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arXiv:math/0703486v1 [math.DG] 16 Mar 2007

K¨AHLER-RICCI FLOW ON A TORIC MANIFOLD

WITH POSITIVE FIRST CHERN CLASS

Xiaohua Zhu∗

Abstract. In this note, we prove that on an n-dimensional compact toric manifold with

positive first Chern class, the K¨ ahler-Ricci flow with any initial (S1)n-invariant K¨ ahler

metric converges to a K¨ ahler-Ricci soliton. In particular, we give another proof for the

existence of K¨ ahler-Ricci solitons on a compact toric manifold with positive first Chern

class by using the K¨ ahler-Ricci flow.

0. Introduction.

Let M be a compact toric manifold with positive first Chern class. Let T∼= (S1)n×Rn

be a maximal torus which acts on M and K0∼= (S1)nbe its maximal compact subgroup.

In this note we discuss a K¨ ahler-Ricci flow with a K0-invariant initial metric on M and

we shall prove

Main Theorem. On a compact toric manifold M with positive first Chern class, the

K¨ ahler-Ricci flow with any initial K0-invariant K¨ ahler metric converges to a K¨ ahler-Ricci

soliton. In particular, it shows that there exists a K¨ ahler-Ricci soliton on any compact

toric manifold with positive first Chern class.

The existence of K¨ ahler-Ricci solitons on a compact toric manifold with positive first

Chern class was proved in [WZ] by using the continuity method. The above theorem

gives another proof for the existence of K¨ ahler-Ricci solitons on such a complex manifold

by using the K¨ ahler-Ricci flow. We note that a more general convergence theorem of

K¨ ahler-Ricci flow on a compact complex manifold which admits a K¨ ahler-Ricci soliton

was recently obtained by Tian and the author in [TZ3]. In that paper the assumption of

the existence of a K¨ ahler-Ricci soliton plays a crucial role. In the case of K¨ ahler-Einstein

manifolds with positive first Chern class the same result was claimed by Perelman ([P2]).

In the present paper we do not need any assumption of the existence of K¨ ahler-Ricci

solitons or K¨ ahler-Einstein metrics and prove the the convergence of K¨ ahler-Ricci flow.

The Ricci flow was first introduced by R. Hamilton in 1982 ([Ha]). Recently G. Perelman

has made a major breakthrough in this area for three-dimensional manifolds ([P1]).

Our proof of the main theorem is to study certain complex Monge-Amp` ere flow instead

of K¨ ahler-Ricci flow. The flow of this type has been studied before by many people (cf.

1991 Mathematics Subject Classification. Primary: 53C25; Secondary: 32J15, 53C55, 58E11.

Key words and phrases. Toric manifold, the K¨ ahler-Ricci flow, K¨ ahler-Ricci solitons.

* Partially supported by NSF10425102 in China and a Huo Y-D fund

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[Ca], [CT1], [CT2]). Indeed, our proof used a deep estimate of Perelman ([P2], also see

[ST]). We combined Perelman’s estimate with estimates on solutions of complex Monge-

Amp` ere flow appeared in [TZ3] and used an argument for C0-estimate on certain real

Monge-Amp` ere equation studied in [WZ].

The organization of this paper is as follows: In Section 1, We describe an unpublished

estimate of Perelman on the time derivative of potential functions of evolved K¨ ahler

metrics along the K¨ ahler-Ricci flow. In Section 2, we reduce the K¨ ahler-Ricci flow to

a real Monge-Amp` ere flow in order to get an upper bound of solutions of potential

functions. Then in Section 3, we use an argument in [TZ3] to get a C0-estimate of

solution. The main Theorem will be proved in Section 4.

1. An estimate of Perelman.

In this section, we first reduce the K¨ ahler-Ricci flow to a fully nonlinear flow on K¨ aher

potentials. Then we discuss a recent and deep estimate of Perelman.

Let (M,g) be an n-dimensional compact K¨ ahler manifold with its K¨ ahler form ωg

representing the first Chern class c1(M) > 0. In local coordinates z1,··· ,zn, we have

√−1

2π

i,j=1

ωg=

n

?

gijdzi∧ dzj, gi¯j= g(∂

∂zi,

∂

∂zj).

Moreover, the Ricci form Ric(ωg) is given by

?

Rij= −∂i∂jlog(det(gkl)),

Ric(ωg) =

2π

√−1

?n

i,j=1Rijdzi∧ dzj.

Since the Ricci form represents c1(M), there exits a smooth function h on M such that

Ric(ωg) − ωg=

√−1

2π

∂∂h.

(1.1)

The Ricci flow was first introduced by R. Hamilton in [Ha]. If the underlying manifold

M is K¨ ahler with positive first Chern class, it is more natural to study the following

K¨ ahler-Ricci flow (normalized),

?

∂g(t,·)

∂t

g(0,·) = g0,

= −Ric(g(t,·))+ g(t,·),

(1.2)

where g0is a given metric with its K¨ ahler class representing c1(M). It can be shown that

(1.2) preserves the K¨ ahler class, so we may write the K¨ ahler form of g(t) at a solvable

time t as

ωϕ= ωg+

√−1

2π

∂∂ϕ

for some smooth function ϕ = ϕ(t,·) = ϕt. This ϕ is usually called a K¨ ahler potential

function associated to the K¨ ahler metric g(t). Using the Maximal Principle, one can

show that (1.2) is equivalent to the following complex Monge-Amp` ere flow for ϕ(t,·),

?

ϕ(0,·) = 0.

2

∂ϕ

∂t= log

det(gij+ϕij)

det(gij)

+ ϕ − h

(1.3)

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Observe that

Differentiating on both sides of (1.3) on t, we have

∂ϕ

∂t|t=0= −h.

∂

∂t

∂ϕ

∂t= ∆′∂ϕ

∂t+∂ϕ

∂t,

where ∆′denotes the Laplacian operator associated to the metric ωϕ. Then it follows

from the standard Maximal Principle,

|∂

∂tϕ(t,·)| ≤ Cet,

and consequently,

|ϕ(t,·)| ≤ Cet.

By using these facts and arguments in deriving the higher order estimates in Yau’s

solution of the Calabi conjecture [Ya], H.D. Cao showed that (1.3) is solvable for all

t ∈ (0,+∞) [Ca].

Using his W-functional and arguments in proving non-collapsing of the Ricci flow [P1],

recently, Perelman proved the following deep estimate [P2] (also see [ST]),

Lemma 1.1. Let ϕtbe a solution of Monge-Amp` ere flow (1.3). Choose ctby the condi-

tion ht= −∂ϕ

?

Then there is a uniform constant A independent of t such that

∂t+ ctsuch that

M

ehtωn

ϕ=

?

M

ωn

g.

|ht| ≤ A.

(1.4)

Lemma 1.1 is crucial in proving our main theorem. Recall that htis defined by (1.1)

with ωgreplaced by ωϕand can be different to a constant. In Section 3 below we will

further prove that ctis uniformly bounded and so

∂ϕ

∂tis.

2. Upper bound of solution.

In this section, we discuss the upper bound of solution of equation (1.3) by reducing it

to a real Monge-Amp` ere flow. We now assume that M is a compact toric manifold with

positive first Chern class ( toric Fano manifold) and g is a K0-invariant K¨ ahler metric

on M. Then under an affine logarithm coordinates (w1,...,wn), its K¨ ahler form ωg is

determined by a convex function u on Rn, namely

√−1

2π

ωg=

∂∂u

on T.

Hence

ωn

g= (1

π)ndet(uij)dx1∧ ... ∧ dxn∧ dΘ,

where wi= xi+√−1θiand dΘ = dθ1∧ ... ∧ dθnis the standard volume form of K0.

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Let Ω ⊂ Rnbe a bounded convex polyhedron associated to the toric Fano manifold

M and denote p(1),··· ,p(m)to be the vertices of Ω. We define a convex function on Rn

by

v0(x) = log?

√−1

m

?

i=1

e?pi,x??.

(2.1)

Then the induced metric ωg0 =

on M [BS]. The expression (2.1) implies that the gradient (moment) mapping Dv0is a

diffeomorphism from Rnto Ω and

2π∂∂v0can be extended as one with c1(M) = [ωg0]

|logdet(v0

ij) + v0| < ∞.

(2.2)

Denote

v(x) = max{x · p(k)| k = 1,··· ,m}.

(2.3)

The graph of v is a convex cone with vertex at the origin. It can be verified that

|v − v0| ≤ C,

(2.4)

namely the graph of v is an asymptotical cone of the graph v0.

Let h0be a smooth function determined by the relation (1.1) associated to the metric

ωg0. Then it is clear,

∂∂[ev0+h0det(v0

ij)] = 0in Rn.

Hence by (2.2) we have, after normalization,

det(v0

ij) = e−h0−v0.

(2.5)

In general, the equation still holds for a convex function u on Rninduced by a K0-

invariant K¨ ahler metric on M since the difference between u and v0can be extended as

a smooth function on M. Note that Im(Du) = Ω, where Im(Du) denotes the image of

gradient map of u in Rn.

Now we consider K¨ ahler-Ricci flow (1.2) with a K0-invariant, initial K¨ ahler metric g

which is induced by a convex function u0on Rn. For simplicity, we may assume that u0

satisfies inf u0= u0(o) = 0. Since K0-invariant preserves under the flow, we can reduce

equation (1.3) to a real Monge-Amp` ere flow as follow,

?

∂u

∂t= logdet(uij) + u,

u(0,·) = u0,

in Rn,

(2.6)

where u = u0+ ϕ.

Lemma 2.1. Let u = ut= u(t,·) be a solution of equation (2.6) and u = ut= ut− ct,

where ctare functions appeared in Lemma 1.1. Let mt= infRn ut(x). Then

|mt| ≤ C

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for some C > 0 independent of t ∈ (0,∞).

Proof. By Lemma 1.1, we have

|∂u

∂t− ct| ≤ A.

It follows by equation (2.6),

c1≤

?

Rne−udx ≤ c2,

for some uniform constants c1and c2. Note that

|Dut| ≤ diam(Ω),

where diam(Ω) denotes the diameter of Ω. Then it is easy to see that mtis uniformly

bounded from below.

To get an upper bound of mt, we use an argument in [WZ]. For any nonnegative

integer k, we denote a set,

Ak= {x ∈ Rn: mt+ k ≤ u(x) ≤ mt+ k + 1}.

Then for any k ≥ 0, set?k

bounded set for any k ≥ 0. By a well-known theorem [Mi], there is a unique ellipsoid E,

called the minimal ellipsoid of A0, which attains minimum volume among all ellipsoids

contain A0, such that

1

nE ⊂ A0⊂ E.

Let B be a linear transformation with det(B) = 1, which leaves the center of E invariant,

such that B(E) is a ball BRwith radius R. Then we have BR/n⊂ B(A0) ⊂ BRfor two

balls with concentrated center.

i=0Ai= {w < mt+ k + 1} is convex. Note that the origin

is contained in Ω. Hence the minimum mtis attained at some point in A0and Akis a

By equation (2.6) and Lemma 1.1, we have

det(uij) =exp{∂u

∂t− ct− u}

≥ ce−u, in Rn.

It follows

det(uij) ≥c

ee−mt,

in A0.

We claim

R ≤

√2n(c

e)−1/2nemt/2n.

(2.7)

Let

v(y) =1

2(c

e)1/ne−mt/n?|y − yt|2− (R

n)2?+ mt+ 1,

where ytis the center of the minimum ellipsoid of A0. Then

det(vij) =c

ee−mt,

in B(A0),

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and v ≥ u on ∂B(A0), where u = u(B−1). Hence by the comparison principle for the

Monge-Amp´ ere operator we have v ≥ u in B(A0). In particular, we have

mt≤ u(yt) ≤ v(yt)

= −1

2(c

e)1/ne−mt/n(R

n)2+ mt+ 1.

Hence (2.7) follows.

By the convexity of u, we have

B(Ak) ⊂ B2(k+1)R(0).

Thus by (2.7), we obtain

?

Rne−udy =

?

?

k

?

?

?

emt

B(Ak)

e−udy

=

k

B(Ak)

e−u|B(Ak)|

≤ ωn

= ωn(2R)n

e−mt−k|2(k + 1)R|n

?(k + 1)n

≤ C1e−mt/2,

ek

where ωnis the area of the sphere Sn−1. We notice that the above integration is invariant

under any linear transformation B with |B| = 1. Returning to the original coordinates

x, by equation (2.6), we have

e−mt/2≥

1

C1

1

C1

?

?

Rne−udx

=

Rnexp{−∂u

?

∂t+ ct}det(uij)dx

Rndet(uij)dx =e−A

≥e−A

C1

C1

|Ω|.

Hence mt≤ C.

Let xtbe the minimal point of utand u = ut(·) = ut(· + xt) − mt. Set ϕ = u − u0.

Then

?

Proposition 2.1.

|sup

M

ϕ| ≤ C.

Proof. The upper bound of supMϕ follows from the fact,

ϕ = u − u0= u − v + v − u0

≤ v − u0= |v − u0|

≤ |v − v0| + |v0− u0| ≤ C.

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From (2.6), we obtain an equation for u(.),

det(uij) = exp{∂u

∂t− ct− u − mt}, in Rn.

(2.8)

It follows that ϕ satisfies an equation,

det(gij+ ϕij) = exp{h +∂u

∂t− ct− ϕ − mt}, in M,

(2.9)

where h is the smooth function determined by the relation (2.5) as v0is replaced by u0.

By (2.9), it is easy to see that

c1≤

?

M

e−ϕ−mtωn

g≤ c2,

(2.10)

for some positive constants c1and c2since∂u

∂t− ctare uniformly bounded. Thus

ϕ + mt≥ −C0and inf

sup

M

Mϕ + mt≤ C0

(2.11)

for some uniform constant C0. Therefore by Lemma 2.1, we get

sup

M

ϕ ≥ C.

The proof is completed.

?

3. Generalized K-energy and C0-estimate.

In this section, we use the monotonicity of generalized K-energy introduced in [TZ2]

and a C0-estimate developed in [TZ3] to get a bound of C0-norm of ϕ.

Let η(M) be a Lie algebra which consists of all holomorphic vector fields on M and

η0(M) be a Lie subalgebra of η(M) induced by torus T. According to [WZ], we see that

there is a holomorphic vector field X ∈ η0(M) on M such that the holomorphic invariant

FX(·) introduced in [TZ2] vanishes , i.e.,

FX(·) ≡ 0, on η(M).

Set a subspace of potential functions space by

MX(ωg) = {ϕ ∈ C∞(M)| ωϕ= ωg+

√−1

2π

∂∂ϕ > 0, Im(X)(ϕ) = 0}.

The generalized K-energy associated to X is a functional defined in MX(ωg) by

˜ µ(ϕ)

= −n√−1

2πV

0M

√−1

2π

?1

?

˙ψ[Ric(ωψ) − ωψ

−(∂∂θX(ωψ) − ∂(hωψ− θX(ωψ)) ∧ ∂θX(ωψ))] ∧ eθX(ωψ)ωn−1

7

ψ

∧ dt,

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where ψ = ψt (0 ≤ t ≤ 1) is a path connecting 0 to ϕ in MX(ωg) and˙ψ =

θX(ωψ) = θX+ X(ψ) with θXdefined by

dψ

dt, and

√−1

2π

∂θX= iX(ωg).

We notice that ˜ µ(ϕ) is just Mabuchi’s K-energy [Ma] when X = 0 in the relation. One

can show that for any σ ∈ Autr(M), ϕσ∈ MX(ωg) and

˜ µ(ϕ) = ˜ µ(ϕσ)(3.1)

where Autr(M) denotes a reductive subgroup of the holomorphic automorphisms group

on M containing T and ϕσis defined by

σ∗ωϕ= ωg+

√−1

2π

∂∂ϕσ.

Let σt= exp{tX} be the one-parameter group generalized by X and ϕ′= ϕ′

family of potential functions given by relations

tbe a

σ∗

tωϕt= σ∗

tωg+

√−1

2π

∂∂(ϕt· σt)

= ωg+

√−1

2π

∂∂ϕ′

t.

Then according to equation (1.3), ϕ′modula constants satisfy the following complex

Monge-Amp` ere flow,

∂ϕ′

∂t

= log

det(gij+ ϕ′

det(gij)

ij)

+ X(ϕ′) + ϕ′− h + θX.

(3.2)

By using this equation, one can show

d˜ µ(ϕ′)

dt

= −1

V

?∞

0

?

M

?∂∂ϕ′

∂t?2eθX+X(ϕ′)(ωϕ′)n.

(3.3)

In particular,

˜ µ(ϕ′) ≤ 0.

Therefore by (3.1), we get

˜ µ(ϕ) = ˜ µ(ϕ + mt) = ˜ µ(ϕ) = ˜ µ(ϕ′) ≤ 0.

(3.4)

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Proposition 3.1.

?ϕ?C0(M)≤ C.

Proof. Recall another functional introduced in [TZ2],

˜F(ϕ)

=˜J(ϕ) −1

V

?

M

ϕeθXωn

g− log(1

V

?

M

eh−ϕωn

g), in MX(ωg),

where

˜J(ϕ) =1

V

?1

0

?

M

˙ψ(eθXωn

g− eθX+X(ψ)ωn

ψ) ∧ dt > 0

is a functional in MX(ωg) which is independent of the choice of path ψt (0 ≤ t ≤ 1)

connecting 0 and ϕ [Zh]. It was proved in [TZ2],

˜ µ(ϕ) ≥˜F(ϕ) − C, ∀ ϕ ∈ MX(ωg),

(3.5)

for some uniform constant C. Thus by (3.4), we have

˜F(ϕ + mt) ≤ C.

(3.6)

By (2.10) and Proposition 2.1, we get from (3.6),

˜J(ϕ) =˜J(ϕ + mt) ≤1

V

?

M

(ϕ + mt)eθXωn

g+ C ≤ C1.

(3.7)

Since

˜J(ϕ) ≥ cI(ϕ)

for some uniform constant c [CTZ], where

I(ϕ) =1

V

?

M

ϕ(ωn

g− ωn

ϕ),

we obtain

I(ϕ) ≤ C2.

(3.8)

On the other hand, applying the argument in the proof of Proposition 3.1 in [TZ3] to

equation (2.9), one can show that

ocsMϕ ≤ C0(1 + I(ϕ))n+1

for some uniform constant C0. Then by (3.8), it follows

ocsMϕ ≤ C3.

Thus by (2.11), it is easy to see

?ϕ + mt?C0(M)≤ ocsM(ϕ + mt) + 2˜C = oscMϕ + 2˜C ≤ C4.

Therefore, By Lemma 2.1, we prove Proposition 3.1.

?

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Corollary 3.1. Let ϕ = ϕtbe a solution of equation (1.3). Then

˜ µ(ϕ) ≥ C.

(3.9)

Proof. By (2.10) and (3.5), we have

˜ µ(ϕ) = ˜ µ(ϕ + mt)

≥˜F(ϕ + mt) − C1

≥ −

M

?

(ϕ + mt)eθXωn

g− C2.

Then (3.9) follows from Lemma 2.1 and Proposition 3.1.

?

By (3.1), (3.3) and Corollary 3.1, we have

1

V

?∞

0

?

M

?∂∂ϕ′

∂t?2eθX+X(ϕ′)(ωϕ′)n∧ dt ≤ C.

It follows

1

V

?∞

0

?

M

?∂∂ϕ′

∂t?2eθX+X(ϕ′)−t(ωϕ′)n∧ dt ≤ C.

(3.10)

The following proposition was proved in [TZ3].

Proposition 3.2. Let h be normalized by adding a suitable constant so that

1

V

?

M

(h − θX)ωn

g= −1

V

?∞

0

?

M

?∂∂ϕ′

∂t?2eθX+X(ϕ′)−t(ωϕ′)n∧ dt.

(3.11)

Then

|ct| ≤ C and |∂ϕ

∂t| ≤ C,

where ctare constants appeared in Lemma 1.1.

4. Convergence of the flow.

In this section, we discuss the higher order estimates for a modified solution of equation

(1.3) and finish the proof of Main Theorem. Let xt∈ Rnbe a family of points determined

in section 2. We observe that

Lemma 4.1. Let i = 0,1..., be any nonnegative integer. Then the distances between xi

and xi+1are uniformly bounded, i.e.,

|xi− xi+1| ≤ C

for some uniform constant C.

Proof. Let v be the convex cone function in Rndefined by (2.3). Then by Proposition

3.1, we have

|ui− v| ≤ |ui− u0| + |u0− v| = |ϕi| + |u0− v| ≤ C,

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and

|ui+1− v| ≤ |ui+1− u0| + |u0− v| = |ϕi+1| + |u0− v| ≤ C.

On the other hand, by Proposition 3.2, we have

|ui(x − xi) − ui+1(x − xi+1)|

= |ui(x − xi) − ui+1(x − xi+1) − mi+ mi+1|

= |ui− ui+1− mi+ mi+1|(x) = |ϕi− ϕi+1− mi+ mi+1|(x) ≤ C.

Thus combining the above relations, we get

|v(x − xi) − v(x − xi+1)| ≤ C.

In particular, by choosing x = xi, we see

|v(xi− xi+1)| = |v(o) − v(xi− xi+1)| ≤ C.

(4.1)

(4.1) implies

|xi− xi+1| ≤ C, ∀ i = 0,1,....

?

With the help of Lemma 4.1, one can choose a family of modified points x′

that

|xt− x′

Let ˜ u(·) = ˜ ut(·) = ut(x′

function on M. Since

tin Rnsuch

t| ≤ C and |dx′

t

dt| ≤ C.

(4.2)

t+·) and ˜ ϕ = ˜ ϕt= ˜ ut−u0. Then ˜ ϕt(·) = ϕt(x′

t+·) is a potential

|˜ ut− ut| ≤ diam(Ω)|xt− x′

t| + |ct| + |mt|,

by Proposition 3.1, we have

?˜ ϕt?C0(M)≤ ?˜ ut− ut?C0(M)+ ?ϕt?C0(M)≤ C.

From (2.6), we see that ˜ u satisfies a parabolic equation

∂˜ u

∂t= logdet(˜ uij) +˜X(˜ u) + ˜ u, in Rn,

(4.3)

where˜ X =˜ Xt=

fields on M. It follows that ˜ ϕ satisfies an equation

dx′

dt. Note that˜ X are corresponding to a family of holomorphic vector

t

∂ ˜ ϕ

∂t

= logdet(gij+ ˜ ϕij) − logdet(gij) +˜ X(˜ ϕ) + ˜ ϕ − h + θ˜

Xt, in M,

(4.4)

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where h is normalized as in Proposition 3.2. Note that θ˜

3.2, we have

Xt=˜Xt(u0). So by Proposition

|∂ ˜ ϕ

∂t| = |∂ϕ

∂t+ θ˜

Xt| ≤ C

Proof of Main Theorem. We shall show that the corresponding K¨ ahler metrics ω˜ ϕasso-

ciated to solution of equation (4.4) converge to a K¨ ahler-Ricci soliton ωKSwith respect

to X. The proof is similar to one of Main theorem in [TZ3]. We give a sketch. First by

modifying Yau’s C2-estimate in [Ya] for certain complex Monge-Amp` ere equation, one

obtains for the solution ˜ ϕ of equation (4.4),

?˜ ϕ?C2(M)≤ C and (gij+ ˜ ϕij) > c0.

Then following Calabi’s C3-estimate for complex Monge-Amp` ere equation [Ya], we fur-

ther get

?˜ ϕ?C3(M)≤ C.

Thus by using regularity theory for the parabolic equation, one sees easily that all Ck-

norms of the solution ˜ ϕ are uniformly bounded. Therefore, we conclude that for any

sequence of functions ˜ ϕt, one can take a subsequence of the sequence which converge

Ck-smoothly to a smooth function ϕ∞on M.

Let σt= exp{tX} and σ′= σ′

corresponding to changes from ϕtto ˜ ϕt. Since

t= ρt· σ−1

t , where ρtare holomorphic transformations

?

M

?∂((σ′)∗∂ϕ′

∂t)?2eθX+X(˜ ϕ)ωn

˜ ϕ=

?

M

?∂∂ϕ′

∂t?2eθX+X(ϕ′)(ωϕ′)n,

then by (3.3), we have

?

M

?∂((σ′)∗∂ϕ′

∂t)?2eθX+X(˜ ϕ)ωn

˜ ϕ= −d˜ µωg(ϕ′)

dt

.

(4.5)

By the lower bound of ˜ µωg(ϕ′) (cf. Corollary 3.1), one sees that there is a sequence of

ti, i = 1,2,..., such that

?

M

?∂((σ′)∗∂ϕ′

∂t|ti)?2ωn

˜ ϕti→ 0, as i → ∞.

(4.6)

On the other hand, by (4.4), we have

√−1

2π

∂∂[(σ′)∗∂ϕ′

∂t] = −Ric(ω˜ ϕ) + ω˜ ϕ+ LXω˜ ϕ.

(4.7)

Then (σ′)∗ ∂

sequence of (σ′)∗ ∂

same indices ti) converge to a constant in the Cksense, and consequently, by (4.7),

K¨ ahler metrics ω˜ ϕticonverge to a K¨ ahler-Ricci soliton (ωg+

the holomorphic vector field X. It remains to prove that the limit (ωg+

is independent of the choice of sequence of ω˜ ϕt. But the last follows from the uniqueness

of K¨ ahler-Ricci solitons proved in [TZ1] and [TZ2]. We leave the details to reader.

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∂tϕ′is Ckuniformly bounded in space, so there exists a convergent sub-

∂tϕ′(ti,·). Hence by (4.6), we conclude that (σ′)∗ ∂

∂tϕ′(ti,·) (still use

√−1

2π∂∂ϕ∞) associated to

√−1

2π∂∂ϕ∞,X)

?

Page 13

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[P2] Perelman, G., unpublished.

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[TZ1] Tian, G. and Zhu, X.H., Uniqueness of K¨ ahler-Ricci solitons, Acta Math., 184 (2000), 271-305.

[TZ2] Tian, G. and Zhu, X.H., A new holomorphic invariant and uniqueness of K¨ ahler-Ricci solitons, Comm.

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[TZ3] Tian, G. and Zhu, X.H., Convergence of K¨ ahler-Ricci flow, 2005, to appear in Jour. of Amer. Math.

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[Zh] Zhu, X.H., K¨ ahler-Ricci soliton type equations on compact complex manifolds with C1(M) > 0,

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Department of Mathematics, Peking University, Beijing, 100871, China

E-mail address: xhzhu@math.pku.edu.cn

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