Stability of line bundle mean curvature flow
aa r X i v : . [ m a t h . DG ] F e b STABILITY OF LINE BUNDLE MEAN CURVATURE FLOW
XIAOLI HAN AND XISHEN JIN
Abstract.
Let (
X, ω ) be a compact K¨ahler manifold of complex dimen-sion n and ( L, h ) be a holomorphic line bundle over X . The line bundlemean curvature flow was introduced by Jacob-Yau in order to find deformedHermitian-Yang-Mills metrics on L . In this paper, we consider the stability ofthe line bundle mean curvature flow. Suppose there exists a deformed Hermit-ian Yang-Mills metric ˆ h on L . We prove that the line bundle mean curvatureflow converges to ˆ h exponentially in C ∞ sense as long as the initial metric isclose to ˆ h in C -norm. Introduction
Let (
X, ω ) be a compact K¨ahler manifold of complex dimension n and L be aholomorphic line bundle over X . Given a Hermitian metric h on L , we define acomplex function ζ on X by ζ := ( ω − F h ) n ω n where F h = − ∂ ¯ ∂ log h is the curvature of the Chern connection with respect tothe metric h . It is easy to see that the average of this function is a fixed complexnumber Z L, [ ω ] := Z X ζ ω n n !depending only on the cohomology classes c ( L ) and [ ω ] ∈ H , ( X, R ). Let ˆ θ bethe argument of Z L, [ ω ] . Definition 1.1.
A Hermitian metric h on L is said to be a deformed Hermitian-Yang-Mills (dHYM) metric if it satisfies(1.1) Im( ω − F ) n = tan(ˆ θ ) Re( ω − F ) n . We define the Lagrangian Phase operator θ : ∧ , X → R by θ ( F ) = n X j =1 arctan λ j where λ j ( j = 1 , · · · , n ) are the eigenvalues of ω − F ∈ End( T , X ). Then accordingto arguments in [9], the equation (1.1) is equivalent to(1.2) θ ( F ) = ˆ θ ( mod 2 π ) . Mathematics Subject Classification.
Key words and phrases.
Deformed Hermitian-Yang-Mills metric, Line bundle mean curvatureflow, Stability. s discussed in [4], we also remark that the constant ˆ θ in (1.2) can be obtained byconsidering the “winding angle” of γ ( t ) = Z X e − t √− ω Ch ( L )as t runs from + ∞ to 1 if γ ( t ) does not cross 0 ∈ C .According to the superstring theory, the spacetime of the universe is constrainedto be the product of a compact Calabi-Yau threefold and a Lorentzian manifoldof four dimension. A ‘duality’ relates the geometry of one Calabi-Yau manifold toanother ‘mirror’ Calabi-Yau manifold. The dHYM equation was first discoveredby Marino et all [13] as the requirement for a D-brane on the B-model of mirrorsymmetry to be supersymmetric. This is explained by Leung-Yau-Zaslow[12] inmathematical language. From a viewpoint of differential geometry, this might bethought of a relationship between the existence of ‘nice’ metrics on the line bun-dle over one Calabi-Yau manifold and special Lagrangian submanifolds in anotherCalabi-Yau manifold. In [12], Leung-Yau-Zaslow showed that the dHYM equationon a line bundle corresponds to the special Lagrangian equation in the mirror. Re-cently, the dHYM metrics have been studied actively(e.g. [2], [4], [5], [7], [8], [9],[14], [15], [16] etc).In [9], Jacob-Yau gave a necessary and sufficient condition for the existence ofdHYM metrics on the K¨ahler surface. More precisely, they proved that: Theorem 1.1 ([9]) . Let L be a holomorphic line bundle on K¨ahler surface X . Thenthere exists one dHYM metric on L if and only if there exists a metric h such that Ω( h ) = cot(ˆ θ ) ω + √− F > . In order to study the existence of dHYM metrics on high dimensional K¨ahlermanifolds, Jacob-Yau [9] introduced a parabolic evolution flow for the metric h onthe line bundle L . Definition 1.2. (Line bundle MCF) Given one Hermitian metric h on L , wedefine a flow of metrics h t = e − u t h by the following equation:(1.3) ddt u t = θ ( F h t ) − ˆ θ where F h t = F h + ∂ ¯ ∂u t is the Chern curvature of L with respect to h t .The flow (1.3) can be regarded as the complex version of the mean curvatureflow for the Lagrangian graph. Thus it is also called line bundle mean curvatureflow (line bundle MCF). As shown in [9], this equation is parabolic and exists ina short time [0 , ε ). They also proved the following theorem on extension of linebundle MCF. Theorem 1.2 ([9]) . Let L be a holomorphic line bundle over the compact K¨ahlermanifold X and h t be a path of metrics on L solving (1.3). Assume that Z L, [ ω ] = 0 and |∇ F t | g ≤ C uniformly in time [0 , T ) , then all derivatives |∇ k F t | g are boundedby some constant C k . Furthermore, if T is finite, then the flow can be extendedto T + ε . If T = ∞ , then the flow subsequently converges to a smooth solution of(1.2). In [9], Jacob-Yau also studied the long time existence and convergence of theline bundle MCF under some assumptions. heorem 1.3 ([9], Theorem 1.3) . Let ( X, ω ) be a K¨ahler manifold with non-negative orthogonal bisectional curvature and L be an ample line bundle. Thenthere exists a natural number k such that L ⊗ k admits a dHYM metric and it isconstructed via a smoothly converging family of metrics along the line bundle MCF.Remark . Indeed, the condition on k in Theorem 1.3 guarantees that the initialdata u satisfies the so-called hypercritical condition, i.e. θ ( F u ) > ( n − π . As aconsequence of this hypercritical condition, F u t > t ≥
0. Then the operator θ ( F u t ) is concave and the Evans-Krylov theory works for higher order estimates.In [2], Collins-Jacob-Yau considered the existence of dHYM metric under as-sumption of C -subsolutions. They also get the following result for the line bundleMCF. Theorem 1.4 ([2], Remark 7.4) . If ˆ θ > ( n − π and u is a subsolution with θ ( F u ) > ( n − π , then the line bundle MCF starting from u converges smoothly toa dHYM metric. In [16], Takahashi proved the collapsing behavior of the line bundle MCF onK¨ahler surfaces with boundary conditions on some cohomology class.Another desirable property of a geometric flow is the stability of stationarypoints. That is, if the initial date is sufficiently close to one stationary point, thenthe flow exists for long time and converges to the stationary point. The stabilityresult gives more evidences that the method of flows will work to find the stationarypoint. There have been a lot of stability results for other geometric flows. In [3],Chen and Li gave some stability results of K¨ahler-Ricci flow with respect to thedeformation of the underlying complex structures under the assumption c ( M ) > C ∞ sense if the initial metric is veryclose to a K¨ahler-Einstein metric on a Fano manifold. In [10], Lotay-Wei provedthat the torsion-free G structures are dynamically stable along the Laplacian flowfor closed G structures. Guedj-Lu-Zeriahi considered the stability of solutions tocomplex Monge-Amp`ere flows in [6].We consider the stability of the line bundle MCF in this paper. That is, ifwe assume that there exists one dHYM metric on holomorphic line bundle over acompact K¨ahler manifold and the initial metric is C close to this dHYM metric,then the flow will admit long-time solution and exponentially converge to this givendHYM metric. Theorem 1.5.
Let ( X, ω ) be a compact K¨ahler manifold of complex dimension n and L be a holomorphic line bundle over X . Assume ˆ h is a dHYM metric on L and h ( t ) = e − u t ˆ h satisfy the line bundle MCF, i.e, ddt u t = θ ( F u t ) − ˆ θ. There exists a constant δ > such that if the smooth initial data u satisfies || D u || L ∞ ≤ δ , then the line bundle MCF exists for long time and u t convergesto a constant exponentially. We do not need any assumption on the phase ˆ θ and the positivity of F u t whichare crucial to guarantee the concavity of the operator θ ( F ). The method to proveTheorem 1.5 is to get uniform estimates of u t . The main step in the proof of this heorem is to obtain that the smallness of D u t is preserved along the line bundleMCF. As an application of this smallness, we get a uniform estimate for ∇ ¯ ∇∇ u t bysome parabolic Calabi-type computation. We should mention that we can not applythe Evans-Krylov theory to get C ,α estimate here as in [9], since we do not have theconcavity condition on the operator θ here. Higher order uniform estimates alongthe line bundle MCF are obtained by the standard parabolic Schauder estimate.We will organize this paper as follows. In Section 2, we review some basics onthe dHYM metrics and give some useful notations and equations. In Section 3, wecompute the evolution equations for u and the derivatives of u up to order 2 alongthe line bundle MCF. In Section 4, we obtain the smallness of D u t along the linebundle MCF. In Section 5, we prove the first part of Theorem 1.5, i.e. the long-timeexistence and convergence of the line bundle MCF subsequently. In Section 6, weprove the exponential convergence of the line bundle MCF which is the second partof Theorem 1.5. 2. Preliminaries
Let (
X, ω ) be a compact K¨ahler manifold of complex dimension n where ω is agiven K¨ahler form. We denote ( L, h ) to be a holomorphic line bundle over X and F to be the Chern curvature with respect to the Hermitian metric h on L . In localcoordinates, we write ω = √− g i ¯ j dz i ∧ d ¯ z j and F = 12 F i ¯ j dz i ∧ d ¯ z j = − ∂ i ∂ ¯ j log( h ) dz i ∧ d ¯ z j . Since ω is K¨ahler, we have the following second Bianchi equality F i ¯ j,k = − ∂ k ∂ ¯ j ∂ i log( h ) + Γ sik F s ¯ j = − ∂ i ∂ ¯ j ∂ k log( h ) + Γ ski F s ¯ j = F k ¯ j,i , i.e. F i ¯ j,k = F k ¯ j,i where “ , k ” is covariant derivative with respect to ω .Using notations above, we introduce a Hermitian (usually not K¨ahler) metric η on X and an endomorphism K of T , ( X ) that are defined by(2.1) η = √− η ¯ kj dz j ∧ d ¯ z k and K := ω − F = g i ¯ j F k ¯ j dz i ⊗ ∂∂z k where η ¯ kj = g ¯ kj + F ¯ kl g l ¯ m F ¯ mj . Then the complex-valued ( n, n )-form ( ω − F ) n can be locally written as( ω − F ) n = n ! det( g ¯ kj + √− F ¯ kj )( √−
12 ) n dz ∧ d ¯ z ∧ · · · ∧ dz n ∧ d ¯ z n = det( g j ¯ k ) det( g ¯ kj + √− F ¯ kj ) ω n = det ( I + √− K ) ω n . Therefore the complex function ζ can be expressed as(2.2) ζ = det( I + √− K ) . sing this expression, we can get the first variation of θ ( F ) (cf. Lemma 3.3 in [9]) δθ ( F ) = Tr(( I + K ) − δK )where the endomorphism I + K of T , X can be expressed locally as I + K = g p ¯ q η ¯ ql ∂∂z p ⊗ dz l . Thus we obtain the following formula on the derivative of θ ( F ) ∂ i θ = Tr(( I + K ) − ∇ i K )= η p ¯ q g ¯ ql ∇ i ( g l ¯ m F ¯ mp )= η p ¯ q ∇ i F ¯ qp = η p ¯ q F p ¯ q,i . (2.3)Now suppose ˆ h the dHYM metric on L and its Chern curvature is denoted byˆ F . Since ˆ h satisfies the equation (1.2), θ ( ˆ F ) = ˆ θ is a constant. Taking derivativesof θ ( ˆ F ), we have the following two equalities0 = Tr(( I + ˆ K ) − ∇ i ˆ K ) = ˆ η p ¯ q ˆ F p ¯ q,i , I + ˆ K ) − ∇ ¯ j ˆ K ) = ˆ η p ¯ q ˆ F p ¯ q, ¯ j (2.4)where { ˆ η p ¯ q } is the inverse matrix of { ˆ η p ¯ q } . Taking derivatives again on the bothsides of (2.4), we obtain the following two equalities for second order derivativesˆ η p ¯ q ˆ F p ¯ q, ¯ ji = − ˆ η p ¯ q,i ˆ F p ¯ q, ¯ j = ˆ η p ¯ t ˆ η s ¯ q ˆ η s ¯ t,i ˆ F p ¯ q, ¯ j = ˆ η p ¯ t ˆ η s ¯ q ˆ F p ¯ q, ¯ j g a ¯ b ( ˆ F s ¯ b,i ˆ F a ¯ t + ˆ F s ¯ b ˆ F a ¯ t,i )(2.5)and ˆ η p ¯ q ˆ F p ¯ q, ¯ j ¯ i = − ˆ η p ¯ q, ¯ i ˆ F p ¯ q, ¯ j = ˆ η p ¯ t ˆ η s ¯ q ˆ η s ¯ t, ¯ i ˆ F p ¯ q, ¯ j = ˆ η p ¯ t ˆ η s ¯ q ˆ F p ¯ q, ¯ j g a ¯ b ( ˆ F s ¯ b, ¯ i ˆ F a ¯ t + ˆ F s ¯ b ˆ F a ¯ t, ¯ i ) . (2.6)These equalities will play significant role in our proof of the main theorem.A very special case is that we take [ F ] = c [ ω ] for some constant c . Then we havea trivial dHYM metric ˆ F = cω . In this case, we haveˆ F p ¯ q,i = 0 , ˆ F p ¯ q, ¯ i = 0 , ˆ F p ¯ q,i ¯ j = 0 and ˆ F p ¯ q, ¯ i ¯ j = 0 . This will make our proof much easier as discussed in Section 3.3.
Evolution inequalities along the line bundle mean curvature flow
In this section, we assume that ˆ h is a given dHYM metric and consider the linebundle MCF (1.3) with the initial metric h = e − u ˆ h . We will give the evolu-tion equations of some quantities related to u t ( x ) along the line bundle MCF. Wealso show some inequalities of these quantities. For convenience, we will omit thesubscript t of u t if there is no confusion.In the following of this paper, we always use the following second order operator∆ η = η p ¯ q ∂ p ∂ ¯ q where η is the Hermitian metric defined in (2.1). We also remark that ∆ η is thelinearization operator of Equation (1.2). t first, we compute the evolution of u along the line bundle MCF. Lemma 3.1.
The evolution equation of u along the line bundle MCF is given by (3.1) ( ∂∂t − ∆ η ) u = 2 u ( θ ( F u ) − ˆ θ − ∆ η u ) − η p ¯ q u p u ¯ q . Proof.
According to the line bundle MCF (1.3), we get that ∂∂t u = 2 u ( θ ( F u ) − ˆ θ )and ∆ η u = 2 u ∆ η u + 2 η p ¯ q u p u ¯ q . Thus we obtain the following equation( ∂∂t − ∆) u = 2 u ( θ ( F u ) − ˆ θ − ∆ η u ) − η p ¯ q u p u ¯ q . (cid:3) We denote ∇ to be the (1 , | · | ω to be the normwith respect to ω . Lemma 3.2.
The evolution equation of |∇ u | ω along the line bundle MCF is givenby ( ∂∂t − ∆ η ) |∇ u | ω = − η p ¯ q g i ¯ j ( u ip u ¯ j ¯ q + u i ¯ q u ¯ jp ) − η p ¯ q g i ¯ j R p ¯ li ¯ q u l u ¯ j + 2 Re( η p ¯ q g i ¯ j ˆ F p ¯ q,i u ¯ j ) . (3.2) Proof.
Firstly, we deal with the derivative with respect to t , ∂∂t |∇ u | ω = ∂∂t ( g i ¯ j u i u ¯ j )= g i ¯ j ( ∂u∂t ) i u ¯ j + g i ¯ j u i ( ∂u∂t ) ¯ j = g i ¯ j θ i u ¯ j + g i ¯ j θ ¯ j u i = g i ¯ j η p ¯ q F p ¯ q,i u ¯ j + g i ¯ j η p ¯ q F p ¯ q, ¯ j u i where we have used Equation (2.3) in the last “=”. Since F p ¯ q = ˆ F p ¯ q + u p ¯ q , we have ∂∂t |∇ u | ω = g i ¯ j η p ¯ q u p ¯ qi u ¯ j + g i ¯ j η p ¯ q u p ¯ q ¯ j u i + g i ¯ j η p ¯ q ˆ F p ¯ q,i u ¯ j + g i ¯ j η p ¯ q ˆ F p ¯ q, ¯ j u i . Secondly, we deal with the “∆ η ”-part,∆ η |∇ u | ω = η p ¯ q ( g i ¯ j u i u ¯ j ) p ¯ q = η p ¯ q g i ¯ j ( u ip u ¯ j ¯ q + u i ¯ q u p ¯ j + u ip ¯ q u ¯ j + u i u ¯ jp ¯ q ) . (3.3)Applying the Ricci identity, we can change orders of derivatives as follow u ip ¯ q = u pi ¯ q = u p ¯ qi + u l R p ¯ li ¯ q ,u ¯ ip ¯ q = u ¯ qp ¯ i = u p ¯ q ¯ i . nserting these two equalities to Equation (3.3), we obtain∆ η |∇ u | ω = η p ¯ q g i ¯ j ( u ip u ¯ j ¯ q + u i ¯ q u p ¯ j )+ η p ¯ q g i ¯ j u ¯ j ( u p ¯ qi + u l R p ¯ li ¯ q ) + η p ¯ q g i ¯ j u i u p ¯ q ¯ j . Therefore, we have the following evolution equation along the line bundle MCF( ∂∂t − ∆ η ) |∇ u | ω = − η p ¯ q g i ¯ j ( u ip u ¯ j ¯ q + u i ¯ q u ¯ jp ) − η p ¯ q g i ¯ j R p ¯ li ¯ q u l u ¯ j + η p ¯ q g i ¯ j ˆ F p ¯ q,i u ¯ j + η p ¯ q g i ¯ j ˆ F p ¯ q, ¯ j u i . Then we finish the proof of the lemma. (cid:3)
Next, let us consider the evolution equations of |∇ ¯ ∇ u | ω and |∇∇ u | ω along theline bundle MCF. For convenience, we first introduce some notations before detailedcomputations Θ = |∇ ¯ ∇ u | ω Θ ′ = |∇∇ u | ω Lemma 3.3.
The evolution equation of Θ along the line bundle MCF is ( ∂∂t − ∆ η )Θ= − η k ¯ l g i ¯ j g p ¯ q ( u i ¯ lp u k ¯ j ¯ q + u i ¯ l ¯ q u k ¯ jp ) − (cid:16) g i ¯ j g k ¯ l η p ¯ b η a ¯ q η a ¯ b, ¯ l F p ¯ q,i u k ¯ j (cid:17) + 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q u k ¯ j ( u a ¯ q R i ¯ ap ¯ l − u a ¯ l R i ¯ ap ¯ q ) (cid:17) + 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q u k ¯ j ˆ F p ¯ q,i ¯ l (cid:17) . (3.4) Proof.
Firstly, we compute the “ ∂∂t ”-part, ∂∂t
Θ = g i ¯ j g k ¯ l ( ∂u∂t ) i ¯ l u k ¯ j + g i ¯ j g k ¯ l ( ∂u∂t ) k ¯ j u i ¯ l = g i ¯ j g k ¯ l θ i ¯ l u k ¯ j + g i ¯ j g k ¯ l θ ¯ jk u i ¯ l = g i ¯ j g k ¯ l ( η p ¯ q F p ¯ q,i ) ¯ l u k ¯ j + g i ¯ j g k ¯ l ( η p ¯ q F p ¯ q, ¯ j ) k u i ¯ l = − (cid:16) g i ¯ j g k ¯ l η p ¯ b η a ¯ q η a ¯ b, ¯ l F p ¯ q,i u k ¯ j (cid:17) + 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q F p ¯ q,i ¯ l u k ¯ j (cid:17) . Secondly, we compute the “∆ η ”-part,∆ η Θ = η p ¯ q ( g i ¯ j g k ¯ l u i ¯ l u k ¯ j ) p ¯ q = g i ¯ j g k ¯ l η p ¯ q (cid:0) u i ¯ lp u k ¯ j ¯ q + u i ¯ l ¯ q u k ¯ jp (cid:1) + 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q u k ¯ j u i ¯ lp ¯ q (cid:17) . Adding these equations together, we get the following evolution equation for Θ,( ∂∂t − ∆ η )Θ= − (cid:16) g i ¯ j g k ¯ l η p ¯ b η a ¯ q η a ¯ b, ¯ l F p ¯ q,i u k ¯ j (cid:17) + 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q u k ¯ j ( F p ¯ q,i ¯ l − u i ¯ lp ¯ q ) (cid:17) − η k ¯ l g i ¯ j g p ¯ q ( u i ¯ lp u k ¯ j ¯ q + u i ¯ l ¯ q u k ¯ jp )= − (cid:16) g i ¯ j g k ¯ l η p ¯ b η a ¯ q η a ¯ b, ¯ l F p ¯ q,i u k ¯ j (cid:17) + 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q u k ¯ j ( u p ¯ qi ¯ l − u i ¯ lp ¯ q ) (cid:17) + 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q u k ¯ j ˆ F p ¯ q,i ¯ l (cid:17) − η k ¯ l g i ¯ j g p ¯ q ( u i ¯ lp u k ¯ j ¯ q + u i ¯ l ¯ q u k ¯ jp ) . (3.5) t last, we deal with the forth order terms appeared above. Indeed, by the Ricciidentity, we have the following formula u p ¯ qi ¯ l − u i ¯ lp ¯ q = u a ¯ q R i ¯ ap ¯ l − u a ¯ l R i ¯ ap ¯ q . Inserting it into Equation (3.5), we get the following equation( ∂∂t − ∆ η )Θ= − η k ¯ l g i ¯ j g p ¯ q ( u i ¯ lp u k ¯ j ¯ q + u i ¯ l ¯ q u k ¯ jp ) − g i ¯ j g k ¯ l η p ¯ b η a ¯ q η a ¯ b, ¯ l F p ¯ q,i u k ¯ j )+ 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q u k ¯ j ( u a ¯ q R i ¯ ap ¯ l − u a ¯ l R i ¯ ap ¯ q ) (cid:17) + 2 Re (cid:16) g i ¯ j g k ¯ l η p ¯ q u k ¯ j ˆ F p ¯ q,i ¯ l (cid:17) which is the result desired. (cid:3) By the same argument, we can get the evolution equations of Θ ′ = |∇∇ u | ω . Forthe completeness, we also list the detail here. Lemma 3.4.
The evolution equation of Θ ′ along the line bundle MCF is given by ( ∂∂t − ∆ η )Θ ′ = − g i ¯ j g p ¯ q η k ¯ b η a ¯ l η a ¯ b,p F k ¯ l,i u ¯ j ¯ q ) + 2 Re( g i ¯ j g p ¯ q η k ¯ l ˆ F k ¯ l,ip u ¯ j ¯ q ) − (cid:16) g i ¯ j g p ¯ q η k ¯ l u ¯ j ¯ q ( u ak R i ¯ ap ¯ l + u ia R k ¯ ap ¯ l + u ap R i ¯ ak ¯ l + u a R i ¯ ak ¯ l,p ) (cid:17) − η k ¯ l g i ¯ j g p ¯ q ( u ipk u ¯ j ¯ q ¯ l + u ip ¯ l u ¯ j ¯ qk ) . (3.6) Proof.
Taking the derivatives of Θ ′ about t , we obtain ∂∂t Θ ′ = g i ¯ j g p ¯ q ( ˙ u ip u ¯ j ¯ q + u ip ˙ u ¯ j ¯ q )= g i ¯ j g p ¯ q ( θ ip u ¯ j ¯ q + u ip θ ¯ j ¯ q )= g i ¯ j g p ¯ q ( η k ¯ l,p F k ¯ l,i + η k ¯ l F k ¯ l,ip ) u ¯ j ¯ q + g i ¯ j g p ¯ q ( η k ¯ l, ¯ q F k ¯ l, ¯ j + η k ¯ l F k ¯ l, ¯ j ¯ q ) u ip = 2 Re (cid:16) g i ¯ j g p ¯ q ( − η k ¯ b η a ¯ l η a ¯ b,p F k ¯ l,i + η k ¯ l F k ¯ l,ip ) u ¯ j ¯ q (cid:17) . We also have the following equation for the “∆ η ”-part,∆ η Θ ′ = η k ¯ l ( g i ¯ j g p ¯ q u ip u ¯ j ¯ q ) ¯ lk = η k ¯ l g i ¯ j g p ¯ q ( u ipk u ¯ j ¯ q ¯ l + u ip ¯ l u ¯ j ¯ qk ) + 2 Re (cid:16) η k ¯ l g i ¯ j g p ¯ q u ¯ j ¯ q u ip ¯ lk (cid:17) . Adding them together, we have the following equation( ∂∂t − ∆ η )Θ ′ = − η k ¯ l g i ¯ j g p ¯ q ( u ipk u ¯ j ¯ q ¯ l + u ip ¯ l u ¯ j ¯ qk ) − (cid:16) η k ¯ l g i ¯ j g p ¯ q u ¯ j ¯ q u ip ¯ lk (cid:17) − (cid:16) g i ¯ j g p ¯ q η k ¯ b η a ¯ l η a ¯ b,p F k ¯ l,i (cid:17) + 2 Re (cid:16) g i ¯ j g p ¯ q η k ¯ l F k ¯ l,ip u ¯ j ¯ q (cid:17) = − η k ¯ l g i ¯ j g p ¯ q ( u ipk u ¯ j ¯ q ¯ l + u ip ¯ l u ¯ j ¯ qk ) + 2 Re (cid:16) η k ¯ l g i ¯ j g p ¯ q u ¯ j ¯ q ( u k ¯ lip − u ip ¯ lk ) (cid:17) − (cid:16) g i ¯ j g p ¯ q η k ¯ b η a ¯ l η a ¯ b,p F k ¯ l,i (cid:17) + 2 Re (cid:16) g i ¯ j g p ¯ q η k ¯ l ˆ F k ¯ l,ip u ¯ j ¯ q (cid:17) . lso by Ricci identity, u k ¯ lip − u ip ¯ lk = u ak R i ¯ a ¯ lp + u ia R k ¯ a ¯ lp + u ap R i ¯ a ¯ lk + u a R i ¯ a ¯ lk,p . Therefore, we have the following evolution equation( ∂∂t − ∆ η )Θ ′ = − g i ¯ j g p ¯ q η k ¯ b η a ¯ l η a ¯ b,p F k ¯ l,i u ¯ j ¯ q ) + 2 Re( g i ¯ j g p ¯ q η k ¯ l ˆ F k ¯ l,ip u ¯ j ¯ q ) − (cid:16) g i ¯ j g p ¯ q η k ¯ l u ¯ j ¯ q ( u ak R i ¯ ap ¯ l + u ia R k ¯ ap ¯ l + u ap R i ¯ ak ¯ l + u a R i ¯ ak ¯ l,p ) (cid:17) − η k ¯ l g i ¯ j g p ¯ q ( u ipk u ¯ j ¯ q ¯ l + u ip ¯ l u ¯ j ¯ qk ) . (cid:3) As an application of the evolution equations above, we can get the followingevolution inequalities (Lemma 3.6 and 3.7) for Θ = |∇ ¯ ∇ u | ω and Θ ′ = |∇∇ u | ω along the line bundle MCF. Before presenting these two lemmas, we recall thefollowing arithmetic-geometric mean inequality for the trace of the positive definiteHermitian matrices Tr(( A − B )( A − B ) T ). Lemma 3.5. If A and B are two Hermitian matrixes, then there holds the followinginequality Tr( A ¯ B T + B ¯ A T ) ≤ Tr( A ¯ A T + B ¯ B T ) . Lemma 3.6.
There exist two positive constants C , C depending only on the ge-ometry of ( X, ω ) and ˆ F such that ( ∂∂t − ∆ η )Θ ≤ η a ¯ q u a ¯ si u s ¯ q ¯ i ( −
12 + C Θ) + C Θ(1 + Θ) . Remark . The key of the proof of this lemma and Theorem 4.2 is that ˆ F is asolution to dHYM and we can apply Equation (2.4) and (2.5). Proof.
For convenience, we will simplify these quantities in the normal coordinatessystem. In fact, we choose normal coordinates near p ∈ X such that g i ¯ j ( p ) = δ i ¯ j , u i ¯ j ( p ) = σ i δ i ¯ j and ∂g i ¯ j ∂z k ( p ) = 0 . We should pay attention that we can diagonalize the Hermitian matrix { u i ¯ j } only, but not the Hermitian matrix η and F . However, in the case [ F ] = c [ ω ], theunique dHYM metric is ˆ F = cω. In this very special case, F is diagonal automatically and we can simplify thecomputation. We leave this very special case to the reader. We deal with thegeneral case in this lemma.Using the notation that F i ¯ j = ˆ F i ¯ j + u i ¯ j , we first rewrite the evolution equationof Θ = |∇ ¯ ∇ u | at p ∈ X as follow, ∂∂t − ∆ η )Θ= − η k ¯ l ( u i ¯ lp u k ¯ i ¯ p + u i ¯ l ¯ p u k ¯ ip ) − (cid:16) η p ¯ b η a ¯ q η a ¯ b, ¯ i F p ¯ q,i σ i (cid:17) + 2 Re (cid:16) σ i ( σ q − σ i ) R i ¯ ip ¯ q η p ¯ q + σ i η p ¯ q ˆ F p ¯ q,i ¯ i (cid:17) = − η k ¯ l ( u i ¯ lp u k ¯ i ¯ p + u i ¯ l ¯ p u k ¯ ip ) − (cid:16) η p ¯ b η a ¯ q η a ¯ b, ¯ i ( ˆ F p ¯ q,i + u p ¯ qi ) σ i (cid:17) + 2 Re (cid:16) σ i ( σ q − σ i ) R i ¯ ip ¯ q η p ¯ q + σ i η p ¯ q ˆ F p ¯ q,i ¯ i (cid:17) = − η k ¯ l ( u i ¯ lp u k ¯ i ¯ p + u i ¯ l ¯ p u k ¯ ip ) − (cid:16) η p ¯ b η a ¯ q η a ¯ b, ¯ i u p ¯ qi σ i (cid:17) + 2 Re (cid:16) σ i η p ¯ q ˆ F p ¯ q,i ¯ i − η p ¯ b η a ¯ q η a ¯ b, ¯ i ˆ F p ¯ q,i σ i (cid:17) + 2 Re (cid:0) σ i ( σ q − σ i ) R i ¯ ip ¯ q η p ¯ q (cid:1) = − η k ¯ l g i ¯ j g p ¯ q ( u i ¯ lp u k ¯ j ¯ q + u i ¯ l ¯ q u k ¯ jp ) + A + A + A (3.7)where we set A = − (cid:16) η p ¯ b η a ¯ q η a ¯ b, ¯ i u p ¯ qi σ i (cid:17) ,A = 2 Re (cid:16) σ i η p ¯ q ˆ F p ¯ q,i ¯ i − η p ¯ b η a ¯ q η a ¯ b, ¯ i ˆ F p ¯ q,i σ i (cid:17) ,A = 2 Re (cid:0) σ i ( σ q − σ i ) R i ¯ ip ¯ q η p ¯ q (cid:1) . Then we will estimate A , A and A respectively. Estimate of A : Since η ¯ kj = g ¯ kj + F ¯ kl g l ¯ m F ¯ mj , we have A = − Re ( η p ¯ b η a ¯ q ( F a ¯ t, ¯ i F t ¯ b + F a ¯ t F t ¯ b, ¯ i ) u p ¯ qi σ i )= − Re ( η p ¯ b η a ¯ q ( ˆ F a ¯ t, ¯ i F t ¯ b + F a ¯ t ˆ F t ¯ b, ¯ i ) u p ¯ qi σ i ) − Re ( η p ¯ b η a ¯ q ( u a ¯ t, ¯ i F t ¯ b + F a ¯ t u t ¯ b, ¯ i ) u p ¯ qi σ i )= B + B where B = − Re ( η p ¯ b η a ¯ q ( ˆ F a ¯ t, ¯ i F t ¯ b + F a ¯ t ˆ F t ¯ b, ¯ i ) u p ¯ qi σ i )and B = − Re ( η p ¯ b η a ¯ q ( u a ¯ t, ¯ i F t ¯ b + F a ¯ t u t ¯ b, ¯ i ) u p ¯ qi σ i ) . Since η is a positive definite Hermitian matrix, we can write η as square of a positivedefinite Hermitian matrix, i.e. there exists a positive definite Hermitian matrix η such that η = η · η . And hence, in local coordinates η p ¯ q = η p ¯ m η m ¯ q . Estimate of B :With notations above, we can rewrite B such that B = − Re ( η p ¯ m η m ¯ b η a ¯ n η n ¯ q ( ˆ F a ¯ t, ¯ i F t ¯ b + F a ¯ t ˆ F t ¯ b, ¯ i ) u p ¯ qi σ i ) . For convenience, we sperate B into two real parts B , and B , where B , = − Re ( η p ¯ m η m ¯ b η a ¯ n η n ¯ q ˆ F a ¯ t, ¯ i F t ¯ b u p ¯ qi σ i ) nd B , = − Re ( η p ¯ m η m ¯ b η a ¯ n η n ¯ q F a ¯ t ˆ F t ¯ b, ¯ i u p ¯ qi σ i ) . According to Lemma 3.5, we obtain that B , = − σ i η p ¯ m u p ¯ qi η n ¯ q · η a ¯ n ˆ F a ¯ t, ¯ i F t ¯ b η m ¯ b ) ≤ η p ¯ b η a ¯ q u p ¯ qi u ¯ bp ¯ i + Cσ i η p ¯ b η a ¯ q F p ¯ t ˆ F t ¯ q,i ˆ F a ¯ s, ¯ i F s ¯ b ≤ η p ¯ b η a ¯ q u p ¯ qi u ¯ bp ¯ i + C Θ F s ¯ b η p ¯ b F p ¯ t ˆ F t ¯ q,i η a ¯ q ˆ F a ¯ s, ¯ i = 1400 η p ¯ b η a ¯ q u p ¯ qi u ¯ bp ¯ i + C Θ X s ¯ t Y t ¯ s where X s ¯ t = F s ¯ b η p ¯ b F p ¯ t and Y t ¯ s = ˆ F t ¯ q,i η a ¯ q ˆ F a ¯ s, ¯ i . It is easy to see that X and Y areall semi-positive definite Hermitian matrix. Since F i ¯ j = ˆ F i ¯ j + u i ¯ j , we have(3.8) − C (1 + √ Θ) I ≤ F ≤ C (1 + √ Θ) I. By the definition of η and the choice of normal coordinates system, we know η = I + F . Thus F and η − communicate in the sense of matrixes multiplication, i.e. η − F = F η − . In this sense, we can rewrite X = η − F and0 ≤ X ≤ C (1 + √ Θ) η − ≤ C (1 + Θ) η − ≤ C (1 + Θ) I. Since Y is also semi-positive definite and η > I , we haveΘ X s ¯ t Y t ¯ s ≤ C (Θ + Θ ) ˆ F s ¯ q,i η a ¯ q ˆ F a ¯ s, ¯ i ≤ C (Θ + Θ ) . Thus we have the following inequality for B , (3.9) B , ≤ η p ¯ b η a ¯ q u p ¯ qi u ¯ bp ¯ i + C (Θ + Θ ) . The similar inequality also holds for B , , i.e.(3.10) B , ≤ η p ¯ b η a ¯ q u p ¯ qi u ¯ bp ¯ i + C (Θ + Θ ) . Adding inequalities (3.9) and (3.10) together, we get that(3.11) B ≤ η p ¯ b η a ¯ q u p ¯ qi u ¯ bp ¯ i + C (Θ + Θ ) . Estimate of B :The estimate of B is similar to that of B . We also sperate B into two realparts B , and B , where B , = − σ i η p ¯ b η a ¯ q u a ¯ t, ¯ i F t ¯ b u p ¯ qi )and B , = − σ i η p ¯ b η a ¯ q F a ¯ t u t ¯ b, ¯ i u p ¯ qi ) . According to Lemma 3.5, we have B , ≤ C ( ε ) σ i η p ¯ b η a ¯ q u p ¯ qi u a ¯ b ¯ i + εη p ¯ b η a ¯ q u a ¯ s ¯ i F s ¯ b u t ¯ qi F p ¯ t ≤ C ( ε )Θ η p ¯ b η a ¯ q u p ¯ qi u a ¯ b ¯ i + εF p ¯ t η p ¯ b F s ¯ b u a ¯ s ¯ i η a ¯ q u t ¯ qi . (3.12) e also denote two semi-positive definite Hermitian matrices X and Y as follow X s ¯ t = F p ¯ t η p ¯ b F s ¯ b and Y t ¯ s = u a ¯ s ¯ i η a ¯ q u t ¯ qi . Then the inequality (3.12) can be written as(3.13) B , ≤ C ( ε )Θ η p ¯ b η a ¯ q u p ¯ qi u a ¯ b ¯ i + ε Tr( XY ) . As discussed in the estimate of B , , the Hermitian matrix X satisfies the followinginequality 0 ≤ X ≤ C (1 + Θ) η − . Combining this inequality with (3.13) and choosing suitable ε , we obtain B , ≤ C ( ε )Θ η p ¯ b η a ¯ q u p ¯ qi u a ¯ b ¯ i + Cε (1 + Θ) η p ¯ b u a ¯ b ¯ i η a ¯ q u p ¯ qi ≤ ( 1400 + C Θ) η p ¯ b η a ¯ q u p ¯ qi u a ¯ b ¯ i . (3.14)Similarly, we also have the following inequality for B , , B , ≤ ( 1400 + C Θ) η p ¯ b η a ¯ q u p ¯ qi u a ¯ b ¯ i . (3.15)Adding (3.14) and (3.15) together, we can get the following inequality for B , B ≤ ( 1200 + C Θ) η p ¯ b η a ¯ q u p ¯ qi u a ¯ b ¯ i . (3.16)Therefore, we have the following estimate for A by adding (3.11) and (3.16), A ≤ ( 1100 + C Θ) η a ¯ q u a ¯ t ¯ k u ¯ qtk + C Θ(1 + Θ) . (3.17) Estimate of A : We rewrite A as follow A = 2 Re( σ i η p ¯ q ˆ F p ¯ q,i ¯ i − σ i η p ¯ b η a ¯ q η a ¯ b, ¯ i ˆ F p ¯ q,i )= 2 Re( σ i ( η p ¯ q − ˆ η p ¯ q ) ˆ F p ¯ q,i ¯ i − σ i η p ¯ b η a ¯ q ( η a ¯ b, ¯ i − ˆ η a ¯ b, ¯ i ) ˆ F p ¯ q,i − σ i η p ¯ b ( η a ¯ q − ˆ η a ¯ q )ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i − σ i ( η p ¯ b − ˆ η p ¯ b )ˆ η a ¯ q ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i − σ i (ˆ η p ¯ b ˆ η a ¯ q ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i − ˆ η p ¯ q ˆ F p ¯ q,i ¯ i ))= 2 Re( σ i ( η p ¯ q − ˆ η p ¯ q ) ˆ F p ¯ q,i ¯ i − σ i η p ¯ b η a ¯ q ( η a ¯ b, ¯ i − ˆ η a ¯ b, ¯ i ) ˆ F p ¯ q,i − σ i η p ¯ b ( η a ¯ q − ˆ η a ¯ q )ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i − σ i ( η p ¯ b − ˆ η p ¯ b )ˆ η a ¯ q ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i )where we have used Equation (2.5) in the last equality. For convenience, we sperate A into four parts and set D = 2 Re( σ i η p ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ q ˆ F p ¯ q,i ¯ i ) ,D = − σ i η p ¯ b η a ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ q ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i ) ,D = − σ i η p ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ b ˆ η a ¯ q ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i ) ,D = − σ i η p ¯ b η a ¯ q ( η a ¯ b, ¯ i − ˆ η a ¯ b, ¯ i ) ˆ F p ¯ q,i ) . ue to the expression of η and ˆ η , we get thatˆ η s ¯ t − η s ¯ t = ˆ F s ¯ b ˆ F b ¯ t − ( ˆ F s ¯ b + u s ¯ b )( ˆ F b ¯ t + u b ¯ t )= u s ¯ b u b ¯ t − ˆ F s ¯ b u b ¯ t − u s ¯ b ˆ F b ¯ t = δ st σ s σ t − ˆ F s ¯ t ( σ t + σ s ) . Hence, ˆ η − η satisfies the following inequality(3.18) − C (Θ + √ Θ) I ≤ ˆ η − η ≤ C (Θ + √ Θ) I for some constant C depending only on ˆ F and ω . With this inequality, we canestimate D , D , D and D respectively.Estimate of D :According to Lemma 3.5, we can estimate D as follow, D =2 Re( σ i η p ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ q ˆ F p ¯ q,i ¯ i )=2 Re( σ i η p ¯ α ˆ F p ¯ q,i ¯ i ˆ η β ¯ q · η α ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ β ) ≤ Cσ i ˆ F p ¯ q,i ¯ i η p ¯ t ˆ F a ¯ t,i ¯ i ˆ η a ¯ q + Cη p ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ q (ˆ η p ¯ q − η p ¯ q ) ≤ C Θ + C (Θ + √ Θ) ≤ C Θ + C Θ (3.19)where we have used the inequality (3.18) in the second ‘ ≤ ’.Estimate of D :Similar to the estimate of D , we have the follow estimate for D , D = − σ i η p ¯ b η a ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ q ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i ) ≤ Cη a ¯ t (ˆ η s ¯ t − η s ¯ t )(ˆ η a ¯ s − η a ¯ s )+ Cσ i η a ¯ t ˆ η a ¯ b,i η p ¯ b ˆ F p ¯ q,i ˆ η s ¯ q ˆ η c ¯ t, ¯ i η c ¯ d ˆ F e ¯ d, ¯ i ˆ η e ¯ s ≤ C (Θ + Θ )(3.20)where we have used the inequality (3.18) and the fact 0 < η − , ˆ η − < I in thesecond “ ≤ ”.Estimate of D Similar to the estimate of D and D , we can estimate D as follow D = − σ i η p ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ b ˆ η a ¯ q ˆ η a ¯ b, ¯ i ˆ F p ¯ q,i ) ≤ Cη p ¯ t (ˆ η s ¯ t − η s ¯ t )(ˆ η p ¯ s − η p ¯ s )+ Cσ i η p ¯ t ˆ F p ¯ q,i ˆ η a ¯ q ˆ η a ¯ b, ¯ i ˆ η s ¯ b ˆ F c ¯ t, ¯ i ˆ η c ¯ d ˆ η e ¯ d,i ˆ η e ¯ s ≤ C (Θ + Θ )(3.21)where we have used the inequality (3.18) and the fact 0 ≤ η − , ˆ η − ≤ I in thesecond “ ≤ ”.Estimate of D :By the definition of η , we know that η a ¯ b, ¯ i − ˆ η a ¯ b, ¯ i = u a ¯ s ¯ i ˆ F s ¯ b + u a ¯ b ¯ i σ b + ˆ F a ¯ b ¯ i σ b + σ a ˆ F a ¯ b, ¯ i + ˆ F a ¯ s u s ¯ b, ¯ i + σ a u a ¯ b, ¯ i . As an application of Lemma 3.5, we have2Re (cid:16) σ i η p ¯ b η a ¯ q ˆ F p ¯ q,i ( u a ¯ s, ¯ i ˆ F s ¯ b + ˆ F a ¯ s u s ¯ b, ¯ i ) (cid:17) ≤ η a ¯ q u a ¯ s, ¯ i u s ¯ q,i + C Θ , Re (cid:16) σ i ( σ a + σ b ) η p ¯ b η a ¯ q ˆ F p ¯ q,i u a ¯ b, ¯ i (cid:17) ≤ η p ¯ b η a ¯ q u a ¯ b, ¯ i u p ¯ q,i + C Θ , (cid:16) σ i ( σ a + σ b ) η p ¯ b η a ¯ q ˆ F p ¯ q,i ˆ F a ¯ b, ¯ i (cid:17) ≤ C Θ . Adding these inequalities together, we obtain that D ≤ η a ¯ q u a ¯ s, ¯ i u s ¯ q,i + 1200 η p ¯ b η a ¯ q u a ¯ b, ¯ i u p ¯ q,i + C (Θ + Θ ) ≤ η a ¯ q u a ¯ s, ¯ i u s ¯ q,i + C (Θ + Θ )(3.22)where we have used the fact that η − < I in the last ‘ ≤ ’.Adding the inequalities (3.19), (3.20), (3.21) and (3.22) together, we get thefollowing estimate for A ,(3.23) A ≤ η a ¯ q u a ¯ si u s ¯ q ¯ i + C (Θ + Θ ) . Estimate of A : At last, we estimate A . Since η − ≤ I and − C ≤ R i ¯ jk ¯ l ≤ C for some positive constant C , we get the following inequality(3.24) A = 2 Re (cid:0) σ i ( σ q − σ i ) R i ¯ ip ¯ q η p ¯ q (cid:1) ≤ C Θ . Adding the inequalities (3.17), (3.23) and (3.24) together and inserting into (3.7),we obtain the inequality desired. (cid:3)
We also have the following estimate for Θ ′ = |∇∇ u | ω : Lemma 3.7.
There exists constants C , C , C depending only on ( X, ω ) and ˆ F such that ( ∂∂t − ∆ η )Θ ′ ≤ ( −
12 + C Θ + C Θ ′ ) η a ¯ l u a ¯ s ¯ p u s ¯ lp + C (Θ + Θ + Θ ′ + Θ ′ + |∇ u | ω ) . Proof.
The proof is similar to the proof of Lemma 3.6. We just give the key estimateand omit the detail here.We choose the same normal coordinate systems near p ∈ X as in Lemma 3.6.We should pay attention that we can not diagonalize u ij at the same time. We canrewrite the evolution equation of Θ ′ = |∇∇ u | ω as follow( ∂∂t − ∆ η )Θ ′ = − η k ¯ l ( u ipk u ¯ i ¯ p ¯ l + u ip ¯ l u ¯ i ¯ pk ) − η k ¯ b η a ¯ l η a ¯ b,p F k ¯ l,i u ¯ i ¯ p ) + 2 Re( η k ¯ l ˆ F k ¯ l,ip u ¯ i ¯ p ) − (cid:16) η k ¯ l u ¯ i ¯ p ( u ak R i ¯ ap ¯ l + u ia R k ¯ ap ¯ l + u ap R i ¯ ak ¯ l + u a R i ¯ ak ¯ l,p ) (cid:17) = − η k ¯ l ( u ipk u ¯ i ¯ p ¯ l + u ip ¯ l u ¯ i ¯ pk ) − η k ¯ b η a ¯ l η a ¯ b,p u k ¯ li u ¯ i ¯ p ) − η k ¯ l u ¯ i ¯ p ( u ak R i ¯ ap ¯ l + u ia R k ¯ ap ¯ l + u ap R i ¯ ak ¯ l + u a R i ¯ ak ¯ l,p ))+ 2 Re( η k ¯ l ˆ F k ¯ l,ip u ¯ i ¯ p − η k ¯ b η a ¯ l η a ¯ b,p ˆ F k ¯ l,i u ¯ i ¯ p )= − η k ¯ l ( u ipk u ¯ i ¯ p ¯ l + u ip ¯ l u ¯ i ¯ pk ) + A ′ + A ′ + A ′ here we set A ′ = − η k ¯ b η a ¯ l η a ¯ b,p u k ¯ li u ¯ i ¯ p ) ,A ′ = − η k ¯ l u ¯ i ¯ p ( u ak R i ¯ ap ¯ l + u ia R k ¯ ap ¯ l + u ap R i ¯ ak ¯ l + u a R i ¯ ak ¯ l,p )) ,A ′ = 2 Re( η k ¯ l ˆ F k ¯ l,ip u ¯ i ¯ p − η k ¯ b η a ¯ l η a ¯ b,p ˆ F k ¯ l,i u ¯ i ¯ p ) . By similar arguments of estimating A , A and A in Lemma 3.3, we get thefollowing estimates for A ′ , A ′ and A ′ A ′ ≤ ( 1100 + C Θ + C Θ ′ ) η k ¯ b u l ¯ bp u k ¯ l ¯ p + C (Θ + Θ + Θ ′ ) A ′ ≤ η a ¯ l u a ¯ s ¯ p u s ¯ lp + C (Θ + Θ + Θ ′ ) A ′ ≤ C (Θ ′ + |∇ u | ω ) . Adding them together, we finish the proof of the lemma. (cid:3) Stability of the line bundle MCF
In this section, we will prove that the C -norm of u ( , t ) keeps small along theline bundle MCF as long as C -norm of u is small enough. For convenience, weset ˜Θ = |∇ ¯ ∇ u | ω + |∇∇ u | ω = Θ + Θ ′ . And we consider the following auxiliary function Q = |∇ ¯ ∇ u | ω + |∇∇ u | ω + K |∇ u | ω + K u − u ( p, = Θ + Θ ′ + K |∇ u | ω + K u − u ( p, where K and K are constant to be determined, p is a fixed point on X . Since X is compact, we have the following lemma according to differential mean valueformula. Lemma 4.1.
There exists a positive constant C depending only on the boundedgeometry ( X, ω ) such that ( Q ( · , ≤ C | D u | L ∞ , at t = 0 Q ( · , t ) ≥ | D u t ( · ) | , at any t ≥ . Remark . We know that D u can be controlled by Q at all time and Q can becontrolled by D u at time t = 0. So in the proof of the following theorem, we willconsider the smallness of Q along the line bundle MCF instead of D u . Theorem 4.2.
There exists a constant δ > such that if || D u || L ∞ ≤ δ , then || D u t || L ∞ ≤ C ω δ along the line bundle MCF for all t ≥ and C ω is a uniformconstant depending only on the bounded geometry of ( X, ω ) . The difference here is that we can not diagonalize the holomorphic Hessian { u ij } and complex { u i ¯ j } at the same time. But this does not cause trouble in our application of Lemma 3.5. roof. According to the arguments above, we have the following evolution inequal-ity for Q ,( ∂∂t − ∆ η ) Q ≤ η p ¯ q ( u p ¯ si u s ¯ q ¯ i + u p ¯ s ¯ i u ¯ qsi )( −
12 + C Θ + C Θ ′ )+ C (Θ + Θ ′ + Θ + Θ ′ + |∇ u | ω ) + K ( u − u ( p, θ ( F u ) − ˆ θ − ∆ η u ) − K η p ¯ q u p u ¯ q − K η p ¯ q R p ¯ qj ¯ i u i u ¯ j − K η p ¯ q ( u ip u ¯ i ¯ q + u i ¯ q u ¯ ip )+ 2 K Re( η p ¯ q ˆ F p ¯ q,i u ¯ i ) . (4.1)For convenience, we set I = 2 K Re( η p ¯ q ˆ F p ¯ q,i u ¯ i ) . We can estimate I in the following way I =2 K Re(( η p ¯ q − ˆ η p ¯ q ) ˆ F p ¯ q,i u ¯ i )=2 K Re( η p ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ q ˆ F p ¯ q,i u ¯ i )=2 K Re( η p ¯ t ( ˆ F s ¯ a ˆ F a ¯ t − F s ¯ a F a ¯ t )ˆ η s ¯ q ˆ F p ¯ q,i u ¯ i )=2 K Re( η p ¯ t ( ˆ F s ¯ t σ t + ˆ F s ¯ t σ s − δ st σ s σ t )ˆ η s ¯ q ˆ F p ¯ q,i u ¯ i ) ≤ K ρ (Θ + Θ ) + C ( ρ ) K |∇ u | ω , where we have used Equation (2.4) in the first “=” and ρ is a constant to bedecided later.To estimate the third term in (4.1), we denote the following endomorphism K u of T , X for every function uK u = g i ¯ j ( ˆ F k ¯ j + u k ¯ j ) ∂∂z i ⊗ dz k . Indeed, we can view K u as a matrix-valued function on X . The function θ ( F su )can be regarded as a function on s ∈ R where F su = ˆ F + s∂ ¯ ∂u. Applying differential mean value theorem to θ ( F su ), we have θ ( F u ) − ˆ θ − ∆ η u = θ ( F u ) − θ ( ˆ F ) − ∆ η u = dθ ( F su ) ds (cid:12)(cid:12)(cid:12)(cid:12) s = ξ ∈ (0 , − ( I + K u ) p ¯ q u p ¯ q =( I + K ξu ) p ¯ q u p ¯ q − ( I + K u ) p ¯ q u p ¯ q =(( I + ˆ K ) − ( ˆ K − K ξu )( I + K ξu ) − ) p ¯ q u p ¯ q = − Tr(( I + ˆ K ) − ( ξ ˆ KU + ξU ˆ K + ξ U )( I + K ξu ) − U ) ≤ C (1 + Θ)Θ here U = { u i ¯ j } is the complex hessian of u and C is a constant depending onlyon ˆ F . Thus, the evolution inequality of Q can be rewritten as follow( ∂∂t − ∆ η ) Q ≤ η p ¯ q ( u p ¯ si u s ¯ q ¯ i + u p ¯ s ¯ i u ¯ qsi )( −
12 + C Θ + C Θ ′ ) − K η p ¯ q u ip u ¯ q ¯ i + C (Θ ′ + Θ ′ ) − K η p ¯ q u i ¯ q u p ¯ i + ( C + K ρ )(Θ + Θ ) + CK ( u − u ( p, )+ ( C + C ( ρ ) K ) |∇ u | ω − K η p ¯ q u p u ¯ q − K η p ¯ q R p ¯ qj ¯ i u i u ¯ j = η p ¯ q ( u p ¯ si u s ¯ q ¯ i + u p ¯ s ¯ i u ¯ qsi )( −
12 + C Θ + C Θ ′ )+ Q + Q + Q + Q (4.2)where we denote Q = − K η p ¯ q u ip u ¯ q ¯ i + C (Θ ′ + Θ ′ ) ,Q = − K η p ¯ q u i ¯ q u p ¯ i + ( C + K ρ )(Θ + Θ ) ,Q = CK ( u − u ( p, ) ,Q = − K η p ¯ q u p u ¯ q + ( C + C ( ρ ) K ) |∇ u | ω − K η p ¯ q R p ¯ qj ¯ i u i u ¯ j . Estimate of Q : We first deal with Q . Indeed, we can estimate Q as follow Q = − K ˆ η p ¯ q u ip u ¯ q ¯ i − K ( η p ¯ q − ˆ η p ¯ q ) u ip u ¯ q ¯ i + C (Θ ′ + Θ ′ ) ≤ − ˆ CK u ip u ¯ p ¯ i − K η p ¯ t (ˆ η s ¯ t − η s ¯ t )ˆ η s ¯ q u ip u ¯ q ¯ i + C (Θ ′ + Θ ′ ) ≤ ( − ˆ CK + C )Θ ′ + C Θ ′ + CK (Θ + √ Θ) η p ¯ s ˆ η s ¯ q u ip u ¯ q ¯ i ≤ ( − ˆ CK + C )Θ ′ + C Θ ′ + CK (Θ + √ Θ)Θ ′ where we apply (3.18) in the second “ ≤ ”. We define K , to be the positive constantsatisfying − ˆ CK , + C = − . Hence, if we take K ≥ K , , then there holds Q ≤ − Θ ′ + C Θ ′ + CK (Θ + √ Θ)Θ ′ = Θ ′ ( − CK (Θ + √ Θ) + C Θ ′ ) . Estimate of Q : We estimate it as follow − K η p ¯ q u i ¯ q u ¯ ip + ( C + K ρ )(Θ + Θ )= − K η p ¯ q u l ¯ q u p ¯ l + ( C + K ρ )(1 + Θ) u p ¯ l u l ¯ q δ pq = u l ¯ q u p ¯ l η p ¯ m (( C + K ρ )(1 + Θ) η q ¯ m − K δ qm )= u l ¯ q u p ¯ l η p ¯ m { (( C + K ρ )(1 + Θ) − K ) δ qm +( C + K ρ )(1 + Θ)( ˆ F q ¯ n + u q ¯ n )( ˆ F n ¯ m + u n ¯ m ) } = u l ¯ q u p ¯ l η p ¯ m { (( C + ρ K − K + ( C + ρ K )Θ)) δ qm + ( C + ρ K )(1 + Θ) ˆ F q ¯ n ˆ F n ¯ m + ( C + ρ K )(1 + Θ)( ˆ F q ¯ n u n ¯ m + ˆ F n ¯ m u q ¯ n + u q ¯ n u n ¯ m ) }≤ u l ¯ q u p ¯ l η p ¯ m { C (1 + Θ) δ qm + ( ρ K − K + CK ρ Θ) δ qm + K ρ ˆ F q ¯ n ˆ F n ¯ m ( C + ρ K )(1 + Θ)( ˆ F q ¯ n u n ¯ m + ˆ F n ¯ m u q ¯ n + u q ¯ n u n ¯ m ) } . Since ˆ F is a bounded (1 , ρ ∈ (0 , ) small enough suchthat the positive definite Hermitian matrix ˆ F ¯ˆ F T satisfies that ρ ˆ F ¯ˆ F T ≤ I. Then we get the following inequality − K η p ¯ q u ip u ¯ i ¯ q + ( C + K ρ )(Θ + Θ ) ≤ u l ¯ q u p ¯ l η p ¯ m { C (1 + Θ) δ qm + ( − K + CK ρ Θ) δ qm + ( C + ρ K )(1 + Θ)( ˆ F q ¯ n u n ¯ m + ˆ F n ¯ m u q ¯ n + u q ¯ n u n ¯ m ) }≤ u l ¯ q u p ¯ l η p ¯ q { ( C − K ) + C (1 + K )(1 + Θ)( √ Θ + Θ) } where we apply the following inequality for the complex Hessian of u − C √ Θ I ≤ { u i ¯ j } ≤ C √ Θ I in the second “ ≤ ”. Similar to the estimate of Q , we choose a positive constant K , such that C − K , = − . So if K ≥ K , , then there holds − K η p ¯ q u ip u ¯ i ¯ q + ( C + K ρ )(Θ + Θ ) ≤ u l ¯ q u p ¯ l η p ¯ q {− C (1 + K )(1 + Θ)( √ Θ + Θ) } . Now we take K = max { K , , K , } . Therefore, we obtain that(4.3) Q ≤ Θ ′ ( − C (Θ + √ Θ) + C Θ ′ ) ≤ Θ ′ ( − CQ )and Q ≤ u l ¯ q u p ¯ l η p ¯ q {− C (1 + Θ)( √ Θ + Θ) }≤ u l ¯ q u p ¯ l η p ¯ q ( − CQ ) . (4.4) Estimate of Q : Before estimate Q , we first deal with Q and choose a certain constant K .Since we have chosen K , we can treat it as a constant. We can rewrite Q asfollow Q = − K η p ¯ q u p u ¯ q + C |∇ u | ω − Cη p ¯ q R p ¯ qj ¯ i u i u ¯ j ≤ − K η p ¯ q u p u ¯ q + C |∇ u | ω , since the curvature R is bounded and 0 < η − < I . Furthermore, we also rewrite |∇ u | ω as follow in the normal coordinates |∇ u | ω = u i u ¯ i = η p ¯ m η q ¯ m u p u ¯ q . Combining with the inequality 0 ≤ η ≤ C (1 + Θ) I and choosing K large enough,we obtain the following inequality for Q , Q ≤ η p ¯ m u p u ¯ q ( − K δ qm + Cη q ¯ m ) ≤ η p ¯ q u p u ¯ q ( − K + C + C Θ) ≤ η p ¯ q u p u ¯ q ( − CQ ) . (4.5) stimate of Q : At last, we estimate Q . Since K is also a given constant and I ≤ η ≤ C (1+Θ) I ,we obtain the following inequality Q = CK ( u − u ( p, ) ≤ CQ (1 + Θ) u i ¯ q u p ¯ i δ pq = CQ (1 + Θ) u i ¯ q u p ¯ i η p ¯ m η q ¯ m ≤ Cη p ¯ q u i ¯ q u p ¯ i Q (1 + Θ ) ≤ η p ¯ q u i ¯ q u p ¯ i (1 + CQ ) . (4.6)Inserting (4.3), (4.4), (4.6) and (4.5) into (4.2), we get that( ∂∂t − ∆ η ) Q ≤ η p ¯ q ( u p ¯ si u s ¯ q ¯ i + u p ¯ s ¯ i u ¯ qsi )( −
12 + C Θ + C Θ ′ )+ η p ¯ q u l ¯ q u p ¯ l ( − CQ + CQ ) + Θ ′ ( − CQ ) + η p ¯ q u p u ¯ q ( − CQ ) ≤ η p ¯ q ( u p ¯ si u s ¯ q ¯ i + u p ¯ s ¯ i u ¯ qsi )( −
12 + C Q )+ η p ¯ q u l ¯ q u p ¯ l ( − C Q ) + (Θ ′ + η p ¯ q u p u ¯ q )( − C Q )(4.7)where C , C , C is constant independent of u . We choose δ ′ > ≤ Q ≤ δ ′ , there holds −
12 + C Q < − C Q < − C Q < . Then we know that Q ≤ δ ′ is preserved along the line bundle MCF by the max-imal principle. And hence, there exists a positive constant δ = δ ′ C ω such that if | D u | L ∞ ≤ δ , then || D u || L ∞ ≤ C ω δ along the line bundle MCF according toLemma 4.1. (cid:3) Long time existence of the line bundle MCF
In this section we will prove the first part of Theorem 1.5, i.e. long-time existenceof the line bundle MCF. In order to prove this, we need to get uniform estimatesfor higher order derivatives of u . According to standard Schauder method, it isenough to obtain the uniform bound of |∇∇ ¯ ∇ u ( , t ) | . Remark . Unlike [9], our assumptions can not guarantee the positivity of F u t along the line bundle MCF. So the operator θ ( F u ) = X arctan λ i ( F u )under our consideration need not be concave and we can not apply Evans-Krylovtheory to get C ,α estimate for u directly. To overcome this difficulty, we will usethe parabolic Calabi type estimate for the line bundle MCF which is new to ourbest knowledge. However, our estimate rely on the smallness of D u along the linebundle MCF. The usual (parabolic)Calabi type estimate under assumption on thebound of || u || C is still an open problem in general case. or convenience, we denote two notations of the third and forth order derivativsΓ = |∇∇ ¯ ∇ u ( , t ) | ω and Ξ = |∇∇ ¯ ∇∇ u | ω + |∇ ¯ ∇ ¯ ∇∇ u | ω . Before proving Theorem 1.5, we first prove the following lemma.
Lemma 5.1. If || u || C is uniformly bounded along the line bundle MCF, then Γ = |∇ ¯ ∇∇ u | ω satisfies the following inequality ( ∂∂t − ∆ η )Γ ≤ C + C Γ − C Ξ where C is a constant dependent only on || u || C , ˆ F , ω and n .Proof. We first compute the evolution equation of Γ along the line bundle MCF.The ∂∂t -part of Γ is given by ∂∂t Γ= g i ¯ a g b ¯ j g k ¯ c ( ∂u∂t ) i ¯ jk u ¯ ab ¯ c + g i ¯ a g b ¯ j g k ¯ c ( ∂u∂t ) ¯ ab ¯ c u i ¯ jk = g i ¯ a g b ¯ j g k ¯ c θ i ¯ jk u ¯ ab ¯ c + g i ¯ a g b ¯ j g k ¯ c θ ¯ ab ¯ c u i ¯ jk = g i ¯ a g b ¯ j g k ¯ c ( η p ¯ q F p ¯ q,i ) ¯ jk u ¯ ab ¯ c + g i ¯ a g b ¯ j g k ¯ c ( η p ¯ q F p ¯ q, ¯ a ) b ¯ c u i ¯ jk =2 Re { ( η p ¯ q, ¯ jk F p ¯ q,i + η p ¯ q, ¯ j F p ¯ q,ik + η p ¯ q,k F p ¯ q,i ¯ j + η p ¯ q F p ¯ q,i ¯ jk ) u ¯ ij ¯ k } =2 Re( η p ¯ q ( ˆ F p ¯ q,i ¯ jk + u p ¯ q,i ¯ jk ) u ¯ ij ¯ k ) − η p ¯ n η m ¯ q ( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,ik u ¯ ij ¯ k ) − η p ¯ n η m ¯ q ( F m ¯ l,k F l ¯ n + F m ¯ l F l ¯ n,k ) F p ¯ q,i ¯ j u ¯ ij ¯ k )+2 Re( η p ¯ b η a ¯ n η m ¯ q ( F a ¯ c,k F c ¯ b + F a ¯ c F c ¯ b,k )( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,i u ¯ ij ¯ k )+2 Re( η p ¯ n η m ¯ b η a ¯ q ( F a ¯ c,k F c ¯ b + F a ¯ c F c ¯ b,k )( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,i u ¯ ij ¯ k ) − η p ¯ n η m ¯ q ( F m ¯ l, ¯ jk F l ¯ n + F m ¯ l, ¯ j F l ¯ n,k + F m ¯ l,k F l ¯ n, ¯ j + F m ¯ l F l ¯ n, ¯ jk ) F p ¯ q,i u ¯ ij ¯ k ) . The ∆ η -part of Γ is given by∆ η Γ= η p ¯ q ( u i ¯ jk u ¯ ij ¯ k ) p ¯ q = η p ¯ q ( u i ¯ jkp u ¯ ij ¯ k ¯ q + u i ¯ jkp ¯ q u ¯ ij ¯ k + u i ¯ jk ¯ q u ¯ ij ¯ kp + u i ¯ jk u ¯ ij ¯ kp ¯ q )=2 Re( η p ¯ q u i ¯ jkp ¯ q u ¯ ij ¯ k ) + η p ¯ q ( u i ¯ jkp u ¯ ij ¯ k ¯ q + u i ¯ jk ¯ q u ¯ ij ¯ kp ) . hus, the evolution formula of Γ is( ∂∂t − ∆ η )Γ=2 Re( η p ¯ q ˆ F p ¯ q,i ¯ jk u ¯ ij ¯ k ) + 2 Re(( u p ¯ q,i ¯ jk − u i ¯ jkp ¯ q ) η p ¯ q u ¯ ij ¯ k ) − η p ¯ n η m ¯ q ( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,ik u ¯ ij ¯ k ) − η p ¯ n η m ¯ q ( F m ¯ l,k F l ¯ n + F m ¯ l F l ¯ n,k ) F p ¯ q,i ¯ j u ¯ ij ¯ k )+2 Re( η p ¯ b η a ¯ n η m ¯ q ( F a ¯ c,k F c ¯ b + F a ¯ c F c ¯ b,k )( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,i u ¯ ij ¯ k )+2 Re( η p ¯ n η m ¯ b η a ¯ q ( F a ¯ c,k F c ¯ b + F a ¯ c F c ¯ b,k )( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,i u ¯ ij ¯ k ) − η p ¯ n η m ¯ q ( F m ¯ l, ¯ jk F l ¯ n + F m ¯ l, ¯ j F l ¯ n,k + F m ¯ l,k F l ¯ n, ¯ j + F m ¯ l F l ¯ n, ¯ jk ) F p ¯ q,i u ¯ ij ¯ k ) − η p ¯ q ( u i ¯ jkp u ¯ ij ¯ k ¯ q + u i ¯ jk ¯ q u ¯ ij ¯ kp )By Ricci identity, we have the following formula while changing order of derivatives u p ¯ qi ¯ jk − u i ¯ jkp ¯ q = u a ¯ jk R i ¯ a ¯ qp + u i ¯ ak R a ¯ j ¯ qp + u i ¯ ja R k ¯ a ¯ qp + u a ¯ jp R i ¯ a ¯ qk + u a ¯ j R i ¯ a ¯ qk,p + u a ¯ qk R i ¯ ap ¯ j + + u a ¯ q R i ¯ ap ¯ j,k − u i ¯ ap R a ¯ j ¯ qk − u i ¯ ak R a ¯ qp ¯ j . Therefore, we obtain that( ∂∂t − ∆ η )Γ=2 Re( η p ¯ q ˆ F p ¯ q,i ¯ jk u ¯ ij ¯ k ) + 2 Re( η p ¯ q u ¯ ij ¯ k ( u a ¯ jk R i ¯ a ¯ qp + u i ¯ ak R a ¯ j ¯ qp + u i ¯ ja R k ¯ a ¯ qp + u a ¯ jp R i ¯ a ¯ qk + u a ¯ j R i ¯ a ¯ qk,p + u a ¯ qk R i ¯ ap ¯ j + + u a ¯ q R i ¯ ap ¯ j,k − u i ¯ ap R a ¯ j ¯ qk − u i ¯ ak R a ¯ qp ¯ j )) − η p ¯ n η m ¯ q ( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,ik u ¯ ij ¯ k ) − η p ¯ n η m ¯ q ( F m ¯ l,k F l ¯ n + F m ¯ l F l ¯ n,k ) F p ¯ q,i ¯ j u ¯ ij ¯ k )+2 Re( η p ¯ b η a ¯ n η m ¯ q ( F a ¯ c,k F c ¯ b + F a ¯ c F c ¯ b,k )( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,i u ¯ ij ¯ k )+2 Re( η p ¯ n η m ¯ b η a ¯ q ( F a ¯ c,k F c ¯ b + F a ¯ c F c ¯ b,k )( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,i u ¯ ij ¯ k ) − η p ¯ n η m ¯ q ( F m ¯ l, ¯ jk F l ¯ n + F m ¯ l, ¯ j F l ¯ n,k + F m ¯ l,k F l ¯ n, ¯ j + F m ¯ l F l ¯ n, ¯ jk ) F p ¯ q,i u ¯ ij ¯ k ) − η p ¯ q ( u i ¯ jkp u ¯ ij ¯ k ¯ q + u i ¯ jk ¯ q u ¯ ij ¯ kp )= T + T + T + T + T + T + T − η p ¯ q ( u i ¯ jkp u ¯ ij ¯ k ¯ q + u i ¯ jk ¯ q u ¯ ij ¯ kp )where we set T = 2 Re( η p ¯ q ˆ F p ¯ q,i ¯ jk u ¯ ij ¯ k ) T = 2 Re( η p ¯ q u ¯ ij ¯ k ( u a ¯ jk R i ¯ a ¯ qp + u i ¯ ak R a ¯ j ¯ qp + u i ¯ ja R k ¯ a ¯ qp + u a ¯ jp R i ¯ a ¯ qk + u a ¯ j R i ¯ a ¯ qk,p + u a ¯ qk R i ¯ ap ¯ j + + u a ¯ q R i ¯ ap ¯ j,k − u i ¯ ap R a ¯ j ¯ qk − u i ¯ ak R a ¯ qp ¯ j )) T = − η p ¯ n η m ¯ q ( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,ik u ¯ ij ¯ k ) T = − η p ¯ n η m ¯ q ( F m ¯ l,k F l ¯ n + F m ¯ l F l ¯ n,k ) F p ¯ q,i ¯ j u ¯ ij ¯ k ) T = 2 Re( η p ¯ b η a ¯ n η m ¯ q ( F a ¯ c,k F c ¯ b + F a ¯ c F c ¯ b,k )( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,i u ¯ ij ¯ k ) T = 2 Re( η p ¯ n η m ¯ b η a ¯ q ( F a ¯ c,k F c ¯ b + F a ¯ c F c ¯ b,k )( F m ¯ l, ¯ j F l ¯ n + F m ¯ l F l ¯ n, ¯ j ) F p ¯ q,i u ¯ ij ¯ k ) T = − η p ¯ n η m ¯ q ( F m ¯ l, ¯ jk F l ¯ n + F m ¯ l, ¯ j F l ¯ n,k + F m ¯ l,k F l ¯ n, ¯ j + F m ¯ l F l ¯ n, ¯ jk ) F p ¯ q,i u ¯ ij ¯ k ) . ince || u || C is bounded and F i ¯ j = ˆ F i ¯ j + u i ¯ j , we can get the following inequalitiesfor T i ( i = 1 , · · · ,
7) according to Cauchy inequality T + T ≤ C + C Γ ,T + T + T ≤ C + C Γ + 1100 Ξ ,T + T ≤ C + C Γ , Hence, we get that ( ∂∂t − ∆ η )Γ ≤ C + C Γ − C Ξ . (cid:3) Now we begin to prove the first part of Theorem 1.5. The exponential conver-gence of will be presented in Section 6.
Proof.
We assume the maximal existence interval of line bundle MCF is [0 , T ).Since Θ = |∇ ¯ ∇ u | ω is small along the line bundle MCF, we have the followingestimate for Θ by choosing δ small enough( ∂∂t − ∆ η )Θ ≤ − C Γ + C, for some positive constants C and C according to Lemma 3.6.Suppose A is a constant to be determined later. Then we have the followinginequality along the line bundle MCF( ∂∂t − ∆ η )( e A Θ Γ)= e A Θ ( ∂∂t − ∆ η )Γ + Ae A Θ Γ( ∂∂t − ∆ η )Θ − Ae A Θ Θ p Γ ¯ q η p ¯ q ) − A e A Θ η p ¯ q Θ p Θ ¯ q Γ ≤ e A Θ ( C + C Γ − C Ξ) + Ae A Θ Γ( − C Γ + C ) − A e A Θ Γ η p ¯ q Θ p Θ ¯ q − Ae A Θ Θ p Γ ¯ q η p ¯ q ) . According to the equation ∇ ( e A Θ Γ) = Ae A Θ ∇ ΘΓ + e A Θ ∇ Γ , we obtain that − Ae A Θ Θ p Γ ¯ q η p ¯ q ) = − Aη p ¯ q Θ p ( e A Θ Γ) ¯ q ) + 2 Re( A e A Θ Θ p Θ ¯ q η p ¯ q Γ) . Inserting it into the evolution inequality above,( ∂∂t − ∆ η )( e A Θ Γ) ≤ e A Θ ( C + C Γ − C Ξ) + Ae A Θ Γ( − C Γ + C ) − A e A Θ Γ η p ¯ q Θ p Θ ¯ q − Aη p ¯ q Θ p ( e A Θ Γ) ¯ q ) + 2 A e A Θ Θ p Θ ¯ q η p ¯ q Γ= e A Θ ( C + C Γ − C Ξ) + Ae A Θ Γ( − C Γ + C ) + A e A Θ Γ η p ¯ q Θ p Θ ¯ q − Aη p ¯ q Θ p ( e A Θ Γ) ¯ q ) . Since |∇ ¯ ∇ u | ω ≤ δ and η − ≤ I , we get thatΘ p Θ ¯ q η p ¯ q = ( u i ¯ jp u j ¯ i + u i ¯ j u j ¯ ip )( u k ¯ l ¯ q u l ¯ k + u k ¯ l u l ¯ k ¯ q ) η p ¯ q ≤ δ Γ , nd 2 Re( A e A Θ Θ p Θ ¯ q η p ¯ q Γ) ≤ δ A Γ e A Θ According to the inequalities above, we get that( ∂∂t − ∆ η )( e A Θ Γ) ≤ − Aη p ¯ q Θ p ( e A Θ Γ) ¯ q )+ e A Θ { ( C − AC + A δ )Γ + AC × Γ + C } . We choose δ small enough such that C − C δ > . Then we can choose A such that − C = C − AC + A δ < . Hence e A Θ Γ is bounded along the line bundle MCF by maximal principle. As aconsequence, Γ is bounded since Θ is bounded, i.e. ∇ ¯ ∇∇ u ( , t ) is uniformly boundedfor all t ∈ [0 , T ).Then we get the uniform estimate of higher order derivatives as follow. Theuniform bound of ∇ ¯ ∇∇ u ( , t ) implies that the C α -norm of ∇ ¯ ∇ u ( , t ) is uniformlybounded for any α ∈ (0 ,
1) and t ∈ [0 , T ). Hence the C α -norm of η is uniformlybounded. The standard parabolic Schauder estimate gives us the uniform higherorder estimate. Then we can extend the line bundle MCF across time T . As aconsequence, we get the long-time existence and convergence of the line bundleMCF in the sense of subsequence. (cid:3) Exponential Convergence
In this section, we prove the exponential convergence as stated in Theorem 1.5.By the line bundle MCF and (2 . θ , ∂∂t θ = Tr(( I + K ) − ∂∂t K ) = η i ¯ j g ¯ jl ∂∂t ( g l ¯ m F ¯ mi ) = η i ¯ j ( ˙ u ) i ¯ j = η i ¯ j θ i ¯ j . Hence according the maximal principle, we know that the maximum and minimumof θ ( · , t ) attains at t = 0, i.e. θ is bounded.6.1. Harnack-type Inequality.
Before considering the exponential convergenceof u ( · , t ), we will first prove a Harnack-type inequality for positive solutions ϕ ofthe following parabolic equation(6.1) ∂v∂t = η i ¯ j v i ¯ j , where η i ¯ j is the Hermitian matrix appeared above dependent on u ( x, t ). ThisHarnack type inequality has been proved by Li-Yau for heat equation in [11]. AndCao proved it for K¨ahler-Ricci flow in [1]. The argument is standard and we givethe details for completeness of this paper.For convenience, we set f = log v and˜ f = t ( η i ¯ j f i f ¯ j − α ˙ f ) The constant δ chosen here may be smaller than that in Lemma 4.2, so the smallness ofHessian is still preserved along line bundle MCF. here α is a constant in (1 , f − η i ¯ j f i ¯ j = η i ¯ j f i f ¯ j and(6.3) ˜ f = − tη i ¯ j f i ¯ j − t ( α −
1) ˙ f .
Lemma 6.1.
There exist constants C and C which depend on the bound of F and the derivatives of F such that the function ˜ f satisfies the following inequality η k ¯ l ˜ f k ¯ l − ˙˜ f ≥ t n ( η i ¯ j f i f ¯ j − ˙ f ) − η i ¯ j f i ˜ f ¯ j ) − ( η i ¯ j f i f ¯ j − α ˙ f ) − C tη i ¯ j f i f ¯ j − C t. Proof.
By direct computation, we have˙˜ f = η i ¯ j f i f ¯ j − α ˙ f + 2 t Re( η i ¯ j f ¯ j ˙ f i ) + t ∂η i ¯ j ∂t f i f ¯ j − αt ¨ f (6.4)and η k ¯ l ˜ f k ¯ l = tη k ¯ l ( η i ¯ j f ik f ¯ j ¯ lF + η i ¯ j f i ¯ l f ¯ jkF + η i ¯ j,k f i ¯ l f ¯ jF + η i ¯ j,k f i f ¯ j ¯ lF + η i ¯ j f ik ¯ l f ¯ jF + η i ¯ j f i f ¯ jk ¯ lF + η i ¯ j, ¯ l f ik f ¯ jF + η i ¯ j, ¯ l f i f ¯ jkF + η i ¯ j,k ¯ l f i f ¯ jF − α ˙ f k ¯ lF )= X i =1 F i . (6.5)For any ε >
0, we have the following inequality | F + F | ≤ tε η i ¯ j f i f ¯ j + 2 εF , | F + F | ≤ tε η i ¯ j f i f ¯ j + 2 εF according to Cauchy inequality. Since η i ¯ j,k ¯ l is bounded along the line bundle MCF,we know that F satisfies | F | ≤ Ctη i ¯ j f i f ¯ j . Furthermore, we can also estimate F + F and F as follow F + F = tη i ¯ j η k ¯ l ( f k ¯ li f ¯ j + f a R k ¯ ai ¯ l f ¯ j + f i f k ¯ l ¯ j ) ≥ − Ctη i ¯ j f i f ¯ j + 2 t Re( η i ¯ j f ¯ j ( η k ¯ l f k ¯ l ) i ) − tη i ¯ j ( η k ¯ l,i f k ¯ l f ¯ j + η k ¯ l, ¯ j f k ¯ l f i ) ≥ − Ctη i ¯ j f i f ¯ j + 2 t Re( η i ¯ j f ¯ j ( η k ¯ l f k ¯ l ) i ) − tε η i ¯ j f i f ¯ j − εtF = − Ctη i ¯ j f i f ¯ j − η i ¯ j f ¯ j ˜ f i ) − t ( α −
1) Re( η i ¯ j f ¯ j ˙ f i ) − tε η i ¯ j f i f ¯ j − εtF = − Ctη i ¯ j f i f ¯ j − η i ¯ j f ¯ j ˜ f i ) − ( α −
1) ˙˜ f + ( α − η i ¯ j f i f ¯ j − α ˙ f )+( α − t ∂η i ¯ j ∂t f i f ¯ j − α ( α − t ¨ f − tε η i ¯ j f i f ¯ j − εtF ≥ − Ctη i ¯ j f i f ¯ j − η i ¯ j f ¯ j ˜ f i ) − ( α −
1) ˙˜ f + ( α − η i ¯ j f i f ¯ j − α ˙ f ) − α ( α − t ¨ f − tε η i ¯ j f i f ¯ j − εtF nd F = − αt ( ˜ ft − ˙˜ ft − ( α −
1) ¨˜ f ) + αtf k ¯ l ∂η k ¯ l ∂t ≥ − Ctε − εF − α ˜ ft + α ˙˜ f + tα ( α −
1) ¨ f .
Adding all inequalities above, we get that η i ¯ j ˜ f i ¯ j ≥ ˙˜ f − η i ¯ j ˜ f i f ¯ j ) − ( η i ¯ j f i f ¯ j − α ˙ f ) + t (1 − ε ) η i ¯ j η k ¯ l f i ¯ l f k ¯ j + t (1 − ε ) η i ¯ j η k ¯ l f ik f ¯ j ¯ l − t ( C + 5 ε ) η i ¯ j f i f ¯ j − Ctε . (6.6)Taking the constant ε small enough and applying the following arithmetic-geometric mean inequality η i ¯ j η k ¯ l f i ¯ l f k ¯ j ≥ n (cid:16) η i ¯ j f i ¯ j (cid:17) = 1 n ( η i ¯ j f i f ¯ j − ˙ f ) , we obtain that η i ¯ j ˜ f i ¯ j − ˙˜ f ≥ t n ( η i ¯ j f i f ¯ j − ˙ f ) − η i ¯ j ˜ f i f ¯ j ) − ( η i ¯ j f i f ¯ j − α ˙ f ) − C tη i ¯ j f i f ¯ j − C t which is the result desired. (cid:3) Lemma 6.2.
There exists constants C and C which depend on F and the deriva-tives of F such that for all t > , the following inequality holds η i ¯ j f i f ¯ j − α ˙ f ≤ C + C t . Proof.
For any fixed T >
0, we assume that ˜ f attained its maximum in X × [0 , T ]at ( x , t ). If t = 0, then we get the required inequality. So we can just considerthe case that t >
0. Then at ( x , t ), by Lemma 6.1,(6.7) t n ( η i ¯ j f i f ¯ j − ˙ f ) − ( η i ¯ j f i f ¯ j − α ˙ f ) ≤ C t η i ¯ j f i f ¯ j + C t . In the case ˙ f ( x , t ) >
0, we have the following inequality t n ( η i ¯ j f i f ¯ j − ˙ f ) − ( η i ¯ j f i f ¯ j − ˙ f ) ≤ C t η i ¯ j f i f ¯ j + C t since α ∈ (1 , x , t ), η i ¯ j f i f ¯ j − ˙ f ≤ C q η i ¯ j f i f ¯ j + C + C t ≤ (cid:18) − α (cid:19) η i ¯ j f i f ¯ j + C + C t . According to α ∈ (1 ,
2) and ˙ f >
0, there holds η i ¯ j f i f ¯ j − α ˙ f ≤ C + C t . By the definition of ˜ f ,˜ f ( x , t ) = t ( η i ¯ j f i f ¯ j − α ˙ f ) ≤ C t + C . Therefore, for all x ∈ M ,˜ f ( x, T ) ≤ ˜ f ( x , t ) ≤ C t + C ≤ C T + C .e. ( η i ¯ j f i f ¯ j − α ˙ f )( x, T ) ≤ C + CT . So we complete the proof in this case.Now let us consider the case when ˙ f ( x , t ) ≤
0. By the inequality (6.7), t n ( η i ¯ j f i f ¯ j ) − η i ¯ j f i f ¯ j + α ˙ f ≤ C t η i ¯ j f i f ¯ j + C t i.e. 12 n ( η i ¯ j f i f ¯ j ) − ( 1 t + C ) η i ¯ j f i f ¯ j ≤ C − α ˙ ft . Then by Cauchy inequality,12 n ( η i ¯ j f i f ¯ j ) − ( 1 t + C ) η i ¯ j f i f ¯ j ≤ C + (cid:18) Ct (cid:19) + ˙ f x , t ),(6.8) η i ¯ j f i f ¯ j ≤ C + Ct − ˙ f . On the other hand, by inequality (6.7), t n ˙ f + α ˙ f ≤ C t η i ¯ j f i f ¯ j + C t + η i ¯ j f i f ¯ j i.e. 1 n ˙ f + α ˙ ft ≤ C η i ¯ j f i f ¯ j + C + η i ¯ j f i f ¯ j t . By Cauchy inequality,1 n ˙ f + α ˙ ft ≤ (cid:16) η i ¯ j f i f ¯ j (cid:17) + (cid:18) Ct (cid:19) + C. Hence, at ( x , t ),(6.9) − ˙ f ≤ Ct + η i ¯ j f i f ¯ j C. By inequalities (6.8) and (6.9), we obtain that η i ¯ j f i f ¯ j ≤ C + Ct + η i ¯ j f i f ¯ j η i ¯ j f i f ¯ j ≤ C + Ct . Asserting this inequality to the inequality (6.9), − ˙ f ≤ C + Ct . Therefore, we get that η i ¯ j f i f ¯ j − α ˙ f ≤ C + Ct . Same argument as in the case ˙ f > (cid:3)
As an application of the previous lemma, we derive the following Harnack-typeinequality of Li-Yau [11] in the case of the line bundle MCF. heorem 6.3. There exists constants C , C and C such that for all < t < t ,we have the following Harnack-type inequality sup x ∈ X v ( x, t ) ≤ inf x ∈ X v ( x, t ) (cid:18) t t (cid:19) C e C t − t + C ( t − t ) . Proof.
Let x, y ∈ X be two arbitrary points and γ be the geodesic with respect tothe background metric ω such that γ (0) = x and γ (1) = y. We also define a curve ξ ( s ) : [0 , → X × [ t , t ] by ξ ( s ) = ( γ ( s ) , (1 − s ) t + st )i.e. ξ is a curve in X × [ t , t ] connecting ( x, t ) and ( y, t ). Then by Lemma 6.2,ln v ( x, t ) v ( y, t ) = − Z ∂∂s f ( ξ ( s )) ds = Z ( − df ( ˙ γ ) − ˙ f ( t − t )) ds ≤ Z ( q η i ¯ j f i f ¯ j − t − t α η i ¯ j f i f ¯ j − ˙ f ( t − t ) + t − t α η i ¯ j f i f ¯ j ) ds ≤ Z (cid:18) − α t − t ) + C ( t − t ) + C ( t − t )(1 − s ) t + st (cid:19) ds = C t − t + C ( t − t ) + C ln t t i.e. v ( x, t ) ≤ v ( y, t ) (cid:18) t t (cid:19) C e C t − t + C ( t − t ) . Since x, y are arbitrary two points in X , we obtain the inequality required. (cid:3) Exponential Convergence.
As a consequence of the Harnack-type inequal-ity above, we first prove the following exponential estimate for˜ u = u − R X uω n R X ω n . Theorem 6.4.
There exist two constants C and C such that (cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ u∂t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C e − C t . Proof.
For convenience, we denote ϕ and ˜ ϕ to be ˙ u and ˙˜ u . It is easy to see that ˜ ϕ and ϕ satisfies Z X ˜ ϕω n = 0 and ∂ϕ∂t = η i ¯ j ϕ i ¯ j . Furthermore, for any fixed t ∈ [0 , ∞ ) and x, y ∈ X , functions ˜ ϕ and ϕ also satisfythe following relation(6.10) | ˜ ϕ ( x, t ) − ˜ ϕ ( y, t ) | = | ϕ ( x, t ) − ϕ ( y, t ) | . t follows from the maximum principle for the parabolic equation that for any0 < t < t , there holds(6.11) sup y ∈ X ϕ ( y, t ) ≤ sup y ∈ X ϕ ( y, t ) ≤ sup y ∈ X ϕ ( y, y ∈ X ϕ ( y, t ) ≥ inf y ∈ X ϕ ( y, t ) ≥ inf y ∈ X ϕ ( y, . Let m be an arbitrary positive integer. For any ( x, t ), we define ξ m ( x, t ) = sup y ∈ X ϕ ( y, m − − ϕ ( x, m − t )and ψ m ( x, t ) = ϕ ( x, m − t ) − inf y ∈ X ϕ ( y, m − . Then according to Equations (6.11) and (6.12), ξ m and ψ m are both non-negativeand satisfy the following parabolic equation ∂ξ m ∂t ( x, t ) = η i ¯ j ( x, m − t )( ξ m ) i ¯ j ( x, t )and ∂ψ m ∂t ( x, t ) = η i ¯ j ( x, m − t )( ψ m ) i ¯ j ( x, t )where η depends on the line bundle mean curvature flow u .In the case that ϕ ( x, m −
1) is constant, the function ϕ ( x, t ) must be constant forall t ≥ m − ϕ is also a constant for all t ≥ m − ϕ vanishes, we obtain that ˜ ϕ ( x, t ) = 0 for all t ≥ m −
1. Then ourtheorem is obvious. Therefore we just need to deal with the case that ϕ ( x, m − ϕ ( x, m −
1) is not constant, ξ m must be positive at some point( x , ξ m ( x, t ) must be positive for all x ∈ X when t >
0. Similarly, we also have ψ m ( x, t ) > x ∈ X when t >
0. Hence,we can apply Theorem 6.3 with t = and t = 1 to obtainsup y ∈ X ϕ ( y, m − − inf y ∈ X ϕ ( y, m −
12 ) ≤ C (sup y ∈ X ϕ ( y, m − − sup y ∈ X ϕ ( y, m )) , sup y ∈ X ϕ ( y, m −
12 ) − inf y ∈ X ϕ ( y, m − ≤ C ( inf y ∈ X ϕ ( y, m ) − inf y ∈ X ϕ ( y, m − C is a positive constant bigger than 1. We also define χ ( t ) to be the oscillationof ϕ ( · , t ), i.e.(6.14) χ ( t ) = sup y ∈ X ϕ ( y, t ) − inf y ∈ X ϕ ( y, t ) . Adding the inequalities (6.13) and (6.14) above gives us χ ( m −
1) + χ ( m −
12 ) ≤ C ( χ ( m − − χ ( m )) . Since χ is a non-negative function and C >
1, there holds χ ( m ) ≤ C − C χ ( m − . y induction,(6.15) χ ( m ) ≤ (cid:18) C − C (cid:19) m χ (0) . According to the inequality (6.11) and (6.12), we also know that χ ( t ) is decreasingin t . Therefore, we conclude from (6.15) that χ ( t ) ≤ C e − C t where C = Cχ (0) C − and C = ln CC − .To obtain the result in the theorem, we observe that there must be a point x t ∈ X such that ˜ ϕ ( x t , t ) = 0 for all t > R X ˜ ϕω n = 0. According to Equation (6.10),for all ( x, t ) ∈ X × [0 , ∞ ), | ˜ ϕ ( x, t ) | = | ˜ ϕ ( x, t ) − ˜ ϕ ( x t , t ) | = | ϕ ( x, t ) − ϕ ( x t , t ) | ≤ χ ( t ) ≤ C e − C t i.e. (cid:12)(cid:12) ˙˜ u (cid:12)(cid:12) ≤ C e − C t . (cid:3) We also have the following exponential convergence result for u in C ∞ norm. Theorem 6.5.
The function ˜ u converges exponentially to smoothly.Proof. Integrating from + ∞ to t and apply Theorem 6.4, we get that ˜ u = u − R X uω n R X ω n converges exponentially to 0 in C .We denote D to be the gradient with respect to ω . For any k ≥
1, we considerthe following inequality ∂∂t Z X | D k ˜ u | ω ω n = Z X D k ˜ u ∗ D k ˙˜ uω n = Z X D k ˜ u ∗ ˙˜ uω n ≤ ( Z X | D k ˜ u | ω n ) ( Z X ˙˜ u ω n ) ≤ C e − C t . Integrating form + ∞ to t , we get that || ˜ u || W k, ( ω ) ≤ C e − C t . Then by the Sobolev embedding theorem, we obtain that || ˜ u || C k ′ ≤ || ˜ u || W k, ( ω ) ≤ C e − C t . (cid:3) Acknowledgement
Both authors are grateful to Professor Xinan Ma, Xi Zhang and Xiaohua Zhufor helpful suggestions on this subject. The second author is supported by theFundamental Research Funds for the Central Universities and the Research Fundsof Renmin University of China. eferences [1] H. D. Cao, Deformation of K¨ahler metrics to K¨ahler-Einstein mdtrics on compact K¨ahlermanifolds, Invent. Math, 81(1985), pp. 359-372.[2] T. Collins, A. Jacob and S.-T. Yau, (1 , H. Yamamoto , A ε -regularity theorem for line bundle mean curvature flow,arxiv:1904.02391, (2019).[8] X. L. Han and X. S. Jin, A rigid theorem for deformed Hermitian-Yang-Mills equation,arxiv:1908.08871, (2019).[9] A. Jacob and S.-T. Yau, A special Lagrangian type equation for holomorphic line bundle,Math. Ann, 369(2017), pp. 869-898.[10] J. D. Lotay and Y. Wei, Stability of torsion free G structures along the Laplacian flow, J.Diff. Geom, 11(2019), pp. 495-526.[11] P. Li and S.-T. Yau, On the parabolic kernel for Schr¨odinger operator, Acta Math, 156(1987),pp. 227-252.[12] N.-C. Leung, S.-T. Yau and E. Zaslow, From special Lagrangian to Hermitian-Yang-Mills viaFourier-Futaki transform, Adv. Theor. Math. Phys, 4(2000), pp. 1319-1341.[13] M. Marino, R. Minasian, G. Moore, A. Stromiger, Nonlinear instantons from supersymmetricp-Branes, arXiv:hep-th/9911206, (1999).[14] V. P. Pingali, The deformed Hermitian-Yang-Mills equation on three-folds, arxiv:1910.01870,(2019).[15] E. Schlitzer and J. Stoppa, Deformed Hermitian-Yang-Mills connections, extended Gaugegroup and scalar curvature, arxiv:1911.10852, (2019).[16] R. Takahashi, Collapsing of the line bundle mean curvature flow on K¨ahler surfaces,arxiv:1912.13145, (2019).[17] X. Zhu, Stability of Kahler-Ricci flow on a Fano manifold, Math. Ann, 356(2013), pp. 1425-1454. Xiaoli Han, Math department of Tsinghua university, Beijing, 100084, China,
E-mail address : [email protected] Xishen Jin, School of Mathematics, Remin University of China, Beijing, 100872, China,
E-mail address : [email protected]@mail.ruc.edu.cn