Supremum of Perelman's entropy and K\"ahler-Ricci flow on a Fano manifold

In this paper, we extend the method in [TZhu5] to study the energy level $L(\cdot)$ of Perelman's entropy $\lambda(\cdot)$ for K\"ahler-Ricci flow on a Fano manifold. Consequently, we first compute the supremum of $\lambda(\cdot)$ in K\"ahler class $2\pi c_1(M)$ under an assumption that the modified Mabuchi's K-energy $\mu(\cdot)$ defined in [TZhu2] is bounded from below. Secondly, we give an alternative proof to the main theorem about the convergence of K\"ahler-Ricci flow in [TZhu3].


Introduction
In this paper, we extend the method in [TZhu5] to study the energy level L(·) of Perelman's entropy λ(·) for Kähler-Ricci flow on an n-dimensional compact Kähler manifold (M, J) with positive first Chern class c 1 (M) > 0 (namely called a Fano manifold). We will show that L(·) is independent of choice of initial Kähler metrics in 2πc 1 (M) under an assumption that the modified Mabuchi's K-energy µ(·) is bounded from below (cf. Proposition 3.1 in Section 3). The modified Mabuchi's K-energy µ(·) is a generalization of Mabuchi's K-energy. It was showed in [TZhu2] that µ(·) is bounded from below if M admits a Kähler-Ricci soliton.
As an application of Proposition 3.1, we first compute the supremum of Perelman's entropy λ(·) in Kähler class 2πc 1 (M) [Pe]. More precisely, we prove that Theorem 0.1. Suppose that the modified Mabuchi's K-energy is bounded from below. Then Here the quantity N X (c 1 (M)) is a nonnegative invariance in K X and it is zero iff the Futaki-invariant vanishes [Fu]. We denote K X to be a class of K X -invariant Kähler metrics in 2πc 1 (M), where K X is an one-parameter compact subgroup of holomorphisms transformation group generated by an extremal holomorphic vector field X for Kähler-Ricci solitons on M [TZhu2]. We note that we do not need to assume an existence of Kähler-Ricci solitons in Theorem 0.1. In fact, if we assume the existence of Kähler-Ricci solitons then we can use a more direct way to prove Theorem 0.1 and that the supremum of λ(·) can be achieved in K X (cf. Section 1). It seems that the supremum of λ(·) can be achieved in the total space of Kähler potentials in 2πc 1 (M) if M admits a Kähler-Ricci soliton. In a special case of small neighborhood of a Kähler-Ricci soliton the positivity has been verified by computing the second variation of λ(·) in [TZhu4].
As another application of Proposition 3.1, we prove the following convergence result for Kähler-Ricci flow.
Theorem 0.2. Let (M, J) be a compact Kähler manifold which admits a Kähler-Ricci soliton (g KS , X). Then Kähler-Ricci flow with any initial Kähler metric in K X will converge to a Kähler-Ricci soliton in C ∞ in the sense of Kähler potentials. Moreover, the convergence can be made exponentially.
We note that that without loss of generality we may assume that a Kähler-Ricci soliton g KS on M is corresponding to the above X (cf. [TZhu1], [TZhu2]). Theorem 0.2 was first proved by Tian and Zhu in [TZhu3] by using an inequality of Moser-Trudinger type established in [CTZ] 1 . Here we will modify arguments in [TZhu5] in our general case that (M, J) admits a Kähler-Ricci soliton to give an alternative proof of this theorem. This new proof does not use such an inequality of Moser-Trudinger type. Moreover, in particular, in case that (M, J) admits a Kähler-Einstein metric this new proof allows us to avoid to use a deep result recentlly proved by Chen and Sun in [CS] for the uniqueness of Kähler-Einsteins in the sense of orbit space to give a self-contained proof to the main theorem in [TZhu5].
The organization of paper is as follows. In Section 1, we discuss an upper bound of λ(·) in general case-without any condition for µ(·) and show that the quantity (2π) −n [nV − N X (c 1 (M))] is an upper bound of λ(·) in K X (cf. Proposition 1.4). In Section 2, we will summarize to give some estimates for modified Ricci potentials of evolved Kähler metrics along Kähler-Ricci flow (cf. Proposition 2.3). In Section 3, we prove Proposition 3.1 and so do Theorem 0.1. Theorem 0.2 will be proved in Section 6. In Section 4, we improve our key Lemma 3.2 in Section 3 independent of time t (cf. Proposition 4.2). Section 5 is a discussion about an upper bound of λ(·) in K Y for a general holomorphic vector field Y ∈ η r (M). Section 7 is an appendix where we discuss the gradient estimate and Laplace estimate for the minimizers of Perelman's W -functional along the Kähler-Ricci flow.
1. An upper bound of λ(·) In this section, we first review Perelman's W -functional for triples (g, f, τ ) on a closed m-demensional Riemannian manifold M (cf. [Pe], [TZhu5]). Here g is a Riemannian metric, f is a smooth function and τ is a constant. In our situation, we will normalize volume of g by and so we can fix τ by 1 2 . Then the W -functional depends only on a pair (g, f ) and it can be reexpressed as follows: where R(g) is a scalar curvature of g and (g, f ) satisfies a normalization condition Then Perelman's entropy λ(g) is defined by It is well known that λ(g) can be attained by some smooth function f (cf. [Ro]). In fact, such a f satisfies the Euler-Lagrange equation of W (g, ·), Following Perelman's computation in [Pe], we can deduce the first variation of λ(g), where Ric(g) denotes the Ricci tensor of g and ∇ 2 f is the Hessian of f . Hence, g is a critical point of λ(·) if and only if g is a gradient shrinking Ricci-soliton which satisfies where f is a minimizer of W (g, ·). The following lemma was proved in [TZhu5] for the uniqueness of solutions (1.4) when g is a gradient shrinking Ricci soliton. In case that (M, J) is an n-dimensional Fano manifold, for any Kähler metric g in 2πc 1 (M), (1.1) is equal to Moreover, (1.6) becomes an equation for Kähler-Ricci solitons, where Ric(ω g ) is a Ricci form of g and L X denotes the Lie derivative along a holomorphic vector field X on M.
where h g is a Ricci potential of g andθ X,ωg is a real-valued potential of X associated to g defined by L X ω g = √ −1∂∂θ X,ωg with a normalization condition (1.9) Mθ X,ωg e hg ω n g = 0.
It was showed that there exists a unique X ∈ η r (M) such that Moreover, F X (v) ≡ 0, for any v ∈ η(M) if (M, J) admits a Kähler-Ricci soliton.
Let K X be an one-parameter compact subgroup of holomorphisms transformation group generated by X. We denote K X to be a class of K Xinvariant Kähler metrics in 2πc 1 (M). Let θ X,ωg be a real-valued potential of X associated to g with a normalization condition Clearly, θ X,ωg =θ X,ωg − c X for some constant c X which is independent of g ∈ K X .
By Jensen's inequality, it is easy to see The equality holds if and only if θ X,ωg = 0. This shows that N X (ω g ) is nonnegative and it is zero if and only if the Futaki-invariant vanishes [Fu]. Moreover, we have Proof. Choose a K-invariant Kähler form ω in 2πc 1 (M). Then for any Kähler metric g in K X there exists a Kähler potential ϕ such that the imaginary part of X(ϕ) vanishes and Kähler form of g satisfies Thus we suffice to prove Here we have used the fact θ X,ωtϕ = θ X,ω 0 + tX(ϕ).
By the above lemma, N X (·) is an invariance on K X , which is independent of choice of g. For simplicity, we denote this invariance by N X (c 1 (M)). The following proposition gives an upper bound of λ(·) in K X related to N X (c 1 (M)).
In fact, by using the facts R(g) = 2n + ∆h g and In the last equality above, we used the relation (1.8). Since X is extremal , we have Thus by (1. 2) for f = −θ X,ω together with Lemma 1.3, one will get (1.11).
In case that M admits a Kähler-Ricci soliton g KS , by Lemma 1.1, a minimizer f of W (g KS , ·) in K X must be −θ X . Thus for any g ∈ K X , by Proposition 1.4, we have Therefore we get the following corollary.
Corollary 1.5. Suppose that (M, J) admits a Kähler-Ricci soliton g KS . Then g KS is a global maximizer of λ(·) in K X and Remark 1.6. Corollary 1.5 implies that a Kähler-Einstein metric is a global maximizer of λ(·) in 2πc 1 (M) even with varying complex structures and Thus Corollary 1.5 also implies that the supremum of λ(·) in case that (M, J) admits a Kähler-Ricci soliton is strictly less than one in case that (M, J) admits a Kähler-Einstein metric.

Estimates for modified Ricci potentials
In this section, we summarize some apriori estimates for modified Ricci potentials of evolved Kähler metrics along Kähler-Ricci flow. Some similar estimates have been also discussed in [TZhu3] and [PSSW], we refer the readers to those two papers. We consider the following (normalized) Kähler-Ricci flow: where Kähler form of g is in 2πc 1 (M). It was proved in [Ca] that (2.1) has a global solution g t = g(t, ·) for all time t > 0. For simplicity, we denote by (g t ; g) a solution of (2.1) with initial metric g. Since the flow preserves the Kähler class, we may write Kähler form of g t as for some Käher potential φ = φ t . Let X ∈ η r (M) be the extremal holomorphic vector field on M as in Section 1 and σ t = exp{tX} an one-parameter subgroup generated by X. Let φ ′ = φ σt be corresponding Kähler potentials of σ ⋆ t ω φt . Then ω φ ′ will satisfy a modified Kähler-Ricci flow, where c is a constant and all Kähler potentials φ ′ = φ ′ t = φ ′ (t, ·) are in a space given by By using the maximum principle to (2.2) or (2.3), we get for some constants c t . Here h φ ′ are Ricci potentials of ω φ ′ which are normalized by The following estimates are due to G. Perelman. We refer the readers to [ST] for their proof.
Lemma 2.1. There are constants c and C depending only on the initial . Then by (2.2), we have Lemma 2.2. There exists a uniform C such that Proof. First we note that θ X,ω φ ′ is uniformly bounded in P X (M, ω) (cf. [Zhu1], [ZZ]). Then by (c) of Lemma 2.1, we have u X,φ ′ C 0 = u X,ω g ′ s C 0 ≤ C, ∀ s > 0 for some uniform constant C. Now we consider the flow (2.3) with zero as an initial Kähler potential and the background Kähler form ω g replaced by ω g ′ s . By an estimate in Lemma 4.3 in [CTZ], we see t ∇u X,ω g ′ In particular, we get Since the above estimate is independent of s, we conclude that the lemma is true.
Now we begin to prove the main result in this section.
Proposition 2.3. Suppose that µ(·) is bounded from below in P X (M, ω). Then we have: Then by (2.6), one sees that there exists a sequence of Thus by using a differential inequality where C is a uniform constant (cf. [PSSW]), we get

The claim immediately implies (a) of Proposition 2.3 by the normalization conditions
To prove the claim, we need to use an inequality 1 2 .
(2.9) can be proved by using the non-collapsing estimate (b) in Lemma 2.
for some uniform constant C. Therefore, by using the following lemma we prove (b) and (c).

Proof of Theorem 0.1
According to [TZhu5], an energy level L(g) of entropy λ(·) along Kähler-Ricci flow (g t ; g) is defined by By the monotonicity of λ(g t ), we see that L(g) exists and it is finite. In this section, our goal is to prove Proposition 3.1. Suppose that the modified Mabuchi's K-energy is bounded from below in K X . Then for any g ∈ K X . c 1 (M)).
The above proposition shows that the energy level L(g) of entropy λ(·) does not depend on the initial Kähler metric g ∈ K X . Thus by using the Kähler-Ricci flow (g t ; g) for any Kähler metric g ∈ K X and the monotonicity of λ(g t ), we will get Theorem 0.1.
To prove Proposition 3.1, we need the following key lemma.
Lemma 3.2. Let f t be a minimizer of W (g t , ·)-functional associated evolved Kähler metric g t of (2.1) at time t and h t a Ricci potential of g t which satisfying the normalization (2.5) . Then there exists a sequence of

Proof. Lemma 3.2 is a generalization of Proposition 4.4 in [TZhu5
]. We will follow the argument there. First by (1.5), it is easy to see that It follows that Since λ(g t ) ≤ W (g t , 0) = (2π) −n nV are uniformly bounded, we see that there exists a sequence of t i ∈ [i, i + 1] such that Note that f t is uniformly bounded [TZhu5]. Hence we see that that (a) of the lemma is true. By (a), we also get This proves (b) of the lemma. It remains to prove (c). Hence, following an argument in the proof of Proposition 4.4 in [TZhu5], we will get estimates where q + t = max{q t , 0} and q − t = min{q t , 0}. Consequently, we derive It follows This completes the proof of claim.
Since equation (1.4) is equivalent to where v t = e −f t 2 , by Lemma 2.1, it is easy to see Then by the standard Moser's iteration, we get from (3.9), This implies (3.8), so we obtain (c) of the lemma.
Proof of Proposition 3.1. Note that R(gt) 2 = n+ 1 2 ∆h t , where ∆ is the Beltrima-Laplacian operator associated to the Riemannian metric g t . Then Thus by (a) of Lemma 3.2, one sees that there exists a sequence of time t i such that On the other hand, since the modified Mabuchi's K-energy is bounded from below, we see that (a) of Proposition 2.3 is true. Then by (c) of Lemma 3.2, it follows Here we used a fact σ ⋆ t θ X,ωg t = θ X,ω g ′ t since X lies in the center of η r (M) (3.13) By combining (3.11) and (3.13), we get Therefore, by using the monotonicity of λ(g t ) along the flow (g t ; g), we obtain (3.1).
It was showed in [TZhu4] that a Kähler-Ricci soliton is a local maximizer of λ(·) in the Kähler class 2πc 1 (M). Together with Corollary 1.5, one may guess that a Kähler-Ricci soliton is a global maximizer of λ(·). More general, according to Theorem 0.1 , we propose the following conjecture.

Improvement of Lemma 3.2
In this section, we use Perelman's backward heat flow to improve estimate (c) in Lemma 3.2 independent of t. Moreover, we show the gradient estimate of f t + h t also holds. Although Lemma 3.2 is sufficient to be applied to prove Theorem 0.1 and Theorem 0.2, results of this section are independent of interests. We hope that these results will have applications in the future.
Fix any t 0 ≥ 1. We consider a backward heat equation in t ∈ [t 0 − 1, t 0 ], with an initial f t 0 (t 0 ) = f t 0 . Clearly, the equation preserves the normalizing condition 1 V M e −ft 0 (t) ω n gt = 1. Moreover, by the maximum principle, we have since △h t are uniformly bounded. Here the constant C(g) depends only on the initial metric g of (2.1). Similarly to (1.5), we can compute (4.3) By using (4.3), we want to prove Lemma 4.1.
Proof. First by (4.3), one sees It follows Thus by using the weighted Poincaré inequality as in (3.4) in last section , we will get where a t = 1 V M h t e ht ω n gt , by a straightforward calculation, we see Then by Lemma 3.2, we get Notice that We can also estimate Hence we derive Therefore, according to (4.5) and (4.6) will implies Consequently, we get (4.4).
Proposition 4.2. (4.7) Proof. With the help of Lemma 4.1, by using same argument in the proof of (c) in Lemma 3.2, we can prove that So we suffice to prove (4.9) ∇(f t + h t ) gt → 0, as t → ∞.
We will use the Moser's iteration to obtain (4.9 ) as in lemma 7.2 in Appendix. We note by (4.8) and Theorem 7.1 that (4.10) where q t = f t + h t satisfies an equation Then substituting the above two inequalities into (4.11), we get By using Zhang's Sobolev inequality [Zha], we deduce where ν = n n−1 . To run the iteration we put p 0 = 1 and p k+1 = p k ν + ν, k ≥ 0. Hence for a constant γ(n) depending only on n, where we have used the fact p k ≤ 2ν k for k ≥ 1. Therefore by (4.10), we prove

5.
Another version of the invariance N X (ω g ) Let Y ∈ η r (M) so that Im(Y ) generates an one-parameter compact subgroup of K. Denote K Y to be a class of K Y -invariant Kähler metrics in 2πc 1 (M). Then according to the proof of Proposition 1.4, we actually prove where X is the extremal vector field as in Section 1.
Proof. Choose a constant c Y so thatθ Y,ωg = θ Y,ωg + c Y satisfies a normalization condition Mθ Y,ωg e hg ω n g = 0. (5.2) Thenθ Y,ωg satisfies an equation ∆θ X,ωg + X(h g ) +θ X,ωg = 0 Thus using the integration by part, we havẽ We compute the first variation of H(Y ) in η r (M). By the definition of Thus we get Therefore, by [TZhu2], we see that there exists a unique critical X ∈ η r (M) of H(·) such that Similarly, we have Thus b(t) ′ ≥ 0. Consequently This means that H(·) is a concave functional on η r (M). It follows that X is a global maximizer of H(·). Therefore we prove the proposition by using the fact H(X) = N X (c 1 (M)).
Corollary 5.2. Let K K be a class of K-invariant Kähler metrics in 2πc 1 (M). Suppose that Here we avoid to use their theorem so that we can generalize the proof to the case of Kähler-Ricci solitons by applying Proposition 3.1. As in [TZhu5], we write an initial Kähler form ω g of Kähler-Ricci flow (2.1) by for a Kähler potential ϕ on M. We define a path of Kähler forms and set I = {s ∈ [0, 1]| (g s t ; g s ) converges to a Kähler-Ricci soliton in C ∞ in sense of Kähler potentials}.
Clearly, I is not empty by the assumption of existence of Kähler-Ricci solitons on M. We want to show that I is in fact both open and closed. Then it follows that I = [0, 1]. This will finish the proof Theorem 0.2.
The openness of I is related to the following stability theorem of Kähler-Ricc flow, which was proved in [Zhu2].
By the continuity of λ(·), we will get (6.6). Now by Corollary 1.4 together with (6.6), we see that ω φ∞ is a global maximizer of λ(·) in K X , so it is a critical point of λ(·). Then it is easy to show that ω φ∞ a Kähler-Ricci soliton with respect to X by computing the first variation of λ(·) as done in (1.5). Thus by the uniqueness result for Kähler-Ricci solitons in [TZhu1], we get where σ ∈ Aut r (M). Since φ ∞ ∈ Λ ⊥ (ω KS ), by Lemma 6.3, φ ∞ must be zero. This is a contradiction to (6.5). The contradiction implies that (6.2) is true.

Appendix
In [TZhu5], it was proved the minimizer f t of W (g t , ·)-functional associated to evolved Kähler metric g t of Kähler-Ricci flow (2.1) is uniformly bounded (see also [TZha]). In this appendix, we show that the gradients of f t are also uniformly bounded, and so are △f t by (1.4). Namely, we prove Theorem 7.1. There is a uniform constant C such that f t + ∇f t + △f t ≤ C, ∀ t > 0.
We will derive ∇f t in Theorem 7.1 by studying a general nonlinear elliptic equation as follows: where the Laplace operator △ is associated to a Kähler metric g in 2πc 1 (M) and F is a smooth function on M ×R + , which satisfies a structure condition: Here 0 ≤ A, B ≤ ∞, 0 ≤ α < 2 n are constants, and H is a proper function on R + which satisfies a growth control at 0: (7.3) lim sup t→0 tH(t) < ∞.
Lemma 7.2. Let w is a positive solution of (7.1). Then (1 + |∇w| 2 )dV g 1/2 , where C s is a Sobelev constant of g and h is a Ricci potential of g.
Proof. We will use the Moser's iteration to L p -estimate of |∇w|. By the Bochner formula, we have