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On moduli spaces of Ricci solitons

We study deformations of shrinking Ricci solitons on a compact manifold M, generalising the classical theory of deformations of Einstein metrics. Using appropriate notions of twisted slices S_f inside the space of all Riemannian metrics on M, we define the infinitesimal solitonic deformations and the local solitonic pre-moduli spaces. We prove the existence of a finite dimensional submanifold of S_f x C^infty(M), which contains the pre-moduli space of solitons around a fixed shrinking Ricci soliton as an analytic subset. We define solitonic rigidity and give criteria which imply it.


Introduction
In this paper we study Ricci solitons on manifolds setting up the theory of their deformations. We recall that a Ricci soliton on a manifold M is a complete Riemannian metric g, with Ricci tensor satisfying the equation Ric + 1 2 L X g − cg = 0 for some complete vector field X and a constant c. Solitons are remarkably interesting metrics as they evolve in a particularly easy way under the Ricci flow, namely by diffeomorphisms and homotheties, and they appear as natural generalisations of Einstein metrics, which are the Ricci solitons with X = 0. The reader is referred to [4,5,11] for a detailed exposition of the theory of Ricci solitons.
It is known that on a compact manifold M every Ricci soliton is of gradient type, i.e., with associated vector field equal to the gradient X = ∇f of a smooth function f . It is then natural to consider the space where M is the space of all Riemannian metrics on the n-dimensional manifold M , and to define the set of normalised Ricci solitons Sol as the set of all pairs (g, f ) ∈ P satisfying the normalised equation Ric + Hess(f ) − g = 0 .
We remark that Sol coincides with the set of all critical points in P of Perelman's entropy functional W (g, f ) or, equivalently, with the zero set of the corresponding Euler-Lagrangian operator S = (S 1 , S 2 ) : P −→ C ∞ (S 2 (T * M )) × C ∞ (M ) (see (2.3)). If we compare all this to the classical theory of Einstein metrics (which can be considered as the critical points of the total scalar curvature H(g), in the space M 1 of Riemannian metrics of unitary volume) we have that Perelman's functional W (g, f ) and the equations S 1 (g, f ) = S 2 (g, f ) = 0 are non trivial analogues of H(g) and Einstein equation, respectively, that are well suited to our purposes.
The aim of this paper is to study the moduli space Sol/D of solitons w.r.t. the natural action of the set D of all diffeomorphisms of M , around a fixed normalised Ricci soliton (g o , f o ). In order to do this, we first define and prove the existence of modified Ebin slices, called f -twisted slices S f , for the D-action on the space M. We then define the solitonic pre-moduli space at (g o , f o ) as the intersection of Sol with the set S fo × C ∞ (M ). Such solitonic pre-moduli space is obviously acted on by the isometry group Iso(g o ) and the orbit space of this action is a local model for the moduli space Sol/D around the class of (g o , f o ).
The space of infinitesimal solitonic deformations is then defined as the linear subspace of the tangent space T (go,fo) P given by the kernel of the differential of S at (g o , f o ). An element (h, a) ∈ T (go,fo) P ≃ C ∞ (S 2 (T * M ))×C ∞ (M ) in such a linear subspace is called essential if the tensor field h belongs to the tangent space of the f o -twisted slice S fo at g o . Our first main result states that the space of all essential infinitesimal solitonic deformations is finite dimensional.
In order to provide a description of the solitonic pre-moduli space, we work in the Hilbert manifold P s (s ≥ [n/2] + 3) given by where H s (M ) is the Sobolev space of order s and M s denotes the space of all Riemannian metrics which are H s -sections of the bundle S 2 (T * M ). This setting is convenient because it allows the use of the Implicit Function Theorem and brings to our main result, which consists in a local description of the pre-moduli space as an analytic subset of a finite dimensional analytic submanifold Z s ⊂ P s through (g o , f o ), whose tangent space is given by the set of essential infinitesimal solitonic deformations.
We then introduce the notion of (weak) solitonic rigidity and provide some criteria that imply it, in particular when the fixed metric g o is Einstein. We analyse the case of compact symmetric spaces of rank one, showing that the complex projective spaces CP n are the only ones with non trivial spaces of essential infinitesimal solitonic deformations. Other general results on solitonic rigidity are given, when the curvature is sufficiently positive or the metric g o is Kähler.
We conclude observing that these results appear to be the natural analogues of the outcomes of the theory of deformations of Einstein metrics, developed by Koiso ([13,15,14,1]), albeit they are based on various non trivial modifications of Koiso's settings and techniques.
The structure of the paper is as follows. In §2, we consider the set of normalised solitons inside P s , define and prove the existence of f -twisted slices and give the notion of solitonic pre-moduli space. In §3, we define the space of essential infinitesimal solitonic deformations, we show that it is finite dimensional and we prove our main result, which gives a local description of the solitonic pre-moduli space. In §4, we introduce the notion of the solitonic rigidity and study the rigidity of the standard metrics of a compact symmetric spaces of rank one, showing that the space of essential infinitesimal solitonic deformations of CP n is non trivial. We also treat the solitonic rigidity of Einstein metrics with certain conditions on the curvature. In §5, we define the weak solitonic rigidity of Einstein metrics and obtain some related results under conditions on the diameter of the manifold or in the Kähler situation.

Notation.
Throughout the paper, M is a compact n-dimensional manifold, F(M ) = C ∞ (M, R) is the space of C ∞ real functions, D is the set of all diffeomorphisms from M into itself and M is the space of all Riemannian metrics on M , i.e., the cone of smooth sections of S 2 (T * M ) that determine positive definite inner products on each tangent space of M .
For a fixed Riemannian metric g on M , we denote by D its Levi Civita connection and by R its Riemann curvature tensor, defined by R XY = D [X,Y ] − [D X , D Y ] for any pair of vector fields X,Y . The Ricci tensor, the scalar curvature and the volume form of g are denoted by Ric, s and µ, respectively. When it is necessary to specify the dependence on the given metric g, a superscript or subscript "g" is sometimes added.
We denote by H s (M ) the Sobolev space of real functions with partial derivatives in L 2 (M ) up to order s. Similarly, if π : E −→ M is a vector bundle, H s (E) is the space of sections of E with square integrable partial derivatives up to order s, so that C ∞ (E) = s≥0 H s (E) is the space of smooth sections of E.
Recall that, by Sobolev embedding's Theorem, H s -differentiability implies C s−[ n 2 ]−1 -differentiability. This allows to consider for any s > n 2 the collection M s of C 0 Riemannian metrics that are in H s (S 2 (T * M )), and the group D s+1 of H s+1 -diffeomorphisms of M , i.e. of C 1 -diffeomorphisms ϕ, with coordinate expressions of ϕ and ϕ −1 both in H s+1 . As it is shown in [9,10], M s and D s+1 are Hilbert manifolds that naturally include M and D, respectively, and D s+1 is a topological group acting on M s via a right action, which extends the standard right action of D on M, namely ϕ * g := ϕ * (g) with ϕ ∈ D s+1 and g ∈ M s . We finally recall that the tangent spaces T g M s and T ϕ D s+1 of M s and D s+1 , respectively, are naturally identifiable with the Hilbert spaces and that the scalar products Let M be an n-dimensional compact manifold. We recall that a Riemannian metric g on M is called (gradient) Ricci soliton if there exist f ∈ F(M ) and a constant c ∈ R so that Ric + Ddf − c · g = 0 . (2.1) If g is a Ricci soliton, there exists a unique c and a unique f , determined up to a constant, satisfying (2.1). The Einstein metrics are Ricci solitons of a special kind, with f constant. The non-Einstein Ricci solitons are called non-trivial. It is known that the non-trivial Ricci solitons on compact manifolds are necessarily shrinking, i.e. with c > 0 (see e.g. [11]), so that any Ricci soliton on M , which is not Ricci flat, can be rescaled to have (2.1) satisfied with the constant c = 1.
In particular, if we consider the set of pairs we have that the equivalence classes up to homotheties of the Ricci solitons that are not Ricci flat, are in one-to-one correspondence with the subset of P defined by We call the elements of this space normalised Ricci solitons.
Consider the map W : P −→ R, given by the restriction to triples The Euler-Lagrange equations for its critical points are (see e.g. [17,5]) so that any such critical point is in Sol (it satisfies the first of (2.3)). Conversely, if (g, f ) ∈ Sol, a well known argument (see e.g. [11]) shows that the function ∆ g f − 1 2 |∇f | 2 g + 1 2 s g +f is equal to a constant, which, by integration, is directly seen to be n + W (g, f ). This means that the following conditions are equivalent: and indicates a useful parallelism between Ricci solitons and Einstein metrics. In the following, we denote by S the Euler-Lagrange operator

Slices and pre-moduli spaces of Ricci solitons.
Let g ∈ M ⊂ M s be a C ∞ Riemannian metric and Iso(g) ⊂ D its isometry group. We recall that Ebin's Slice Theorem ( [9,10]) states that, for any s ≥ n 2 + 3, there exist -a D s+1 -invariant neighbourhood U s ⊂ M s of g, -a Hilbert submanifold S s ⊂ U s through g (called the Ebin slice), -a neighbourhood V s+1 ⊂ D s+1 / Iso(g) of the coset o = Iso(g) and a local cross section χ : determines a homeomorphism between S s × V s+1 and U s .
We remark that the Ebin slice S s is in fact the image under the exponential map exp : which is complementary to the tangent space T g (D s+1 * g). Such complementary subspace can be shortly described as follows. Consider the operators where L X is the Lie derivative along the vector field X and (e i ) is an arbitrary g x -orthonormal basis of T x M , x ∈ M . One can check that α g has injective symbol and that −δ g is the adjoint α From this, one gets the <, > g -orthogonal decomposition ( [10], Cor. 6.9) , which is then the tangent space T g S s at g.
Imitating Ebin's construction, for any Ricci soliton (g, f ) ∈ P we now want to construct a special submanifold S s f ⊂ M s with the same property of the Ebin slice, but with a tangent space T g S s f ⊂ T g M s characterised in a different way, more convenient for our purposes.
Let g ∈ M s and assume that f ∈ F(M ) is a smooth function, which is Iso(g)-invariant (this surely occurs when (g, f ) is a Ricci soliton). The corresponding twisted divergence is the operator (2.7) When g is considered as known, we will shortly write δ f instead of δ (g,f ) .
Notice that, in analogy with −δ, the operator −δ f is the formal adjoint of α g w.r.t. the "twisted" inner product on We have that and there exists a submanifold S s f ⊂ M s , which satisfies the same conditions (1) and (2) We follow the line of arguments of [10], Thm.7.1. Since α has injective symbol and −δ (g,f ) is the formal adjoint of α g w.r.t. the inner product <, > (g,f ) , one can check that δ (g,f ) • α g is elliptic and infer that (2.8) holds (see e.g. [10], Cor. 6.9).
Consider now the orbit O s := D s+1 * g in M s and the restricted tangent bundle T (M s )| O s over O s . For any g ′ = ϕ * (g) ∈ O s , consider the subspace of T g ′ M s defined by We claim that π : ν → O s is a smooth subbundle of π : T M s | O s → O s . In fact, one can consider the smooth family of inner products <, > (g ′ ,f ′ ) , determined by g ′ = ϕ * (g) ∈ O s and f ′ = ϕ * (f ), and the corresponding family of orthogonal projectors where, for shortness, we denote by α := α g and δ f ′ := δ (g ′ ,f ′ ) . The map P g ′ is actually well defined because the composition is a single valued map: to see this, one needs to observe that and that the self-adjoint elliptic operator Consider now the projection P : T M s | O s → T O s , given at any g ′ ∈ O s by the operator P g ′ and observe that ν = ker P . Following exactly the same arguments in [10], p. 31-32, one can check that P is a smooth map and hence that π : ν = ker P → O s is a smooth bundle, as claimed. Now, if exp : T M s −→ M s is the exponential map relative to the weak Riemannian metric (1.2), there exists a neighbourhood V ⊂ ν of the zero section, such that exp | V : V −→ U ⊂ M s is a diffeomorphism onto a D s+1invariant neighbourhood U of O s . By the same arguments of [9], p. 32-33, the submanifold S s f = exp(V ∩ ν| g ) satisfies all required conditions.
We call S s f the f -twisted slice at g in M s ; when g is smooth, we set S ∞ f := S s f ∩ M. As in [9] Thm. 7.4, one can check that also the set S ∞ f satisfies the conditions (1) and (2) of Ebin slices. For any s ≥ n 2 + 3, we denote by P s the Hilbert manifold which naturally extends P and we obviously extend (2.1) to the operator We also set Sol s := (S s ) −1 (0). Since any Ricci soliton of class C 2 is necessarily real analytic ( [8]), by standard arguments one gets that Sol s = Sol for every s ≥ n 2 + 3, in complete analogy with the Einstein case (see e.g. [15], Lemma 2.5).
Let (g, f ) ∈ Sol and recall that f is Iso(g)-invariant, so that one can consider the f -twisted slice S s f ⊂ M s through g and the subset S ∞ f = S s f ∩M of M. In analogy with [15] (see also [1], Ch. 12), we introduce the following: where S s f is a f -twisted slice through g and s ≥ n 2 + 3. Note that A (g,f ) is invariant under the natural action of Iso(g) on P s . Moreover, from the properties of slices, for any s ≥ n 2 + 3, the quotient (Sol∩U s )/D s+1 of a sufficiently small neighbourhood Sol∩U s of (g, f ) ∈ Sol s can be naturally identified with the quotient A (g,f ) / Iso(g) of the corresponding solitonic pre-moduli space A (g,f ) . This means that the local behavior of the moduli space Sol/D is determined by the quotients of the pre-moduli spaces by the actions of (finite dimensional!) groups of isometries.

The solitonic pre-moduli spaces are real analytic sets in finite-dimensional manifolds
Consider a C 1 -curve (g t , f t ) in Sol passing through a fixed Ricci soliton (g 0 , f 0 ) = (g, f ). The tangent vector (h =ġ| t=0 , a =ḟ | t=0 ) is necessarily in the kernel of the linearized operator dS (g,f ) of the operator (2.1). If furthermore (g t , f t ) takes values in a pre-moduli space A (g,f ) , its tangent vector (h, a) satisfies the additional condition δ (g,f ) h = 0. These observations lead to the following definition.
Definition 3.1. For a given (g, f ) ∈ Sol, the elements of the subspace ker dS (g,f ) ⊂ T (g,f ) P are called infinitesimal solitonic deformations (shortly, i.s.d.) of (g, f ). The i.s.d.'s (h, a) such that δ (g,f ) h = 0 are called essential.
Let us denote by Z (g,f ) the space of the essential i.s.d.'s at (g, f ) and set We will shortly see that Z s (g,f ) is actually identifiable with Z (g,f ) . In the next lemma, we give the explicit expression for the linearization dS (g,f ) of the operator S = (S 1 , S 2 ) at a normalised Ricci soliton (g, f ). In the following formulas, ∆ f is the twisted Laplacian (also called Bakry-Emery Laplacian or Witten Laplacian), acting on symmetric 2-tensors as

Lemma 3.2. For any
Proof.
By the classical formulas for the variations of Ricci tensors and Hessians (see e.g. [1,25]) we have that and Ric •h denotes the (2, 0)-tensor, associated by the metric g to the composition of Ric and h, viewed as (1, 1)-tensors. From Ricci soliton equation, the definition of δ f h and the fact that L (ı ∇f h) ♯ g = [Dh · ∇f ], equation (3.1) follows. Equation (3.2) follows from the computations in [17], p. 3332, together with the fact that dW | (g,f ) = 0 because (g, f ) ∈ P is a soliton.
From Lemma 3.2, we directly get the following We remark here that Z s (g,f ) = Z (g,f ) . Indeed the same argument as in the proof of Theorem 3.3 shows that Z s (g,f ) consists of elements (h, a) in T (g,f ) P s such that 2a = Tr(h) and 1 2 ∆ f h + R(h) = 0, hence h and a are smooth. The next theorem is an analogue of [15], Thm. 3.1 on Einstein metrics and is a crucial property of solitonic pre-moduli spaces.
, the claim follows from the Implicit Function Theorem in Hilbert spaces and the same arguments of the proof of [15], Thm. 3.1. Now, we observe that the tangent space We also note that dS s (V s (g,f ) ) = F s (W s (g,f ) ) , where F denotes the elliptic differential operator Hence, the proof reduces to showing that F s (W (g,f ) ) is a closed subspace of H s−2 (S 2 T * M ) × H s−2 (M ).
Consider the linear differential operator Lemma 3.5. For every g ∈ M s and f ∈ H s (M ), with s ≥ n 2 + 3, Proof. The claim is a consequence of the contracted second Bianchi identity 2 δ Ric = ds and the equalities As a corollary we have that for every (g ′ , u) ∈ P s So, taking the differential of (3.6) at (g, f ) and by the fact that S(g, f ) = 0, Using Lemma 3.2, the equality (3.7) can be rewritten as where we indicate by G s the elliptic operator Notice also that ker β g ∩ Im F s is a closed subspace, because F s is elliptic.
On the other hand, given (k, v) ∈ ker β g ∩ Im F s , we may write (k, v) = F s (h, u) for some (h, u) ∈ H s (S 2 T * M ) × H s (M ). Using (2.6) we can find some X ∈ H s+1 (T M ) such that h = α(X) + h 1 with α(X) := L X g and δ f h 1 = 0. Moreover we can decompose u = u o + u 1 , with u o ∈ R and u 1 satisfying M u 1 e −f µ g = 0. Since (k, v) ∈ ker β g , we have that G s (δ f k) = 0 and hence G s (δ f (α(X))) = 0, i.e. X ∈ V := ker(G s • δ f • α) ⊂ H s+1 (T M ), which is a finite dimensional space being G s elliptic. Substituting, we get that where F s (α(V ) × R) is a finite dimensional space. From (3.9) and (3.10), it follows that F s (W (g,f ) ) has finite codimension in the closed subspace ker β g ∩ Im F s . Since F s (W (g,f ) ) is the image of a bounded linear operator, a standard argument shows that F s (W (g,f ) ) is closed (see e.g. [20], p. 119).

Solitonic rigidity
In the following, we constantly assume s ≥ n 2 + 3. Definition 4.1. A normalised Ricci soliton (g, f ) is said to be solitonic rigid or, shortly, sol-rigid (resp. solitonic rigid in P s ) if there exists a neighbourhood U ⊂ P (resp. U ⊂ P s ) of (g, f ) such that U ∩ Sol consists only of the D-orbit (resp. D s+1 -orbit) of (g, f ).
By a classical result of Palais ([19]; see also [13] p. 53) if two normalised Ricci solitons are in the same D s+1 -orbit, they both lie in the same D-orbit. So, if (g, f ) is solitonic rigid in P s , it is automatically sol-rigid.
Example 4.2. The constant curvature metrics of the standard sphere S n and of the real projective space RP n are sol-rigid. In fact, by a result of Böhm and Wilking ( [3]), the shrinking Ricci solitons with 2-positive curvature have constant sectional curvature, so that any Ricci soliton on S n or RP n , which is close to the standard metric g o , is surely isometric to g o .
The next proposition is a direct consequence of the notion of pre-moduli spaces and Theorem 3.4 and gives a useful tool for proving sol-rigidity.  Let (g, f ) be a normalised Ricci soliton. If the space Z (g,f ) of essential i.s.d.'s is trivial, the soliton (g, f ) is sol-rigid.
Recall now that an Einstein metric g on M with Einstein constant c = 1 corresponds to a normalised Ricci soliton (g, f ) ∈ Sol with f constant and equal to f = log vol(M,g) (2π) n/2 . So, the set E of all (normalised) Einstein metrics can be identified with the subset of P given by For any (g, f ) ∈ E, we call space of essential infinitesimal Einstein deformations of (g, f ) the subspace of T (g,f ) P  b) If M = CP n = SU n+1 /S(U 1 × U n ), n ≥ 2, the space of essential infinitesimal Einstein deformations E (go,fo) is trivial, while the space of essential infinitesimal solitonic deformations Z (go,fo) is SU n+1equivariantly isomorphic to su n+1 .
Proof. (a) The sol-rigidity of M = S n and M = RP n have been already discussed in Example 4.2. For the cases M = HP n and M = Ca P 2 , by the results in [13], we know that E (go,fo) = 0. We claim that 2 ∈ Spec(−∆, F(M )) and this immediately implies (a) by Proposition 4.4 and Theorem 3.4. This claim can be checked using the results in [6], where the spectrum of the Laplacian is computed for every compact rank one symmetric space. However, in that paper, the Laplacian is computed w.r.t. the invariant metric g induced by the Cartan-Killing form of the Lie algebra of isometries. Since the Ricci tensor of such a metric is equal to 1 2 g o , the needed check corresponds to verify that 1 ∈ Spec(−∆ g , C ∞ (N )) for N = HP n or CaP 2 . From [6], we have that and the claim follows.
(b) When M = CP n = SU n+1 /S(U 1 × U n ), from [13], we have that E (go,fo) = 0. It remains to determine Z (go,fo) and this can be done using standard arguments of Representation Theory and some other results in [13], as follows. Note that the same arguments determine Z (go,fo) for other Hermitian symmetric spaces.
Let G = SU n+1 , K = S(U 1 × U n ) and consider the Cartan decomposition g = k ⊕ m of g = su n+1 , in which m is naturally identified with the tangent space T eK M at the origin eK ∈ G/K. Denote by B the Cartan-Killing form of g and by C the Casimir element (w.r.t. B) of the universal enveloping algebra U (g) of g. Recall that the G-invariant Riemannian metric g with g| eK = −B| m×m is the multiple of the Fubini-Study metric g o that satisfies Ric = 1 2 g. Since any G-homogeneous vector bundle π : E −→ M , with fiber W = E| eK , can be naturally identified with the bundle π : In particular, Z (go,fo) ≃ ker T , where T is the operator T : C ∞ (G, S 2 (m * )) K −→ C ∞ (G, S 2 (m * )) K , corresponding to ∆ + 2R : Lemma 4.6. The finite-dimensional G-module ker T is such that (4.5)

Proof.
By [13], Prop. 5.3, for any p ≥ 1, the action of the Casimir element C ∈ U (g) on the G-module C ∞ (G, p (m * )) K is identifiable with the differential operator C = −∆ − 2R + p 2 Id. This means that C| ker T = Id ker T and, in particular, that ker T has no trivial summand. On the other hand, it is known that, for any compact simple Lie algebra k, the Casimir operator acts as the identity on an irreducible complex k-module V if and only if V ≃ k C (see e.g. [14], p.654). Hence, (ker T ) C ∼ = m · g C for some integer m. By Frobenius reciprocity, m = dim C Hom K (g C , S 2 (m * ) C ) and, by standard arguments of Representation Theory, we have m = 2.
Consider now the Hodge-Laplacian eigenspace F = {f ∈ F(M )| − ∆f = f } and recall that, by standard facts on compact Kähler-Einstein manifolds (see e.g. [1], Ch. 2), the map ı : F(M ) −→ X(M ), ı(f ) = J(∇f ), gives a G-equivariant isomorphism between F and the Lie algebra g = su n+1 of Killing vector fields of (M = CP n , g o ). Consider the maps ψ 1 , ψ 2 : F −→ C ∞ (S 2 T * M ) defined by (see also [14], p.659). We want to show that ker T = Im ψ 1 ⊕ Im ψ 2 and that from which (b) will immediately follow. We first note that Im ψ 1 ⊂ ker T by definitions, while Im ψ 2 ⊂ ker T because of the following argument. Given a local orthonormal frame field {e i } i=1,...,n with D e i e j | xo = 0 at a fixed point Secondly, if h ∈ Im ψ 1 ∩ Im ψ 2 (i.e. h = Ddf 1 = f 2 g o for some f 1 , f 2 ∈ F ), since If n ≥ 2, it follows that df 1 = df 2 = 0, so that f 1 = f 2 = 0 and Im ψ 1 ∩ Im ψ 2 = {0}. From Lemma 4.6 and (4.6), we see that ker T = Im ψ 1 ⊕ Im ψ 2 and that all elements of ker T ∩ ker δ are of the form h = Ddf + 1 2 f g o for some f ∈ F . Remark 4.7. Theorem 3.4 (b) shows that in a neighborhood of the standard metric g o of M = CP n , the moduli space Sol/D has at most dimension n = dim su n+1 /SU n+1 . A detailed study of the solitonic pre-moduli space at the Fubini-Study metric g o will be the content of some of our future investigations.
The next proposition is another consequence of Proposition 4.4 and shows that Einstein metrics with sufficiently large positive curvatures are sol-rigid.

Other rigidity properties of Ricci solitons
Definition 5.1. An Einstein metric g is said to be weakly solitonic rigid if there is a neighbourhood U of g in M such that every Ricci soliton in U is Einstein.
The following proposition is a consequence of a deep recent result in [12].
Proposition 5.2. Let g be an Einstein metric with Einstein constant c > 0 and the diameter d. If then g is weakly solitonic rigid.
Proof. In [12] it is proved that a shrinking Ricci soliton (g, f ), satisfying Ric + Ddf − cg = 0 for some c > 0, is Einstein whenever its diameter d satisfies (5.1). If D : M −→ R is the continuous map where d(g ′ ) and Vol(g ′ ) are the diameter and the volume of g ′ , respectively, then U = D −1 ((−∞, 2( √ 2 − 1)π)) works in the definition of weak rigidity.
Remark 5.4. The blow-up of the complex projective plane at ν generic points, with ν ≥ 5, is a compact Kähler Einstein manifold ( [23]) with no non trivial holomorphic vector fields and H 2 (M, Θ) = 0 (see e.g. [16]). We do not know whether it also satisfies the above condition on the spectra of the Laplacian.