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Stability of K\"ahler-Ricci flow

We prove the convergence of K\"ahler-Ricci flow with some small initial curvature conditions. As applications, we discuss the convergence of K\"ahler-Ricci flow when the complex structure varies on a K\"ahler-Einstein manifold.


The motivation
In [17], R. Hamilton introduced the famous Ricci flow which deforms any Riemannian metric in the direction of negative Ricci curvature. If the Ricci flow exists globally, this will lead to the existence of some canonical geometric structure (either Einstein metrics or some non-trivial solitons). Unfortunately, the flow usually will develop singularities in finite time. One key issue is to study the formation of singularities over finite time [19]. According to R. Hamilton, the singularity can be divided roughly into two types: the fast forming one (Type I) and the slow forming one (Type II). From the PDE point of view, the Type I singularity is more "gentle" where the curvature is controlled by 1 T −t . We will focus our discussions on the "fast forming" singularities now.
Let g be any Riemannian metric and we start the Ricci flow from g. Suppose the maximal existence time is T (g) < ∞. Suppose further that this is a Type I singularity. One intriguing question one may ask is: for any metric in a small neighborhood of g, will the maximal existence time be T (g) also? Will that be Type I? Note that Einstein metrics with positive scalar curvature, or Ricci solitons (either stable or shrinking), develop exactly finite time Type I singularities. The question above can be simplified to Question 1.1. [5] Let g be a stable or shrinking Ricci soliton. Does there exist a small neighborhood where Ricci flow will still develop only Type I singularities? Will the Ricci flow asymptotically converge to some Ricci soliton?
This is a very interesting question. Unfortunately, the answer in full generality should be negative: Even if we assume a metric g is C ∞ close to an Einstein metric with positive scalar curvature, it is not clearly at all whether the flow will only develop Type I singularities, and whether the flow will converge to an Einstein metric after proper re-scaling. Nonetheless, it is an intriguing question to find a set of suitable geometric conditions so that this question has an affirmative answer. This type of problem is called linear stabilities and there are lots of research on this topic (cf. [24], [11], [16], [36] etc).
One way to approach this problem is to associate certain functional to Ricci flow. In general setting, the first choice of course will be Perelman's entropy functional. Unfortunately, Perelman's entropy functional depends on the blowing up time T and we have no idea how T will vary when we vary initial metric.
It is now clear why we should study this important problem on Kähler manifolds first. In any compact manifold, for the Ricci flow initiating from any Kähler metric, the blowing up time is not only a priori determined, but more importantly it can be explicitly computed in terms of the first Chern class and the Kähler class. Therefore, Question 1.1 will be highly interesting and feasible with present technology when restricted to the Kähler setting. A more modest question one may ask is Question 1.2. [5] Let (M, g, J) be a Kähler-Einstein manifold. Does there exist a small neighborhood of the Kähler-Einstein metric such that the Kähler-Ricci flow will converge to a Kähler-Einstein metric? Theorem 1.4. Let (M, g KE , J KE ) be a Kähler-Einstein manifold with c 1 (M) > 0, and no nonzero holomorphic vector fields. For any Kähler metric ω g ∈ 2πc 1 (M)∩H 1,1 (M, J) with possibly different complex structure J satisfying (1.1)

On the convergence of Kähler-Ricci flow
In our previous paper [7], we proved some convergence theorems for the Kähler-Ricci flow with certain initial energy and curvature conditions. Here we will refine those arguments and prove the following type of stability results : For any γ, Λ > 0, there exists a small positive constant ǫ(γ, Λ) > 0 such that for any metric g in the subspace of Kähler metrics where λ 1 (g) is the first eigenvalue of the metric ω g and Ca(ω g ) denotes the (normalized) Calabi energy, the Kähler-Ricci flow with the initial metric ω g will converge exponentially fast to a Kähler-Einstein metric.
The assumption (1.2) implies that (M, J) has no non-zero holomorphic vector fields. Note that we don't assume the existence of Kähler-Einstein metrics on (M, J).
One might tempt to think that condition (1.2) implies the existence of Kähler-Einstein metrics. That might be true (one need to address a potential collapsing issue with (1.2)), except that it is a Kähler-Einstein metric with possibly different complex structures. The difficulty is really created by the fact the space of complex structure modulo diffeomorphisms is not a Hausdorff space.
As in Theorem 1.6, when (M, J) has non-zero holomorphic vector fields, we need to assume the pre-stable condition. 3) The proof of the main theorems is more tricky than that in [7], but the ideas are the same. First by Sprouse's result in [30] the smallness of the Calabi energy implies that the L ∞ norm of the traceless Ricci curvature is small after a short time(cf. Proposition 4.1), which further implies that the eigenvalue is strictly great than 1 (cf. Lemma 3.1). The eigenvalue estimates can be used to prove the exponential decay of the traceless Ricci curvature for a short time (cf. Theorem 3.6 and 3.7), which implies the full curvature tensor is uniformly bounded for a short time by Yau's estimates (cf. Theorem 3.8). However, the boundedness of the full curvature tensor in turn implies the rough curvature estimates for the next time interval, and we can repeat the previous arguments. Using this "iteration" idea we can actually prove that the full curvature tensor is uniformly bounded for all time.
In our subsequent papers, we will remove the condition on the Futaki invariant and the bound of the full curvature tensor, and give more general results on the relation between the pre-stable condition and the convergence of Kähler-Ricci flow. Recently, Tian-Zhu [34] proved very interesting and much stronger results on stability of Kähler-Ricci flow by using Perelman's W -functional. the continuous support and encouragement during the course of this work. The second named author would also like to thank Professor W. Y. Ding and X. H. Zhu for their help and some enlightening discussions.

Preliminaries
Let M be a compact Kähler manifold with c 1 (M) > 0. Choose an initial Kähler metric g with the Kähler form ω g ∈ 2πc 1 (M). By the Hodge theorem, any Kähler form in the same Kähler class can be written as for some real potential function ϕ on M. The Kähler-Ricci flow (cf. [17]) on a Kähler manifold M is of the form ∂g ij ∂t = −R ij + g ij , ∀ i, j = 1, 2, · · · , n.
(2.1) It follows that on the level of Kähler potentials, the Kähler-Ricci flow becomes where h g is defined by Let R be the the average of the scalar curvature, which is a constant depending only on the Kähler class and the underlying complex structure. Then the normalized Calabi energy (cf. [1] [2]) is defined by Since the Kähler metric ω ϕ is in the canonical class, we can check that Define the Futaki invariant by for any holomorphic vector field X on M. It is well-known that the Futaki invariant doesn't depend on the particular representative we choose in the Kähler class.
We recall some basic results from our previous papers. First, by the tensor maximum principle we have the following basic lemma: [7]) Suppose that the curvature of the initial metric satisfies the following condition there exists a constant T (Λ) > 0, such that we have the following bound for the evolving Kähler metric (2.5) Lemma 2.1 is slightly different from Lemma 12 in [6], but the idea of the proof is the same. We remind the readers that T doesn't depend on the bound of the traceless Ricci curvature, which is useful for the proofs of the main theorems. Now we state a parabolic version of Moser iteration argument (cf. [8]). Theorem 2.2. Suppose the Poincare constant and the Sobolev constant of the evolving Kähler metrics g(t) are both uniformly bounded by σ, and the scalar curvature R(g(t)) has a uniform lower bound Λ. If a nonnegative function u satisfying the following inequality where |f | L p (M,g(t)) is uniformly bounded by some constant c for some p > n 2 , then for any t ∈ (a, b) and τ ∈ (0, b − a), we have 1

Remark 2.3.
Recently, Q. Zhang [38]and R. Ye [37] proved that the Sobolev constant is uniformly bounded along the Kähler-Ricci flow without any assumptions. Here we don't need to use this result.
The parabolic Moser iteration theorem is the main tool in the paper. It can be applied to control the pointwise norm of the traceless Ricci curvature, provided that the Calabi energy is sufficiently small. Once we have the bound of the traceless Ricci curvature, we can estimate the potential function and the full curvature tensor along the Kähler-Ricci flow by Yau's estimates: [7], [9], [35]) For any positive constants Λ, B > 0 and small η > 0, there exists a constant C 1 depending only on Λ, B, η such that if the background metric ω satisfies and the potential function |ϕ(t)|, |φ(t)| ≤ B, then We state the following well-known result on the estimate of the Sobolev constant. The readers are referred to [10] [22] for details.

Estimates
In this section, we will prove several results which will be useful in the proof of the main theorems.

The first eigenvalue of the Laplacian operator
To prove the exponential decay of the traceless Ricci curvature, we need to estimate the first eigenvalue of the evolving Laplacian operator. The calculation of the eigenvalue along the Ricci flow is well-known in literatures(cf. [4] for example).
2. If |Ric − ω|(t) ≤ ǫe −αt for some α > 0 and for all t ∈ [0, T ], then Taking the derivative with respect to t, we have we calculate the derivative of λ 1 (t) Applying (3.1), we have that The assumption (1) implies Hence, we get the inequalities The first part of the lemma follows immediately. Similarly we can prove the second part.

The pre-stable condition
In this section, we estimate the first eigenvalue when M has non-zero holomorphic vector fields.
Here we follow closely the argument in [7]. First, we recall the following definition in [7]: The complex structure J of M is called pre-stable, if no complex structure in the closure of its orbit of diffeomorphism group contains larger (reduced) holomorphic automorphism group.
Remark 3.3. The "pre-stable" condition was defined in [7], but it is well-known in previous literatures. In the statement of Theorem 1.8 of [6], the first named author used this condition to study the convergence to Kähler-Ricci flow, and in [26] Phong-Sturm defined a stability condition as "condition (B)". These two definitions are essentially the same and we called it "pre-stable" in [7].
Now we recall some basic facts of the first and second eigenvalues of the Laplacian operator acting on functions. For any smooth function where λ 1 is the first eigenvalue of ∆ g . If we assume the function f is perpendicular to the first eigenspace of ∆ g then we get These inequalities can be proved by the eigenvalue decomposition of the function f . Note that for a Kähler-Einstein manifold (M, ω KE ) with non-zero holomorphic vector fields, it is wellknown that the first eigenvalue λ 1 = 1 and the first eigenspace, which we denote by η(M), is isomorphic to the space of holomorphic vector fields. Now we need the following convergence result of a sequence of Kähler metrics, which is well-known in literature (cf. [26], [31]).

Proposition 3.4.
Let M be a compact Kähler manifold. Let (g(t), J(t)) be any sequence of metrics g(t) and complex structures J(t) such that g(t) is Kähler with respect to J(t). Suppose the following is true: is uniformly bounded for any integer l(0 ≤ l < k); 2. The injectivity radii i(M, g(t)) are all bounded from below; 3. There exist two uniform constant c 1 and c 2 such that 0 < c 1 ≤ Vol(M, g(t)) ≤ c 2 .
Then there exists a subsequence of t j , and a sequence of diffeomorphism F j : M → M such that the pull-back metricsg(t j ) = F * j g(t j ) converge in C k,α (∀ α ∈ (0, 1)) to a C k,α metric g ∞ . The pull-back complex structure tensorsJ(t j ) = F * j J(t j ) converge in C k,α to an integral complex structure tensorJ ∞ . Furthermore, the metric g ∞ is Kähler with respect to the complex structurẽ J ∞ .
These being understood, we have the: then for any smooth function f satisfying we have the following where γ > 0 depends only on ǫ, Λ 0 and Λ 1 .
Proof. The inequality (3.7) was proved in [7], and the argument also works for (3.8). For the readers' convenience, we give the details here. Suppose not, for any positive numbers ǫ m → 0, there exists a sequence of Kähler metrics where the smooth functions f m satisfy where 0 < γ m → 0. Without loss of generality, we may assume that which means Note that the complex structure J ∞ lies in the closure of the orbit of diffeomorphisms, while ω ∞ is a Kähler-Einstein metric of (M, J ∞ ). By the standard deformation theorem in complex structures, we have dim Aut r (M, J) ≤ dim Aut r (M, J ∞ ).
By abusing notation, we can write Now, let f ∞ be the W 3,2 limit of f m , then by (3.12) and (3.13) we have Thus, f ∞ is a non-trivial function. Since ω ∞ is a Kähler-Einstein metric, we have This implies that f ∞ is perpendicular to the first eigenspace of △ ω∞ . In other words, there is a δ > 0 such that However, this contradicts the following fact: The lemma is then proved.

The exponential decay of traceless Ricci curvature
In this section, we will use the estimates of the first eigenvalue in the previous subsection to prove the exponential decay of traceless Ricci curvature.
Theorem 3.6. Suppose for any time t ∈ [0, T ], Then for fixed τ < T , there exists ǫ 0 (H, γ) > 0 such that if ǫ ∈ (0, ǫ 0 ) we have Proof. First, we prove that the Calabi energy decays exponentially (3.14) In fact, direct calculation shows If we choose ǫ sufficiently small, the inequality (3.14) follows immediately. Now applying the parabolic Moser iteration, the estimate of the Calabi energy implies the exponential decay of the traceless Ricci curvature. In fact, since the traceless Ricci tensor u = |Ric − ω| 2 (t) satisfies the following inequality: By the parabolic Moser iteration Theorem 2.2, for t ∈ [τ, T ] we have Here we have used the fact that the Sobolev constant is uniformly bounded since the traceless Ricci curvature is small by the assumption. Note that at the initial time Ca(0) ≤ H 2 ǫ 2 , the above inequality implies The theorem is proved.
When M has non-zero holomorphic vector fields, we need to use the pre-stable condition to get the exponential decay of the Ricci curvature.

Theorem 3.7. Suppose that M is pre-stable and the Futaki invariant vanishes, and for any time
Then for fixed τ < T , there exists ǫ 0 (H, γ) > 0 such that if ǫ ∈ (0, ǫ 0 ) we have Here γ is the constant obtained in Theorem 3.5.
Proof. The argument is essentially the same as the proof of Theorem 3.6. In fact, by Shi's results in [29] all the derivatives of the curvature tensor are bounded after a short time t = τ 2 , where τ is a fixed number as in Theorem 3.6. Therefore the assumption (3.5) in Theorem 3.5 is satisfied for t ∈ [ τ 2 , T ] and we have the inequalities (3.7) and (3.8). Note that the function Therefore, we have the inequality where γ > 0 is the constant obtained in Theorem 3.5. Then by the same argument in the proof of Theorem 3.6 we have the decay of the Calabi energy Applying the parabolic Moser iteration as in the proof of Theorem 3.6, we get the exponential decay of the traceless Ricci curvature. The theorem is established.

The curvature tensor
In this section, we use the exponential decay of the traceless Ricci curvature to estimate the curvature tensor along the Kähler-Ricci flow. then the full curvature tensor is uniformly bounded where Λ is the bound of the curvature tensor with respect to the background metric ω g .

Remark 3.9. Phong-Song-Sturm-Weinkove proved a similar result in their paper (cf. Lemma 6 in [27]). However, the proof here is elementary and we don't need to use Perelman's estimates.
First we recall a useful normalization for the solution ϕ(t) of the Kähler-Ricci flow. Observe that for any solution ϕ(t) of Kähler Ricci flow the functionφ(t) = ϕ(t) + Ce t also satisfies the above equation for any constant C. Since ∂φ ∂t (0) = ∂ϕ ∂t (0) + C, we havec(0) = c(0) + C, wherẽ Thus we can normalize the solution ϕ(t) such that the average ofφ(0) is any given constant.
Recall that in Chen-Tian's paper [8], if the K-energy is bounded from below, we can normalize the solution such that c(t) is uniformly bounded for all t > 0. However, in our case we have no information about the K-energy. To overcome this difficulty, we estimate c(t) as follows: Lemma 3.10. (cf. [8])Suppose that for t ∈ [0, T ] for some constant C 4 > 0. Then we can normalize the solution ϕ(t) so that Then for all time t ∈ [0, T ], we have the following estimates

Proof. A simple calculation yields
Now we normalize our initial value of c(t) as From the equation for c(t), we have Thus, we have Proof of Theorem 3.8. We will use Theorem 2.4 to bound the full curvature tensor. It suffices to bound ϕ, ∂ϕ ∂t for time t ∈ [0, T ]. Here we need to normalize ϕ(t) such that its average c(t) has good estimates. Note that the normalization in Lemma 3.10 depends on T , which is different from [8].
Here we omit the Sobolev constant in the constant Λ 0 because the traceless Ricci curvature is very small.

Proof of Theorem 1.8
In this section, we follow the idea of our previous paper [7] to prove Theorem 1.8. The proof needs the technical condition that the first eigenvalue of the initial metric is strictly greater than 1, which will be removed in Section 6 by the pre-stable condition.

The traceless Ricci curvature
In this subsection, we will prove that under the assumption of Theorem 1.8 the traceless Ricci curvature is small after a short time. In fact, we have the following proposition: Proposition 4.1. Given a Kähler metric ω g satisfying the properties for some constants λ, Λ > 0 and sufficiently small ǫ > 0. Then for the solution ω(t) of the Kähler-Ricci flow with the initial metric ω g , there exists T (Λ) and ǫ 0 (γ, Λ) > 0 such that if ǫ ∈ (0, ǫ 0 ) then at time T the metric ω(T ) satisfies
First, we will prove that the Sobolev constant is uniformly bounded if the Calabi energy is small and the Ricci curvature has a lower bound. We will follow an approach taken by [30] and also [9]. Recall that C. Sprouse proved the following lemma in [30]:   Proof. By the curvature assumption and (3.15) we get The lemma follows immediately.
Since the initial Calabi energy is small, it follows from Lemma 4.5 and (4.6) that So it remains to prove λ 1 (T ) > 1 + γ 2 provided ǫ is sufficiently small. We remind the readers that it cannot be proved by Lemma 3.1 since the traceless Ricci curvature may not be small near t = 0. However, we have the lemma: where ǫ = 3C(Λ, n)σǫ Combining (4.14)-(4.16) with (4.12), we get Applying the maximum principle, we get Now using Lemma 4.6, we have the estimate The first eigenvalue λ 1 (T ) > 1 + γ 2 if ǫ is small enough. The proposition is established.

The iteration argument
Thanks to Proposition 4.1, we only need to prove the convergence of Kähler-Ricci flow under the assumption that the traceless Ricci curvature is sufficiently small. In other words, Theorem 1.8 follows immediately from the following: the Kähler-Ricci flow with the initial metric ω g will converge exponentially fast to a Kähler-Einstein metric.
Proof. The proof consists of several parts.

STEP 1.
In this step we give estimates near the initial time. Consider the Kähler-Ricci flow (2.2) with the normalization condition (2.3) and the assumption (4.17). By Lemma 2.1, the maximum principle implies that there exists T 1 (Λ) > 0 such that (4.18) By Lemma 3.1, for sufficiently small ǫ we have the estimate Since we have the eigenvalue estiamte, by Theorem 3.6 the traceless Ricci curvature decays exponentially where α = γ 6 . Since the traceless Ricci curvature is small for t ∈ [0, τ ] by (4.18), we can assume (4.19) holds for all t ∈ [0, T 1 ]. Now we choose ǫ sufficiently small such that By Theorem 3.8, the full curvature tensor has the estimate

STEP 2.
Recall that in step 1 we have the estimates at time t = T 1 , (4.20) By Lemma 2.1, the maximum principle implies that there exists T 2 (Λ 1 ) > 0 such that By Lemma 3.1 for sufficiently small ǫ the first eigenvalue has the estimate As in the proof of Theorem 3.6, the parabolic Moser iteration implies the traceless Ricci curvature where we have used the estimates (4.21) and (4.23). Set H 2 = max{H 1 , C 2 (2Λ 1 , γ 2 , τ )H 1 e −α T 1 2 }. Then we get the inequality then there exists k 0 ∈ N such that if k ≥ k 0 and ǫ is sufficiently small (depending on k 0 ), the above estimates still hold for all t ∈ [T 1 + (k − 1)T 2 , T 1 + kT 2 ].
Proof. The maximum principle and the definition of T 2 imply that Then by Lemma 3.1 the first eigenvalue if we choose ǫ sufficiently small (depending on H k ). Here we have used the inequality Then by (3.16) the Calabi energy decays exponentially where H k+1 is defined by Then there exists k 0 ∈ N such that for all k ≥ k 0 Hence, we have the estimate for k ≥ k 0 , (4.28) We choose ǫ small such that H k 0 ǫ < 1. Therefore, by Theorem 3.8 and the definition of Λ 1 we have The lemma is proved. STEP 4. By Lemma 4.8, the bisectional curvature is uniformly bounded and the traceless Ricci curvature decays exponentially, and therefore the W 1,2 norm of ∂ϕ ∂t − c(t) decays exponentially. Then following the argument in [8], the Kähler-Ricci flow will converge exponentially fast to a Kähler-Einstein metric. This theorem is proved.

Proof of Theorem 1.4
In this section, we prove Theorem 1.4. The key observation is that if a Kähler metric g with possibly different complex structure J is sufficiently close to a Kähler-Einstein metric (g KE , J KE ), the conditions in Theorem 1.8 or Theorem 4.7 are automatically satisfied, and Theorem 1.4 follows immediately.
Proof of Theorem 1.4. Let (g, J) be a Kähler metric with for sufficiently small ǫ > 0, we need to check that (g, J) satisfies the assumption (4.17) of Theorem 4.7. In fact, since (g KE , J KE ) has no holomorphic vector fields, the first eigenvalue of the Laplacian ∆ KE is strictly greater than 1, and by Lemma 5.1 λ 1 (g) ≥ λ 1 (g KE )(1 − ǫ) > 1.
6 Proof of Theorem 1.6 and 1.10 In this section, we will use the pre-stable condition to drop the assumptions that M has no nonzero holomorphic vector fields, and the dependence of the initial first eigenvalue of the Laplacian. The idea of the proof is similar to that in section 4.
First, we will prove Theorem 1.10. As in Proposition 4.1 we prove that the traceless Ricci curvature is small along the Kähler-Ricci flow after a short time. for some constant C(Λ).