aa r X i v : . [ m a t h . DG ] M a y HOMOGENEOUS RICCI SOLITONS ARE ALGEBRAIC
MICHAEL JABLONSKI
Abstract.
In this short note, we show that homogeneous Ricci solitons are algebraic. As anapplication, we see that the generalized Alekseevskii conjecture is equivalent to the Alekseevskiiconjecture. Introduction
A Riemannian manifold (
M, g ) is said to be a Ricci soliton if it satisfies the equation(1.1) ric g = cg + L X g for some c ∈ R and some smooth vector field X ∈ X ( M ). Such metrics are of interest as theycorrespond to self-similar solutions of the Ricci flow ∂∂t g = − ric g That is, g is the initial value of a solution to the Ricci flow of the form g t = c ( t ) ϕ ∗ t g , where c ( t ) ∈ R and ϕ t ∈ Diffeo ( M ). In this way, Ricci solitons are geometric fixed points of the flow and so arespecial metrics.Homogeneous Ricci solitons arise naturally as limits under the Ricci flow [15, 14] and, indepen-dently, hold a distinguished place apart from other homogeneous metrics. For example, nilmanifoldscannot admit Einstein metrics, but do often admit Ricci solitons [9, 7], Ricci solitons on nilmani-folds are precisely the minima of a natural geometric functional [13], and Ricci solitons are metricsof maximal symmetry on certain solvmanifolds [5].One natural kind of example arises as follows. Consider a homogeneous space G/K where K isclosed and connected. For every derivation D ∈ Der ( g ) such that D : k → k , we have a well-definedmap D g / k : g / k → g / k . Denote such derivations of g by Der ( g / k ). A homogeneous Ricci soliton( G/K, g ) is called G -semi-algebraic if the (1 ,
1) Ricci tensor is of the form(1.2)
Ric = cId + 12 ( D g / k + D g / k t )on g / k ≃ T e G/K , for some c ∈ R and some D ∈ Der ( g / k ). This definition is motivated by the ideaof taking our family of diffeomorphisms { ϕ t } above to come from automorphisms of the group G which leave K invariant, see [6] or [12] for more details.If our semi-algebraic Ricci soliton satisfies the seemingly stronger condition that D g / k is sym-metric, then it is called a G -algebraic Ricci soliton . Up to this point, all known examples ofsemi-algebraic Ricci solitons were in fact algebraic and isometric to solvmanifolds. (This followsfrom [6] together with [11].) Further, it was known that every homogeneous Ricci soliton must besemi-algebraic relative to its full isometry group [6]. We now present our main result. Theorem 1.
Every G -semi-algebraic Ricci soliton is necessarily G -algebraic. Corollary 2.
Let ( M, g ) be a homogeneous Ricci soliton. There exists a transitive group G , ofisometries, such that M = G/K is a G -algebraic Ricci soliton. This work was supported in part by NSF grant DMS-1105647. he theorem above resolves questions raised by Lafuente-Lauret [12] and He-Petersen-Wylie [4].In these works, it was shown that one can always extend a simply-connected, algebraic solitonto an Einstein metric on a larger homogeneous space. There the goal was to relate the classicalAlekseevskii conjecture on Einstein metrics to a more general version for Ricci solitons. Moreprecisely, they showed that (among simply-connected manifolds) the Alekseevkii conjecture forEinstein metrics is equivalent to the (apriori) more general conjecture in the case of algebraic Riccisolitons. We state these conjectures for completeness. Alekseevskii Conjecture:
Every homogeneous Einstein metric with negativescalar curvature is isometric to a simply-connected solvmanifold.
Generalized Alekseevskii Conjecture:
Every expanding homogeneous Riccisoliton is isometric to a simply-connected solvmanifold.Until now, it was not clear if these conjectures were equivalent. Applying [12] or [4] in thesimply-connected case together with [8] and the results here, we now know the following.
Theorem 3.
The generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture.
Remark.
It is important to note that the Alekseevskii conjecture stated above is a more modern,geometric version than that given in [2] . The version given in [2] has the weaker, topologicalconclusion that a non-compact, homogeneous, Einstein space is only diffeomorphic to R n . It isstill an open question as to whether the classical version stated in [2] is equivalent to the strongerversion we pose above.Acknowledgments: It is our pleasure to thank Ramiro Lafuente for providing useful commentson a draft of this manuscript. 2.
Ricci solitons by type
The analysis of (homogeneous) Ricci solitons varies depending on which of the following categoriesthe metric falls into. A Ricci soliton is called shrinking, steady, or expanding (respectively) if thecosmological constant c appearing in Eqn. 1.1 satisfies c > c = 0, or c < Shrinking solitons.
The simplest example of a non-Einstein, homogeneous, shrinker is obtainedby considering a compact homogeneous Einstein space M ′ (which necessarily has positive scalarcurvature) and taking a product with R n , i.e. M = M ′ × R n . Here the vector field X ∈ X ( M )appearing in Eqn. 1.1 generates a family of diffeomorphisms which simply dilate the R n factor.Examples of this type are called trivial Ricci solitons and a result of Petersen-Wylie [16] says thatevery homogeneous shrinking Ricci soliton is finitely covered by a trivial one. Observe that suchspaces are algebraic Ricci solitons. Steady solitons.
A homogeneous steady soliton is necessarily flat. This well-known fact is provedas follows. Along the Ricci flow of any homogeneous manifold, the scalar curvature sc evolves bythe ODE ddt sc = 2 | Ric | As the scalar curvature of a steady soliton does not change along the flow, we see that the homo-geneous, steady solitons are Ricci flat and so flat by [1]. Such spaces are trivially algebraic Riccisolitons. xpanding solitons. Every homogeneous, expanding Ricci soliton is necessarily non-compact,non-gradient and all known examples of such spaces are isometric to solvable Lie groups with left-invariant metrics. While there is no characterization in this case as nice as the previous two cases,new structural results have recently appeared in [12]. The results obtained there are essential inour proof and we briefly recall those which we need.We first observe that it suffices to prove the theorem for simply-connected manifolds. Nowconsider a simply-connected, expanding, semi-algebraic Ricci soliton on
G/K . As
G/K is endowedwith a G -invariant metric, Ad ( K ) is contained in a compact subgroup of Aut ( G ) and so we have adecomposition g = p ⊕ k , where p is an Ad ( K )-complement to k . We fix the point p = eK ∈ M = G/K and naturally identify p with T p M as follows X ∈ p ↔ dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 exp ( sX ) · p = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 exp ( sX ) K. Although there is more than one choice of p that one can make, we apply the work [12] in thesequel and so we choose, as they do, to have B ( k , p ) = 0, where B is the Killing form of g .As G/K admits an expanding Ricci soliton, we know from [12] that the group G decomposes as N ⋊ U where N is the nilradical and U is a reductive subgroup which contains the stabilizer K .Thus the underlying manifold of M may be considered as N × U/K and we naturally identify thepoint p = eK ∈ G/K with ( e, eK ) ∈ N × U/K . The subalgebra u contains a subspace h whichis complementary to k , and so we have T p M ≃ p = n ⊕ h . Furthermore, n and h are orthogonalsubspaces of T p M . For more details, see [12].Denote the restriction of our metric g to p ≃ T e G/K by h· , ·i . Denote by H ∈ p the ‘meancurvature vector’ of G/K defined by h H, X i = tr ( ad X ) for all X ∈ p Observe that H ∈ h . It is a useful fact that the subspace h of u is ( ad H )-stable [12, Prop. 4.1]. If D is the soliton derivation appearing Eqn. 1.2, then we have D = − ad H + D where D is the derivation which vanishes on u and restricts to the nilsoliton derivation on n .In [12, Prop. 4.14], several conditions are given for when a semi-algebraic Ricci soliton is actuallyalgebraic. One of those conditions is(2.1) S ( ad H | h ) = 0where S ( A ) = ( A + A t ). This is the technical result that we will prove, from which the theoremfollows. 3. The proof of theorem 1
The soliton inner product h· , ·i on T p M above gives rise to a natural inner product on theendomorphisms of T p M given by h A, B i = tr ( AB t ), where B t denotes the metric adjoint of B relative to h· , ·i . Lemma 4.
Using the above inner product on endomorphisms we have h (0 , ad H | h ) , Ric i = 0 where (0 , ad H | h ) is the map on T p M defined as on n and ad H | h on h . Remark.
As has been observed by R. Lafuente [10] , our proof of the lemma holds more generally.In fact, one simply needs the group to satisfy G = U ⋉ N with N nilpotent, U reductive, and K < U ,the metric to satisfy N ⊥ U/K at eK , and the element H may be replaced by any Y ∈ u satisfying [ Y, k ] ⊂ k . efore proving the lemma, we use it to verify that Eqn. 2.1 holds. Verification of 2.1.
Consider the mean curvature vector H ∈ u . As u is reductive, ad H | u istraceless. Furthermore, since ad H vanishes on the stabilizer k (see Eqn. 26 of [12]) and u = k ⊕ h ,we see that tr ad H | h = 0. Together with the above lemma we have0 = h (0 , ad H | h ) , Ric i = h (0 , ad H | h ) , cId − S ( ad H ) + D i = h ad H | h , cId | h − S ( ad H | h ) i = c tr ( ad H | h ) − tr S ( ad H | h ) = 0 − tr S ( ad H | h ) Thus S ( ad H | h ) = 0, as claimed. (cid:3) We now prove the lemma by considering a certain deformation of the metric g on M . As ad H vanishes on k and K is connected, the family of automorphisms Φ t = C exp ( tH ) ∈ Aut ( U ) is theidentity on K and hence gives rise to well-defined diffeomorphisms φ t on U/K given by φ t ( uK ) = Φ t ( u ) K for u ∈ U Note that (Φ t ) ∗ = Ad ( exp ( tH )) = e t ad H ∈ Aut ( u ). On the manifold M = N × U/K , we considerthe family of diffeomorphisms given by ϕ t = ( id, φ t ) on N × U/K
The deformations of g of interest are g t = ϕ t ∗ g .As ϕ t fixes the point p := eK = ( e, eK ) ∈ M = N × U/K , and scalar curvature is an invariant,we have ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 sc ( ϕ t ∗ g ) p = 0We use this in the following general equation which holds for any family of metrics { g t } withvariation h = ∂∂t g t (see [3, Lemma 3.7])(3.1) ∂∂t sc = − ∆ H + div ( div h ) − h h, ric i where in local coordinates we have(3.2) ∆ H = g ij g kl ∇ i ∇ j h kl and(3.3) div ( div h ) = g ij g kl ∇ i ∇ k h jl Observe, at the point p := eK = ( e, eK ) of M we have ∂∂t | t =0 ( ϕ t ) ∗ = (0 , ad H | h ) and so the lemmafollows from Eqn. 3.1 (evaluated at p ) upon showing the terms ∆ H and div ( div h ) vanish. Remark.
Recall that, in local coordinates, we define the metric inverse g ij as the function satisfying δ li = g ij g jl . By choosing a frame which is g -orthonormal at every point, one would have that both g ij and g ij are the identity. We make such a choice below. To ease computational burden, we build a frame which is g -orthonormal at every point andexploits the property that our metric g is G -invariant. We start with an orthonormal basis of T p M .As T p M = n ⊕ h , we may choose a basis { e i } which is the union of an orthonormal basis of n together with an orthonormal basis of h . ext, we extend the basis { e i } to a local frame nearby to p ∈ M . To do this, we first considera slice S of the right K action on G through e ∈ G . That is, we have a submanifold S of G containing e such that dim S = dim G/K and the map s sK s ∈ S is a diffeomorphism of a neighborhood of e ∈ S to a neighborhood of eK ∈ G/K . Now, for q ∈ M nearby to p , there exists s ∈ S such that q = s · p and we define e i ( q ) = s ∗ e i , where s ∗ denotes the differential of the translation s : p q . We note that the frame is well-definedas our choice of s ∈ S is unique, since S is a slice. Furthermore, our frame is g -orthonormal as g is G -invariant.Using the above choice of frame nearby to p ∈ M , we now study Eqns. 3.2 and 3.3. We beginby computing the variation h of g t = ϕ t ∗ g in terms of { e i } . For a point q ∈ M near p ,(3.4) h ij ( q ) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( g t ) ij ( q ) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( g t )( e i ( q ) , e j ( q )) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 g (( ϕ t ) ∗ e i ( q ) , ( ϕ t ) ∗ e j ( q ))Next we compute ( ϕ t ) ∗ v q for a vector v q ∈ T q M .As G = N U , there exist n ∈ N and u ∈ U such that s ∈ S may be written as s = nu and q = ( nu ) · p . Furthermore, there exists X ∈ p = n ⊕ h such that v q = ( nu ) ∗ dds (cid:12)(cid:12) s =0 exp ( sX ) · p . Tounderstand Eqn. 3.4, we analyze separately the cases when X is an element of n or of h .For X ∈ n , we have( ϕ t ) ∗ v q = ( ϕ t ) ∗ ( nu ) ∗ X = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ϕ t ( nu exp ( sX ) · p )= dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ϕ t ( n u exp ( sX ) u − u · p )= dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ϕ t ( n u exp ( sX ) u − , uK )(3.5) = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ( n exp ( sAd u X ) , Φ t ( u ) K )= dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ( n Φ t ( u ) Φ t ( u ) − exp ( sAd u X )Φ t ( u ) K )= dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ( n Φ t ( u ) exp ( sAd Φ t ( u ) − Ad u X ) K )= ( n Φ t ( u )) ∗ Ad Φ t ( u ) − u X Here we have used that N is normal in G . Note also that Ad Φ t ( u ) − u X ∈ n . n the case when X ∈ h ⊂ u , we have( ϕ t ) ∗ v q = ( ϕ t ) ∗ ( nu ) ∗ X = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ϕ t ( nu exp ( sX ) · p )= dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ϕ t ( n u exp ( sX ) K )(3.6) = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ( n Φ t ( u exp ( sX )) K )= dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ( n Φ t ( u ) exp ( s (Φ t ) ∗ X )) K )= ( n Φ t ( u )) ∗ (Φ t ) ∗ X Observe that since ad H preserves h ([12] Eqn. 32), (Φ t ) ∗ X ∈ h and so the the last line is consistentwith our identification of p = n ⊕ h with T p M .From Eqns. 3.4, 3.5, and 3.6 we see that(i) If e i ∈ n and e j ∈ h , then g ij ( q ) = 0.(ii) If e i ∈ n and e j ∈ h , then h ij ( q ) = 0.(iii) If e i , e j ∈ h , then h ij ( q ) does not depend on n and u , and so is constant in q .(iv) If e i , e j ∈ n , then h ij ( q ) does not depend on n , but does depend on u .Using these observations, we see that the only possible non-zero terms of div ( div h ) = g ij g kl ∇ i ∇ k h jl are when e j , e l ∈ n and e i , e k ∈ h . However, ( g αβ ) = Id implies ( g αβ ) = Id and so g kl = 0. Thisyields div ( div h ) = 0Next we study ∆ H = g ij g kl ∇ i ∇ j h kl . As above, the only possible non-zero terms occur when e k , e l ∈ n and e i , e j ∈ h . Further, as our frame is orthonormal, we have∆ H ( q ) = g ii ( q ) g kk ( q )( ∇ i ∇ i h kk )( q ) = X i ∇ i ∇ i X k h kk ! ( q )where the first sum is over the frame from h and the second is over the frame from n . From Eqns. 3.4and 3.5 we have h kk ( q ) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 g (( ϕ t ) ∗ e k ( q ) , ( ϕ t ) ∗ e k ( q ))= ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 h Ad Φ t ( u ) − u ( e k ) , Ad Φ t ( u ) − u ( e k ) i = 2 h e k , ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Ad Φ t ( u ) − u )( e k ) i = 2 h e k , ad M ( e k ) i where M = ddt (cid:12)(cid:12) t =0 Φ t ( u ) − u . To see that this last line makes sense, observe that Φ t ( u ) − u is acurve in U with Φ ( u ) − u = e and thus ddt (cid:12)(cid:12) t =0 Φ t ( u ) − u ∈ u . Remark.
Although M is a function of u , we suppress this detail as it does not impact the rest ofour proof. e claim that ad M | n is traceless. To see this, we use that fact that U being reductive andconnected implies U = [ U, U ] Z ( U ), where Z ( U ) is the center of U . Thus, we may write u = u u where u ∈ [ U, U ] and u ∈ Z ( U ). As u is central and Φ t is an inner automorphism, Φ t ( u ) = u and Φ t ( u ) − u = Φ t ( u ) − u ∈ [ U, U ]This gives ad M ∈ ad [ u , u ] from which our claim immediately follows.Putting the above computations together,∆ H ( q ) = X i ∇ i ∇ i X k h kk ! ( q )= 2 X i ∇ i ∇ i tr ad M | n = 0which completes the proof of the lemma. References [1]
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