On the linear stability of K\"ahler-Ricci solitons

This is a short note proving that K\"ahler-Ricci solitons with $\dim H^{(1,1)}(M)\ge 2$ are linearly unstable. This extends the results of Cao-Hamilton-Ilmanen in the K\"ahler-Einstein case.

This result generalises the result of Cao-Hamilton-Ilmanen who gave a simple argument in [4] for Kähler-Einstein metrics satisfying dim H (1,1) (M ) ≥ 2. The generalisation to Kälher-Ricci solitons is also suggested in their paper. We begin by recalling some facts about the ν-functional, then proceed to compute the second variation of ν and define 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 considered by Tian and Zhu [10]. They prove that if one considers variations in Kahler metrics in the fixed class c 1 (M ) then the ν-energy is maximised at a Kahler-Ricci soliton. There is also the recent work of Cao and Meng-Zhu [5] who also give a detailed calculation of the second variation of ν.

The Perelman ν-functional
Throughout this paper (M, g) will denote a closed Riemannian manifold. Most of the material here is contained in some form in [11] and [2] and of course, was originally taken from [9]. Perelman defined the following functional on triples (g, f, τ ), where g is a Riemannian metric, f a smooth function and τ > 0 a constant. Definition 2.1. Let f be a smooth function on M and τ > 0 a real number. Then the W -functional is given by where R is the scalar curvature of the metric g.
Some authors [10] absorb the constant τ into the other two terms and it is not hard to see that W(g, f, τ ) = W(τ −1 g, f, 1). The W-functional is also invariant under diffeomorphisms i.e. W(g, f, τ ) = W(φ * g, φ * f, τ ) for any diffeomorphism φ : M → M . Fix a compatibility condition for the triple (g, f, τ ) by requiring This leads to the definition of the ν-functional: Definition 2.2. Let f be a smooth function on M and τ > 0 a real number. Then the W -functional is given by We will not comment on the existence theory except to say that for fixed g there exists a τ > 0 and smooth f that attain the infimum in the above definition. The pair (f, τ ) satisfy the equations 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 M evolving via the Ricci-flow. Then ν(g(t)) is montone increasing, unless g is a Riccisoliton in which case ν is stationary.
We also record the first variation of ν which makes it clear that stationary points of the functional are shrinking gradient Ricci solitons.
Theorem 2.4 (Perelman [9]). The first variation of ν in the direction h, D g ν(h) is given by

The second variation of ν
The aim of this section give a self-contained proof of the second variation formula of ν at a Ricci soliton with potential function f . This is not new and has been known to experts for some time but we include it here for completeness. We work with the scaled L 2 -inner product on tensors This inner product is adapted to the following operators for any one-form α and function F . Obviously these reduce to the usual divergence and Laplacian when the metric is Einstein. The operator ∆ f is often referred to as the Bakry-Émery Laplacian. The sign convention for the curvature tensor we adopt is Z and Rm(X, Y, W, Z) = g(R(X, Y )W, Z). In index notation we have R(∂ i , ∂ j , ∂ k , ∂ l ) = Rm ijkl . For h ∈ s 2 (T M * ), define the symmetric curvature operator Rm(h, ·) ∈ s 2 (T M * ) by Rm(h, ·) ij = R kilj h kl . We will also need the curvature operator on 2-forms, usually denoted by R The convention for divergence we adopt is div(h) = tr 12 (∇h). The reader should note that this definition is the opposite sign to the divergence operator considered in [11]. When restricted to forms we also have the codifferential δ which, with this convention, satisfies δ(σ) = −div(σ). If we denote by div * f the adjoint to div f with respect to the scaled L 2 -inner product ·, · f , then, as remarked in [3], div * f = div * . Here div * is the adjoint to div with respect to the usual inner product on tensors.
Theorem 3.1 (Cao-Hamilton-Ilmanen). Let g be an Ricci-Soliton with potential function f satisfying Ric(g) + Hess(f ) = 1 2τ g. For h ∈ s 2 (T M * ), consider variations g(s) = g + sh. Then the second variation of the ν-energy is where N is given by Here v h is the solution of and C(h, g) is a constant depending upon h and g.
Remark : This theorem was first stated in [3] but with an error in the term C(h, g). This has subsequently been corrected in the recent work of Cao and Meng-Zhu [5]. They find that Definition 3.2. A soliton is linearly stable if the operator N is non-positive definite and 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 . Much of what is required is explained extremely well in [11] and so many proofs are omitted. Ric(g(s)) Hess(tr(h)).
Let R(s) = tr g(s) Ric(g(s)) be the scalar curvature of g(s) then It is necessary to know how geometric quantities associated to functions f : (−ǫ, ǫ)× M → R vary. The convention we adopt is that ∆f = tr(Hess(f )) = −δdf , so the Laplacian has negative eigenvalues.

Proof. The Hessian of a function f is given by
By Proposition 2.3.1 in [11] g( d ds We also have Hence the variation is given by and the result follows. Proof. From the definition of div f we have div f (h)(·) = div(h)(·) − h(∇f, ·) so we need to compute The term div(h(∇f, ·)) = div(h)(∇f ) + h, Hess(f ) , and so If the soliton varies by δg = h, then this induces a variation in the pair (f, τ ) which is denoted (δf, δτ ).
Proof. Consider the variation in the equation which yields As we are at a soliton, δν = 0 and R = n 2τ − ∆f .

This yields
The left hand side is div f div f (h), by Lemma 3.7, and the right hand side may be written as At a Ricci soliton, the scalar curvature R is given by The second equation is simply the variation in the equation We proceed to the proof of Theorem 3.1: Proof. The first variation is given by g e −f dV g so it is sufficient to compute the variation in the term This is given by δτ (Ric(g) + Hess(f )) − 1 2 h + τ (δRic + δHess(f )).

The variation in the Hessian is given by
Putting this together the variation is given by which may be rewritten as The result follows on taking δτ τ = C(h, g).
Taking f constant recovers: Corollary 3.8 (Einstein Case [4]). Let g be an Einstein metric with Ric(g) = 1 2τ g. For h ∈ s 2 (T M * ), consider variations g(s) = g + sh. Then the second variation of the ν-energy at is where N is given by Here v h is the solution of

The instability of Kähler-Ricci solitons
We first establish the following simple Lemma comparing curvature operators on a Kahler manifold: Proof. We calculate using an adapted orthonormal basis {e i , Je i }. From the definition of R, the Bianchi identity and the J-invariance of the curvature tensor; The proof of the main result will be modeled on the Kahler-Einstein case which is outlined in [4]. Proof. Choose a trace-free harmonic (1, 1)-form σ ∈ H (1,1) (M ) that induces a perturbation σ J = σ(·, J·). As the complex structure is parallel div(σ J ) = 0 and (∇ * ∇σ J ) = (∇ * ∇σ) J . We can use a Weitzenbock formula on 2-forms to compare the rough Laplacian with the Hodge Laplacian ∆ H = −(d + δ) 2 . The Weitzenbock formula for an Einstein metric with Einstein constant 1 2τ is where R is the curvature operator for 2-forms. Our formula for the variation becomes We will refer to ∆ f,H as the twisted Laplacian. Forms σ satisfying ∆ f,H (σ) = 0 are said to be twisted harmonic forms. Modifications of the Hodge Laplacian similar to this are also found in the seminal work of Witten [13] on the Morse inequalities. The following Lemma gives some crucial properties of this Laplacian: The operator ∆ f,H has the following properties: (1) ∆ f,H = ∆ H − L ∇f .
(3) ∆ f,H satisfies a Weitzenbock identity for 2-forms σ: Proof. From the definition we have Using Cartan's magic formula this yields The second claim follows from the fact that ∇f is a holomorphic vector field. For the Weitzenbock formula recall that for 2 forms where R is the curvature operator for 2-forms. A simple computation yields L ∇f (σ) − (∇ ∇f )(σ) = (Hess(f ) · σ + σ · Hess(f )). Hence To conclude the proof of Theorem 1.1 is presented: Proof. The first step is to choose a twisted harmonic form σ ∈ H (1,1) (M ). If we denote the space of twisted harmonic forms by H f then we have the natural map π : H f → H 2 (M ) given by π(σ) = [σ].
The proof that this map is an isomorphism follows that of the usual Hodge theorem except that the 'energy' in our case is given by The twisted Laplacian also preserves the (p, q)-decomposition of forms so we get the required isomorphism of vector spaces The hypothesis on the dimension of H (1,1) (M ) means that there exist θ 1 , θ 2 ∈ H (1,1) f and λ, µ ∈ R, not both zero, satisfying where ρ is the Ricci form. Choosing σ = λθ 1 + µθ 2 and computing gives and the theorem follows.
Corollary 1.2 follows immediately from this theorem.

Remark:
In the recent work [5] the authors consider the operator They prove that the Ricci form ρ is twisted harmonic and that it satisfiesL(Ric) = 1 2τ Ric. Our proof here can be phrased as showing that the multiplicity of this eigenspace is at least dim H (1,1) (M ). One may then apply their Proposition 3.1 to conclude that Kähler-Ricci solitons are linearly unstable.
It would be most interesting to determine whether the perturbations σ J are in fact eigentensors of the operator N . It is clear that if v σJ = 0 then σ J is an eigentensor. We observe that the recent work of Futaki-Sano [6] and Ma [8] gives a spectral gap for the Bakry-Émery Laplacian when the Bakry-Émery Ricci curvature is bounded below. They show: Clearly this bound is weaker that the Lichnerowicz bound in the Einstein case. Hence one may not deduce that v σJ = 0 for the unstable variations of Theorem 1.1. However, if v σJ = 0 such a soliton would prove that the above estimate is in some sense sharp. We suspect however that v σj = 0 and hence that these variations are eigentensors of N . Implicit in the calculation of the second variation formula is the fact that the potential function f (once normalised) is an eigenfunction of ∆ f with eigenvalue − 1 τ (Futaki and Sano also prove this). It would be interesting to know if ∆ f has any eigenvalues in the interval (− 1 τ , − 1 2τ ] and the authors are currently numerically investigating this problem for the Koiso-Cao and Wang-Zhu solitons.