A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KA¨HLER CLASS (II) HAOZHAOLI1 ANDYALONGSHI2 5 1 0 2 Abstract. Inthispaper,wegivearesultonthepropernessoftheK-energy,whichanswers n aquestionofSong-Weinkove[11]inanydimensions. Moreover,weextendourpreviousresult a J onthepropernessofK-energyin[9]tothecaseofmodifiedK-energyassociatedtoextremal 7 Ka¨hler metrics. ] G Contents D . 1. Introduction 1 h t 2. Proof of Theorem 1.1 3 a m 3. Proof of Theorem 1.2 4 [ 3.1. The modified K-energy µ˜ and J˜functional 4 3.2. The existence of critical points of J˜ 6 1 v 3.3. Proof of Theorem 1.2 11 6 References 13 1 5 1 0 . 1 1. Introduction 0 5 1 This paper is a continuation of our previous work [9]. In [9], we give a criterion for the : properness of the K-energy in a general Ka¨hler class of a compact Ka¨hler manifold by using v i Song-Weinkove’s resultonJ-flow in[11], which extendstheworksof Chen[1], Song-Weinkove X [11] and Fang-Lai-Song-Weinkove [6]. In [11], Song-Weinkove showed that the K-energy is r a properonaKa¨hlerclass [χ ]ofan-dimensionalKa¨hlermanifoldM withc (M) < 0whenever 0 1 there are Ka¨hler metrics ω πc (M) and χ [χ ] such that 1 ′ 0 ∈− ∈ πc (M) [χ ]n 1 (cid:16)−n 1 [χ0·]n 0 − χ′−(n−1)ω(cid:17)∧χ′n−2 > 0. (1.1) Moreover, Song-WeinkoveaskedwhethertheK-energyisboundedfrombelowiftheinequality (1.1) is not strict (Remark 4.2 of [11]). In [6] Fang-Lai-Song-Weinkove studied the J-flow on the boundary of the Ka¨hler cone and gave an affirmative answer in complex dimension 2. In [9], we give a partial answer to this question, which says that the K-energy is proper if 1Research partially supported byNSFC grant No. 11131007. 2Research partially supported byNSFC grants No. 11101206. 1 2 HAOZHAOLI1 ANDYALONGSHI2 c (M) < 0 and the Ka¨hler class [χ ] satisfies 1 0 c (M) [χ ]n 1 1 0 − n · [χ ]+(n 1)c (M) 0. − [χ ]n 0 − 1 ≥ 0 The first main result in this paper is the following theorem, which answers the question of Song-Weinkove in any dimensions. Theorem 1.1. Let M be a n-dimensional compact K¨ahler manifold with c (M) < 0. If 1 the K¨ahler class [χ ] satisfies the property that there are two K¨ahler metrics χ [χ ] and 0 ′ 0 ∈ ω πc (M) such that 1 ∈ − πc (M) [χ ]n 1 (cid:16)−n 1 [χ0·]n 0 − χ′−(n−1)ω(cid:17)∧χ′n−2 ≥ 0, (1.2) then the K-energy is proper on the K¨ahler class [χ ]. 0 Our second main result is to extend [9] to the case of extremal Ka¨hler metrics. To state our main results, we recall Tian’s α-invariant for a Ka¨hler class [χ ]: 0 α ([χ ]) = sup α > 0 C > 0, e α(ϕ supϕ)χn C, ϕ (M,χ ) , M 0 n (cid:12)∃ ZM − − 0 ≤ ∀ ∈ H 0 o (cid:12) where (M,χ ) denotes the s(cid:12)pace of Ka¨hler potentials with respect to the metric χ . For 0 0 H any compact subgroup G of Aut(M), and a G-invariant Ka¨hler class [χ ], we can similarly 0 define the α invariant by using G-invariant potentials in the definition. M,G Theorem 1.2. Let M be a n-dimensional compact K¨ahler manifold and X an extremal vector field of the K¨ahler class [χ ] with potential function θ := θ (χ ). Assume ImX generates 0 X X 0 a compact group of holomorphic automorphisms1, and L χ = 0. If the Ka¨hler class [χ ] ImX 0 0 satisfies the following conditions for some constant ǫ : (1) 0 ǫ < n+1α ([χ ]), ≤ n M 0 (2) πc (M) < (ǫ+minθ )[χ ], 1 X 0 (3) πc (M) [χ ]n 1 1 0 − n · +minθ +ǫ [χ ]+(n 1)πc (M) > 0, (cid:16)− [χ0]n M X (cid:17) 0 − 1 then the modified K-energy is proper on (M,χ ), where (M,χ ) is the subspace of X 0 X 0 H H (M,χ ) with the extra condition ImX(ϕ) = 0. If instead of (1), we assume [χ ] is G- 0 0 H invariant for a compact subgroup G of Aut(M) , and 0 ǫ < n+1α ([χ ]), then the ≤ n M,G 0 modified K-energy is proper on the space of G-invariant potentials. For the definitions of extremal vector field and modified K-energy, see section 3. Note that the minθ here is actually an invariant of the Ka¨hler class according to [10] and Appendix of X [17]. We can also replace the condition (3) of Theorem 1.2 by some weaker assumptions as in Theorem1.1,however wepreferthisversionsince(3)iseasier tocheck. TheproofofTheorem 1.2 relies on the study of the modified J-flow, which is an extremal version of the usual J flow defined by Donaldson [5] and Chen [1]. Here we modify the proof of Song-Weinkove [11] 1If [χ ]=c (M), then this is always true,see Theorem F of [8]. 0 1 3 to get the existence of critical metrics of the modified J functional and then we apply the argument in [9] to get the properness of the modified K-energy. In a recent interesting paper [4], Dervan gives a different sufficient condition on the proper- ness of the K energy on a general Ka¨hler class by direct analyzing the expression of the K energy, which gives better results in some examples (cf. [9][4]). While Dervan’s condition is useful mainly when M is Fano, our theorem applies on more general manifolds. Whether one can improve both results is still an interesting problem. In section 2, we prove Theorem 1.1, which is a strengthen of the Main Theorem of [9]. Then in section 3, we study the modified J-flow and prove Theorem 1.2. 2. Proof of Theorem 1.1 In this section we prove Theorem 1.1. Here we use the notations in our previous work [9]. Proof of Theorem 1.1. By the assumption (1.2), for sufficiently small ǫ > 0 we have (nc+ǫ)χ (n 1)ω χn 2 > 0, (2.1) ′ ′ − (cid:16) − − (cid:17)∧ where πc (M) [χ ]n 1 1 0 − c = − · . [χ ]n 0 We can write (2.1) as n(c+ǫ)χ (n 1)(ω+ǫχ) χn 2 > 0, (2.2) ′ ′ ′ − (cid:16) − − (cid:17)∧ Since ω and χ are Ka¨hler metrics, we have ω + ǫχ > 0. by Song-Weinkove’s result (cf. ′ ′ Theorem 1.1 in [11]) there exists a Ka¨hler metric χ [χ ] such that 0 ∈ (ω+ǫχ) χn 1 = (c+ǫ)χn. ′ − ∧ Thus,thefunctionalJˆω+ǫχ′,χ0 isboundedfrombelowon[χ0].Sinceχ′ ∈ [χ0],bytheargument of [13]2 [9] there is a uniform constant C > 0 such that for any ϕ (M,χ ), 0 ∈ H |Jˆω+ǫχ′,χ0(ϕ)−Jˆω+ǫχ0,χ0(ϕ)| ≤ C. Therefore, Jˆ (ϕ) is bounded from below and we have ω+ǫχ0,χ0 Jˆ (ϕ) ǫ I (ϕ) J (ϕ) C, ϕ (M,χ ). (2.3) ω,χ0 ≥ − (cid:16) χ0 − χ0 (cid:17)− ∀ ∈ H 0 Now using Tian’s α-invariant we have (see Lemma 4.1 of [11], also [14] page 95 ) χn χn ϕ ϕ log αI (ϕ) C Z χn n! ≥ χ0 − X 0 n+1 α (I (ϕ) J (ϕ)) C, ϕ (M,χ ) (2.4) ≥ n · χ0 − χ0 − ∀ ∈ H 0 2The authors would like to thank G. Sz´ekelyhidi for telling them this fact, which they overlooked when preparing [9]. 4 HAOZHAOLI1 ANDYALONGSHI2 for any α (0,α ([χ ])). Set ω := Ric(χ ) > 0. Combining the inequalities (2.3)-(2.4) we M 0 0 0 ∈ − have χn χn µ (ϕ) = log ϕ ϕ +Jˆ (ϕ) χ0 Z χn n! ω0,χ0 X 0 χn χn log ϕ ϕ +Jˆ (ϕ) C ≥ Z χn n! ω,χ0 − X 0 n+1 α ǫ I (ϕ) J (ϕ) C. ≥ (cid:16) n − (cid:17)(cid:16) χ0 − χ0 (cid:17)− Therefore, for sufficently small ǫ the K energy is proper. (cid:3) 3. Proof of Theorem 1.2 3.1. The modified K-energy µ˜ and J˜ functional. We first recall some notations in [9]. Let (M,χ ) be a n-dimensional compact Ka¨hler manifold with a Ka¨hler form 0 √ 1 χ0 = 2− hi¯jdzi ∧dz¯j. We denote by (M,χ ) the space of Ka¨hler potentials 0 H √ 1 (M,χ0) = ϕ C∞(M,R) χϕ = χ0+ − ∂∂¯ϕ> 0 . H { ∈ | 2 } A metric χ is called “extremal” if the gradient of the scalar curvature R(χ) is a holomorphic vector field, i.e. R(χ) R θ (χ) = 0, X − − where R is the integral mean value of R(χ) (which is a topological number) and θ (χ) is the X normalized holomorphic potential of a holomorphic vector field X with respect to the metric χ. Namely, θ (χ) satisfies the equalities X √ 1 χn L χ = − ∂∂¯θ (χ), θ (χ) = 0. X X X 2 Z n! M Such a holomorphic vector field X is called an “extremal vector field”. Futaki and Mabuchi proved that “extremal vector field” makes sense in a general Ka¨hler manifold and is unique [8]. Here we always assume that L χ = 0, hence θ is real-valued. For such an extremal ImX X vector field X, we modified the space of Ka¨hler potentials accordingly: (M,χ )= ϕ (M,χ ) ImX(ϕ) = 0 . X 0 0 H { ∈ H | } For any ϕ (M,χ ), the potential θ (χ ) is also real-valued (Since we always have X 0 X ϕ ∈ H θ (χ ) = θ (χ)+X(ϕ).). Then we can define the modified K-energy on (M,χ ) by the X ϕ X X 0 H variational formula χn ϕ δµ˜ (ϕ) = δϕ(R(χ ) R θ (χ )) . χ0 −Z ϕ − − X ϕ n! M Then the critical point of µ˜ is just an extremal Ka¨hler metric in [χ ]. χ0 0 5 The J˜functional with respect to a reference closed (1,1)-form ω (not necessarily positive) is defined by the formula 1 ∂ϕ χn J˜ (ϕ) = Jˆ (ϕ)+ tθ (ϕ ) ϕtdt ω,χ0 ω,χ0 Z Z ∂t X t n! 0 M 1 ∂ϕ dt 1 ∂ϕ χn = t(ω χn 1 cχn ) + tθ (ϕ ) ϕtdt, (3.1) Z Z ∂t ∧ ϕ−t − ϕt (n 1)! Z Z ∂t X t n! 0 M 0 M − where [ω][χ ]n 1 0 − c = . [χ ]n 0 When we choose ω = Ric(χ ), then a direct computation shows that 0 0 − χn χn µ˜ (ϕ) = log ϕ ϕ +J˜ (ϕ). (3.2) χ0 Z χn n! ω0,χ0 M 0 Note that the modified K-energy and J-functional actually make sense on the larger space (M,χ ), though the value may be not real. However the J˜functional enjoys the following 0 H interesting property as the usual J functional: Proposition 3.1. When ω is positive, the real part of J˜ is strictly convex along any C1,1 ω,χ0 geodesics in (M,χ ), and its imaginary part is linear. 0 H Proof. Since theusual Jˆfunctional is real valued and strictly convex along any C1,1 geodesics by the work of Chen, we only need to compute the second order derivative of the additional term. Suppose the geodesic is C2, then d χn d χn ϕ˙ θ (ϕ ) ϕt = ϕ˙ (θ (χ )+X(ϕ )) ϕt dt Z t X t n! dt Z t X 0 t n! M M = ϕ¨ θ (ϕ )+X(1ϕ˙2) χnϕt + ϕ˙ θ (ϕ )√−1∂∂¯ϕ˙ χϕn−t 1 t X t t X t t ZM h 2 i n! ZM 2 ∧ (n 1)! − χn 1 χn = ϕ¨ < ∂ϕ˙ ,∂ϕ˙ > θ (ϕ ) ϕt + X( ϕ˙2) ϕt t t t X t ZM h − i n! ZM 2 n! √−1∂θ (ϕ ) ∂¯(1ϕ˙2) χϕn−t 1 −Z 2 X t ∧ 2 t ∧ (n 1)! M − χn 1 χn = ϕ¨ < ∂ϕ˙ ,∂ϕ˙ > θ (ϕ ) ϕt + L ϕ˙2 ϕt ZM h t− t t i X t n! ZM X(cid:16)2 t n! (cid:17) χn = ϕ¨ < ∂ϕ˙ ,∂ϕ˙ > θ (ϕ ) ϕt = 0. t t t X t ZM h − i n! Then we approximate a general C1,1 geodesic by C2 geodesics as Chen-Tian [3]. So we conclude that ReJ˜is also strictly convex along any C1,1 geodesics, and ImJ˜is linear. (cid:3) A direct corollary of Proposition 3.1 is the following result: Corollary 3.2. If J˜ has a critical point ϕ (M,χ ) and ω > 0, then J˜ is bounded ω,χ0 ∈ HX 0 ω,χ0 from below on (M,χ ). X 0 H 6 HAOZHAOLI1 ANDYALONGSHI2 Proof. This is a minor modification of Chen’s proof of Proposition 3 in [1]. We just connect any ψ (M,χ ) with ϕ by a C1,1 geodesic in (M,χ ). Then since both J˜ (ϕ) and ∈ HX 0 H 0 ω,χ0 J˜ (ψ)arerealvalued,byProposition3.1,J˜ isrealvaluedalongthisgeodesic, andhence ω,χ0 ω,χ0 strictly convex. The rest of the proof is identical to that of Chen in [1], so we omit it. (cid:3) 3.2. The existence of critical points of J˜. In this subsection, we always assume ω is a closed positive (1,1)-form, i.e. a Ka¨hler form. We want to find out the critical point of J˜ . ω,χ0 By definition, a critical point ϕ (M,χ ) satisfies X 0 ∈ H 1 ω χn 1 = c+ θ (χ ) χn. (3.3) ∧ ϕ− (cid:16) n X ϕ (cid:17) ϕ We have a similar theorem as Song-Weinkove, saying that the existence of a “subsolution” (in a suitable sense) to the above Euler-Lagrange equation will actually leads to a solution: Theorem 3.3. If there is a metric χ [χ ] satisfying ′ 0 ∈ (ncχ (n 1)ω) χn 2+θ (χ)χn 1 > 0, (3.4) ′ ′ − X ′ ′ − − − ∧ and L ω = 0, then there is a smooth K¨ahler metric χ = χ + √ 1∂∂¯ϕ [χ ] satisfying Im X ϕ 0 2− ∈ 0 the equation (3.3), and the solution χ is unique. ϕ Notethat(3.4)automaticallyimpliesθ (χ)isrealvalued. TheuniquenesspartofTheorem X ′ 3.3 follows directly from Proposition 3.1. We only need to study the existence problem. Without loss of generality, we may assume that the initial metric χ satisfies (3.4). To prove 0 Theorem 3.3, we introduce the following flow, called “modified J-flow”: ∂ϕ ω χϕn−1 1 = c ∧ + Re θ (χ ) ∂t − χn n X ϕ ϕ 1 = nc+Re θ (χ ) Λ ω . (3.5) n(cid:16) X ϕ − χϕ (cid:17) Denote the right hand side operator by L(ϕ), then it is easy to see that the linearization of L is given by ∆˜ + 1Re X, where n ∆˜f = 1hk¯l∂ ∂ f, hk¯l = χk¯jχi¯lg . n k ¯l i¯j Since ∆˜ is strictly elliptic, we always have short time solution to the flow equation (3.5). A modified J-flow starts with an element of (M,χ ) will remain in this space: X 0 H Lemma 3.4. If ϕ = ϕ satisfies (Im X)(ϕ ) = 0, and L ω = 0, then along the 0 t=0 0 Im X | modified J-flow, we always have (Im X)(ϕ) = 0. Proof. Denote Im X by Y. By the assumption L ω = 0 = L χ , we have Y Y 0 Y(Λ ω) = χα¯jχiβ¯(L χ ) g +χi¯j(L ω) χϕ − ϕ ϕ Y ϕ αβ¯ i¯j ϕ Y i¯j = χα¯jχiβ¯ L χ +Y(ϕ) g − ϕ ϕ Y 0 αβ¯ i¯j (cid:0) (cid:1) = χα¯jχiβ¯(Y(ϕ)) g = n∆˜Y(ϕ) − ϕ ϕ αβ¯ i¯j − 7 Form (3.5), we have ∂Y(ϕ) 1 1 = Y(Λ ω)+ Y θ (χ )+Re X(ϕ) ∂t −n χϕ n X 0 (cid:0) (cid:1) 1 1 = ∆˜Y(ϕ)+ (Re X) Y(ϕ) + Y θ (χ ) , X 0 n n (cid:0) (cid:1) (cid:0) (cid:1) where the last equality follows from the fact that the real part and imaginary part of a holomorphic vector field always commute.3 Claim: Wealways haveY θ (χ ) = 0,thusY(ϕ)satisfiesaverygoodparabolicequation X 0 and we conclude from maxim(cid:0)um prin(cid:1)ciple that Y(ϕ) = 0 along the flow. To prove the claim, just note that from the definition of θ (χ ), we always have X 0 Xiχ = ∂ θ (χ ). i¯j ¯j X 0 So we have X θ (χ ) = XjX¯iχ = X 2 . X 0 j¯i | |χ0 (cid:0) (cid:1) Since both X θ (χ ) and θ (χ ) are real, we have (Im X) θ (χ ) = Im X θ (χ ) = X 0 X 0 X 0 X 0 0. (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) (cid:3) From the above lemma, we see that actually we can rewrite our equation as ∂ϕ ω χϕn−1 1 = c ∧ + θ (χ ) ∂t − χn n X ϕ ϕ 1 = nc+θ (χ ) Λ ω . (3.6) n(cid:16) X ϕ − χϕ (cid:17) Differentiating (3.6) with respect to t, we have ∂ ∂ϕ ∂ϕ 1 ∂ϕ = ∆˜ + X . ∂t ∂t ∂t n (cid:16)∂t(cid:17) The maximum principle implies that ∂ϕ ∂ϕ ∂ϕ min max . M ∂t(cid:12)t=0 ≤ ∂t ≤ M ∂t(cid:12)t=0 (cid:12) (cid:12) In particular, (cid:12) (cid:12) Λ ω maxΛ ω+maxθ (χ ) minθ (χ ). χ ≤ M χ0 M X ϕ − M X 0 Since both χ and χ are ImX-invariant by Zhou-Zhu [17], the term max θ (χ ) 0 ϕ M X ϕ − min θ (χ ) is uniformly bounded. Thus, Λ ω has uniform positive upper bound along the M X 0 χ flow. In particular, there is a uniform constant c >0 such that χ c ω (3.7) ϕ ≥ as long as the flow exists. Lemma 3.5. There is a uniform constant C > 0 such that for any (x,t) we have Λωχ CeA(ϕ−infM×[0,t]ϕ), ≤ and ϕ C as long as the flow exists. C0 | | ≤ 3Note that a holomorphic vector field is always of the form Z −iJZ, where Z is real-holomorphic, i.e. L J =0. So we have[Z,JZ]=L (JZ)=JL Z =0. Z Z Z 8 HAOZHAOLI1 ANDYALONGSHI2 Proof. Following Song-Weinkove [12], in normal coordinates of ω, we have ∆˜(Λ χ)= 1hk¯lR i¯j(g)χ + 1hk¯lgi¯j∂ ∂ χ , (3.8) ω n k¯l i¯j n k ¯l i¯j where R i¯j(g) denotes the curvature tensor of g. By the equation of the modified J-flow, we k¯l have ∂ 1 1 Λ χ = gi¯j∂ ∂ (χk¯lg )+ gi¯j∂ ∂ (θ (χ )) ∂t ω −n i ¯j k¯l n i ¯j X ϕ = 1 gi¯jhpq¯∂ ∂ χ gi¯jhrq¯χps¯∂ χ ∂ χ gi¯jhps¯χrq¯∂ χ ∂ χ +χk¯lR (g) n(cid:16) i ¯j pq¯− i rs¯ ¯j pq¯− i rs¯ ¯j pq¯ k¯l (cid:17) 1 + gi¯j∂ ∂ (θ (χ )). n i ¯j X ϕ Therefore, we have ∂ (∆˜ )log(Λ χ) ω − ∂t ∆˜(Λ χ) ˜(Λ χ)2 ∂ ω ω = |∇ | log(Λ χ) Λ χ − (Λ χ)2 − ∂t ω ω ω = 1 hk¯lR i¯j(g)χ +gi¯jhrq¯χps¯∂ χ ∂ χ +gi¯jhps¯χrq¯∂ χ ∂ χ nΛωχ(cid:16) k¯l i¯j i rs¯ ¯j pq¯ i rs¯ ¯j pq¯ −χk¯lRk¯l(g)−n|∇˜(ΛΛωωχχ)|2 −gi¯j∂i∂¯jθX(χϕ)(cid:17) 1 hk¯lR i¯j(g)χ χk¯lR (g) gi¯j∂ ∂ θ (χ ) , ≥ nΛωχ(cid:16) k¯l i¯j − k¯l − i ¯j X ϕ (cid:17) where we used the inequality by Lemma 3.2 in [15] n ˜(Λ χ)2 (Λ χ)gi¯jhrq¯χps¯∂ χ ∂ χ . (3.9) ω ω i rs¯ ¯j pq¯ |∇ | ≤ On the other hand, we have 1 X(logΛωχ) = ΛωχX(cid:16)gi¯j(χ0,i¯j +ϕi¯j)(cid:17) 1 = X(gi¯jχ )+X(gi¯jϕ ) Λωχ(cid:16) 0,i¯j i¯j (cid:17) 1 = X(gi¯jχ )+∆ (X(ϕ)) Xkϕ , Λωχ(cid:16) 0,i¯j g − ,i k¯i(cid:17) where we used the fact that X(gi¯jϕ )= Xkϕ = Xkϕ = gi¯j(X(ϕ)) Xkϕ . i¯j i¯ik k¯ii i¯j ,i k¯i − 9 Combining the above identities, we have 1 ∂ ∆˜ + X log(Λ χ) ω (cid:16) n − ∂t(cid:17) 1 hk¯lR i¯j(g)χ χk¯lR (g) gi¯j∂ ∂ θ (ϕ)+X(gi¯jχ )+∆ (X(ϕ)) Xkϕ ≥ nΛωχ(cid:16) k¯l i¯j − k¯l − i ¯j X 0,i¯j g − ,i k¯i(cid:17) = 1 hk¯lR i¯j(g)χ χk¯lR (g) Xkϕ ∆ θ +X(gi¯jχ ) nΛωχ(cid:16) k¯l i¯j − k¯l − ,i k¯i − g X 0,i¯j (cid:17) = 1 hk¯lR i¯j(g)χ χk¯lR (g) Xkχ +Xkχ ∆ θ +X(gi¯jχ ) , nΛωχ(cid:16) k¯l i¯j − k¯l − ,i k¯i ,i 0,k¯i− g X 0,i¯j (cid:17) where θ is the holomorphic potential of X with respect to χ . Note that X 0 ∆˜ + 1X ∂ ϕ = 1 hk¯lϕ +χi¯jg nc θ (cid:16) n − ∂t(cid:17) n(cid:16) k¯l i¯j − − X(cid:17) 1 = 2χi¯jg hi¯jχ nc θ . n(cid:16) i¯j − 0,i¯j − − X(cid:17) Thus, we have 1 ∂ n ∆˜ + X log(Λ χ) Aϕ ω (cid:16) n − ∂t(cid:17)(cid:16) − (cid:17) 1 hk¯lR i¯j(g)χ χk¯lR (g) Xkχ +C(χ ,ω,X) ≥ Λωχ(cid:16) k¯l i¯j − k¯l − ,i k¯i 0 (cid:17) 2Aχi¯jg +Ahi¯jχ +ncA+Aθ . − i¯j 0,i¯j X By the assumption (3.4), we can choose ǫ >0 sufficiently small such that (ncχ (n 1)ω) χn 2+θ (χ )χn 1 > 2ǫχn 1. (3.10) 0− − ∧ 0− X 0 0− 0− Moreover, since χ is uniformly bounded from below, we can choose A large such that ϕ −AΛ1ωχ(cid:16)hk¯lRk¯l i¯j(g)χi¯j −χk¯lRk¯l(g)−X,kiχk¯i+C(ω,X)(cid:17) ≤ ǫ, then at the maximum point (x ,t ) of log(Λ χ) Aϕ, we have 0 0 ω − nc+θ +hi¯jχ 2χi¯jg ǫ. X 0,i¯j − i¯j ≤ We choose normal coordinates for the metric χ so that the metric χ is diagonal with entries 0 λ , ,λ . We denote the diagonal entries of ω by µ , ,µ . Thus, we have 1 n 1 n ··· ··· n n µ µ i i nc+θ (x )+ 2 ǫ, X 0 λ2 − λ ≤ Xi=1 i Xi=1 i which implies that for any fixed index k, we have the inequality n n 1 2 µ µ k k ǫ µ 1 µ + 2 +nc+θ (x ) ≥ i=X1,i=k i(cid:16)λi − (cid:17) −i=X1,i=k i λ2k − λk X 0 6 6 n µ k nc+θ (x ) µ 2 . (3.11) X 0 i ≥ − − λ X k i=1,i=k 6 10 HAOZHAOLI1 ANDYALONGSHI2 On the other hand, by (3.10) we have for any k, (ncχ (n 1)ω) χn 2 β +θ (χ )χn 1 β > 2ǫχn 1 β , 0− − ∧ 0− ∧ k X 0 0− ∧ k 0− ∧ k where β := √ 1dzk dz¯k. This means k − ∧ n nc+θ (x ) µ > 2ǫ. X 0 i − X i=1,i=k 6 Combining the above inequalities, we have λk < 2 and there is a constant C = C(n,ǫ) such µk ǫ that at the point (x ,t ), 0 0 Λ χ C. ω ≤ Thus, at any point (x ,t ) we have the estimate 0 0 Λωχ CeA(ϕ−infM×[0,t]ϕ). ≤ The passage from this C2 estimate to C0 estimate does not use the equation and hence is identical to Song-Weinkove[11] and Weinkove [15][16], so we omit it. (cid:3) Corollary 3.6. The modified J-flow exists for any t [0, ). ∈ ∞ Proof. Suppose the solution exists only in [0,T) with T < . We will derive a contradiction. ∞ By the above lemma, we have uniform C0 and C2 estimates. By interpolation, we also have uniform C1 estimate on [0,T). By Evans-Krylov estimate, we also have uniform C2,α estimate. Then we can take limit of ϕ(,t ) as t T to get a ϕ . Since χ is uniformly i i T ϕ · → bounded from below, χ + √ 1∂∂¯ϕ is a Ka¨hler form. So the solution can extend beyond T, 0 2− T a contradiction! (cid:3) Now we can use the modified J-flow to finish the proof of Theorem 3.3: Proof of Theorem 3.3. By our above discussion, the modified J-flow has a unique solution ϕ(,t) for t [0, ). By the proof of the above theorem, we also have a uniform C2,α · ∈ ∞ estimate. By the standard bootstrap argument, the solutions are uniformly bounded with respect to any Ck norm. Then for any sequence t , we can find a subsequence, also i → ∞ denoted by t such that ϕ(,t ) ϕ in C . We shall prove that ϕ solves (3.3). i i ∞ · → ∞ ∞ To show this, we define an energy functional associated with the modified J-functional following Chen [2]: χn χn E (ϕ) := θ (χ ) Λ ω 2 ϕ = σ2 ϕ, X,χ0 Z X ϕ − χϕ n! Z n! M (cid:0) (cid:1) M