STUART J. HALL AND THOMAS MURPHY. (Communicated by Jianguo Cao)

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1 PROCEEDINGS OF THE AERICAN ATHEATICAL SOCIETY Volume 139, Number 9, September 2011, Pages S (2011) Article electronically publishe on arch 9, 2011 ON THE LINEAR STABILITY OF KÄHLER-RICCI SOLITONS STUART J. HALL AND THOAS URPHY (Communicate by Jianguo Cao) Abstract. We show that Kähler-Ricci solitons with im H (1,1) () 2are linearly unstable. This extens the results of Cao-Hamilton-Ilmanen in the Kähler-Einstein case. 1. Introuction A Ricci soliton is a complete Riemannian metric g, a vector fiel X an a real number c satisfying Ric(g)+L X g cg =0. The soliton is calle shrinking, steay or expaning if c > 0,c = 0 or c < 0 respectively. If X = f for a smooth function f the soliton is sai to be a graient Ricci soliton with potential function f. If X = 0 we recover the efinition of an Einstein metric. Solitons are important in the theory of the Ricci flow as they occur as fixe points of the flow, up to the action of the iffeomorphism group. Perelman s grounbreaking observation [9] was that solitons are the critical points of a certain functional (the ν-functional which we efine below). An interesting question is whether the functional is locally maximise at these points (i.e. linearly stable) as this affects the behaviour of the Ricci flow nearby to a soliton. It is this question that we take up in this article for a special class of graient shrinking solitons, namely Kähler-Ricci solitons. These are solitons on Fano manifols which have the aitional property that g is a Kähler metric an the vector fiel f is holomorphic. The only known examples of compact Ricci solitons which are not proucts or Einstein are the Koiso-Cao soliton [7], [1], a U(2)-invariant soliton explicitly constructe on CP 2 CP 2 an a class of toric Kähler metrics constructe on Fano manifols by Wang an Zhu [12]. The precise theorem we wish to prove is the following: Theorem 1.1. If (,g,j) is a Kähler-Ricci soliton with im H (1,1) () 2, then the soliton is linearly unstable. Receive by the eitors August 21, athematics Subject Classification. Primary 53C44; Seconary 53C25. Key wors an phrases. Ricci solitons, Perelman s ν-functional, linear stability. This work forms part of the first author s Ph.D thesis fune by the EPSRC. He woul like to thank his avisor, Professor Simon Donalson, for his comments an encouragement uring the course of this work. The secon author was supporte by an IRCSET postgrauate fellowship. The authors woul also like to thank Professor Huai-Dong Cao for useful communications c 2011 American athematical Society Reverts to public omain 28 years from publication

2 3328 STUART J. HALL AND THOAS URPHY Corollary 1.2. The Koiso-Cao soliton on CP 2 CP 2 an the Wang-Zhu soliton on CP 2 2CP 2 are both linearly unstable. This result generalises the result of Cao-Hamilton-Ilmanen who gave a simple argument in [4] for Kähler-Einstein metrics satisfying im H (1,1) () 2. The generalisation to Kähler-Ricci solitons is also suggeste in their paper. We begin by recalling some facts about the ν-functional, then procee to compute the secon variation of ν an efine linear stability precisely. We finally show how one can construct unstable variations in a manner similar to [4]. The question of linear stability for Kähler-Ricci solitons has also been consiere by Tian an Zhu [10]. They prove that if one consiers variations in Kähler metrics in the fixe class c 1 (), then the ν-energy is maximise at a Kähler-Ricci soliton. There is also the recent work of Cao an Zhu [5], who also give a etaile calculation of the secon variation of ν. 2. The Perelman ν-functional Throughout this paper (,g) will enote a close Riemannian manifol. ost of the material here is containe in some form in [11] an [2] an, of course, was originally taken from [9]. Perelman efine the following functional on triples (g, f, ), where g is a Riemannian metric, f a smooth function an >0aconstant. Definition 2.1. Let f be a smooth function on an >0areal number. Then the W -functional is given by W (g, f, ) = [(R + f 2 )+f n](4π) n 2 e f V g, where R is the scalar curvature of the metric g. Some authors [10] absorb the constant into the other two terms, an it is not har to see that W(g, f, ) =W( 1 g, f, 1). The W-functional is also invariant uner iffeomorphisms; i.e., W(g, f, ) =W(φ g, φ f,) for any iffeomorphism φ :. Fix a compatibility conition for the triple (g, f, ) by requiring e f V (4π) n g =1. 2 This leas to the efinition of the ν-functional: Definition 2.2. Let f be a smooth function on an >0areal number. Then the W -functional is given by ν(g) = inf{w(g, f, ) :(g, f, ) is compatible}. We will not comment on the existence theory except to say that for fixe g there exists a >0ansmoothf that attain the infimum in the above efinition. The pair (f,) satisfy the equations ( 2Δf + f 2 R) f + n + ν =0an(4π) n 2 fe f V g = n 2 + ν. The first important result about the ν-function is the following: Theorem 2.3 (Perelman [9]). Let g(t) be a family of Riemannian metrics on evolving via the Ricci flow. Then ν(g(t)) is montone increasing, unless g is a Ricci soliton, in which case ν is stationary.

3 ON THE LINEAR STABILITY OF KÄHLER-RICCI SOLITONS 3329 We also recor the first variation of ν which makes it clear that stationary points of the functional are shrinking graient Ricci solitons. Theorem 2.4 (Perelman [9]). The first variation of ν in the irection h, D g ν(h), is given by (4π) n 2 (Ric(g)+Hess(f)) 1 2 g, h e f V g. 3. The secon variation of ν The aim of this section is to give a self-containe proof of the secon variation formula of ν at a Ricci soliton with potential function f. This is not new an has been known to experts for some time, but we inclue it here for completeness. We work with the scale L 2 -inner prouct on tensors, f =, e f V g. This inner prouct is aapte to the following operators: iv f (h) =e f iv(e f h)anδ f (h) =Δ(h) ( f h), in the sense that iv f (α)e f V g =0an Δ f (F )e f V g =0 for any one-form α an function F. Obviously these reuce to the usual ivergence an Laplacian when the metric is Einstein. The operator Δ f isoftenreferreto as the Bakry-Émery Laplacian. The sign convention for the curvature tensor we aopt is R(X, Y )Z = Y X Z X Y Z + [X,Y ] Z an Rm(X, Y, W, Z) =g(r(x, Y )W, Z). In inex notation we have R( i, j, k, l ) = Rm ijkl. For h s 2 (T ), efine the symmetric curvature operator Rm(h, ) s 2 (T )by Rm(h, ) ij = R kilj h kl. We will also nee the curvature operator on 2-forms, usually enote by R: R :Λ 2 () Λ 2 (), R(σ) ij = Rm ijkl σ kl. The convention for ivergence we aopt is iv(h) =tr 12 ( h). The reaer shoul note that this efinition is the opposite sign to the ivergence operator consiere in [11]. When restricte to forms we also have the coifferential δ which, with this convention, satisfies δ(σ) = iv(σ). If we enote by ivf the ajoint to iv f with respect to the scale L 2 -inner prouct, f, then, as remarke in [3], ivf = iv. Here iv is the ajoint to iv with respect to the usual inner prouct on tensors. Theorem 3.1 (Cao-Hamilton-Ilmanen). Let g be a Ricci soliton with potential function f satisfying Ric(g)+Hess(f) = 1 2 g. For h s2 (T ), consier variations g(s) = g + sh. Then the secon variation of the ν-energy is 2 s 2 ν(g(s)) = h, Nh e f V (4π) n g, 2 where N is given by N(h) = 1 2 Δ f (h)+rm(h, )+iv f iv f (h)+ 1 2 Hess(v h) C(h, g)ric.

4 3330 STUART J. HALL AND THOAS URPHY Here v h is the solution of (3.1) Δ f (v h )+ 1 2 v h = iv f iv f (h), an C(h, g) is a constant epening upon h an g. Remark 3.2. This theorem was first state in [3] but with an error in the term C(h, g). This has subsequently been correcte in the recent work of Cao an Zhu [5]. They fin that C(h, g) = Ric, h e f V g, Re f V g where R is the scalar curvature of g. Definition 3.3. A soliton is linearly stable if the operator N is non-positive-efinite an linearly unstable otherwise. We now collect some formulae for how various geometric quantities vary through a family of Riemannian metrics g(t) evolving via g t = h, whereh s 2 (T ). uch of what is require is explaine extremely well in [11], an so many proofs are omitte. Lemma 3.4 (Propositions an in [11]). Let g(s) =g + sh be a family of Riemannian metrics on. Then s Ric(g(s)) = 1 2 (h) Rm(h, )+ 1 (Ric h + h Ric) 2 iv iv(h) 1 2 Hess(tr(h)). Let R(s) =tr g(s) Ric(g(s)) be the scalar curvature of g(s). Then s R(s) = h, Ric + iviv(h) Δ(tr(h)). It is necessary to know how geometric quantities associate to functions f : ( ɛ, ɛ) R vary. The convention we aopt is that Δf = tr(hess(f)) = δf, so the Laplacian has negative eigenvalues. Lemma 3.5 (Prop in [11]). Let g(s) =g + sh be a family of Riemannian metrics on an let f(s, x) :( ɛ, ɛ) R be a family of smooth functions. Then s Δ(f) =Δ g ( f) 1 (tr(h)),f h, Hess(f) iv(h),f, 2 s f 2 = h( f, f)+2 f, f. The final object whose variation must be compute is the Hessian of the functions f(s, x). Lemma 3.6. Let g(s) =g + sh be a family of Riemannian metrics on an let f(s, x) :( ɛ, ɛ) R be a family of smooth functions. Then s Hess(f) = Hess( f)+ 1 2 (Hess(f) h+h Hess(f))+1 2 ( f h)+iv (h( f, )).

5 ON THE LINEAR STABILITY OF KÄHLER-RICCI SOLITONS 3331 Proof. The Hessian of a function f is given by Hess(X, Y )=g( X f,y ). Hence s Hess(f)(X, Y )=h( X f,y )+g( s ( X f),y) = h( X f,y )+g( s ( X) f,y )+g( X f,y ). s By Proposition in [11], g( s ( X) f,y )= 1 2 [( f h)(x, Y )+( X h)(y, f) ( Y h)(x, f)]. We also have g( X s f,y )+h( X f,y )=g( X f,y) ( X h)(y, f) = Hess( f)(x, Y ) ( X h)(y, f). Hence the variation is given by Hess( f)(x, Y )+ 1 2 ( f h)(x, Y )+ 1 2 [h( X f,y )+h( Y f,x)] 1 2 [( Xh( f, ))(Y )+( Y h( f, ))(X)], an the result follows. Lemma 3.7. For h s 2 (T ),then iv f iv f (h) =iviv(h)+h( f, f) 2 f, iv(h) h, Hess(f). Proof. From the efinition of iv f we have iv f (h)( ) =iv(h)( ) h( f, ), so we nee to compute iv f (iv(h)( ) h( f, )) = e f iv(e f (iv(h)( ) h( f, )) = iviv(h) iv(h)( f) iv(h( f, )) + h( f, f). The term iv(h( f, )) = iv(h)( f)+ h, Hess(f), anso iv f iv f (h) =iviv(h)+h( f, f) 2 f, iv(h) h, Hess(f). If the soliton varies by δg = h, then this inuces a variation in the pair (f,), which is enote by (δf, δ). Lemma 3.8. Let (g, f, ) escribe a Ricci soliton an consier a variation g(s) = g + sh inucing variations (δf, δ) in f an. Ifv h =(tr(h) 2δf 2δ (f ν)), then v h satisfies Δ f (v h )+ v h 2 = iv f iv f (h) an ( nδ 2 f + δf(1 f)+f 1 2 tr(h))e f V g =0. Proof. Consier the variation in the equation ( 2Δf + f 2 R) f + n + ν =0, which yiels δ( 2Δf + f 2 R)+( 2δ(Δf)+δ f 2 δr) δf + δν =0.

6 3332 STUART J. HALL AND THOAS URPHY As we are at a soliton, δν =0anR = n 2 Δf. Hence 2Δf + f 2 R = Δf + f 2 n 2 = Δ f f n 2. Lemmas 3.4 an 3.5 imply that δ( 2Δf) =Δ( 2δf)+ tr(h), f +2 h, Hess(f) +2 iv(h),f an δr = h, Ric iviv(h)+ Δ(tr(h)) = h, Hess + 1 tr(h) iviv(h)+δ(tr(h)). 2 This yiels δ ( 2Δf + f 2 R)+Δ(tr(h) 2δf)+ (tr(h) 2δf), f + 1 (tr(h) 2δf) =iviv(h)+h( f, f) h, Hess 2 iv(h),f. 2 The left han sie is iv f iv f (h), by Lemma 3.8, an the right han sie may be written as δ ( 2Δf + f 2 R)+Δ f (tr(h) 2δf)+ 1 (tr(h) 2δf). 2 At a Ricci soliton, the scalar curvature R is given by R = Δf + n 2,so Δ f (f) =Δ(f) f 2 =2Δ(f) f 2 + R n 2 = f + n 2 + ν, whence Δ f f + f 2 = f 2 + n 2 + ν. Setting f = 2δ (f ν), Δ f ( f)+ f 2 = δ (f ) 2δ ( n 2 + ν )+δν 2 Now letting v h = tr(h) 2δf 2 δ(f ν), Δ f (v h )+ v h 2 = iv f iv f (h). The secon equation is simply the variation in the equation (4π) n 2 fe f V g = n 2 + ν. We procee to the proof of Theorem 3.1: = δ ( 2Δf + f 2 R). Proof. The first variation is given by s ν(g) =(4π) n 2 h, (Ric + Hess(f)) 1 2 g e f V g, so it is sufficient to compute the variation in the term (Ric)+Hess(f)) 1 2 g. This is given by δ(ric(g)+hess(f)) 1 h + (δric + δhess(f)). 2

7 ON THE LINEAR STABILITY OF KÄHLER-RICCI SOLITONS 3333 Using the previous results, δric = 1 2 (h) Rm(h, )+ 1 2 (Ric h + h Ric) iv iv(h) 1 2 Hess(tr(h)), which, using the fact we are at a soliton, yiels δric = 1 2 (h) Rm(h, ) 1 2 (Hess(f) h + h Hess(f)) + h 2 iv iv(h) 1 2 Hess(tr(h)). The variation in the Hessian is given by Hess( f)+ 1 2 (Hess(f) h + h Hess(f)) ( f h)+iv (h( f, )). Putting this together, the variation is given by δric(g) 2 (Δ(h) ( f h)) Rm(h, ) iv (iv(h) h( f, )) + 2 Hess(2δ f +2δf tr(h)), which may be rewritten as = δric(g)+( 1 2 Δ f (h) Rm(h, ) iv f iv f (h) 1 2 Hess(v h)). The result follows on taking δ = C(h, g). Taking f constant recovers: Corollary 3.9 (Einstein Case [4]). Let g be an Einstein metric with Ric(g) = 1 2 g. For h s 2 (T ), consier variations g(s) =g + sh. Then the secon variation of the ν-energy at g is 2 s ν(g(s)) = h, Nh V g, vol(g) where N is given by N(h) = 1 2 (h)+rm(h, )+iv iv(h)+ 1 2 Hess(v g h) tr(h)v g. 2nvol(g) Here v h is the solution of Δ(v h )+ 1 2 v h = iviv(h). 4. The instability of Kähler-Ricci solitons We first establish the following simple lemma comparing curvature operators on akähler manifol: Lemma 4.1. Let (,g,j) be a Kähler manifol. If σ A (1,1) (), then2rm(σ J, ) R(σ) J =0.

8 3334 STUART J. HALL AND THOAS URPHY Proof. We calculate using an aapte orthonormal basis {e i,je i }. From the efinition of R, the Bianchi ientity an the J-invariance of the curvature tensor, R(σ)(X, JY )= R(e i,e j,x,jy)σ(e i,e j ) i,j = i,j (R(X, e i,e j,jy)+r(e j,x,e i,jy)) σ(e i, e j ) = i,j = i,j = i,j (R(X, e i,je j,y)+r(e j,x,je i,y)) σ(e i,e j ) 2R(e i,x,je j,y)σ(e i,e j ) 2R(e i,x,je j,y)σ(e i,j(je j )) =2Rm(σ J, )(X, Y ). The proof of the main result will be moelle on the Kähler-Einstein case, which is outline in [4]. Proposition 4.2. Let (,g,j) be a Kähler-Einstein manifol with im H (1,1) () 2. Then g is linearly unstable. Proof. Choose a trace-free harmonic (1, 1)-form σ H (1,1) () that inuces a perturbation σ J = σ(,j ). As the complex structure is parallel, iv(σ J )=0an ( σ J )=( σ) J. We can use a Weitzenbock formula on 2-forms to compare the rough Laplacian with the Hoge Laplacian Δ H = ( + δ) 2. The Weitzenbock formula for an Einstein metric with Einstein constant 1 2 is Δ H = R+ 1 i, where R is the curvature operator for 2-forms. becomes Our formula for the variation 2N(σ J )=Δ H (σ) J R(σ) J + 1 σ J +2Rm(σ J, )+Hess(v σj ) = R(σ) J + 1 σ J +2Rm(σ J, )+Hess(v σj ). It follows from Equation 3.1 that the function v σj is an eigenfunction of the Laplacian. Recall that the first non-zero eigenvalue of the Laplacian on an Einstein manifol with Einstein constant c>0 satisfies λ 1 n n 1 c by the famous Lichnerowicz boun. Hence v σj = 0. From Lemma 4.1, 2Rm(σ J, ) R(σ) J = 0, so we have exhibite an eigentensor for N with eigenvalue 1 2. Hence metrics of this type are unstable. Consiering the ivergence operator δ :Ω() k Ω() k 1 as the ajoint of the exterior erivative on forms, then, with our conventions, iv(σ) = δ(σ) for any k-form σ. Set δ f (σ) =e f δ(e f σ)=δ(σ)+ι f σ an Δ f,h (σ) = (δ f + δ f )(σ).

9 ON THE LINEAR STABILITY OF KÄHLER-RICCI SOLITONS 3335 We will refer to Δ f,h as the twiste Laplacian. Formsσ satisfying Δ f,h (σ) =0are sai to be twiste harmonic forms. oifications of the Hoge Laplacian similar to this are also foun in the seminal work of Witten [13] on the orse inequalities. The following lemma gives some crucial properties of this Laplacian: Lemma 4.3. The operator Δ f,h has the following properties: (1) Δ f,h =Δ H L f. (2) Δ f,h preserves the ecomposition into (p, q)-forms. (3) Δ f,h satisfies a Weitzenbock ientity for 2-forms σ: Δ f (σ) Δ f,h (σ) = R(σ)+ 1 σ. Proof. From the efinition we have Δ f,h (σ) =Δ H (σ) ( ι f + ι f )σ. Using Cartan s magic formula this yiels Δ f,h (σ) =Δ H (σ) L f (σ). The secon claim follows from the fact that f is a holomorphic vector fiel. For the Weitzenbock formula recall that for 2-forms (σ) Δ H (σ) = R(σ)+Ric σ + σ Ric = R(σ) (Hess(f) σ + σ Hess(f)) + 1 σ, where R is the curvature operator for 2-forms. A simple computation yiels L f (σ) ( f )(σ) =(Hess(f) σ + σ Hess(f)). Hence (Δ f Δ f,h )(σ) = R(σ)+ 1 σ. To conclue, the proof of Theorem 1.1 is presente. Proof. The first step is to choose a twiste harmonic form σ H (1,1) (). If we enote the space of twiste harmonic forms by H f, then we have the natural map π : H f H 2 () givenbyπ(σ) =[σ]. The proof that this map is an isomorphism follows that of the usual Hoge theorem except that the energy in our case is given by σ 2 e f V g. The twiste Laplacian also preserves the (p, q)-ecomposition of forms, so we get the require isomorphism of vector spaces H (1,1) f = H (1,1) (). The hypothesis on the imension of H (1,1) () means that there exist θ 1,θ 2 H (1,1) f an λ, μ R, not both zero, satisfying λθ 1 + μθ 2,ρ f = λθ 1 + μθ 2,ρ e f V g =0,

10 3336 STUART J. HALL AND THOAS URPHY where ρ is the Ricci form. Choosing σ = λθ 1 + μθ 2 an computing gives N(σ J ),σ J f = 1 2 σ J 2 > 0, an the theorem follows. Corollary 1.2 follows immeiately from this theorem. Remark 4.4. In the recent work [5] the authors consier the operator ˆL = 1 2 Δ f + Rm(h, ). They prove that the Ricci form ρ is twiste harmonic an that it satisfies ˆL(Ric) = 1 2 Ric. Our proof here can be phrase as showing that the multiplicity of this eigenspace is at least im H (1,1) (). One may then apply their Proposition 3.1 to conclue that Kähler-Ricci solitons are linearly unstable. It woul be most interesting to etermine whether the perturbations σ J are in fact eigentensors of the operator N. Itisclearthatifv σj =0,thenσ J is an eigentensor. We observe that the recent work of Futaki-Sano [6] an a [8] gives a spectral gap for the Bakry-Émery Laplacian when the Bakry-Émery Ricci curvature is boune below. They show: Theorem 4.5 (a, Futaki-Sano). Let φ be a smooth function on an suppose that the Bakry-Émery Ricci curvature, efine as Ric φ = Ric(g)+Hess(φ), satisfies the boun Ric φ c, for some number c>0. Then the first non-zero eigenvalue λ 1 of the Bakry-Émery Laplacian Δ φ := Δ φ satisfies λ 1 c. Clearly this boun is weaker than the Lichnerowicz boun in the Einstein case. Hence one may not euce that v σj = 0 for the unstable variations of Theorem 1.1. However, if v σj 0, such a soliton woul prove that the above estimate is in some sense sharp. We suspect however that v σj = 0 an hence that these variations are eigentensors of N. Implicit in the calculation of the secon variation formula is the fact that the potential function f (once normalise) is an eigenfunction of Δ f with eigenvalue 1 (Futaki an Sano also prove this). It woul be interesting to know if Δ f has any eigenvalues in the interval ( 1, 1 2 ], an the authors are currently numerically investigating this problem for the Koiso-Cao an Wang-Zhu solitons. References [1] Huai-Dong Cao, Existence of graient Kähler-Ricci solitons, Elliptic an parabolic methos in geometry (inneapolis, N, 1994), A K Peters, Wellesley, A, 1996, pp R (98a:53058) [2], Geometry of Ricci solitons, Chinese Ann. ath. Ser. B 27 (2006), no. 2, R (2007f:53036) [3], Recent avances in geometric analysis, Av. Lect. ath. (A.L..) 11 (2009), [4] Huai-Dong Cao, Richar S. Hamilton, an Tom Ilmanen, Gaussian ensities an stability for some Ricci solitons, arxiv:math/ (2004).

11 ON THE LINEAR STABILITY OF KÄHLER-RICCI SOLITONS 3337 [5] Huai-Dong Cao an eng Zhu, On secon variation of Perelman s Ricci shrinker entropy, arxiv: v1 (2010). [6] Akito Futaki an Yuji Sano, Lower iameter bouns for compact shrinking Ricci solitons, arxiv: (2010). [7] Norihito Koiso, On rotationally symmetric Hamilton s equation for Kähler-Einstein metrics, Recent topics in ifferential an analytic geometry, Av. Stu. Pure ath., vol. 18, Acaemic Press, Boston, A, 1990, pp R (93:53057) [8] Li a, Eigenvalue estimates an L1 energy on close manifols, arxiv: v1 (2009). [9] Grisha Perelman, The entropy formula for the Ricci flow an its geometric applications, arxiv: (2002). [10] Gang Tian an Xiaohua Zhu, Perelman s w-functional an stability of Kähler-Ricci flow, arxiv: v1 (2008). [11] Peter Topping, Lectures on the Ricci flow, Lonon athematical Society Lecture Note Series, vol. 325, Cambrige University Press, Cambrige, R (2007h:53105) [12] Xu-Jia Wang an Xiaohua Zhu, Kähler-Ricci solitons on toric manifols with positive first Chern class, Av.ath.188 (2004), no. 1, R (2005:53074) [13] Ewar Witten, Supersymmetry an orse theory, J. Differential Geom. 17 (1982), no. 4, (1983). R (84b:58111) Department of athematics, Imperial College, Lonon, SW7 2AZ, Unite Kingom aress: stuart.hall06@imperial.ac.uk School of athematical Sciences, University College Cork, Irelan aress: tommy.murphy@ucc.ie