Stability of the Calabi flow near an extremal metric

Stability of the Calabi flow near an extremal metric

HONGNIANHUANG ANDKAIZHENG

Abstract. We prove that on a K¨ahler manifold admitting an extremal metricω and for any K¨ahler potentialϕ₀close toω, the Calabi flow starting atϕ₀exists for all time and the modified Calabi flow starting atϕ₀will always be close toω. Fur- thermore, when the initial data is invariant under the maximal compact subgroup of the identity component of the reduced automorphism group, the modified Cal- abi flow converges to an extremal metric nearωexponentially fast.

Mathematics Subject Classification (2010):53C44 (primary); 32Q15, 32Q26 (secondary).

1. Introduction

LetMbe a K¨ahler manifold and#be the K¨ahler class inH²(M,R)∩H^1,1(M,C).

By the∂∂¯-lemma, any K¨ahler metricω_ϕin#can be written as ω_ϕ =ω+√

−1∂∂ϕ¯

for some smooth real-valued K¨ahler potential ϕ. The space of K¨ahler metrics is defined by

H= {ϕ∈C^∞(M,R)|ω+√

−1∂∂ϕ¯ >0}.

Donaldson [17], Mabuchi [20] and Semmes [21] independently defined a Weil- Peterson type-metric onH, under whichHbecomes a non-positively curved infinite dimensional symmetric space. Chen [5] proved that any two points in H ^{can be} connected by aC^1,1 geodesic and thatHis a metric space, which verifies two of Donalson’s conjectures.

In order to tackle the existence of a constant scalar curvature K¨ahler metric (cscK) problem, Calabi [2, 3] introduced a well-known functional

Ca(ϕ)=

!

M

S(ϕ)²ωⁿ_ϕ,

Received June 3, 2010; accepted in revised form November 9, 2010.

whereS(ϕ)is the scalar curvature ofω_ϕ. The critical point of this Calabi functional is called an extremal K¨ahler metric. Calabi discovered that an extremal K¨ahler metric is a cscK if and only if the Calabi-Futaki invariant is equal to zero. Later, he suggested that one may study the gradient flow of the K-energy to search for the cscK. This flow is defined as

∂ϕ

∂t = S−S (1.1)

and it decreases the Calabi energy. Since (1.1) is a fourth order equation, the maxi- mal principle fails. In [18], Donaldson proposed a programme to study the conver- gence of the Calabi flow. On a Riemannian surface, P. Chru´sciel [16] proved that the flow exists for all time and converges to a cscK metric by using the Bondi mass.

Later Chen [6] and Struwe [22] gave a different proof assuming the uniformization theorem. In Chen-Zhu [15], they removed the assumption of the uniformization the- orem. For higher dimensions, the Calabi flow has been studied in Chen-He [8–10]

and Tosatti-Weinkove [23]. In Chen-He [8], they proved that the Calabi flow can start from aC^3,αK¨ahler potential and become smooth immediately ast >0.

One defines the little H¨older spacec^k,αto be the closure of smooth functions in the usual H¨older normC^k,α.

Theorem 1.1 ([8]). If ω_ϕ0 = ω + √

−1∂∂ϕ¯ ₀ satisfies |ϕ₀|_c^3,α_(M,g) ≤ K, and λω < ω₀ = ω_ϕ0 < 'ω, where K,λ,'are positive constants, then the Calabi flow initiating fromϕ₀admits a unique solution

ϕ(t)∈C([0,T],c^3,α(M,g))∩C((0,T],c^4,α(M,g))

for small T = T(λ,',K,ω). More specifically, for anyt ∈ (0,T], there is a constantC =C(λ,',K,ω)such that

t^1/4(| ˙ϕ(t)|_c^0,α(M)+ |ϕ(t)|_c^4,α_(M))≤C,|ϕ(t)|_c^3,α_(M)≤C.

Remark 1.2. He [19] shows that the Calabi flow can start fromω_ϕ where ϕ ∈ c^2,α(M).

Theorem 1.3 ([8]). The solution obtained above belongs to

C⁰([0,T],c^3,α(M))∩C⁰((0,T],C^∞(M)).

Chen and He then use an energy argument to show that when K¨ahler manifolds admits a cscK ω and the initial K¨ahler potential is C^3,α small, the Calabi flow converges exponentially fast to a cscK nearby. In He [19], he improved this result forC^2,αsmall initial K¨ahler potentials.

In this short note, we prove a parallel theorem for extremal K¨ahler metrics using a different method from Chen-Ding-Zheng [7]. In that paper, they defined a flow called the pseudo-Calabi flow. They proved the short time existence from c^2,αinitial K¨ahler potentials, the long time existence under uniform Ricci bounds

and the stability near a cscK. Since the linearized operator of the pseudo-Calabi flow is not self-adjoint, they set up a unified frame to tackle the stability problem of the K¨ahler Ricci flow (cf. Zheng [24]), the Calabi flow and the pseudo-Calabi flow. This method strongly relies on the geometric structure of the space of K¨ahler metrics.

First, we will prove the long time existence of the Calabi flow.

Theorem 1.4. On K¨ahler manifolds admitting an extremal metric ω and for any positive constantK^,λ, there is a small constant(depending onω,K^,λ, such that for any K¨ahler potentialϕ₀, if

|ϕ₀|C^2,α(M)<K^{, λω<ω}ϕ0,

!

M|ϕ₀|²ωⁿ <(, then the Calabi flow exists for all time.

Next we want to study the modified Calabi flow. LetKbe a maximal compact subgroup in the reduced automorphism group. Denote the corresponding Lie alge- bra ofK byh0(M), which is the ideal of holomorphic vector fields with zeros. For any holomorphic vector fieldY˜ ∈h0(M), denoteY˜ =Y −√

−1J Y. Then there is a real functionθ_Y such that

L_Y˜ω=LYω=√

−1∂∂θ¯ _Y(t)

and !

M

θ_Yωⁿ =0.

For an arbitrary metric

ω_ϕ=ω+√

−1∂∂ϕ,¯ the correspondingθ_Y˜(ϕ)is

θ_Y˜(ϕ)=θ_Y+ ˜Y(ϕ).

Following Futaki-Mabuchi [1], suppose X,˜ Y˜ ∈h0(M), then the bilinear form B(X,˜ Y˜)=

!

M

θ_X(ϕ)˜ θ_Y(ϕ)˜ ωⁿ_ϕ is independent of the choice ofω_ϕin the K¨ahler class#= [ω].

Let ϕ(t) be a one parameter of K¨ahler potentials satisfying the Calabi flow equation and let σ(t)be the holomorphic group generated by X, the real part of the extremal vector field X. Then˜ σ^∗(t)ω =ω+i∂∂ρ(t¯ )satisfies the Calabi flow equation since

∂σ^∗(t)ω

∂t =L_Xω(t)=i∂∂¯(S(t)−S).

Hence we can choose ρ(t)to be a parameter of K¨ahler potentials satisfying the Calabi flow equation starting from 0.

Letψ(t)=σ(−t)^∗(ϕ(t)−ρ(t)). Notice that, by definition, X = σ_∗⁻¹(_∂t^∂σ).

So we obtain the modified Calabi flow,

∂ψ

∂t =−X(ψ(t))+σ(−t)^∗

"

∂ϕ(t)

∂t −∂ρ(t)

∂t

#

=−X(ψ(t))+σ(−t)^∗(S(ϕ(t))−S)−(S(ω)−S)

= Sψ −S−θ_X −X(ψ)

= Sψ −S−θ_X(ψ).

Theorem 1.5. On K¨ahler manifolds admitting an extremal metric ω, for any K^- invariant K¨ahler potentialϕ₀close toω(in the sense of Theorem(1.4)), the modified Calabi flow exponentially converges to a nearby extremal metric.

ACKNOWLEDGEMENTS. The authors want to thank Professor X. X. Chen for many useful discussions on the stability problem of Calabi flow. They also want to express their gratitude to Professor V. Apostolov for his interest and help in this problem.

The second author wants to thank his advisor Professor W. Y. Ding, also Professor X. H. Zhu and Professor H. Z. Li for their help and encouragement. The first author would like to thank Professor P. F. Guan for valuable discussions.

2. Long time existence

First of all, we would like to give a rough estimate of the geodesic distance between any two K¨ahler potentialsϕ₀,ϕ₁whenωϕ₀,ωϕ₁ <'ω.

Lemma 2.1. d(ϕ0,ϕ₁) <C(')$%

M|ϕ₀−ϕ₁|²ωⁿ&¹₂ . Proof. Letϕ_t =(1−t)ϕ₀+tϕ₁for 0≤t ≤1. Then

d(ϕ0,ϕ₁)≤L(γt)=

! ₁

0

'!

M

"

∂γ_t

∂t

#2

ωⁿ_γt (¹₂

dt

≤

! ₁

0

"!

M

(ϕ₀−ϕ₁)²ωⁿ_γt

#¹₂ dt

≤C(')

"!

M|ϕ₀−ϕ₁|²ωⁿ

#¹₂ . We are ready to give a proof of Theorem 1.4.

Proof. Suppose that the conclusion fails, then there exist positive constantsK^,λ,' and a sequence ofϕ_s⁰such that

|ϕ_s⁰|C^2,α <K^{, λω<ω}ϕ^s₀<'ω,

!

M|ϕ_s⁰|²ωⁿ < 1

s s=1,2,3· · · . By virtue of the short time existence theorem, we get a sequence of solutionsϕ_s(t) satisfying the flow equation (1.1) withϕ_s(0)=ϕ⁰_s. LetTsbe the first time such that

|ϕ_s(Ts)−ρ(Ts)|c^2,α(ω(Ts))=2C holds or

λω(Ts) <ω_ϕs(Ts)<'ω(Ts) fails,

where the constantC is from Theorem 1.1. ThenT_s is bounded from below for sufficiently larges. Otherwise there is a subsequence ofϕ_s(T_s)converging toϕ_∞ in the C^2,α((ω) sense, whereα⁽ < α. Notice that λω ≤ ω_ϕ∞ ≤ 'ω, but that λω<ω_ϕ∞<'ωfails.

On the other hand, Lemma 2.1 shows thatd(0,ϕ_s⁰)→0 ass → ∞. Since the distance function decreases under the Calabi flow, we have

d(ρ(T_s),ϕ_s(T_s))→0

ass → ∞. Letϕ_∞(t)be one parameter potentials satisfying the Calabi flow equa- tion initiating fromϕ_∞. Then fort0≥t,

d(ρ(t0),ϕ_∞(t0)) ≤d(ρ(t),ϕ_∞(t))

≤d(ρ(t),ρ(T_s))+d(ρ(T_s),ϕ_s(T_s))+d(ϕ_s(T_s),ϕ_∞(t)).

By Lemma 2.1, d(ϕs(Ts),ϕ_∞(t)) → 0 ass → ∞ andt → 0. Henceρ(t0) = ϕ_∞(t0), which implies 0=ϕ_∞, a contradiction.

Moreover, from Theorem 1.3, we obtain the higher order uniform bounds of the sequence of the solutions:

|ϕ_s(Ts)−ρ(Ts)|C^k,α(ω(Ts))≤C(k), ∀k≥0.

Therefore we can choose a subsequence ofφ_s=σ(−Ts)^∗(ϕ_s(Ts)−ρ(Ts))so that φ_s→φ_∞inC^k,α(ω),∀k≥0,

and |φ_∞|_C^2,α_(ω)=2C (or λω<ω_φ∞<'ωfails).

However, this contradicts the fact thatd(0,φ_∞)=0.

Corollary 2.2. Given a K¨ahler potentialϕ₀close to0in the sense of Theorem1.4, then the modified Calabi flow stays in a neighborhood of0. Ifϕ₀is K-invariant, then the modified Calabi flow converges to an extremal metric nearby.

Proof. That the modified Calabi flow stays in a neighborhood of 0 can be easily seen from the regularity Theorem 1.3. Notice that the Calabi flow decreases the Calabi energy,i.e.

∂

∂tCa(ωϕ)=−2

!

MLϕ(Sϕ)Sϕωⁿ_ϕ,

whereLϕis the Lichnerowicz operator with respect to ω_ϕ. It follows that we can take a sequence oft_j → ∞such that

jlim→∞

!

MLϕ(tj)(Sϕ(tj))Sϕ(tj) ωⁿ_ϕ(tj)=0.

Then there is a subsequence of tj such thatψ(tj)converges to a potentialψ_∞ in

C^∞and !

MLψ_∞(Sψ_∞)Sψ_∞ωⁿ_ψ∞=0.

Hence ω_ψ∞ is an extremal metric. Ifϕ₀ is K-invariant, then ψ_∞ is a fixed point under the modified Calabi flow and the modified Calabi flow decreases the geodesic distance betweenψ(t)andψ_∞. Hence the flow converges toψ_∞.

3. Exponential decay

We define the modified Calabi energy as Ca(ψ)) =

!

M

(S(ψ)−S−θ_X(ψ))²ωⁿ_ψ.

The evolution of the modified Calabi energy along the modified Calabi flow is

∂_t

!

M

ψ˙²ωⁿ_ψ =

!

M

(2ψ˙ψ¨ + ˙ψ²,ψψ)ω˙ ⁿ_ψ

=2

!

M

˙

ψ(S˙ψ −θ˙_X(t)−ψ˙_iψ˙ⁱ)ω_ψⁿ

=2

!

M

ψ(˙ −Lψ˙ + ˙ψⁱSi−θ˙_X(t)−ψ˙_iψ˙ⁱ)ωⁿ_ψ

=2

!

M

˙

ψ(−Lψ˙ + ˙ψⁱ(ψ˙_i +θ_X(t)_i)−θ˙_X(t)−ψ˙_iψ˙ⁱ)ωⁿ_ψ

=2

!

M

ψ(˙ −Lψ˙ + ˙ψⁱψ˙_i +X(ψ˙)−X(ψ)˙ −ψ˙_iψ˙ⁱ)ωⁿ_ψ

=−2

!

M

ψ˙Lψω˙ ⁿ_ψ. In this computation we use the identities

ψ˙ⁱ(θ_{X i} +(X(ψ))_i)= ˙ψⁱ(θ_X(t))_i = X(ψ˙).

The modified Calabi-Futaki invariant F(˜ Y˜)= F(Y˜)−B(X,˜ Y˜)=

!

M

θ_Y˜(ψ)(S−S−θ_X˜(ψ))ωⁿ_ψ (3.1) is equal to zero whenMadmits an extremal metricω. This shows that the direction of the modified Calabi flow is always perpendicular to the kernel of the Lichnerow- icz operator. To obtain exponential convergence, one needs to give a uniform lower bound of the the first eigenvalue of L_t along the modified Calabi flow. More pre- cisely, we have the following lemma which is similar to Chen-Li-Wang [11].

Lemma 3.1. Along the modified Calabi flow, there is a positive constant λ > 0 such that for sufficiently larget and for any

f ∈A_t = {f ∈C^∞_R (M)|

!

M

fωⁿ_ψ(t)=0and

!

M

θ_Y(t)fω_ψⁿ_(t)=0,∀Y˜ ∈h₀(M)},

we have !

M

Lt(f)fωⁿ_ψ(t) ≥λ

!

M

f²ωⁿ_ψ(t).

Proof. If not, there must be a sequenceψ_s=ψ(s)and fssuch that

!

M|(fs)_{i j}|²ωⁿ_ψs < 1 s;

!

M

f_s²ω_ψⁿ_s =1;

!

M

fsωⁿ_ψs =0. (3.2) Since the C^l norm of ψ_s is uniformly bounded for any l ≥ 0. Using the Ricci identity

!

M|(fs)_ij¯|²ωⁿ_ψs =

!

M|(fs)_{i j}|²ωⁿ_ψs +

!

M

R^i¯j(fs)_i(fs)_j¯ωⁿ_ψs

and the interpolation inequality, we conclude that fs are uniformlyW^2,2bounded.

So we can pass to the limit and get

!

M|(f_∞)_{i j}|²ωⁿ_ψ

∞ =0;

!

M

f_∞²ωⁿ_ψ

∞=1;

!

M

f_∞ωⁿ_ψ

∞ =0. (3.3) Since in local coordinates,↑∂¯f_∞is holomorphic in the weak sense, f_∞is smooth indeed. From the assumption of A_t we have

!

M

θ_Y(ψ_∞)f_∞ωⁿ_ψ

∞=0,∀Y˜ ∈h₀(M).

In particular, we may chooseY˜ =↑∂¯f_∞∈h₀(M). Hence,

!

M

f_∞²ωⁿ_∞=0.

This contradicts (3.3).

It is easy to see thatCa(ψ(t) ))≤Ce^−λt. To get exponential convergence ofψ(t), we calculate the evolution formula for%

M|∇^k(ψ(t)−ψ_∞)|²ωⁿ:

∂

∂t

!

M|∇^k(ψ(t)−ψ_∞)|²ωⁿ

=

!

M∇^k(S−S−θ_X(ψ))∗ ∇^k(ψ(t)−ψ_∞)ωⁿ

=

!

M

(S−S−θ_X(ψ))∗ ∇^2k(ψ(t)−ψ_∞)ωⁿ

≤ "!

M

(S−S−θ(X))²ωⁿ

#1/2"!

M|∇^2k(ψ(t)−ψ_∞)|²ωⁿ

#1/2

≤ C/S−S−θ(X)/L²(ω)

≤ C/S−S−θ(X)/L²(ωt)

≤ Ce^−λ2t.

By the Sobolev embedding, we conclude that

/ψ_t−ψ_∞/_C^l(ω)≤ /ψ_t −ψ_∞/_W^k,2_(ω)≤ Ce^−λ2t. Hence we obtain the result stated in Theorem 1.5.

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Centre interuniversitaire de recherches en g´eom´etrie et topologie

Universit´e du Qu´ebec `a Montr´eal Case postale 8888, Succursale centre-ville Montr´eal (Qu´ebec), H3C 3P8, Canada hnhuang@gmail.com

Academy of Mathematics and Systems Sciences Chinese Academy of Sciences Beijing, 100190, P.R. China kaizheng@amss.ac.cn