aa r X i v : . [ m a t h . DG ] F e b RIGIDITY OF SU n -TYPE SYMMETRIC SPACES WAFAA BATAT, STUART JAMES HALL, THOMAS MURPHY,AND JAMES WALDRON
Abstract.
We prove that the bi-invariant Einstein metric on SU n +1 is isolated in the moduli space of Einstein metrics, even though it admitsinfinitesimal deformations. This gives a non-K¨ahler, non-product exam-ple of this phenomenon adding to the famous example of CP n × CP found by Koiso. We apply our methods to derive similar solitonic rigid-ity results for the K¨ahler–Einstein metrics on ‘odd’ Grassmannians. Wealso make explicit a connection between non-integrable deformationsand the dynamical instability of metrics under Ricci flow. Introduction
Rigidity of Einstein metrics.
For a fixed manifold M , a centralobject of study is the set of Einstein metrics viewed as a subset of all Rie-mannian metrics on M . In general, giving a full description of the set ofEinstein metrics on M is an intractable project; a more reasonable goal isto produce a local description of this set around some fixed Einstein metric g .The foundational work on the local structure theory of the set of Einsteinmetrics was carried out by Koiso [16] who showed that the premoduli space ofEinstein metrics about g has the structure of an analytic subset of a smoothmanifold Z . Furthermore, the tangent space of Z at g is naturally isomor-phic to a certain eigenspace of the Lichnerowicz Laplacian. Hence, if theeigenspace is zero-dimensional, g must be isolated in the premoduli space;isolated Einstein metrics are often referred to as being rigid . For spaceswhere the dimension of the eigenspace is positive, an interesting problemarises: do any of the tangent vectors (so-called infinitesimal deformations ofthe Einstein metric) come from genuine deformations of g through Einsteinmetrics?Koiso investigated this question for compact symmetric spaces in [14] and[15]. We will explain Koiso’s findings in detail in Section 2. A key result isthat all but one of the symmetric spaces where the dimension of the relevanteigenspace of the Lichnerowicz Laplacian is non-zero are associated to SU n ;they include the bi-invariant metric on SU n for n >
2, and the K¨ahler–Einstein metric on the Grassmannians SU n /S ( U n − k × U k ) for n > k >
1. To investigate the rigidity of these metrics we need to investigatewhether any infinitesimal deformations integrate up to genuine curves of instein metrics. Koiso developed an obstruction to integrability and usedthis to show that the canonical metric on CP n × CP is isolated in themoduli space but admits infinitesimal deformations, none of which are inte-grable. This was the first such example of this phenomenon in the literature.One difficulty in computing Koiso’s obstruction is finding a concrete descrip-tion of the space of infinitesimal deformations beyond its formal definitionas an eigenspace of the Lichnerowicz Laplacian. For the SU n -type spaces,Gasqui and Goldschmidt found such descriptions in [8] and [9]. Exploitingthese methods leads to the main result. Theorem A.
The bi-invariant Einstein metric on SU n +1 is isolated in themoduli space of Einstein metrics. This theorem provides an example of a family of non-product and non-K¨ahler Einstein metrics admitting infinitesimal deformations, none of whichare integrable. In the course of the proof of Theorem A we will show thatalmost all infinitesimal variations of SU n are obstructed and give a precisecharacterisation of those which are not. However, we suspect the unob-structed variations are not integrable and that a higher order obstructionwill demonstrate this.To prove Theorem A we follow the representation theoretic methods used byKoiso [14], [15], and Gasqui–Goldschmidt [9]. For a symmetric space G/K ,one can view Koiso’s obstruction as an element of the space Hom G ( s ( g ) , g );an infinitesimal deformation is unobstructed if it is a zero of the associatedmap. As the relevant Hom-spaces are one or two dimensional, it is possibleto argue that the obstruction is a multiple of a particular map. The theo-rem follows by showing that the map has no zeros and the obstruction is anon-zero multiple of the map.The strategy employed in the proof is quite general and should be applicableto similar rigidity questions for the symmetric spaces SU n /SO n , SU n /Sp n ,and SU p + q /S ( U p × U q ) which have been open for forty years since Koiso’swork. We give a brief discussion of this question and related issues in Section6.1.2. Rigidity of Ricci solitons.
Einstein metrics can be thought of asfixed points of the Ricci flow ∂g∂t = − g ) . Viewed thus, the Einstein metrics are a special case of a type of metric calleda Ricci soliton. One can then ask whether a given Einstein metric g can bedeformed through Ricci solitons. The structure theory of the moduli spaceof Ricci solitons was developed by Podesta and Spiro in [25] and we referthe reader to Section 5 for more details. All Hermitian symmetric spaces dmit infinitesimal solitonic deformations. We are able to prove that for the‘odd’ Grassmannians, none of these deformations can be integrated to givea non-Einstein Ricci soliton. Theorem B.
The K¨ahler–Einstein metric on the Grassmannian Gr k ( C n +1 ) is weakly solitonically rigid. Our method also yields a new proof of a theorem of Kr¨oncke. This strength-ens the previous result in the case when k = 1. It concerns the solitonicrigidity of the Fubini–Study metric on CP n . Theorem C (Kr¨oncke, [18]) . The Fubini–Study metric on CP n is isolatedin the moduli space of Ricci solitons. Stability of the Ricci flow.
The relationship between infinitesimaldeformations of Einstein metrics, their integrability, and the dynamical sta-bility of the Ricci flow has been known for quite some time through the worksof Sesum [30], Haslhofer and M¨uller [12], and Kr¨oncke [17]. In Section 6 wecompute the third variation of Perelman’s ν -functional for infinitesimal de-formations of an Einstein metric (this is related to the calculation madefor conformal variations by Kr¨oncke [20] - see also the calculation of Knopfand Sesum [13]). The next theorem shows that the variation is a multiple ofKoiso’s obstruction to integrability I (see Section 2 for the precise definitionof this obstruction). Theorem D.
Let ( M m , g ) be an Einstein metric with positive Einstein con-stant τ > and let h ∈ s ( T ∗ M ) be an EID. Then if g ( s ) := g + sh , d νds (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − τ πτ ) m I ( h ) . This yields a new proof of the following result which was first demonstratedby Cao and He in [4].
Theorem E (Cao–He [4]) . For n > , the bi–invariant metric on SU n isdynamically unstable as a fixed point of the Ricci flow. Note that this result does not depend upon the parity of n . This is because,to show instability, we do not require all infinitesimal Einstein deformationsto be obstructed, but just a single one. We can find such a deformation forall n > Existing work on rigidity and stability.
As mentioned already, thefoundational work on the rigidity of Einstein metrics on symmetric spaceswas carried out by Koiso in the papers [14], [15], and [16]. In particular,Koiso classified the compact irreducible spaces admitting infinitesimal Ein-stein deformations. Geometric constuctions of the infinitesimal deformationswere given by Gasqui and Goldschmidt in [8] and [9]. Recently, Derdzinskiand Gal demonstrated the bi-invariant Einstein metric on SU n is isolatedwithin the space of left-invariant Einstein metrics on SU n [6]; Theorem A emoves the restriction that the deformations are through left-invariant met-rics in the case when n is odd.The calculation of the second variation of Perelman’s ν -entropy and thelink between the spectrum of the Lichnerowicz Laplacian and the stabilityof the Ricci flow appeared in [3] (with a detailed proof given in [5]). Thedynamical stability of compact symmetric spaces was considered by Caoand He in [4] where the authors also addressed the stability of compact ir-reducible symmetric spaces with respect to the Einstein–Hilbert functional(where deformations are through TT tensors). The work of Semmelmannand Weingart [29] and Schwahn [26] now gives a complete understanding ofthe Einstein–Hilbert picture.There have also been other interesting developments concerning deformabil-ity of Einstein metrics and stability questions. For example, there is recentwork by Kr¨oncke on sine-cones [19]. The interaction of rigidity and stabilityquestions with other geometric stuctures (e.g. special holonomy) has beentaken up by Wang and Wang in [32] and [33] as well as the joint workswith Semmelmann [27] [28] . The relationship between deformability of G structures and Einstein metrics is investigated by Nagy and Semmelmannin [22].1.5. Conventions.
All the manifolds we consider in this paper will besmooth and closed. We follow the convention that the rough Laplacianis ∆ h = tr ( ∇ h ) and it therefore has a non-positive spectrum. The cur-vature tensor isR( X, Y ) Z = ∇ X,Y Z = ∇ [ X,Y ] Z + ∇ Y ∇ X Z − ∇ X ∇ Y Z. The curvature operator on symmetric tensors Rm : s ( T ∗ M ) → s ( T ∗ M ) isdefined by Rm( h )( X, Y ) = X k h ( R ( X, E k ) Y, E k ) , where { E i } is a local orthonormal frame. When referring to symmetricspaces we adopt the convention of only referring to the manifold with theRiemannian symmetric space metric implicitly understood. Occasionally wewill perform tensor calculations using indices and we will use the Einsteinsummation convention that repeated indices are to be summed over. Acknowledgements:
We would like to thank Paul-Andi Nagy for his interest in our work, discus-sions about the paper [25], and for bringing the reference [6] to our attention. . Deformations of Einstein metrics
General Theory.
We give the essential definitions and theory in thissection. Almost all of what is written here is expanded upon and wellexplained in the book [1].
Definition 2.1 (Premoduli space) . Let (
M, g ) be an Einstein manifoldscaled to have unit volume, let M be the space of all metrics on M with unitvolume, and let S g ⊂ M be the slice to the action of the diffeomorphismgroup D on M . The subset of Einstein metrics in S g is the premoduli spaceof Einstein structures about g . We denote this set by E ( g ). Definition 2.2 (Einstein Infinitesimal Deformation (EID)) . Let (
M, g ) bean Einstein manifold and let h ∈ s ( T ∗ M ) satisfytr( h ) = 0 , (2.1)div( h ) = 0 , (2.2)and ∆ h + 2Rm( h ) = 0 . (2.3)The tensor h is referred to as an essential Einstein infinitesimal deformation (EID). We denote the space of such h by ε ( g ).As Equation (2.3) is elliptic, we see that the space ε ( g ) is always finite dimen-sional. Tensors that satisfy Equations (2.1) and (2.2) are often referred toas transverse trace-free or TT tensors. The foundational structure theoremfor E ( g ) is due to Koiso. Theorem 2.3 (Koiso [16]) . Let ( M, g ) be an Einstein manifold. Then withinthe slice S g there exists a finite dimensional real analytic submanifold Z suchthat:(i) The tangent space of Z at g is the space of essential Einstein infini-tesimal deformations ε ( g ) .(ii) The manifold Z contains the premoduli space E ( g ) as a real analyticsubset. To try to understand what spaces arise as E ( g ), we consider whether theelements of ε ( g ) are the genuine tangents to a curve of metrics in E ( g )passing through the Einstein metric g . To make this precise, we introducethe Einstein operator E : M → s ( T ∗ M ) given by E ( g ) = Ric( g ) − (cid:18) R M S( g ) d Vol g dim( M ) (cid:19) g, (2.4)where Ric( g ) and S( g ) are respectively the Ricci curvature and scalar cur-vature of g . When dim( M ) >
2, solutions of the equation E ( g ) = 0 areEinstein metrics. f g is an Einstein metric and g ( t ) is a curve of metrics in M such that g (0) = g and ˙ g (0) = h ∈ ε ( g ), it is straightforward to check that ddt E ( g ( t )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 . Definition 2.4 (Integrability of EIDs) . Let (
M, g ) be an Einstein manifoldand let k ∈ N . We say that an element h ∈ ε ( g ) is integrable to order k ifthere exist h , h , . . . , h k ∈ s ( T ∗ M ) such that the curve g k ( t ) := g + th + j = k X j =2 t j j ! h j , satisfies d j dt j E ( g k ( t )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 , for all j = 1 , . . . , k . An element h ∈ ε ( g ) is formally integrable if there is aformal power series g ( t ) := g + th + ∞ X j =2 t j j ! h j , such that E ( g ( t )) = 0.An immediate consequence of Koiso’s structure Theorem 2.3 is that if h ∈ ε ( g )is formally integrable then there is a smooth curve of Einstein metrics in E ( g )such that ˙ g (0) = h (in other words a convergent power series solution can beconstructed). It is also clear that if h ∈ ε ( g ) is formally integrable, then itmust be integrable to order k for all k ∈ N . Hence it is natural to investigatethe obstruction to EIDs being integrable to order two. Formally we see that d dt E ( g ( t )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = E ′′ ( h, h ) + E ′ (¨ g (0)) , where E ′ and E ′′ are the first and second derivatives (suitably interpreted)of the Einstein operator E defined by (2.4). Koiso proved a useful way tocheck integrability to order two. Lemma 2.5 (Koiso, Lemma 4.7 in [15]) . Let ( M, g ) be an Einstein manifold.Then h ∈ ε ( g ) is integrable to order two if and only if E ′′ ( h, h ) ∈ ε ( g ) ⊥ .Here the orthogonal complement is with respect to the L -inner product on s ( T ∗ M ) induced by g . Using this result, we see that a necessary condition for h to be integrableis the vanishing of the quantity h E ′′ ( h, h ) , h i L . This quantity was alsocomputed by Koiso. Proposition 2.6 (Koiso [15]) . Let ( M, g ) be an Einstein metric with Ein-stein constant λ > and let h ∈ ε ( g ) . Then an obstruction to the integra-bility of h to order two is given by the nonvanishing of the quantity I ( h ) := 2 λ h h ki h kj , h ij i + 3 h∇ i ∇ j h kl , h ij h kl i − h∇ i ∇ j h kl , h ik h jl i , (2.5) here each of the brackets denotes the L -inner product induced by the metric g on the appropriate bundle. Deformations of compact symmetric spaces.
For compact irre-ducible symmetric spaces, the space of Einstein infinitesimal deformations ε ( g ) for the canonical Einstein metrics can be computed using representationtheoretic methods. Theorem 2.7 (Koiso, Theorem 1.1 in [14] - see also [8] Proposition 2.40) . Let ( M, g ) be a compact irreducible symmetric space. Then the space of EIDs ε ( g ) is { } , except in the following cases:(i) M = SU n with n ≥ , here ε ( g ) ∼ = su ( n ) ⊕ su n ,(ii) M = SU n /SO n with n ≥ , here ε ( g ) ∼ = su n ,(iii) M = SU n /Sp n with n ≥ here ε ( g ) ∼ = su n ,(iv) M = SU p + q /S ( U p × U q ) with p ≥ q ≥ , here ε ( g ) ∼ = su p + q ,(v) M = E /F , here ε ( g ) ∼ = e . For the spaces with non-zero ε ( g ), if M = G/K with G the isometry group of g , then ε ( g ) ∼ = g . Of course, this isomorphism is really just a statement aboutthe dimension of ε ( g ) and part of the difficulty in determining whether theinfinitesimal deformations are integrable is describing them geometrically.Fortunately, for spaces (i)-(iii) in Theorem 2.7, there is a straightforwardway describing the tensors in ε ( g ) due to Gasqui and Goldschmidt. Thisconstruction is detailed in the memoir [9]; we give explicit differential geo-metric proofs to show the construction really does produce EIDs.Let M be a symmetric space of compact type; M is thus the coset space ofa Riemannian symmetric pair ( G, K ) where G is a compact, semi-simple Liegroup and K is a closed subgroup of G . We denote the Lie algebras of G and K by g and k respectively and note that the inner product on g inducedby the Killing form yields the Ad K -invariant decomposition g = k ⊕ p . The subspace p can be identified with the tangent space of M at the iden-tity coset eK ∈ G/K . For an integer p ≥ s p ( p ∗ ) K theAd K -invariant symmetric p -tensors (or equivalently the Ad K -invariant sym-metric degree p polynomials over the space p ). An element q ∈ s p ( p ∗ ) K gives rise to a G -invariant symmetric p -tensor field σ ( q ) that coincides with q at the identity coset. Conversely, every such G -invariant symmetric tensorfield arises this way.We now assume we have a given q ∈ s ( p ∗ ) K and a Killing vector field η ∈ iso ( M ), and we define a symmetric 2-tensor h η by h η := ι η σ ( q ) . (2.6) e will henceforth forget the dependence on q and write σ for σ ( q ). Thefollowing important result is due to Gasqui and Goldschmidt; we give a newand direct proof. Lemma 2.8 (Gasqui–Goldschmidt) . Let M = G/K be a compact irreduciblesymmetric space with its canonical Einstein metric g . Then the tensor field h η ∈ ε ( g ) .Proof. To show that h η satisfies Equation (2.1) we consider the linear mapΦ : p → R defined by Φ( v ) := tr( ι v σ ) | ( e · K ) . As Φ is Ad K -invariant, a non-trivial kernel would be an Ad K -invariant sub-space of p . Hence Φ must vanish identically as the symmetric space M is ir-reducible. For a ∈ G , the trace of h η at a coset ( a · K ) is Φ( a − · η ( a · K )) = 0(where we note that the non-vanishing component of a − · η ( a · K ) is in p ).To demonstrate Equation (2.2) we recall the result that any G -invarianttensor field on a symmetric space is parallel. Thus( ∇ · h η )( · , · ) = σ ( ∇ · η, · , · ) . Taking the trace we see thatdiv( h η )( Y ) = X j σ ( ∇ E j η, E j , Y ) , for an orthonormal frame { E j } . However, after using the metric to producesections of End( T M ), this is the trace of the composition of the symmetric2-tensor ι Y σ and the skew-symmetric tensor ∇ · η (the skew-symmetry fol-lows from the fact that η is a Killing field). Hence the trace vanishes and h η is divergence-free.To demonstrate Equation (2.3) we use the fact that, as the metric is ho-mogeneous, we only need to check the equation∆( h η )( X, Y ) + 2Rm( h η )( X, Y ) = 0 , for Killing fields X, Y . We also consider a local orthonormal frame { E i } with ∇ E i vanishing at the point at which we want to compute the identity.To start with, we note∆ h η ( X, Y ) = ( ∇ E i ,E i h η )( X, Y ) = σ ( ∇ E i ,E i η, X, Y ) , and Rm( h η )( X, Y ) = σ ( η, E i , R ( X, E i ) Y ) = σ ( η, E i , R ( Y, E i ) X ) . In general, given a Killing field ξ , we have the identity σ ( ∇ W ξ, Y, Z ) + σ ( W, ∇ Y ξ, Z ) + σ ( W, Y, ∇ Z ξ ) = 0 , (2.7)In particular σ ( ∇ X η, E i , Y ) + σ ( ∇ E i η, X, Y ) + σ ( ∇ Y η, E i , X ) = 0 . aking the derivative with respect to E i and using the fact that we havedemonstrated tensors of the form ι ξ σ are divergence free we obtain, σ ( ∇ E i ∇ X η, E i , Y ) + σ ( ∇ E i ∇ E i η, X, Y ) + σ ( ∇ E i η, ∇ E i X, Y )+ σ ( ∇ E i η, X, ∇ E i Y ) + σ ( ∇ E i ∇ Y η, E i , X ) = 0 . Applying Equation (2.7) on the terms σ ( ∇ E i η, ∇ E i X, Y ) and σ ( ∇ E i η, X, ∇ E i Y )yields σ ( ∇ E i η, ∇ E i X, Y ) = − σ ( ∇ ∇ Ei X η, E i , Y ) . Hence σ ( ∇ E i ,X η, E i , Y ) + σ ( ∇ E i ,E i η, X, Y ) + σ ( ∇ E i ,Y η, E i , X ) = 0 . (2.8)We now expand σ ( η, E i , ∇ X,E i Y ) = X · σ ( η, E i , ∇ E i Y ) − σ ( ∇ X η, E i , ∇ E i Y ) = 0which implies that Equation (2.8) becomes σ ( R ( X, E i ) η, E i , Y ) + σ ( ∇ E i ,E i η, X, Y ) + σ ( R ( Y, E i ) η, E i , X ) = 0 . (2.9)We note that we obtain a similar equation simply by swapping η and Xσ ( R ( η, E i ) X, E i , Y ) + σ ( ∇ E i ,E i X, η, Y ) + σ ( R ( Y, E i ) X, E i , η ) = 0 . (2.10)The next ingredient is an identity (see Lemma 33 in [24] but note the differingsign convention) true for any Killing field ξ on an arbitrary Riemannianmanifold: ∇ X,Y ξ = R ( ξ, X ) Y. Hence σ ( ∇ E i ,E i X, η, Y ) = σ ( ∇ E i ,E i η, X, Y ) = − λσ ( η, X, Y ) . Thus subtracting Equation (2.9) from Equation (2.10) along with an appli-cation of the first Bianchi identity yields σ ( R ( X, η ) E i , E i , Y ) = σ ( R ( Y, E i ) η, E i , X ) − σ ( R ( Y, E i ) X, E i , η ) . The quantity σ ( R ( X, η ) E i , E i , Y ) vanishes as it is the trace of an antisym-metric map (as in the proof above that h η satisifes Equation (2.2)). Hencewe can reinterpret Equation (2.9) as σ ( R ( Y, E i ) X, E i , η ) + σ ( ∇ E i ,E i η, X, Y ) + σ ( R ( X, E i ) Y, E i , η ) = 0 . This can be rewritten as(∆ h η )( X, Y ) + 2Rm( h η )( X, Y ) = 0 , as required. (cid:3) Irreducible symmetric spaces with G -invariant symmetric 3-tensors form avery short list. Proposition 2.9 (Gasqui–Goldschmidt, [9] Proposition 2.1) . Let M be anirreducible simply-connected symmetric space of compact type. The space s ( p ) K vanishes unless M is one of the following spaces: SU n , with n ≥ ;ii SU n /SO n , with n ≥ ;iii SU n /Sp n , where n ≥ ;iv E /F .For the cases (i)-(iv), then the space s ( p ) K is one-dimensional. Note the slight difference in the spaces between the previous propositionand Theorem 2.7. This is because the infinitesimal deformations of theGrassmannians are not given by ι v σ , since the space s ( p ) K vanishes for theGrassmannians. We shall discuss this in Section 6.3. Input from Representation Theory
Let M = G/K be a symmetric space of type (i)-(v) from Theorem 2.7. Ifwe denote by ψ ( h, h ) the projection of the quantity E ′′ ( h, h ) to the spaceof EIDs ε ( g ) then ψ can be seen to be a symmetric bilinear G -equivariantmap. In other words, ψ ∈ Hom G ( s ( g ) , g ). Lemma 2.5 can be restated inthe following manner:an EID h ∈ ε ( g ) is integrable to order 2 if and only if ψ ( h, h ) = 0 . The strategy we employ to show that infinitesimal deformations are notintegrable can by summarised as follows:(1) Demonstrate that ψ ∈ V ⊂ Hom G ( s ( g ) , g ) where V is a one dimen-sional subspace of Hom G ( s ( g ) , g ); hence ψ = C Ψ for some C ∈ R and Ψ a generator of V .(2) Compute the exact form of the map Ψ and show that if Ψ( x, x ) = 0then x = 0.(3) Find some h ∗ ∈ ε ( g ) so that the obstruction in Proposition 2.5 I ( h ∗ ) = h ψ ( h ∗ , h ∗ ) , h ∗ i L = 0 . (4) Conclude that C = 0 and so ψ ( h, h ) = 0 for any h ∈ ε ( g ) with h = 0.In this paper we will consider this strategy for G = SU n × SU n and G = SU n .In this section we carry out steps (1) and (2) for these groups. We expectthis material is standard and considerations such as these are implicit inmuch of the works of Koiso, and Gasqui and Goldschmidt; we include theproofs for completeness. Lemma 3.1.
Let G = SU n × SU n with n ≥ , identify g := Lie( G ) with su n ⊕ su n , and consider g as a representation of G via the adjoint repre-sentation. Then the dimension of real vector space Hom G (cid:0) s ( g ) , g (cid:1) is givenby: dim Hom G (cid:0) s ( g ) , g (cid:1) = ( n = 22 n ≥ . roof. Let G C = SL n ( C ) × SL n ( C ) and identify g C := Lie G C with sl n ⊕ sl n .Consider g C as a representation of G C and g C via the adjoint representation.It follows from the fact that G is maximal compact subgroup in G C thatdim R Hom G (cid:0) s ( g ) , g (cid:1) = dim C Hom G C (cid:0) s ( g C ) , g C (cid:1) . If T is a maximal torus in G C , W = N G C /T is the associated Weyl group,and t = Lie T then by Theorem 1 in [2] there is an isomorphism of gradedvector spaces (in fact of graded C [ g ] G C ∼ = C [ t ] W -modules)Hom G C (S g C , g C ) ∼ = Hom W (S t , t )and in particular an isomorphism of complex vector spacesHom G C (cid:0) s ( g C ) , g C (cid:1) ∼ = Hom W (cid:0) s ( t ) , t (cid:1) . This discussion shows that it suffices to prove thatdim C Hom W (cid:0) s ( t ) , t (cid:1) = ( n = 22 n ≥ . (3.1)Fix T = T ′ × T ′ where T ′ is the standard maximal torus in SL n ( C ). Then W is naturally identified with S n × S n . Denote by C the trivial representation of S n , by U the one dimensional alternating representation, by V the standard n − n ≥
3, by V ( d − , theSpecht module corresponding to the partition ( d − , M and M representations of S n we use M ⊠ M to denote the representation M ⊗ M of S n × S n in which the first (respectively the second) copy of S n acts onthe first (respectively the second) tensor factor. See for example Chapter 4of [7] for further details about representations of symmetric groups.The S n × S n representation t decomposes into irreducible S n × S n represen-tations as t ∼ = ( V ⊠ C ) ⊕ ( C ⊠ V ) . (3.2)There is the decomposition of the representation s ( t ) as s ( t ) ∼ = s (( V ⊠ C ) ⊕ ( C ⊠ V )) ∼ = s ( V ⊠ C ) ⊕ s ( C ⊠ V ) ⊕ ( V ⊠ C ) ⊗ ( C ⊠ V ) ∼ = (cid:0) s ( V ) ⊠ C (cid:1) ⊕ (cid:0) C ⊠ s ( V ) (cid:1) ⊕ ( V ⊠ V ) . (3.3)If d = 2 then V = U , s ( V ) = V ⊗ = C , and (3.3) decomposes intoirreducible S n × S n representations as( C ⊠ C ) ⊕ ( C ⊠ C ) ⊕ ( V ⊠ V ) . (3.4)If d ≥ S n represen-tations s ( V ) ∼ = U ⊕ V ⊕ V ( d − , , and so (3.3) decomposes into irreducible S n × S n representations as( U ⊠ C ) ⊕ ( V ⊠ C ) ⊕ (cid:0) V ( d − , ⊠ C (cid:1) ⊕ ( C ⊠ U ) ⊕ ( C ⊠ V ) ⊕ (cid:0) C ⊠ V ( d − , (cid:1) ⊕ ( V ⊠ V ) . (3.5) quation (3.1) then follows from Schur’s Lemma and comparing (3.2) with(3.4) and (3.2) with (3.5). (cid:3) The map ψ can be given explicitly. Lemma 3.2.
Let G and g be as in Lemma 3.1. For n ≥ the vector space Hom G (cid:0) s ( g ) , g (cid:1) has a basis consisting of the following two functions: ψ (cid:0) ( X, Y ) , (cid:0) X ′ , Y ′ (cid:1)(cid:1) := √− (cid:18)(cid:0) XX ′ + X ′ X (cid:1) − n tr (cid:0) XX ′ + X ′ X (cid:1) Id , (cid:19) ψ (cid:0) ( X, Y ) , (cid:0) X ′ , Y ′ (cid:1)(cid:1) := √− (cid:18) , (cid:0) Y Y ′ + Y ′ Y (cid:1) − n tr (cid:0) Y Y ′ + Y ′ Y (cid:1) Id (cid:19) Furthermore, the subspace of
Hom G (cid:0) s ( g ) , g (cid:1) consisting of maps invariantunder the involution on su n ⊕ su n defined by ( x, y ) → ( y, x ) is one dimen-sional and is spanned by the map Ψ := ψ + ψ .Proof. It is straightforward to check that ψ and ψ are linearly independentsymmetric bilinear maps from g × g to g . The result then follows from Lemma3.1. The second assertion is easily checked directly. (cid:3) Lemma 3.3.
Let Ψ be as in Lemma 3.2. Then, for n odd, the zero set of Ψ is { } ; for n even, the zero set of Ψ is the G -orbit of the plane spanned bythe elements (Λ , and (0 , Λ) of g = su n ⊕ su n , where Λ is the block diagonalsum of n copies of the matrix (cid:18) √− −√− (cid:19) . Proof.
Suppose that Ψ ( x ) = 0. Write x = ( X, Y ) where
X, Y ∈ su n . ThenΨ ( x ) = 2 √− (cid:18) X − n tr( X )Id , Y − n tr( Y )Id (cid:19) . As Ψ is G -equivariant we can assume that X and Y are diagonal. The kernelof the linear map T : Mat n ( C ) → Mat n ( C ) given by T ( X ) = X − n tr( X )Id , is the set of multiples of the identity matrix Id. It follows that Ψ ( x ) = 0 ifand only if each of X and Y satisfy the condition that the squares of theireigenvalues are all equal. For an element Z of su n (the elements of whichare traceless with imaginary eigenvalues) this condition is only satisfied ifthe eigenvalues are either all zero, or if n is even and the eigenvalues are ofthe form ai and − ai for some a ∈ R , with each occurring with multiplicity n , in which case Z is SU n -conjugate to a multiple of Λ. (cid:3) Finally in this section, we note that a very similar proof to that in Lemma3.1 yields the following result. emma 3.4. Let G = SU n with n ≥ , identify g := Lie( G ) with su n , andconsider g as a representation of G via the adjoint representation. Then thedimension of real vector space Hom G (cid:0) s ( g ) , g (cid:1) is given by: dim Hom G (cid:0) s ( g ) , g (cid:1) = ( n = 21 n ≥ . Calculations for SU n In this section we give the proof of the Theorem A using the strategy outlinedin the previous section; in this section we focus on step (3) — demonstratingthat the obstruction map does not vanish identically. It is important tofix conventions at this stage to ensure the correct calculation of variousconstants. The Lie algebra su n is the space of skew-Hermitian n × n matrices.We endow su n with the Euclidean inner product h A, B i := − tr( AB ) . This inner product is (1 / n ) times the one induced by the Killing form andso the resulting bi-invariant Einstein metric has Einstein constant λ = n/ / σ ∈ s ( su ( n )) SU n . Upto scale, there is only one such tensor given by σ ( X, Y, Z ) := √− XY Z ) + tr(
XZY )) , (4.1)where X, Y, Z ∈ su n . The following result allows us to reduce computing Koiso’s obstruction (2.5)to a pointwise calculation (rather than having to integrate over the wholemanifold).
Lemma 4.1.
Let η ∈ su n and denote by ˜ η the corresponding left-invariant(resp. right-invariant) extension to a vector field on SU n . Then the tensor h η is left-invariant (resp. right invariant).Proof. We shall demonstrate the left-invariant statement. Let g, k ∈ SU n and let X, Y ∈ X ( SU n ). We compute L ∗ g ( h η )( X, Y )( k ) = h η ( L g ∗ X, L g ∗ Y )( gk ) = σ (˜ η ( gk ) , L g ∗ X, L g ∗ Y )( gk ) , and thus as ˜ η is left-invariant we obtain L ∗ g ( h η )( X, Y )( k ) = L ∗ g ( σ )(˜ η, X, Y )( k ) = σ (˜ η, X, Y )( k ) = h η ( X, Y )( k ) . (cid:3) .1. Results in su n . If we use the previous lemma to produce a left-invariant tensor h η and choose the identity as the point in SU n at whichto perform calculations, the algebra we need to do takes place in su ( n ). Inthis subsection we collect the relevant results about the Euclidean vectorspace ( su n , h· , ·i ). All of the following lemmata are proved directly usingstraightforward (if tedious) matrix algebra; most of the proofs are omitted.In order to perform calculations, we fix an orthonormal basis of su n . Lemma 4.2.
For ≤ k ≤ n − , define matrices T k = √− p k ( k + 1) Diag , , . . . , | {z } k terms , − k, , . . . , , and, for ≤ k < l ≤ n , define matrices E r ( k, l ) and E c ( k, l ) by ( E r ( k, l )) pq = 1 √ δ kp δ lq − δ kq δ lp ) , and ( E c ( k, l )) pq = √− √ δ kp δ lq + δ kq δ lp ) . Then the matrices B = { T , T , . . . , T n − } ∪ { E r ( k, l ) , E c ( k, l ) } ≤ k X, Y ∈ su ( n ).To do this, we introduce a small variation on the notation for the basisgiven in Lemma 4.2. Given an unordered pair { k, l } , if a = min { k, l } and b = max { k, l } , then we define E r ( { k, l } ) = E r ( a, b ) and E c ( { k, l } ) = E c ( a, b ) . Lemma 4.3. Let T k , E r ( k, l ) , and E c ( k, l ) be the orthonormal basis of ( su n , h· , ·i ) defined in Lemma 4.2. Then(i) If p = q , then T p T q + T q T p = 2 p ( p + 1) Diag − , − , . . . , − | {z } p terms , − p , , . . . , (ii) If p > q , then T p T q + T q T p = 2 p pq ( p + 1)( q + 1) Diag − , − , . . . , − | {z } q terms , q, , . . . , (iii) T p E r ( k, l ) + E r ( k, l ) T p = (( T p ) kk + ( T p ) ll ) E r ( k, l ) iv) T p E c ( k, l ) + E c ( k, l ) T p = (( T p ) kk + ( T p ) ll ) E c ( k, l ) (v) If { k, l } 6 = { p, q } , then E r ( k, l ) E r ( p, q ) + E r ( p, q ) E r ( k, l ) = √− √ δ ql E c ( { p, k } ) − δ qk E c ( p, l ) − δ lp E c ( k, q ) + δ pk E c ( { l, q } )) . (vi) If k = p and l = q , then E r ( k, l ) E r ( p, q ) + E r ( p, q ) E r ( k, l ) =Diag , , . . . , | {z } k − , − , , , . . . , | {z } l − k − , − , , , . . . , (vii) If { k, l } 6 = { p, q } , E r ( k, l ) E c ( p, q ) + E c ( p, q ) E r ( k, l ) = √− √ ± δ ql E r ( { p, k } ) + δ pl E r ( k, q ) + δ kq E r ( p, l ) ± δ pk E r ( { q, l } )) . (viii) If k = p and l = q , then E r ( k, l ) E c ( p, q ) + E c ( p, q ) E r ( k, l ) = 0 .(ix) If { k, l } 6 = { p, q } , , then E c ( k, l ) E c ( p, q ) + E c ( p, q ) E c ( k, l ) = √− √ δ ql E c ( { p, k } ) + δ pl E c ( k, q ) + δ kq E c ( p, l ) + δ pk E c ( { q, l } )) (x) If k = p and l = q , then E c ( k, l ) E c ( p, q ) + E c ( p, q ) E c ( k, l ) =Diag , , . . . , | {z } k − , − , , , . . . , | {z } l − k − , − , , , . . . We will see that the precise signs of the terms in (vii) will not matter; thereal utility of Lemma 4.3 is that, if we choose basis elements X, Y, ∈ B with X = Y , then √− XY + Y X ) ⊥ T n − . Thus choosing η to be a convenient multiple of T n − turns out to greatlysimplify calculations. Lemma 4.4. Let η = √− , , . . . , | {z } n − , − ( n − and let H ∈ s ( g ∗ ) bedefined by H ( X, Y ) = √− η ( XY + Y X )) . Then H is diagonal with respect to the orthonormal basis B defined in Lemma4.2. Furthermore H ( T p , T p ) = 2 for p < n − , (4.2) H ( T n − , T n − ) = − n − , (4.3) H ( E r ( k, l ) , E r ( k, l )) = H ( E c ( k, l ) , E c ( k, l )) = 2 for l < n, (4.4) H ( E r ( k, n ) , E r ( k, n )) = H ( E c ( k, n ) , E c ( k, n )) = − ( n − . (4.5) roof. We begin by noting that η = p n ( n − T n − and H ( X, Y ) = h η, √− XY + Y X ) i . Using Lemma 4.3 we see that all the non-diagonal terms of H vanish. Theremaining diagonal terms are easily calculated. (cid:3) In order to compute the terms of Koiso’s obstruction (2.5) that containcovariant derivatives we will need information about Lie brackets [ · , η ]. Thefollowing is again straightforward algebra. Lemma 4.5. Let η be as in Lemma 4.4. Then for any X ∈ Mat( C ) n × n wehave [ X, η ] = n √− . . . − X n . . . − X n ... ... . . . ... ... . . . − X ( n − n X n X n . . . X n ( n − . The fact that H is diagonal will mean we need only compute very few Lie-bracket-type terms. The following collects what we will need. Lemma 4.6. Let η be as in Lemma 4.4. Then(i) for < k < l < n [ E r ( k, l ) , [ E r ( k, l ) , η ]] = [ E c ( k, l ) , [ E c ( k, l ) , η ]] = 0 , (ii) [ E r ( k, n ) , [ E r ( k, n ) , η ]] = n √− , , . . . , | {z } k − , − , , , . . . , , , (iii) [ E c ( k, n ) , [ E c ( k, n ) , η ]] = n √− , , . . . , | {z } k − , − , , , . . . , , . Proof. Part (i) follows immediately from Lemma 4.5 as, for any 0 ≤ j ≤ n ,( E r ( k, l )) jn = ( E c ( k, l )) jn = 0 , if l < n . Parts (ii) and (iii) are also straightforward after using Lemma 4.5to compute the first Lie bracket. (cid:3) Computing I . In this section we consider the symmetric 2-tensor h η that is generated by the inclusion of the left-invariant extension of η in σ defined by Equation (4.1). We proceed by computing each of the terms ofthe obstruction I given in Equation (2.5). Lemma 4.7. Let η be as in Lemma 4.4 and let h η = ι ˜ η σ , where ˜ η is the left-invariant extension of η and σ is the symmetric 3-tensor defined in Equation(4.1). Then h ( h η ) ki ( h η ) kj , ( h η ) ij i L = 2( n − n − − n )Vol( SU n ) , here Vol( SU n ) > is the volume of the bi–invariant metric on SU n inducedthe inner product h· , ·i on su ( n ) .Proof. The tensor h η is left-invariant by Lemma 4.1. Hence we can computeat the identity and so h ( h η ) ki ( h η ) kj , ( h η ) ij i L = h H ki H kj , H ij i Vol( SU n ) , where H is the symmetric bilinear form defined in Lemma 4.4 (we also use h· , ·i to denote the inner product on s ( su ( n )) induced by the one on su ( n )).As H is diagonal with respect to the orthonormal basis B defined in Lemma4.2 we can compute h H ki H kj , H ij i = tr( H ) , where the trace on the right-hand side is the usual trace of a matrix. Usingequations (4.2)-(4.5) we can computetr( H ) = (( n − × − (8( n − ) + (( n − n − × − (( n − × ( n − ) . The result follows after simplification. (cid:3) Lemma 4.8. Let h η be as in Lemma 4.7. Then h∇ i ∇ j ( h η ) kl , ( h η ) ij ( h η ) kl i L = [( n − ( n − n ( n + 2)]Vol( SU n ) . Proof. As σ is parallel we have (computing at the identity) ∇ i ∇ j ( h η ) kl = σ ( ∇ i ∇ j ˜ η, E k , E l ) , where { E i } is any basis of g . We are using bi-invariant metric on SU n so,using left-invariant extensions of the basis { E i } , we have ∇ i ∇ j ˜ η = 14 [ E i , [ E j , η ]] . Thus h∇ i ∇ j ( h η ) kl , ( h η ) ij ( h η ) kl i L = 14 σ ([ E i , [ E j , η ]] , E k , E l ) H ij H kl Vol( SU n ) . If we take { E i } = B , and as H is diagonal with respect to the basis B , weneed only consider E i = E j and E k = E l and so h∇ i ∇ j ( h η ) kl , ( h η ) ij ( h η ) kl i L = 14 σ ([ E i , [ E i , η ]] , E k , E k ) H ii H kk . From Lemma 4.6 we see the right-hand side of the previous equation is non-zero only when E i = E r ( k, n ) or E i = E c ( k, n ); in either case, H ii = − n − h [ E i , [ E j , η ]] , H ij H kl i = − (cid:18) n ( n − (cid:19) n − X k =1 h σ ( v k , · , · ) , H ( · , · ) i , where v k = √− , , . . . , | {z } k − , − , , , . . . , , . e split the sum up as follows: h σ ( v k , · , · ) , H ( · , · ) i = n − X i =1 σ ( v k , T i , T i ) H ( T i , T i )+ 2 X ≤ i Lemma 4.9. Let h η be as in Lemma 4.7. Then h∇ i ∇ j h kl , h ik h jl i L = − 12 [( n − ( n − n ( n + 2)]Vol( SU n ) . Proof. As in the proof of Lemma 4.8, the fact that σ is parallel means that,for any local frame { E i } , ∇ i ∇ j h kl = σ ( ∇ i ∇ j η, E k , E l ) . omputing at the identity in the orthonormal basis B , Lemma 4.4 yields h η = H is diagonal and so, to compute h∇ i ∇ j h kl , h ik h jl i , we need onlyconsider terms of the form σ ( ∇ i ∇ j η, E i , E j ) = 14 σ ([ E i , [ E j , η ]] , E i , E j ) . Noting that σ is Ad-invariant and we can write σ ( ∇ i ∇ j η, E i , E j ) = − σ ([ E j , η ] , E i , [ E i , E j ]) . Using the definition of σ given by Equation(4.1) we have − σ ([ E j , η ] , E i , [ E i , E j ]) = − √− 14 tr([ E j , η ]( E i [ E i , E j ] + [ E i , E j ] E i ))= − √− 14 tr([ E j , η ]( E i E j − E j E i ))= − √− h [ E j , η ] , [ E j , E i ] i = √− h [ E j , [ E j , η ]] , E i i , where the final equality follows as the inner product h· , ·i is also Ad-invariant.Thus we have (relabelling indices as needed) h∇ i ∇ j h kl , h ik h jl i L = − √− 14 tr([ E i , [ E i , η ]] E k ) H ii H kk Vol( SU n )= − σ ([ E i , [ E i , η ]] , E k , E k ) H ii H kk Vol( SU n ) . = − h∇ i ∇ j ( h η ) kl , ( h η ) ij ( h η ) kl i L , = − 12 [( n − ( n − n ( n + 2)]Vol( SU n ) , where Lemma 4.8 was applied to deduce the final equality. (cid:3) We now have everything needed to prove the main theorem in this paper. Proof of Theorem A. As 2 n + 1 is odd, the obstruction to integrability tosecond order ψ has no non-trivial zeroes by Lemma 3.3. Thus, to show thatnone of the elements of ε ( g ) are integrable to second order we simply needto find h ∗ ∈ ε ( g ) such that I ( h ∗ ) = 0. We take h ∗ = h η where η is as inLemma 4.4, and then, using Lemmas 4.7, 4.8, and 4.9, and the fact that theEinstein constant λ = n ( h ∗ ) =2 λ h h ki h kj , h ij i + 3 h∇ i ∇ j h kl , h ij h kl i − h∇ i ∇ j h kl , h ik h jl i = (cid:16) · n (cid:2) n − n − − n ) (cid:3) +3 (cid:2) ( n − ( n − n ( n + 2) (cid:3) +6 12 [( n − ( n − n ( n + 2)] (cid:19) Vol( SU n ) . Simplifying, this yields I ( h ∗ ) = 4 n ( n − n − SU n ) = 0 . Hence all the infinitesimal variations are non-integrable and so the Einsteinmetric is isolated in its moduli space. (cid:3) Applications to Solitonic Rigidity Ricci Solitons. Einstein metrics are fixed points (up to diffeomor-phism and homothethic scaling) of the Ricci flow ∂g∂t = − g ) . (5.1)There are also non-Einstein fixed points of Equation 5.1 known as Riccisolitons which are metrics that solveRic( g ) + 12 L X g = λg, (5.2)for a vector field X ∈ Γ( T M ) and constant λ ∈ R . A soliton is known as expanding, steady, or shrinking depending upon whether λ < , λ = 0 , or λ > 0. If one can find a function f such that X = ∇ f then Equation (5.2)becomes Ric( g ) + Hess( f ) = λg, (5.3)and the metric g is known as a gradient Ricci soliton with potential func-tion f . Perelman [23] demonstrated that, on compact manifolds, any non-Einstein Ricci soliton is a shrinking gradient Ricci soliton. Therefore weonly consider shrinking gradient Ricci solitons on compact manifolds in thesequel.5.2. Ricci soliton moduli. The study of moduli spaces for Ricci solitonswas initiated by Podest`a and Spiro in [25]. It is clear from Equation (5.3)that, given a gradient Ricci soliton g with potential function f , adding aconstant to f yields admissible potential function. Hence, in order to studymoduli of solitons, the potential functions should be normalised.If M denotes the space of all Riemannian metrics on a manifold M m ,Podest`a and Spiro introduce the set P ⊂ M × C ∞ ( M ) defined by P = (cid:26) ( g, f ) (cid:12)(cid:12)(cid:12)(cid:12) Z M e − f d Vol g = (2 π ) m/ (cid:27) . hey define the set S ol ⊂ P to be the pairs solving Equation (5.3) afterfixing λ = 1. Given a Ricci soliton ( g, f ) ∈ S ol they define an ‘ f -twisted’slice inside M for the action of the diffeomorphism group and the premodulispace about ( g, f ) is the intersection of the slice (or rather the product ofthe slice and C ∞ ( M )) and the set S ol . We will not directly need the theorydeveloped and so we refer the reader to the paper [25] for further details.There is also a solitonic notion of infinitesimal variation generalising that ofEIDs given in Definition 2.2. Podest`a and Spiro use a different definition ofthe space of essential infinitesimal soliton deformations and then show thefollowing is equivalent (see Theorem 3.3 in [25]). Definition 5.1 (Essential infinitesimal solitonic variation) . Let ( g, f ) ∈ S ol be a shrinking Ricci soliton and let h ∈ s ( T ∗ M ) satisfytr( h ) = α, (5.4)div f ( h ) := div( h ) − ι ∇ f h = 0 , (5.5)and ∆ f h + 2Rm( h ) = 0 , (5.6)where α ∈ C ∞ ( M ) and ∆ f ( · ) := ∆( · ) − ∇ ∇ f ( · ). Then the pair ( h, a ) isreferred to as an (essential) infinitesimal solitonic deformation (ISD). Wedenote the space of such ( h, α ) by σ ( g, f ).In [25], Podest`a and Spiro prove the solitonic analogue of Koiso’s Theorem2.3: given ( g, f ) ∈ S ol , there is a real analytic submanifold Z of the productof the f -twisted slice and C ∞ ( M ) such that the tangent space to Z at ( g, f )is σ ( g, f ) and the premoduli space for ( g, f ) is a real-analytic subset of Z .If the Ricci soliton is an Einstein metric then the potential function f isconstant and f = log (Vol( M, g )) − m (cid:18) πλ (cid:19) . In this case the space of infinitesimal solitonic deformations decomposes asthe direct sum of the infinitesimal Einstein deformations and an eigenspacefor a particular eigenvalue of the Laplacian on functions. Lemma 5.2 (Podesta–Spiro, Proposition 4.4 in [25] - see also Kr¨onckeLemma 6.2 in [17]) . Let ( M, g ) be an Einstein manifold with Einstein con-stant λ . Let EID be the space of infinitesimal Einstein deformations andISD the space of infinitesimal solitonic deformations. Then ISD has thefollowing decomposition: ISD = EID ⊕ (cid:26) λv · g + Hess( v ) (cid:12)(cid:12)(cid:12)(cid:12) v ∈ C ∞ ( M ) , ∆ v = − λv (cid:27) . We will denote the − λ -eigenspace of the Laplacian by V − λ . As with in-finitesimal Einstein deformations, a natural question is whether an elementof ISD is the tangent to a genuine family of Ricci solitons. Kr¨oncke provedthe following analogy to Lemma 2.5. heorem 5.3 (Kr¨oncke, Theorem 5.7 in [18]) . Let ( M, g ) be an Einsteinmetric with Einstein constant λ > and let v ∈ V − λ .Then h = λv · g + ∇ v ∈ ISD is not integrable to second order if there existsa function w ∈ V − λ such that Z M v w d Vol g = 0 . An equivalent formulation of this theorem is to say that, if v ∈ V − λ , thenthe corresponding ISD h is integrable to second order if and only if the L -projection of v to V − λ vanishes.Let M = G/K be a compact irreducible Hermitian symmetric space; aclassical result due to Matsushima [21] gives a G -equivariant isomorphismΦ : g → V − λ . The methods of Section 4 thus give a proof of the followingtheorem of Kr¨oncke. Proof of Theorem C. The space of infinitesimal Einstein deformations ε ( g )vanishes [15] and so ISD ∼ = su n +1 .The obstruction given by Theorem 5.3 is an element of Hom SU n +1 ( s ( g ) , g ),where g = su n +1 which is one dimensional by Lemma 3.4. The obstructioncan be understood to be a multiple of the mapΨ( X, Y ) = √− (cid:18) XY + Y X − tr( XY + Y X )2 n + 1 Id (cid:19) , and as 2 n + 1 is odd, this map does not have any non-trivial zeroes. It ispossible to find an eigenfunction v ∈ V − λ such that Z CP n v d Vol g = 0 , (see [11], [13], or [20]). Hence the obstruction is a non-zero multiple of Ψand so does not vanish for any perturbation induced by v ∈ V − λ . (cid:3) Podest`a and Spiro also considered the following notion of rigidity for anEinstein metric. Definition 5.4 (Weak solitonic rigidity - Definition 5.1 in [25]) . A compactEinstein metric g is said to be weakly solitonic rigid if there is a neighbour-hood U ⊂ M of g such that every Ricci soliton in U is Einstein.We can now prove Theorem B which is a generalisation of Theorem C inthis weaker setting. Proof of Theorem B. The structure theorem of Podest`a–Spiro for the soli-tonic premoduli space guarantees that in a small enough neighbourhood of g , all solitons come from integrable deformations of g .For the symmetric space metric g on Gr k ( C n +1 ) the space of soliton de-formations splits as in Lemma 5.2. As in the proof of Theorem C, the igenspace V − λ is SU n +1 -equivariantly isomorphic to su n +1 . In [11], theauthors show that there exists an eigenfunction f ∈ V − λ with Z Gr k ( C n +1 ) f d Vol g = 0 . Hence Kr¨oncke’s obstruction in Theorem 5.3 does not vanish and is givenby a multiple of the map Ψ from the proof of Theorem B which has nonon-trivial zeroes on su n +1 . Hence the only possibly integrable definitionsare EIDs. (cid:3) Applications to stability of the Ricci flow and to otherrigidity problems Applications to dynamical stability. For a general compact Rie-mannian manifold ( M m , g ), Perelman’s ν -entropy [23] is given by ν ( g ) = inf (cid:26) W ( g, f, τ ) : f ∈ C ∞ ( M ) , τ > , (4 πτ ) − m Z M e − f d Vol g = 1 (cid:27) , where W ( g, f, τ ) = Z M [ τ (S( g ) + |∇ f | ) + f − m ] e − f d Vol g . For any metric g , the infimum is always achieved by a pair ( f, τ ). Perel-man demonstrated the following formula for the first variation of ν ; let g ( s ) := g + sh for some h ∈ s ( T ∗ M ) and s ∈ ( − ε, ε ) then dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ν ( s ) = − τ (4 πτ ) − m Z M (cid:28) h, Ric( g )+Hess( f ) − τ g (cid:29) e − f d Vol g . (6.1)What Equation (6.1) demonstrates is that Ricci solitons are critical pointsof ν . Furthermore, from Equation (6.1), it can be easily deduced that the ν -entropy is increasing along the Ricci flow except at these critical points;hence, variations that increase the entropy are destabilising.We return to the case where g is an Einstein metric but, in order to keepour formulae in line with others in the literature, we will write the Einsteinconstant λ = τ . The second variation variation of ν at an Einstein metricwas computed by Cao, Hamilton and Ilmanen [3] (see [5] for a proof inthe general case of a Ricci soliton). They showed that the second variationon TT tensors is controlled by the spectrum of the Lichnerowicz Laplacianand that the second variation vanishes if h is an EID. Thus, to determinewhether an EID h is destabilising, we must compute the third derivative of ν ( g ( s )) at s = 0. To do this we will follow the method used by Knopf and esum [13] to compute the third variation in conformal directions and write dν ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 = A Z M (cid:28) h, Ric( g ) + Hess( f ) − τ g (cid:29) e − f d Vol g = A Z M (cid:18) g ip g jq h pq (cid:19)| {z } B (cid:18) Ric ij + Hess( f ) ij − τ g ij (cid:19)| {z } C (cid:18) e − f d Vol g (cid:19)| {z } D , where A = − τ (4 πτ ) − m . In the following proof we will also use the conven-tion of [13] and write A ′ , A ′′ , etc. for the derivative with respect to s andevaluated at s = 0 of any quantity A . Proof of Theorem D. Clearly, when s = 0 we have that C = 0. HenceLemma 2.4 of [5] implies that τ ′ = 0 and so A ′ = 0. Using the formulaefor Ric ′ (see [31] for example), the fact that h is an EID and that f (0) isconstant, we have C ′ = − Hess( f ′ ) . Cao and Zhu give a charactersiation of f ′ for variations at general Riccisoliton (see the proof of Lemma 2.4 in [5]). It is clear that at an Einsteinmetric, if h is TT then f ′ = 0 and so C ′ = 0. Elementary calculus thenyields d ds ν ( g ( s )) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = A Z M BC ′′ D = − τ (4 πτ ) − m Z M (cid:28) h, Ric( g ) ′′ + (Hess( f )) ′′ (cid:29) e − f d Vol g . As f ′ = 0, (Hess( f )) ′′ = Hess( f ′′ ) . The TT tensors are L -orthogonal to Lie derivatives of the metric and so,as Hess( f ′′ ) = L ∇ f ′′ g , we have d ds ν ( g ( s )) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − τ (4 πτ ) − m Z M (cid:28) h, Ric( g ) ′′ (cid:29) e − f d Vol g . The result then follows from Proposition 4.2 and Lemma 4.3 in [15]. (cid:3) Proof of Theorem E. For any n > h suchthat I ( h ) = 0. For the family of metrics g ( s ) := g + sh , the first and secondderivatives of ν ( g ( s )) but, by Theorem D, the third derivative of ν ( g ( s ))does not vanish. Hence g is not a local maximum of the ν -entropy and so g is dynamically unstable. (cid:3) Applications to other rigidity problems. For the symmetric spaces SU n +1 /SO n +1 , if one can demonstrate that, for some h η , the quantity I does not vanish then none of the EID on these spaces is integrable. It wouldbe particularly interesting to compute I given Theorem D, as the dynamicalstability of the five-dimensional space SU /SO is currently unknown. ur methods show almost all the EIDs on SU n are not integrable. Higherorder obstructions need to be considered for the remaining cases.Koiso demonstrated in [15] that Hom E ( s ( e ) , e ) = 0 and so all the EIDson this space are integrable to second order.The Grassmannians have EIDs, but these do not arise from the constructionoutlined in Section 2. For Grassmannians SU p + q /S ( U p × U q ), where K = S ( U p × U q ) and p = p × p Z − Z t q × q ! , for Z ∈ M p × q ( C ). As the product of any three matrices in p is trace-free, wesee directly that the space s ( p ) K vanishes in this case. The construction ofinfinitesimal variations for the Grassmannians is detailed in Chapter VIII ofthe book [8]. The method of generating EIDs uses some of the complex differ-ential geometry specific to the case of the Grassmannians and in particular,the generalised Euler sequence. Again, for the spaces SU n +1 /S ( U n − k × U k ),if one such deformation is obstructed then they are all obstructed. We hopeto investigate this construction in a future work.Finally, we note that the bi-invariant metric on the compact Lie group G isknown to admit infinitesimal solitonic deformations [4]. In [10], the secondauthor demonstrated that there exist deformations that are not integrableto second order and hence the Einstein metric is dynamically unstable. 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