote the matrix associated to −ad(e3). The study of Lorentzian non-unimodular Lie algebras splits into three different situations depending on the restriction of the scalar product ,of gto the unimodular kernel u, namely the cases in which ,|u×uis Lorentzian, Riemannian or degenerate. In each of them there exists an adapted basis so that the Lie algebra is given as in table II. By rescaling {e1,e 2,e 3}one may assume that tr ad(e3) = 2 and thus one works with a representative of the homothety class of the initial metric. Moreover, for https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 9 type IV.3 Lie algebras, taking ˆu1=u1,ˆu2=α+δ 2u2and ˆu3=2 α+δu3, one has that tr ad(u3) = 2 and the new basis is still orthonormal, so one remains in the same isometry class. In the Riemannian situation (see [19]) one may rotate the orthonormal basis {e1,e 2}so that ad(e3)(e1) is orthogonal to ad(e3)(e2). A straightforward calculation shows that such normalization remains valid in case IV.2 (when the restriction of the inner product to the unimodular kernel is positive definite). Hence one may assume that the structure constants satisfy αγ +βδ =0andα+δ= 2 in this case. This is due to the fact that the self-adjoint part of the endomorphism ϕis diagonalizable, a fact that cannot be assumed in the other two cases. The explicit calculations in §4.5 (remark 4.15)and4.7 (remark 4.23) show that the analogous normalizations considered in [12] for cases IV.1 and IV.3 impose restrictions in the corresponding families of metrics. 3.3. Homogeneous pp-waves In the subsequent analysis of homogeneous critical metrics, we will see that some of the families that show up admit a parallel null line field, as occurs in theorem 2.1-(1). More specifically, some of these examples are pp-waves. Apart from Cahen–Wallach symmetric spaces (CWε), there are other two families of homogeneous pp-waves. A three-dimensional homogeneous pp-wave admits local adapted coordinates (u, x, y) where the metric is given by g(∂y,∂ y)=−2f(x, y), g(∂y,∂ u)=g(∂x,∂ x) = 1. The function fdetermines the type of space, which is locally isometric to one of the following models [13]: •Nbis defined by taking f(x, y)=b−2ebx, with b=0, •Pcis defined by taking f(x, y)=1 2x2α(y), with α=cα3/2and α>0, •CWεis defined by taking f(x, y)=εx2, with ε=±1. The geometries Pc,andCWεare plane waves and, jointly with Nb, cover all possible homogeneous pp-wave classes. Theorem 3.2. Let (M, g)be a three-dimensional homogeneous pp-wave. Then one of the following holds: (1) If (M, g)is a plane wave modelled on CWεor Pcthen it is critical for all quadratic curvature functionals. (2) If (M, g)is a pp-wave modelled on Nbthen it is S-critical but not Ft-critical for any t∈R. Proof. For metrics that belong to the families Pcand CWε, a direct calculation shows that Δρ,R[ρ], ρand τvanish identically. Hence, equation (2.3) holds for all t∈Rand these metrics are critical for all quadratic curvature functionals. For manifolds modelled on Nb, the terms R[ρ], ρand τvanish, but Δρ(∂y,∂ y)=b2ebx = 0, since b= 0. Therefore, equation (2.3) is not satisfied for any t. https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 16 M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo 4.4. Type III critical metrics Gis unimodular and there exists a pseudo-orthonormal basis {u1,u 2,u 3}, with u1,u 2=u3,u 3= 1, such that the Lie brackets of the Lie algebra gare given by [u1,u 2]=u1+λu3,[u1,u 3]=−λu1,[u2,u 3]=λu2+u3,(4.6) with λ∈R. There are no Einstein metrics in this family. Moreover, the scalar curvature is given by τ=−3 2λ2, so metrics defined by (4.6)areS-critical if and only if λ=0. Theorem 4.10. Let Gbe a type III Lie group with a left-invariant metric g.Then gis Ft-critical if and only if Gis isomorphic to E(1,1) and gis isometric to a metric given by (4.6)with λ=0. In this case,the metric is critical for all t∈R. Proof. With respect to the pseudo-orthonormal basis {u1,u 2,u 3},Ftis determined by the following expressions: Ft 22 = 6(2 −t)λ2and Ft 23 =(1−3t)λ3. Thus, Ftvanishes identically if and only if λ=0.  Remark 4.11. Left-invariant metrics in theorem 4.10 have a null parallel line field L= span{u1}. Moreover, the Ricci operator is two-step nilpotent, and thus they are pp-waves. Since no metric in theorem 4.10 is locally symmetric, they correspond to metrics given in theorem 3.2-(1), so they are locally isometric to plane waves in the family Pc. 4.5. Type IV.1: critical metrics with Lorentzian unimodular kernel Gis non-unimodular and the induced metric on the unimodular kernel uis Lorentzian. Then there exists an orthonormal basis {e1(−),e 2(+),e 3(+)}such that the Lie algebra gis given by [e1,e 2]=0,[e1,e 3]=αe1+βe2,[e2,e 3]=γe1+δe2,(4.7) with α, β, γ, δ ∈R. We work with a representative of the homothety class so that α+δ= 2. These metrics are Einstein for the following values of α,βand γ: (i) α=1andγ=β, (ii) β=±αand γ=±(2 −α). Note that the case A=ad(e3) = Id is included in (i). Also, note that the structure given by (β, γ)=(α, 2−α) is isometric to (β, γ)=(−α, α −2) since the replacement e1→−e1provides an isometric isomorphism interchanging (β, γ) and (−β, −γ). The scalar curvature has the expression τ=1 2(−4(α−2)α+(β− γ)2−16). Thus metrics given by (4.7)areS-critical if and only if α=1± 1 2(β−γ)2−12 or the manifold is Einstein. https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 17 Theorem 4.12. Let Gbe a type IV.1 Lie group. A non-Einstein left-invariant metric gon Gis Ft-critical if and only if it is homothetic to a metric given by (4.7) with structure constants as follows: (1) α=1+1 2(β−γ). In this case,gis Ft-critical for t=−1 3(det A+ √1−det A)and det A⩽1. (2) det A=0. In this case,gis Ft-critical for t=−3(β+γ)2−8 (β+γ)2−16 . (3) The derivation ad(e3)is self-adjoint,i.e. β=−γ. In this case,gis Ft-critical for t=−2−det A 4−det A. Moreover, det Atakes all possible real values and Arealizes all possible Jordan normal forms. Proof. The symmetric (0,2)-tensor field Ft=Δρ+2(R[ρ]−1 3ρ2g)+2tτ(ρ− 1 3τg) is given, with respect to the basis {e1,e 2,e 3}, by the following components: Ft 11 =−1 3(8 det A2+ 4(6α−β2+4βγ +5γ2−12) det A +(6α−β2+βγ +2γ2−8)(3(β+γ)2−8) +t((β+γ)2+ 4(det A−4))(2 det A+6α−β2+βγ +2γ2−8)), Ft 12 =−((α−2)β+αγ)(3(β+γ)2−8) + 4((3 −2α)β+(1−2α)γ) det A −t((α−2)β+αγ)((β+γ)2+ 4(det A−4)), Ft 22 =1 3(8 det A2−46α−5β2−4βγ +γ2det A −(3(β+γ)2−8) 6α−2β2−βγ +γ2−4 +t(β+γ)2+ 4 det A−16−6α+2β2+βγ −γ2+ 2 det A+4 ), Ft 33 =−1 3(4(det A−1) + (β+γ)2) ×(4(det A−2) + 3(β+γ)2+t(4(det A−4) + (β+γ)2)). A straightforward calculation shows that Ft 33 vanishes if and only if det A=1− (β+γ)2 4or t=−4(det A−2)+3(β+γ)2 4(det A−4)+(β+γ)2. Firstly, we assume det A=1−(β+γ)2 4, then α=1±1 2(β−γ). Since (β, γ)→ (−β, −γ) provides an isometry, we assume without loss of generality that α= 1+1 2(β−γ). The components of Ftsimplify to Ft 11 =−Ft 12 =Ft 22 =−(2 det A+6t+β+γ)(β+γ−2)(β−γ). If β=γ, then α= 1 and the metric is Einstein. If γ=2−β, then α=βand the metric is also Einstein. Hence t=−1 6(2 det A+β+γ), which corresponds to case (1). Secondly, we assume t=−4(det A−2)+3(β+γ)2 4(det A−4)+(β+γ)2. The components of Ftreduce to Ft 11 =Ft 22 = 2 det A(β+γ)(β−γ),and Ft 12 =−4 det A(β+γ)(α−1). Thus, the tensor Ftvanishes if det A= 0, which corresponds to case (2), if β= −γ, which corresponds to case (3), or if α=1 and β=γ, which is an Einstein metric.  https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 18 M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo Figure 2. This diagram shows the values of tfor Ft-critical metrics in each family of non-unimodular Lie groups following the notation in theorems 4.12,4.16,4.20. Remark 4.13. Metrics in theorem 4.12 provide examples of Ft-critical metrics for values of tin R\{−1}. The attained values in each of the cases are illustrated in figure 2and are described as follows: •Case (1) provides Ft-critical metrics for t∈(−5 12 ,∞) with energy ρ2+tτ2= 3(β+γ)(β+γ−2). •Case (2) provides Ft-critical metrics for t∈(−∞,−3) ∪(−1 2,+∞) with energy ρ2+tτ2=6(β+γ)2. •Case (3) provides Ft-critical metrics for all t∈R\{−1}with zero energy. For any non-Einstein critical metric, observe that the energy of the corresponding functional Ftis zero if and only if β=−γ. Remark 4.14. The Ricci operator of critical metrics in theorem 4.12 is determined as follows: •Critical metrics corresponding to case (1) have a single Ricci curvature equal to −2, which is a double root of the minimal polynomial. •Critical metrics in case (2) have diagonalizable Ricci operator with two-distinct Ricci curvatures Ric = −diag[1 2(β+γ)2,4−1 2(β+γ)2,4−1 2(β+γ)2]. •The Ricci operator of critical metrics corresponding to case (3) has eigenvalues {−2(2 −det A),−2(1 ±√1−det A}, which may be real or complex depending on the value of det A. Furthermore if det A= 1, there is a single eigenvalue which is a double root of the minimal polynomial. Remark 4.15. Theorem 4.12 shows that the normalization ad(e3)(e1)⊥ad(e3)(e2) considered in [12] is not always possible. Indeed, left-invariant metrics in theorem 4.12-(3), where Ais self-adjoint, satisfy the above normalization only if Adiagonalizes. Thus, the cases with complex eigenvalues and a double root of the minimal polynomial are missed if one assumes the possibility of normalizing as above. Indeed, if one considers metrics given by (4.7) with αγ −βδ =0,noFt-critical metric is found with t∈(−3,−1 2). https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 19 4.6. Type IV.2: critical metrics with Riemannian unimodular kernel Gis non-unimodular and the induced metric on the unimodular kernel uis Riemannian. Then there exists an orthonormal basis {e1(+),e 2(+),e 3(−)}such that the Lie algebra gis given by [e1,e 2]=0,[e1,e 3]=αe1+βe2,[e2,e 3]=γe1+δe2,(4.8) with α, β, γ, δ ∈Rsatisfying α+δ= 2. A direct calculation shows that these metrics are Einstein if and only if α=1andγ=−β(which includes the case ad(e3)= Id). Moreover, the scalar curvature is given by τ=1 2(4(α−2)α+(β+γ)2+ 16) and satisfies τ>0. Hence the only S-critical metrics within this family are the Einstein ones. Theorem 4.16. Let Gbe a type IV.2 Lie group. A non-Einstein left-invariant metric gon Gis Ft-critical if and only if it is homothetic to a metric given by (4.8) with structure constants as follows: (1) det A=0. In this case,gis Ft-critical for t=−8+3(β−γ)2 16+(β−γ)2. (2) The derivation ad(e3)is self-adjoint,i.e. β=γ. In this case,gis Ft-critical for t=−2−det A 4−det Aand det A<1. Proof. On the pseudo-orthonormal basis {e1,e 2,e 3}, the a priori non-vanishing components of Ftare given by the following expressions: Ft 11 =1 3(8(det A)2+ 4(6α+β2+4βγ −5γ2−12) det A −(6α+β2+βγ −2γ2−8)(3(β−γ)2+8) −t(2 det A+6α+β2+βγ −2γ2−8)(4 det A−(β−γ)2−16)), Ft 12 =β(8α3−28α2+α(9γ2+ 32) −2(γ2+ 8)) + 3(α−2)β3−αβ2γ +αγ(4α−3γ2−16 + 8 det A)+t((α−2)β−αγ)((β−γ)2−4 det A+ 16), Ft 22 =1 3(8(det A)2−4(6α+5β2−4βγ −γ2) det A + (3(β−γ)2+ 8)(6α+2β2−βγ −γ2−4)) +t(4 det A−16 −(β−γ)2)(−6α−2β2+βγ +γ2+ 2 det A+4), Ft 33 =1 3(4 det A−4−(β−γ)2) ×(4 det A−8−3(β−γ)2+t(4 det A−16 −(β−γ)2)). A straightforward calculation shows that Ft 33 vanishes if and only if det A=1+ 1 4(β−γ)2or t=−4detA−8−3(β−γ)2 4detA−16−(β−γ)2. Since det A=−α2+2α−βγ, the equation det A=1+1 4(β−γ)2has real solutions only if β=−γ, in which case α= 1. Hence, this condition leads to an Einstein metric. https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 20 M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo Hence, we assume t=−4detA−8−3(β−γ)2 4detA−16−(β−γ)2. Simplifying the components of Ft,we get: Ft 11 =−Ft 22 =−2 det A(β−γ)(β+γ), Ft 12 = 4 det A(β−γ)(α−1). Thus the tensor Ftvanishes if det A= 0, which corresponds to case (1), if β=γ, which corresponds to case (2), or if α=1andβ=−γ, which corresponds to an Einstein metric.  Remark 4.17. Notice that the family of metrics in theorem 4.16-(2) gives critical metrics for all t∈(−1,−1 3) with zero energy, whereas the family of metrics in theorem 4.16-(1) gives critical metrics for all t∈(−3,−1 2] with energy given by ρ2+tτ2=−6(β−γ)2. The value t=−1 2for metrics with β=γcorresponds to critical metrics which are Einstein (α=1, β=γ= 0). These values of tare illustrated in Fig. 2. Remark 4.18. The Ricci operator of critical metrics obtained in theorem 4.16 is diagonalizable with eigenvalues (1) 4+1 2(β−γ)2,4+1 2(β−γ)2,−1 2(β−γ)2, (2) 2(2 −det A),2(1 −√1−det A),2(1 + √1−det A). Remark 4.19. The Lie algebra given by (4.8) corresponds to a semi-direct product Rr2where the metric restricted to r2has positive definite signature. Therefore, the results are analogous to those obtained for three-dimensional Riemannian nonunimodular Lie groups (cf. [5]). 4.7. Type IV.3: critical metrics with degenerate unimodular kernel Gis non-unimodular and the restriction of the metric to the unimodular kernel uis degenerate. Then there exists a pseudo-orthonormal basis {u1,u 2,u 3}of the Lie algebra, with u1,u 1=u2,u 3= 1, such that [u1,u 2]=0,[u1,u 3]=αu1+βu2,[u2,u 3]=γu1+δu2,(4.9) with α, β, γ, δ ∈R,andα+δ= 2. These metrics are Einstein only if they are flat, which occurs if γ=0andα=0,1. Moreover, the scalar curvature is τ=γ2 2,so metrics given by (4.9)areS-critical if and only if γ=0. Theorem 4.20. Let Gbe a type IV.3 Lie group. A non-Einstein left-invariant metric gon Gis Ft-critical if and only if it is homothetic to a metric given by (4.9) with structure constants as follows: (1) det A=0. In this case,gis Ft-critical for t=−3. (2) The derivation ad(e3)is self-adjoint,i.e.,γ=0. In this case, gis Ft-critical for all t∈R. https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 21 Proof. The a priori non-vanishing components of the tensor field Ftare given, with respect to the {u1,u 2,u 3}basis, by Ft 11 =2 3(3 + t)γ4,Ft 13 =−(3 + t)αγ3, Ft 23 =−1 3(3 + t)γ4,Ft 33 =γ2(3α2−det A+t(α2−det A)). Ft 11 vanishes if and only if γ=0 or t=−3. If γ= 0, the tensor field vanishes identically with independence of the value of t. This corresponds to case (2). Now, we assume γ=0andt=−3. The only non-vanishing component is Ft 33 = 2γ2det A, that vanishes if and only if det A= 0. This corresponds to case (1).  Remark 4.21. The Ricci operator of metrics corresponding to theorem 4.20-(1) is diagonalizable with eigenvalues {−γ2 2,γ2 2,γ2 2}from where it also follows that the energy vanishes in this case (ρ2−3τ2= 0). Remark 4.22. Left-invariant metrics in theorem 4.20-(2) have a parallel degenerate line field L= span{u2}. Moreover, the Ricci operator is two-step nilpotent, so they are pp-waves. In the non-flat case, if α= 2, then they are locally symmetric and, hence, a Cahen–Wallach symmetric space CWε. Otherwise, they are plane waves Pc(see theorem 3.2). Furthermore, since the Ricci operator is two-step nilpotent, all functionals Fthave zero energy in this case. Remark 4.23. The Ricci operator of any metric (4.9) satisfies Ric = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −γ2 20αγ αγ γ2 2det A−α2 00 γ2 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ with eigenvalues −γ2 2,γ2 2,γ2 2. Considering the subfamily given by α2= det A one has that the minimal polynomial of the Ricci operator is (λ+γ2 2)(λ−γ2 2) if αγ = 0, but it is (λ+γ2 2)(λ−γ2 2)2if αγ = 0. Hence there are metrics within this subfamily which are not isometric to any metric obtained with the normalization αγ = 0 proposed in [12]. Therefore, the normalization ad(e3)(e1)⊥ ad(e3)(e2) (equivalently αγ =0)in(4.9) cannot be applied if one intends to consider representatives of all metrics. 5. Special cases 5.1. Critical metrics for the L2-norm of the curvature tensor In dimension three a metric is critical for the curvature functional g→ MRg2dvolgif and only if it is Ft-critical for t=−1 4. The first Riemannian homogeneous example of a non-Einstein F−1/4-critical metric was given by Lamontagne [17], and it was shown in [5] that no other possibilities may occur in the https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 22 M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo positive definite setting. The Lorentzian situation allows other non-Einstein examples which, in addition to plane waves given in theorem 4.10 and theorem 4.20-(2), are homothetic to the following: •The unimodular Lie group SL(2,R) with metric (4.1) given by (λ1,λ 2,λ 3)= (1,1,4 11 )or(λ1,λ 2,λ 3)=(4 11 ,1,1) as in theorem 4.1-(1). The Ricci operators are Ric = −4 121 diag[9,9,2] and Ric = −4 121 diag[2,9,9], respectively. •The unimodular Lie group SL(2,R) with metric (4.4) homothetic to a metric given by α=1andβ2=1 12 (7λ2+4λ+ 16) where λis the only real solution of 3λ3−20λ2−20λ−32 = 0 as in theorem 4.4-(1). The Ricci operator has complex eigenvalues in this case as shown in remark 4.6. •Non-unimodular Lie groups with Lorentzian unimodular kernel corresponding to theorem 4.12 as follows: The non-unimodular Lie group with Lie algebra determined by tr A=2and det A=1 4(1 −2√2) equipped with a left-invariant metric are homothetic to those given in theorem 4.12-(1) with β+γ=1+√2. The non-unimodular Lie group with Lie algebra determined by tr A=2and det A= 0 equipped with a left-invariant metric homothetic to those given in theorem 4.12-(2) with β+γ=4 √11 . The non-unimodular Lie group with a Lie algebra determined by tr A=2and det A=4 3equipped with a left-invariant metric homothetic to those given in theorem 4.12-(3) with α=1±γ2−1 3. The Ricci operator corresponding to the above non-unimodular groups has been discussed in remark 4.14. Proceeding in an analogous way one may explicitly give all homogeneous metrics which are critical for functionals with a geometric or physical meaning, such as F−3/8or F−23/64. 5.2. Locally conformally flat homogeneous critical metrics Homogeneous locally conformally flat three-dimensional manifolds were classified in [16], from where it follows that they are locally symmetric or, otherwise, they correspond to one of the following homothetic classes: •A type Ib Lie group (4.4) with α=−1 2,β=√3 2and λ=1. •A type III Lie group (4.6) with λ=0. •A type IV.1 Lie group (4.7) with β+γ=1andα=1 2(2 + β−γ), γ=1 2. •A type IV.3 Lie group (4.9) with γ=0andα/∈{0,1,2}. Type Ib metrics above have Ricci curvatures {−2,1±√3√−1}. Hence they are S-critical, since the scalar curvature vanishes, but they are not critical for any quadratic curvature functional Ft. This is in sharp contrast with the curvature https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 23 homogeneous case (see remarks 2.4 and 2.5). Type IV.1 metrics above are F−5/12critical and have a single Ricci curvature μ=−2, which is a double root of the minimal polynomial. Metrics corresponding to types III and IV.3 are critical for all quadratic curvature functionals and have two-step nilpotent Ricci operator. 5.3. Homogeneous F−1/2-critical metrics and semi-symmetric spaces All symmetric spaces are critical for the functional F−1/2(see subsection 3.1). These spaces are generalized by the so-called semi-symmetric spaces. They are manifolds whose curvature tensor coincides with that of a symmetric space at each point, where the symmetric model may change from point to point. Consequently, a unimodular Lie group is semi-symmetric if and only if it is symmetric or it corresponds to a pp-wave locally modelled on Nb(of type II) or Pc(of type III). Results in §4show that a unimodular non-Einstein Lie group is F−1/2-critical if and only if it is of type III as in theorem 4.10 (and thus semi-symmetric) or it is a type Ib Lie group determined by (4.4) with β2=2α2+αλ +3 4λ2and 16α3+ 12α2λ+8αλ2−λ3= 0. Furthermore, the Ricci operator has complex eigenvalues in this case and, hence, it does not correspond to any semi-symmetric space. A non-unimodular Lie group is semi-symmetric if and only if it is symmetric or it corresponds to a type IV.3 Lie group given by (4.9) with γ= 0 (in which case it is a pp-wave modelled on Pcor CWε). Moreover, results in §4show that a nonunimodular Lie group is F−1/2-critical if and only if it is symmetric or type IV.3 given by (4.9) with γ=0. A three-dimensional semi-symmetric homogeneous Lorentzian manifold which is critical for a quadratic curvature functional is S-critical or F−1/2-critical. Conversely, any homogeneous F−1/2-critical metric is semi-symmetric unless it corresponds to the type Ib Lie group above with complex Ricci curvatures. 5.4. Critical metrics and algebraic Ricci solitons ARicci soliton is a triple (M, ,,X), where (M, ,) is a pseudo-Riemannian manifold and Xis a vector field on Mthat satisfies the differential equation LX,+ρ=μ,,(5.1) where Ldenotes the Lie derivative and μ∈R. Ricci solitons are not only generalizations of Einstein metrics, but they correspond to self-similar solutions of the Ricci flow. A Ricci soliton is said to be trivial if the pseudo-Riemannian metric g is Einstein. In the context of Lie groups with left-invariant metric, a Ricci soliton is said to be left-invariant if equation (5.1) holds for a left-invariant vector field X. Moreover, (G, ,) is said to be an algebraic Ricci soliton if the Ricci operator satisfies Ric = μId +Dfor some derivation Dof the Lie algebra (see [18]). This condition for the Ricci operator implies that equation (5.1) holds (see [18]). The aim of this section is to relate Lie groups endowed with a left-invariant Ft-critical metric and those that are algebraic Ricci solitons. Following the classification in §3.2 we consider unimodular Lie groups and analyse when Ric −μId is a derivation of the Lie algebra. After some straightforward calculations (we omit details in the interest of brevity), we obtain the following (see also [1]). https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press 24 M. Brozos-V´azquez, S. Caeiro-Oliveira and E. Garc´ıa-R´ıo Lemma 5.1. Let (G, ,)be a three-dimensional unimodular Lorentzian Lie group. If (G, ,)is a non-Einstein algebraic Ricci soliton,then it is isometric to one of the following: (1) The Heisenberg group with Lie algebra (4.1)of type Ia given by (λ1,λ 2,λ 3)=(0,0,λ),or (λ1,λ 2,λ 3)=(λ, 0,0). (2) The Euclidean group E(2) with associated Lie algebra (4.1)of type Ia given by (λ1,λ 2,λ 3)=(0,λ,−λ). (3) The Poincar´egroupE(1,1) with (a) A type Ia Lie algebra (4.1)given by (λ1,λ 2,λ 3)=(λ, −λ, 0). (b) A type Ib Lie algebra (4.4)given by α=λ=0,or (c) A type III Lie algebra (4.6)with λ=0. The corresponding analysis for the non-unimodular case gives the following. Lemma 5.2. A three-dimensional non-unimodular Lorentzian Lie group is an algebraic Ricci soliton if and only if it is Einstein or the operator ad(e3)is self-adjoint. Remark 5.3. The classification of algebraic Ricci solitons in [1] misses some possibilities in the non-unimodular case due to the normalization problem already pointed out in remark 4.15. Indeed, there exist algebraic Ricci solitons with nondiagonalizable Ricci operator, which do not correspond to those listed in [1](see lemma 5.2 and remark 4.14). The following result shows the relation between algebraic Ricci solitons and critical metrics for quadratic curvature functionals. Theorem 5.4. Let (G, ,)be a three-dimensional Lorentzian Lie group with left invariant metric. If (G, g)is an algebraic Ricci soliton,then it is critical for a quadratic curvature functional with zero energy. Conversely,if (G, ,)is critical for a quadratic curvature functional with zero energy, then it is an algebraic Ricci soliton,except if (G, ,)is isometric to a type IV.3 non-unimodular Lie group given by (4.9)with det A=0and γ=0. Proof. If (G, ,) is Einstein, then it is a trivial algebraic Ricci solitons and, moreover, it is critical for all quadratic curvature functionals. For unimodular Lie groups, a direct analysis of the energy ρ2+tτ2of metrics in theorems 4.1,4.4,4.7 and 4.10 shows that those with zero energy are precisely the algebraic Ricci solitons given in lemma 5.1. The same analysis is carried out for non-unimodular Lie groups. As a result, all algebraic Ricci solitons in lemma 5.2 are critical for some functional with zero energy. The converse is also true with only one exception: metrics in theorem 4.20- (1). These metrics are critical for the functional F−3=(ρ2−3τ2)dvol gwith energy ρ2−3τ2= 0 (see remark 4.21). However, a straightforward calculation https://doi.org/10.1017/prm.2022.44 Published online by Cambridge University Press Curvature homogeneous Lorentzian critical metrics 25 shows that Ric −μId acts as a derivation only if γ= 0 (see lemma 5.2), so they are algebraic Ricci solitons only in this case, which is the intersection with the subfamily in theorem 4.20-(2).  Acknowledgments Supported by projects PID2019-105138GB-C21(AEI/FEDER, Spain) and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain). References 1 W. Batat and K. Onda. Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. J. Geom. Phys.114 (2017), 138–152. 2 M. Berger. 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