Kähler-Ricci flow with unbounded curvature

Kähler-Ricci flow with unbounded curvature

Shaochuang Huang, Luen-Fai Tam

American Journal of Mathematics, Volume 140, Number 1, February 2018, pp. 189-220 (Article)

Published by Johns Hopkins University Press DOI:

For additional information about this article

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https://doi.org/10.1353/ajm.2018.0003

https://muse.jhu.edu/article/683727/summary

By SHAOCHUANGHUANGand LUEN-FAITAM

Abstract. Letg(t)be a smooth complete solution to the Ricci flow on a noncompact manifold such thatg(0)is K¨ahler. We prove that if|Rm(g(t))|g(t)is bounded bya/tfor somea >0, theng(t) is K¨ahler fort >0. We prove that there is a constanta(n)>0 depending only onnsuch that the following is true: Supposeg(t)is a smooth complete solution to the K¨ahler-Ricci flow on a non- compactn-dimensional complex manifold such thatg(0)has nonnegative holomorphic bisectional curvature and|Rm(g(t))|g(t)≤a(n)/t, theng(t)has nonnegative holomorphic bisectional curva- ture fort >0. These generalize the results by Wan-Xiong Shi. As applications, we prove that (i) any complete noncompact K¨ahler manifold with nonnegative complex sectional curvature and maximum volume growth is biholomorphic toCⁿ; and (ii) there is(n)>0 depending only onnsuch that if (Mⁿ,g0)is a complete noncompact K¨ahler manifold of complex dimensionnwith nonnegative holo- morphic bisectional curvature and maximum volume growth and if(1+(n))⁻¹h≤g0≤(1+(n))h for some Riemannian metrichwith bounded curvature, thenMis biholomorphic toCⁿ.

1. Introduction. In [25], Simon proved that there is a constant (n)>0 depending only onnsuch that if(Mⁿ, g₀)is a completen-dimensional Riemannian manifold and if there is another metrichwith curvature bounded byk₀and

(1+(n))⁻¹h≤g0≤(1+(n))h,

then the so-calledh-flow has a smooth short time solutiong(t)such that

|Rm(g(t))|g(t)≤C/t.

(1.1)

Hereh-flow is exactly the usual Ricci-DeTurck flow. We call ith-flow as in [25]

for emphasizing the background metrich. For the precise definition ofh-flow, see Section 5. The method by Schn¨urer-Schulze-Simon [21] can be carried over to construct Ricci flow using the above solution to theh-flow. On the other hand, in [2], Cabezas-Rivas and Wilking proved that if(M, g0)is a complete noncompact Riemannian manifold with nonnegative complex sectional curvature, and if the volume of geodesic ballB(x,1)of radius 1 with center atxis uniformly bounded below away from 0, then the Ricci flow has a smooth complete short time solution with nonnegative complex sectional curvature so that (1.1) holds. Recall that a Riemannian manifold is said to have nonnegative complex sectional curvature if R(X, Y,Y ,¯ X¯)≥0 for any vectorsX, Y in the complexified tangent bundle.

Manuscript received June 15, 2015; revised May 15, 2016.

Research of the second author supported in part by Hong Kong RGC General Research Fund #CUHK 1430514.

American Journal of Mathematics 140 (2018), 189–220. c2018 by Johns Hopkins University Press.

189

It is natural to ask the following:

QUESTION. Suppose g₀ is K¨ahler. Are the above solutions g(t) to the Ricci flow also K¨ahler fort >0?

It is well known that ifMis compact or if the flow has bounded curvature, the answer to the above question is yes by [11, 24]. On the other hand, the case where the curvature ofg₀may be unbounded has also been studied before. It was proved by Yang and Zheng [28] that for aU(n)-invariant initial K¨ahler metric onCⁿ, the solution constructed by Cabezas-Rivas and Wilking is K¨ahler fort >0, under some additional technical conditions. In this paper, we want to prove the following:

THEOREM 1.1. If (Mⁿ, g₀) is a complete noncompact K¨ahler manifold with complex dimensionnand ifg(t)is a smooth complete solution to the Ricci flow on M×[0, T],T >0, withg(0) =g₀such that

|Rm(g(t))|g(t)≤a t for somea >0, theng(t)is K¨ahler for all 0≤t≤T.

This gives an affirmative answer to the above question. The result is related to previous works on the existence of K¨ahler-Ricci flows without curvature bound, see [3, 4, 10, 28] for example.

We may apply the theorem to the uniformization conjecture by Yau [29] which states that a complete noncompact K¨ahler manifold with positive holomorphic bi- sectional curvature is biholomorphic to Cⁿ. A previous result by Chau and the second author [5] says that the conjecture is true if the K¨ahler manifold has maxi- mum volume growth and has bounded curvature, see also [6, 17]. Combining this with the above theorem, we have:

COROLLARY 1.1. Let(Mⁿ, g0)be a complete noncompact K¨ahler manifold with complex dimensionnand nonnegative complex sectional curvature. Suppose Mⁿhas maximum volume growth. ThenMⁿis biholomorphic toCⁿ.

For K¨ahler surface (n=2), sectional curvature being nonnegative is equiv- alent to complex sectional curvature being nonnegative [30]. Hence in particular, any complete noncompact K¨ahler surface with nonnegative sectional curvature and maximum volume growth is biholomorphic toC². We should mention that recently Liu [15, 14] proves that a complete noncompact K¨ahler manifold with nonnegative holomorphic bisectional curvature and maximum volume growth is diffeomorphic to the Euclidean space and is biholomorphic to an affine algebraic variety, gener- alizing a result of Mok [17]. Moreover, if the volume of geodesic balls are close to the Euclidean balls with same radii or if the complex dimension is less than or equal to 3, then the manifold is biholomorphic toCⁿ.

By Theorem 1.1, we know that from the solution to theh-flow constructed by Simon [25] one can construct a solution to the K¨ahler-Ricci flow ifg0is K¨ahler. In view of the conjecture of Yau, we would like to know whether the nonnegativity of holomorphic bisectional curvature will be preserved by the solutiong(t)to the K¨ahler-Ricci flow. The second result in this paper is the following:

THEOREM 1.2. There is 0 < a(n) < 1 depending only on n such that if g(t) is a smooth complete solution to the K¨ahler-Ricci flow on M×[0, T] with sup_x∈M|Rm(x, t)| ≤ ^a_t and if g(0) has nonnegative holomorphic bisectional curvature, whereM is ann-dimensional noncompact complex manifold, theng(t) also has nonnegative holomorphic bisectional curvature for allt∈[0, T].

We should mention that in [28], Yang and Zheng proved that the nonnega- tivity of bisectional curvature is preserved under the K¨ahler-Ricci flow forU(n)- invariant solution onCⁿwithout any condition on the bound of the curvature.

Following exactly the same method as in [25], one can prove that for anya >0, if(n, a)>0 is small in the result of Simon, then curvature of the solution to the h-flow will be bounded by a/t. However, given the results in [25], one may also obtain this estimate using an interpolation inequality by Schn¨urer-Schulze-Simon [22]. The authors would like to thank the referee for pointing out this fact.

Hence as a corollary to Theorem 1.2, using [5] again, we have:

COROLLARY 1.2. There exists (n) >0 depending only on n such that if (Mⁿ, g0)is a complete noncompact K¨ahler manifold of complex dimensionnwith nonnegative holomorphic bisectional curvature and maximum volume growth, and if there is a Riemannian metrich on M with bounded curvature satisfying (1+ (n))⁻¹h≤g₀≤(1+(n))h, thenMis biholomorphic toCⁿ.

By a result of Xu [27], we also have the following corollary which says that the condition that the curvature is bounded in the uniformization result in [5] can be relaxed to the condition that the curvature is bounded in some integral sense.

Namely, we have:

COROLLARY 1.3. Let (Mⁿ, g₀) be a complete noncompact K¨ahler manifold of complex dimensionn≥2 with nonnegative holomorphic bisectional curvature and maximum volume growth. Suppose there isr₀>0 and there isC >0 such that

1 Vx(r0)

Bx(r0)|Rm|^p _p¹

≤C

for somep > nfor allx∈M. ThenM is biholomorphic toCⁿ.

The paper is organized as follows: In Section 2, we prove a maximum principle and apply it in Section 3 to prove Theorem 1.1. In Section 4, we prove Theorem 1.2.

In Section 5, we will construct solutions to the K¨ahler-Ricci flow with nonnegative holomorphic bisectional curvature through theh-flow.

Acknowledgments. The second author would like to thank Albert Chau for some useful discussions and for bringing our attention to the results in [27]. The authors would also like to thank the referee for some useful suggestions so that the proofs of Lemmas 5.1 and 5.2 can be greatly simplified.

Added in proof. For more recent results on Yau’s conjecture, see [13, 16].

2. A maximum principle. In this section, we will prove a maximum prin- ciple, which will be used in the proof of Theorem 1.1.

Let(Mⁿ, g0)be a complete noncompact Riemannian manifold. Letg(t)be a smooth complete solution to the Ricci flow onM×[0, T],T >0 withg(0) =g0, i.e.,

⎧⎪

⎨

⎪⎩

∂

∂tg=−2 Ric, onM×[0, T]; g(0) =g₀.

(2.1)

Let Γ and ¯Γ be the Christoffel symbols of g(t) and ¯g=g(T) respectively. Let A= Γ−Γ¯. ThenAis a(1,2)tensor. In the following, lower casec, c1, c2, . . .will denote positive constants depending only onn.

LEMMA 2.1. With the above notation and assumptions, suppose the curva- ture satisfies |Rm(g(t))|g(t) ≤at⁻¹ for some positive constant a. Then there is a constantc=c(n)>0, such that

(i)

T t

₋ca

¯

g≤g(t)≤ T

t ca

¯ g;

(ii) |∇Rm| ≤Ct⁻³² for some constant C=C(n, a)>0 depending only on n, a;

(iii)

|A|g¯ ≤Ct^−12−ca,

for some constantC=C(n, T, a)>0 depending only onn, T anda.

Proof. (i) follows from the Ricci flow equation.

(ii) is a result in [23], see also [9, Theorem 7.1].

To prove (iii), in local coordinates:

∂

∂tA^k_ij=−g^kl ∇iR_jl+∇jR_il−∇lR_ij .

At a point where ¯gij =δij such thatgij =λiδij, by (i) and (ii), we have ∂

∂t|A|²g¯

≤C₁t^−c1a|∇Ric|g¯|A|g¯

≤C₂t^−c2a−32|A|g¯

for some constantsC1, C2 depending only onn, T, aandc1, c2depending only on

n. From this the result follows.

Under the assumption of the lemma, sinceg(T)is complete and the curvature of ¯g=g(T)is bounded bya/T, we can find a smooth functionρonMsuch that

d_g¯(x, x₀) +1≤ρ(x)≤C(d_g¯(x, x₀) +1), |∇ρ|g¯+|∇²ρ|g¯≤C, (2.2)

where∇is covariant derivative with respect to ¯gandC>0 is a constant depending onnanda/T, see [24, 26].

LEMMA 2.2. With the same assumptions and notation as in the previous lemma,ρ(x)satisfies

|∇ρ| ≤C1t^−ca and

|Δρ| ≤C2t^−12−ca

whereC1,C2depend only onn, T, aandc >0 depends only onn. Here∇andΔ are the covariant derivative and Laplacian ofg(t)respectively.

Proof. The first inequality follows from Lemma 2.1(i). To estimate Δρ, at a point where ¯gij =δij andgij is diagonalized, we have

Δρ−Δρ=g^ij∇i∇jρ−g¯^ij∇i∇jρ

≤g^ij

∇i∇j−∇i∇j

ρ+ g^ij−g¯^ij

∇i∇jρ

≤ |g^ijA^k_ijρ_k|+C3t^−c1a

≤C₄t^−12−c2a

for some constantsC₃,C₄depending only onn, T, a, andc₁, c₂depending only on n. By the estimates ofΔρ, the second result follows.

LEMMA 2.3. Let (Mⁿ, g) be a complete noncompact Riemannian manifold with dimensionnand letg(t)be a smooth complete solution to the Ricci flow on M×[0, T],T >0 withg(0) =g₀such that the curvature satisfies|Rm| ≤at⁻¹for somea >0.

Letf ≥0 be a smooth function onM×[0, T]such that (i)

∂

∂t−Δ

f≤ a tf; (ii) ^∂_∂t^kk^f_t

=0=0 for allk≥0;

(iii) sup_x∈Mf(x, t)≤Ct^−l, for some positive integerlfor some constantC.

Thenf≡0 onM×[0, T].

Proof. We may assume that T ≤1. In fact, if we can prove that f ≡0 on M×[0, T1]whereT1=min{1, T}, then it is easy to see thatf≡0 onM×[0, T] becausef and the curvature ofg(t)are uniformly bounded onM×[T1, T].

Letp∈M be a fixed point, and let d(x, t) be the distance betweenp, xwith respect tog(t). By [19] (see also [8, Chapter 18]), for allr0, ifd(x, t)> r0, then

∂₋

∂td(x, t)−Δd(x, t)≥ −C0

t⁻¹r0+ 1 r0

(2.3)

in the barrier sense, for someC0=C0(n, a)depending only onnanda. Here

∂₋

∂td(x, t) =lim inf

h→0⁺

d(x, t)−d(x, t−h)

h .

(2.4)

The inequality (2.3) means that for any >0, there is a functionσ(y)nearxsuch thatσ(x) =d(x, t),σ(y)≥d(y, t)nearx, such thatσisC²and

∂₋

∂td(x, t)−Δσ(x)≥ −C₀

t⁻¹r₀+ 1 r0

−. (2.5)

In the following, we always take=T⁻²¹.

Letf be as in the lemma. First we want to prove that for any integer k > a there is a constantBksuch that

sup

x∈Mf(x, t)≤B_kt^k. (2.6)

LetF=t^−kf, then ∂

∂t−Δ

F ≤ −k−a t F ≤0.

(2.7)

Let 1≥φ≥0 be a smooth function on[0,∞)such that φ(s) =

1, if 0≤s≤1;

0, ifs≥2,

and such that −C1 ≤φ ≤0, |φ| ≤C1 for some C1 >0. Let Φ =φ^m, where m >2 will be chosen later. ThenΦ =1 on[0,1]andΦ =0 on[2,∞), 1≥Φ≥0,

−C(m)Φ^q≤Φ≤0, |Φ| ≤C(m)Φ^q, where C(m)>0 depends on mand C1, andq=1−_m².

For anyr >>1, letΨ(x, t) = Φ(^d(x,t_r ⁾). Let θ(t) =exp(−αt^1−β), whereα >0, 0< β <1 will be chosen later.

We claim that one can choosem,αandβsuch that for allr >>1 H(x, t) =θ(t)Ψ(x, t)F≤1

onM×[0, T]. If the claim is true, then we have thatFis bounded. Hencef(x, t)≤ B_kt^k.

First note thatΨ(x, t)has compact support inM×[0, T]. By assumption (ii) and the factf is smooth, we conclude thatH(x, t) is continuous on M×[0, T]. Moreover, by (ii) again,H(x,0) =0. SupposeH(x, t)attains a positive maximum at(x₀, t₀)for somex₀∈M,t₀>0. Supposed(x₀, t₀)< r, then there is a neigh- borhoodU ofxandδ >0 such thatd(x, t)< rforx∈U and|t−t₀|< δ. For such (x, t),H(x, t) =θ(t)F(x, t). SinceH(x0, t0)is a local maximum, we have

0≤ ∂

∂t−Δ

H

=θF+θ ∂

∂t−Δ

F

≤θF

<0 which is a contradiction.

Hence we must haved(x₀, t₀)≥r. Ifr >>1, thenr≥T¹², and at(x₀, t₀),

∂₋

∂t|t=t0d(x, t)− t0σ(x)≥ −C₀t⁻

1 2

0 , by takingr0=t

1 2

0. HereC0>0 is a constant depending onnanda,σ(x)is a barrier function nearx0.

LetΨ( x) = Φ(^σ(_r^x)), and let

H˜(x, t) =θ(t)Ψ(˜ x)F(x, t) which is defined nearx₀for allt. Moreover,

H(x˜ 0, t0) =H(x0, t0)

and

H˜(x, t₀)≤H(x, t₀)

nearx0 because σ(x)≥d(x, t0) nearx0 and Φ≤0. HenceH(x, t0) has a local maximum at(x0, t0)as a function ofx. So we have

∇H˜(x0, t0) =0 (2.8)

and

ΔH˜(x0, t0)≤0.

(2.9) At(x₀, t₀),

0≥Δ

θ(t)Ψ( x)F(x, t)

=θΨΔ F+θFΔΨ+ 2θ ∇F,∇Ψ

=θΨΔ F+θF 1

rΦΔσ+ 1

r²Φ|∇σ|²

−2θ|∇Ψ| ² Ψ F

≥θΦΔF+θF 1

rΦΔσ+ 1

r²Φ|∇σ|²

− 2 r²θΦ²

Φ F (2.10)

where we have used the fact thatσ(x)≥d(x, t0)nearx0andσ(x0) =d(x0, t0)so that|∇σ(x₀)| ≤1.Φand the derivativesΦ andΦare evaluated at^d(x_r^0,t0).

On the other hand, 0≤lim inf

h→0⁺

H(x0, t0)−H(x0, t0−h) h

=θΨF+θΨ∂

∂tF+θFlim inf

h→0⁺

−Ψ(x0, t0−h) + Ψ(x0, t0)

h .

Now

−Ψ(x₀, t₀−h) + Ψ(x₀, t₀) =−Φ(d(x₀, t₀−h)

r ) + Φ(d(x₀, t₀)

r )

=1

rΦ(ξ)(d(x0, t0)−d(x0, t0−h)), for someξbetween ¹_rd(x₀, t₀−h)and ¹_rd(x₀, t₀)which implies

lim inf

h→0⁺

−Ψ(x₀, t₀−h) + Ψ(x₀, t₀)

h ≤lim sup

h→0⁺

−Ψ(x₀, t₀−h) + Ψ(x₀, t₀) h

= 1 rΦ∂₋

∂td(x₀, t)|t=t0

because Φ≤0, whereΦ is evaluated at ¹_rd(x0, t0). In the following,Ci will de- note positive constants independent ofα, β. Combining the above inequality with (2.10), we have at(x0, t0):

0≤θΦF+θΦ∂

∂tF+θF1 rΦ∂₋

∂td(x₀, t₀)

−θΨΔF−θF 1

rΦΔσ+ 1

r²Φ|∇σ|²

+ 2 r²θΦ²

Φ F

≤θΦF+C₂

t⁻

1 2

0 Φ^q+ Φ^2q−1

θF

≤ −α(1−β)t⁻₀^βθΦF +C₃θ

t⁻

1 2

0 t₀^{−(1−q)(k+l)}(ΦF)^q+t⁻

1 2

0 t⁻₀^{2(1−q)(k+l)}(ΦF)^2q−1

≤θ

−α(1−β)t⁻₀^βΦF+C4t⁻

1

2−2(1−q)(k+l)

0 (ΦF)^q+ (ΦF)^2q−1

where Φ,Φ,Φ are evaluated at d(x0, t0)/r. Now first choose m large enough depending only onk, lso that¹₂+2(1−q)(k+l) =β <1. Then chooseαsuch that α(1−β)>2C₄. Then one can see that we must haveΦF≤1. HenceH=θΦF≤1 at the maximum point ofH(x, t). This completes the proof of the claim.

Next, letF =t^−af. Then ∂

∂t−Δ

F ≤0.

Letρbe the function in Lemma 2.2, we have

|Δρ| ≤C₅t^−b

for someb >1. Letη(x, t) =ρ(x)exp(^2C₁₋⁵_bt^1−b). Note thatη(x,0) =0.

∂

∂t−Δ

η=exp 2C₅

1−bt^1−b 2C₅t^−bρ−Δρ

≥C5t^−bexp 2C5

1−bt^1−b

>0.

where we have used the fact thatρ≥1. SinceF≤C6t²inM×[0, T]. In particular it is bounded. Then for any >0

∂

∂t−Δ

(F−η−t)<0.

There ist₁>0 depending only on, C₆such thatF−t <0 fort≤t₁. Fort≥t₁, F−η <0 outside some compact set. Hence ifF−η−t >0 somewhere, then

there exist x0 ∈M, t0>0 such that F−η−t attains maximum. But this is impossible. SoF−η−t≤0. Let→0, we haveF =0.

3. Preservation of the K ¨ahler condition. In this section, we want to prove Theorem 1.1 and give some applications. Recall Theorem 1.1 as follows:

THEOREM 3.1. If (Mⁿ, g₀) is a complete noncompact K¨ahler manifold with complex dimension nand if g(t) is a smooth complete solution to the Ricci flow (2.1) onM×[0, T],T >0, withg(0) =g₀such that

|Rm(g(t))|g(t)≤a t for somea >0, theng(t)is K¨ahler for all 0≤t≤T.

We will use the setup as in [24, Section 5]. LetT_CM=T_RM⊗_RCbe the com- plexification ofT_RM, whereT_RMis the real tangent bundle. Similarly, letT_C^∗M= T_R^∗M⊗RC, whereT_R^∗M is the real cotangent bundle. Letz={z¹, z², . . . , zⁿ}be a local holomorphic coordinate onM, and

z^k=x^k+√

−1x^k+n

x^k∈R, x^k+n∈R, k=1,2, . . . , n.

In the following:

• i, j, k, l, . . .denote the indices corresponding to real vectors or real covectors;

• α, β, γ, δ, . . . denote the indices corresponding to holomorphic vectors or holomorphic covectors,

• A, B, C, D, . . .denote bothα, β, γ, δ, . . .and ¯α,β,¯ γ,¯ δ, . . ..¯

Extendgij(t),Rijkl(t)etc.C-linearly to the complexified bundles. We have:

gAB =gA¯B¯, RABCD=RA¯B¯C¯D¯.

In our convention, R₁₂₂₁=R(e₁, e₂, e₂, e₁) is the sectional curvature of the two- plane spanned by orthonormal paire₁, e₂.R_ABCDhas the same symmetry asR_ijkl and it satisfies the Bianchi identities.

Letg^AB := (g⁻¹)^AB, it meansg^ABg_BC=δ^A_C, and let R_AB=g^CDR_ACDB onM×[0, T]. Then we have

∂

∂tg_AB =−2R_AB (3.1)

and

∂

∂tRABCD

=R_ABCD−2g^EFg^GHR_EABGR_{F HCD}

−2g^EFg^GHR_EAGDR_{F BHC}+2g^EFg^GHR_EAGCR_{F BHD}

−g^EF(REBCDRF A+RAECDRF B+RABEDRF C+RABCERF D) (3.2)

onM×[0, T], see [24].

We begin with the following lemma:

LEMMA3.1. Let(M, g0)be a K¨ahler manifold, andg(t)be a smooth solution to the Ricci flow withg(0) =g0. In the above set up, we have

∂^k

∂t^kR_ABγδ|t=0=0 at each point ofM for allk≥0 and for allA, B, γ, δ.

Proof. Letp∈M with holomorphic local coordinatez. In the following, all computations are at(z,0)unless we have emphasis otherwise. We will prove the lemma by induction. Consider the following statement:

H(k)

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

H1(k): ∂^k

∂t^kRABγδ =0 H₂(k): ∂^k

∂t^kg_AB=0 ifA, Bare of the same type H₃(k): ∂^k

∂t^kR_AB =0 ifA, Bare of the same type H4(k): ∂^k

∂t^kΓ^C_AB=0 unlessA, B, Care of the same type H5(k): ∂^k

∂t^kRABγδ;E =0 H₆(k): ∂^k

∂t^kR_ABγδ;EF =0 H₇(k): ∂^k

∂t^kR_ABγδ=0.

Here we denote covariant derivative with respect to g(t) by “;” and the partial derivative by “,”. IfHi(k)are true for alli=1, . . . ,7, we will say thatH(k)holds.

As usual:

Γ^CAB=1

2g^CD(g_AD,B+g_DB,A−g_AB,D).

We first consider the case thatk=0. Since the initial metric is K¨ahler, it is easy to see thatH(0)holds. Now we assumeH(i)holds for alli=0,1,2, . . . , k. We want to showH(k+1)holds. We first see that

∂^k+1

∂t^k+1RABγδ

= ∂^k

∂t^k(R_ABγδ)−

m+n+p+q=k 0≤m,n,p,q≤k

2(g^EF)m(g^GH)n(REABG)p(R_{F Hγδ})q

−

m+n+p+q=k 0≤m,n,p,q≤k

2(g^EF)m(g^GH)n(R_EAGδ)p(R_{F BHγ})q

+

m+n+p+q=k 0≤m,n,p,q≤k

2(g^EF)m(g^GH)n(REAGγ)p(RF BHδ)q

−

m+n+p=k 0≤m,n,p≤k

(g^EF)m(RAF)n(REBγδ)p−

m+n+p=k 0≤m,n,p≤k

(g^EF)m(RBF)n(RAEγδ)p

−

m+n+p=k 0≤m,n,p≤k

(g^EF)m(R_γF)n(R_ABEδ)p−

m+n+p=k 0≤m,n,p≤k

(g^EF)m(R_δF)n(R_ABγE)p.

Here(·)p=_∂t^∂pp(·).

Suppose(g_AB)p =0 att=0 ifA, B are of the same type forp=0,1, . . . , k, then it is also true that(g^AB)p=0 ifA, Bare of the same type forp=0,1, . . . , k.

On the other hand, in the R.H.S. of the above equality, the derivative of each term with respect totis only up to orderk, by the induction hypothesis,H₁(k+1)holds.

Now

∂

∂tgAB=−2RAB,

it is easy to see thatH2(k+1)holds becauseH3(k)holds.

Since

∂^k+1

∂t^k+1Rαβ =

m+n=k+1 0≤m,n≤k+1

(g^CD)m(RαCDβ)n,

and since thatH₁(k+1)and H₂(k+1) hold, we conclude thatH₃(k+1) holds.

Here we have used the symmetries ofR_ABCD.

Since

∂^k+1

∂t^k+1Γ^α_Aβ¯=−

m+n=k 0≤m,n≤k

(g^αD)m RβD;A¯ +R_{AD; ¯β}−R_Aβ;D¯

n

=−

m+n=k 0≤m,n≤k

(g^ασ¯)m Rβ¯σ;A¯ +R_{A¯σ; ¯β}−R_Aβ; ¯¯σ

n,

by the induction hypothesis. IfA=γ¯, then each term on the R.H.S. is zero by the induction hypothesis. IfA=γ, then

(Rβ¯σ;γ¯ )n= (Rβ¯σ,γ¯ )n−(Γ^Eγσ¯R_Eβ¯)n−(Γ^E_γβ¯REσ¯)n, so it vanishes becausen≤k. On the other hand,

R_{γσ; ¯¯ β}−R_γβ; ¯¯σ=g^CD(R_{γCDσ; ¯¯β}−R_γCDβ; ¯¯σ)

=g^CD(R_{γCDσ; ¯¯β}+R_{γCσD; ¯¯ β}+R_γCβ¯σ;D¯ )

=g^CDR_γCβ¯σ;D¯ . So

R_{γσ; ¯¯β}−R_γβ; ¯¯σ

n=0 forn≤kby the induction hypothesis. Thus,

∂^k+1

∂t^k+1Γ^α_Aβ¯=0

att=0. SinceΓ^C_AB= Γ^C_BAandΓ^C_AB= Γ^CA^¯¯B¯, it is easy to see thatH₄(k+1)holds.

Next,

R_ABγδ;E =R_ABγδ,E−Γ^GEAR_GBγδ−Γ^GEBR_AGγδ

−Γ^GEγR_ABGδ−Γ^GEδRABγG. ByH₁(k+1), we have

∂^k+1

∂t^k+1R_ABγδ,E=

∂^k+1

∂t^k+1R_ABγδ

E=0.

SinceH1(i)andH4(i)are true for 0≤i≤k+1,H5(k+1)is true. SinceH1(i), H4(i)andH5(i)are true for 0≤i≤k+1,H6(k+1)is true. FinallyH6(i)is true for 0≤i≤k+1 implies thatH7(k+1)holds. Therefore,H(k+1)holds.

Now we use the Uhlenbeck’s trick to simplify the evolution equation of the complex curvature tensor. We pick an abstract vector bundleV overM which is

isomorphic toT_CM and denote the isomorphismu0:V →T_CM. We take{eA:= u⁻₀¹(_∂z^∂A)}as a basis ofV. We also consider the metrichonV given byh:=u^∗₀g0. We letu0evolute by

⎧⎨

⎩

∂

∂tu(t) =Ric◦u(t), u(0) =u₀. In local coordinate, we have

⎧⎪

⎨

⎪⎩

∂

∂tu^A_B=g^ACRCDu^D_B, u^A_B(0) =δ_B^A.

Consider metrich(t):=u^∗(t)g(t)onV for eacht∈[0, T]. It is easy to see that

∂t∂h(t)≡0 for allt, soh(t)≡hfor allt. We useu(t)to pull the curvature tensor onT_CM back toV:

Rm(e_A, e_B, e_C, e_D):=R(u(e_A), u(e_B), u(e_C), u(e_D)). In local coordinate, we have

R˜_ABCD=R_{EF GH}u^E_Au^F_Bu^G_Cu^H_D onM×[0, T]. One can also check that

h_AB =hA¯B¯, R˜_ABCD=R˜A¯B¯C¯D¯.

Define a connection on V in the following: For any smooth section ξ on V, X∈T_CM,

D_X^t ξ=u⁻¹(∇^t_X(u(ξ))).

One can check thatD^th=0 andD^tu=0. We defineacting on any tensor onV by

:=g^EFD_E^tD_F^t. Then by (3.2), the evolution equation of ˜Ris:

∂

∂tR˜ABCD=R˜ABCD−2h^EFh^GHREABGRF HCD

−2h^EFh^GHR˜EAGDR˜F BHC+2h^EFh^GHR˜EAGCR˜F BHD

(3.3)

whereh^AB = (h⁻¹)^AB.

LEMMA3.2. With the above notations, we have

∂^k

∂t^kR˜_ABγδ=0 att=0 for allA, Bandγ, δ.

Proof. Note that we have:

∂^k

∂t^kR_ABγδ=

m+n+p+q+r=k 0≤m,n,p,q,r≤k

(u^E_A)m(u^F_B)n(u^G_γ)p(u^H_δ )q(R_{EF GH})r.

By Lemma 3.1, in order to prove the lemma, it is sufficient to prove that _∂t^∂kku^α_β¯=0 and _∂t^∂kku^α_β^¯ =0 for allkfor allα, βatt=0.

Recall that

⎧⎪

⎨

⎪⎩

∂

∂tu^A_B=g^ACR_CDu^D_B, u^A_B(0) =δ_B^A.

Henceu^α_β¯=0 andu^α_β^¯=0. By induction, Lemma 3.1, and the fact thatu^A_B(0) =δ^A_B, one can prove that show _∂t^∂kku^α_β¯ =0 and _∂t^∂kku^α_β^¯ =0 for all k. This completes the

proof of the lemma.

Proof of Theorem 3.1. As in [24], define a smooth function ϕon M×[0, T] by

ϕ(z, t) =h^αξ¯h^βζ¯h^γσ¯h^δη¯R˜αβγδR˜ξ¯ζ¯σ¯η¯+h^αξ¯ h^βζ¯ h^γσ¯h^δη¯R˜_α¯βγδ¯ R˜ξζσ¯η¯

+h^αξ¯ h^βζ¯h^γσ¯h^δη¯R˜αβγδ¯ R˜_ξζ¯σ¯η¯+h^αξ¯h^βζ¯ h^γσ¯h^δη¯R˜_αβγδ¯ R˜ξζ¯ σ¯η¯. (3.4)

One can checkϕis well defined (independent of coordinate changes onM) and is nonnegative. The evolution equation ofϕis (See [24]):

∂

∂t−

ϕ=R˜_CDEF∗R˜_GHαβ∗R˜_ABγδ−2g^EFR˜_ABγδ;ER˜_ABγδF. (3.5)

As the real case, define the norm of the complex curvature tensor by:

|R_ABCD(t)|²_g(t)=g^AEg^BFg^CGg^DHR_ABCDR_{EF GH}. Then we have

|R_ABCD(t)|=|R_ijkl(t)| ≤a t

onM×[0, T]by assumption. By the definition ofRABCD, we also have:

|RABCD(t)|=|RABCD(t)| ≤ a t. Combining with (3.5), we have

∂

∂t−

ϕ≤C1

t ϕ onM×[0, T]for some constantC₁. Moreover,

ϕ≤ |R_ABCD(t)|²≤a²/t².

On the other hand, by (3.4), Lemma 3.2 and the fact thathis independent oft, we conclude that att=0,

∂^k

∂t^kϕ=0,

for allk. By Lemma 2.3, we conclude that ϕ≡0 onM×[0, T]. As in [24], we

conclude thatg(t)is K¨ahler for allt >0.

COROLLARY 3.1. Let(Mⁿ, g₀)be a complete noncompact K¨ahler manifold of complex dimensionnwith nonnegative complex sectional curvature. Suppose

pinf∈M{V_p(1)|p∈M}=v₀>0.

whereV_p(1)is the volume of the geodesic ball with radius 1 and center atpwith respect tog₀. Then there is T >0 depending only on n, v₀ such that the K¨ahler- Ricci flow has a smooth complete solution onM×[0, T]with initial datag(0) =g₀ and such that g(t) has nonnegative complex sectional curvature. Moreover the curvature satisfies:

|Rm(g(t))|g(t)≤c t wherecis a constant depending only onn, v₀.

Proof. The corollary follows immediately from the result of Cabezas-Rivas

and Wilking [2], and Theorem 3.1.

COROLLARY 3.2. Let(Mⁿ, g0)be a complete noncompact K¨ahler manifold with complex dimensionnand nonnegative complex sectional curvature. Suppose Mⁿhas maximum volume growth. ThenMⁿis biholomorphic toCⁿ.

Proof. By volume comparison, we have

pinf∈M{V_p(1)|p∈M}=v₀>0.