Convergence of the J-flow on Kähler Surfaces

Transcription

1 communications in analysis and geometry Volume 1, Number 4, , 004 Convergence of the J-flow on Kähler Surfaces Ben Weinkove Donaldson defined a parabolic flow of potentials on Kähler manifolds which arises from considering the action of a group of symplectomorphisms on the space of smooth maps between manifolds. One can define a moment map for this action, and then consider the gradient flow of the square of its norm. Chen discovered the same flow from a different viewpoint and called it the J-flow, since it corresponds to the gradient flow of his J-functional, which is related to abuchi s K-energy. In this paper, we show that in the case of Kähler surfaces with two Kähler forms satisfying a certain inequality, the J-flow converges to a zero of the moment map. 1. Introduction. In [Do], Donaldson described how a number of geometric situations fit into a general framework of diffeomorphism groups and moment maps. In the Kähler setting, he used this framework to define a natural parabolic flow, as follows. Suppose that (, ω) isacompactkähler manifold of dimension n and let χ 0 be another Kähler form on, in a different Kähler class. Consider the infinite-dimensional manifold of diffeomorphisms f :, homotopicto the identity. carries a natural symplectic form Ω defined by Ω f (v, w)= ω(v, w) χn 0 n!, for sections v, w of f (T). The group G of exact χ 0 -symplectomorphisms of acts on by composition on the right, preserving Ω. We can identify the Lie algebra of G with the space of functions on of integral zero with respect to the volume form induced by χ 0. A moment map µ : Lie(G) for the group action is given by µ(f) = f (ω) χ n 1 0 χ n 0 ω χn 1 0, χn 0 whereweareusingthel inner product to identify Lie(G) with its dual. It is natural to look for solutions of µ(f) =0 (modg). (1.1) 949

2 950 Ben Weinkove These points form the symplectic quotient. Under certain conditions, one would hope that the gradient flow f t of the function µ on would converge to give a solution of (1.1). The gradient flow can be rewritten as a flow of Kähler forms (ft ) 1 (χ 0 )on. This defines a parabolic flow on the space of Kähler potentials and is the object of study of this paper. At around the same time, Chen [C1] independently discovered the same flow as the gradient flow of his J-functional. He later called it the J-flow [C]. He showed in [C1] that the J-functional is related to the abuchi K-energy [a], which plays a key role in the study of Kähler geometry and stability in the sense of geometric invariant theory (see [Y], [T], [T3] and [PS] for example). Explicitly, the J-flow is defined as follows. Let c be the constant given by c = ω χn 1 0, χn 0 and let H be the space of Kähler potentials 1 H = {φ C () χ φ = χ 0 + φ > 0}. The J-flow is the flow on H given by φ t = c ω χn 1 φ t χ n. φ t φ 0 = 0. (1.) A critical point of the J-flow gives a Kähler metric χ satisfying ω χ n 1 = cχ n. (1.3) Donaldson [Do] asked whether one can find a solution to (1.3) in the class [χ 0 ] under certain assumptions. He noted that a necessary condition is that [ncχ 0 ω] beakähler class, and conjectured that this condition be sufficient. Chen [C1] confirmed this conjecture in the case n =, without using the J-flow, by observing that (1.3) reduces to a onge-ampère equation which can be solved by the well-known result of Yau [Y1]. The conjecture is still open for n>. Chen [C1] shows that Donaldson s conjecture would imply a result on the lower bound of the abuchi K-energy for compact Kähler manifolds with negative first Chern class. Namely, if ω c 1 () withω>0, then for Kähler classes [χ 0 ] satisfying nc[χ 0 ] [ω] > 0,

3 Convergence of the J-flow on Kähler Surfaces 951 the abuchi K-energy would have a lower bound in the class [χ 0 ]. Solutions of the J-flow exist for a short time by general theory, since the flow is parabolic. In [C], Chen showed that the flow always exists for all time for any smooth initial data. He also showed that if the bisectional curvature of ω is non-negative then the J-flow converges to a critical metric. In general, the behaviour of the flow is not known. In this paper, we deal with the case n = with no curvature restrictions. Our main result is as follows. ain Theorem Suppose that (, ω) has dimension n =and that ncχ 0 ω>0. Then the J-flow (1.) converges in C to a smooth critical metric. The outline of the paper is as follows. In section we state some preliminary facts about the flow and introduce notation. In section 3, the maximum principle is used to derive an estimate on the second derivatives of φ in terms of φ itself. In section 4, a C 0 estimate for φ is given. The argument uses the second order estimate, a oser iteration argument applied to the exponential of φ and the result of Tian [T1] (see also [TY]) on the existence of constants α>0andc such that e αφ χn 0 n! C, for all φ in H with sup φ = 0. In section 5, the proof of the main theorem is completed.. Preliminaries and notation. From now on, assume that ω has been scaled so that c =1/n. We will work in local coordinates, and write 1 1 ω = g ij dzi dz j, χ 0 = χ 0 ij dzi dz j, and 1 1 χ = χ ij dzi dz j = (χ 0 ij + i j φ)dz i dz j, where χ = χ φ (suppressing the t-subscript.) The operators Λ ω and Λ χ act on (1, 1) forms α = 1 α ij dzi dz j by Λ ω α = g ij α ij, and Λ χ α = χ ij α ij.

4 95 Ben Weinkove The J-flow (1.) can be written Differentiating with respect to t gives ( ) φ φ = 1 n (1 Λ χω) φ t=0 = 0. (.1) = where the operator acts on functions f by f = 1 n χkj χ il g ij k l f. ( ) φ, (.) For convenience, write h kl = χ kj χ il g ij. The tensor h kl is positive definite and its inverse defines a Hermitian metric on. The operator is, up to a constant factor, the Laplacian associated to this Hermitian metric. By the maximum principle for parabolic equations, (.) implies that which gives a lower bound for χ, inf (Λ χ 0 ω) Λ χ ω sup(λ χ0 ω), (.3) χ The J-functional [C1] is defined by 1 ω. (.4) sup (Λ χ0 ω) J ω,χ0 (φ) = 1 0 φ t ω χ n 1 φ t (n 1)! dt, where {φ t } is a path in H between 0 and φ. The functional is independent of the choice of path. We will need the following formula for the functional in the case n =. Taking the path φ t = tφ, weseethat J ω,χ0 (φ) = 1 φω (χ 0 + χ). (.5)

5 Convergence of the J-flow on Kähler Surfaces 953 Chen also makes use of the I-functional, 1 I ω,χ0 (φ) = 0 This is a well-known functional in Kähler geometry (see [a]). Notice that I(φ) =0alongtheflow. Forn =, this functional is given by I ω,χ0 (φ) = 1 φ (χ 0 + χ χ 0 + χ ). (.6) 6 Inthecourseofthepaper,C 0,C 1,... will denote constants depending only on the initial data ω and χ 0. Curvature expressions such as R ijkl will always refer to the metric g ij. φ t χ n φ t n! dt. 3. Second order estimate. We use the maximum principle to obtain an estimate on the second derivative of φ in terms of φ. We choose to calculate the evolution of (log Λ ω χ Aφ) for some constant A (compare to [Y1], [Au] or [Si] for the analogous estimate for the well-known onge-ampère equation, and [Ca] for the Kähler-Ricci flow.) Theorem 3.1. Suppose that (, ω) has dimension n =and that χ 0 ω>0. (3.1) Let φ = φ t be a solution of the J-flow (.1) on[0, ). Then there exist constants A>0 and C>0depending only on the initial data such that for any time t 0, χ = χ φt satisfies Proof. We will calculate Λ ω χ Ce A(φ inf [0,t] φ). (3.) ( )(log(λ ωχ) Aφ). Using normal coordinates for ω, firstcalculate (Λ ω χ) = 1 n hkl k l (g ij χ ij ) = 1 n hkl R ij χ kl ij + 1 n hkl g ij k l χ ij.

6 954 Ben Weinkove And (Λ ωχ) = (gij i j φ) = 1 n gij i j (χ kl g kl ) = 1 n (gij i (χ pl j χ pq χ kq )g kl + g ij χ kl R ijkl ) = 1 n (gij h pq i j χ pq g ij h rq χ ps i χ rs j χ pq g ij h ps χ rq i χ rs j χ pq + χ kl R kl ). Now log(λ ω χ)= (Λ ω χ) Λ ω χ (Λ ω χ) (Λ ω χ), where (Λ ω χ) = 1 n hkl k (Λ ω χ) l (Λ ω χ). Note that by the Kähler property of χ, wehave i j χ kl = k l χ ij. Then ( )log(λ ωχ) = 1 nλ ω χ (hkl R ij kl χ ij n (Λ ω χ) Λ ω χ + g ij h ps χ rq i χ rs j χ pq χ kl R kl ). + g ij h rq χ ps i χ rs j χ pq We need the following lemma to deal with the second term on the right hand side. Lemma 3.. n (Λ ω χ) (Λ ω χ)g ij h rq χ ps i χ rs j χ pq.

7 Convergence of the J-flow on Kähler Surfaces 955 Proof. Using normal coordinates for ω in which χ is diagonal, and making use of the Cauchy-Schwartz inequality, we obtain n (Λ ω χ) = i,j,k χ kk χ kk k χ ii k χ jj = = ( ) 1/ ( ) 1/ (χ kk ) k χ ii (χ kk ) k χ jj i,j k k ( ) 1/ (χ kk ) k χ ii i k ) 1/ (χ kk ) χ ii k χ ii i χii ( k i χ ii (χ kk ) χ ii k χ ii i,k = (Λ ω χ) i,k (χ kk ) χ ii k χ ii k χ ii = (Λ ω χ) i,k (χ kk ) χ ii i χ ki i χ ik (Λ ω χ) i,j,k(χ kk ) χ ii j χ ki j χ ik = (Λ ω χ)g ij h rq χ ps i χ rs j χ pq. Let C 0 be a constant satisfying R ij kl C 0 g kl g ij. Then, ( )log(λ ωχ) 1 nλ ω χ ( C 0h kl g kl g ij χ ij χ kl R kl ) = 1 n ( C 0h kl g kl 1 Λ ω χ χkl R kl ).

8 956 Ben Weinkove Now calculate ( )φ = 1 n (hkl k l φ + χ ij g ij 1) = 1 n (χkj χ il g ij χ kl h kl χ 0 kl + χ ij g ij 1) = 1 n (χij g ij h kl χ 0 kl 1). At this point we must choose our value of A. From our assumption (3.1), we can choose 0 <ɛ<1/3 to be sufficiently small so that χ 0 (1 + 3ɛ)ω. (3.3) Let A be given by A = C 0 ɛ. Fix a time t>0. There is a point (x 0,t 0 )in [0,t]atwhichthe maximum of (log(λ ω χ) Aφ) is achieved. We may assume that t 0 > 0. At this point, we have 0 ( )(log(λ ωχ) Aφ) 1 n ( C 0h kl g kl 1 Λ ω χ χkl R kl Aχ ij g ij + Ah kl χ 0 kl + A) 1 n ( C 0h kl g kl 1 Λ ω χ χkl R kl Aχ ij g ij +(1 ɛ)ah kl χ 0 kl + ɛah kl g kl + A) = 1 n ( 1 Λ ω χ χkl R kl Aχ ij g ij +(1 ɛ)ah kl χ 0 kl + A). From the lower bound (.4) on χ kl,thetermχ kl R kl is bounded above and hence at (x 0,t 0 ), we have From (3.3), we get 1+(1 ɛ)h kl χ 0 kl χ ij g ij C 1 (Λ ω χ). 1+(1+ɛ)h kl g kl χ ij g ij C 1 (Λ ω χ). (3.4) We will compute in normal coordinates at x 0 for ω in which χ is diagonal and has eigenvalues λ 1,λ. From (.4), λ 1 and λ are bounded below by

9 Convergence of the J-flow on Kähler Surfaces 957 a positive constant. We want to show that they are also bounded above. First, observe that for n =, 1 Λ χ ω = det χ (det ω)(λ ω χ), and by (.3), this is bounded along the flow. ultiplying (3.4) by (det χ/ det ω) gives, λ 1 λ +(1+ɛ)( λ λ 1 + λ 1 λ ) (λ 1 + λ ) C. From (.3), we may suppose that one of the eigenvalues, say λ, is bounded from above. Rewrite the inequality as λ 1 (λ +(1+ɛ) 1 λ ) + (1 + ɛ) λ λ 1 λ C. Then, since the function f :(0, ) R defined by f(x) =x +(1+ɛ) 1 x, is bounded below by a small positive constant depending on ɛ, weseethat λ 1 must also be bounded above. Hence at the point (x 0,t 0 ), there exists C depending only on the initial data such that Then, on [0,t], Exponentiating gives completing the proof of the theorem. Λ ω χ C. log(λ ω χ) Aφ log C A inf φ. [0,t] Λ ω χ Ce A(φ inf [0,t] φ), 4. Zero order estimate. WeproveanestimateontheC 0 norm of φ using a oser iteration method applied to the exponential of the solution rather than a power of the solution (compare to [Y1]) and the estimate of Theorem 3.1.

10 958 Ben Weinkove Theorem 4.1. Suppose that (, ω) has dimension n =and that χ 0 ω>0. Let φ t be a solution of the J-flow (.1)on[0, ). Then there exists a constant C depending only on the initial data such that φ t C 0 () C. Proof. Suppose first that inf φ t is bounded from below uniformly in time. We will show that this implies the above estimate. Since the functional J ω,χ0 decreases along the flow, there exists a constant C 0 such that φ t ω (χ 0 + χ φt ) C 0, using (.5). Let C 1 be a positive constant satisfying Then φ t ω = C 1 ω C 1 ω χ 0. (φ t inf φ t)ω + inf φ t ω C 1 C 0 C 1 (φ t inf φ t)ω χ 0 +inf φ t φ t ω χ φt +inf φ t ω ( = C 1 C 0 C 1 (φ t inf φ t)ω χ φt ( ) +inf φ t ω C 1 ω χ 0 ( ω C 1 C 1 C 0 +inf φ t ω χ 0 ). ω C 1 ω χ 0 ) This gives an upper bound for φ t ω depending on the lower bound for inf φ t. Since ω φ t > Λ ω χ 0 along the flow, it follows from the existence of a lower bound on the Green s function of ω that sup φ t is bounded from above, giving us the required estimate. Now suppose that no such lower bound for inf φ t exists. Then we can assume that there is a sequence of times t i such that (i) inf φ ti =inf t [0,ti ] inf φ t

11 Convergence of the J-flow on Kähler Surfaces 959 (ii) inf φ ti. We will seek a contradiction. For a fixed i, write ψ ti = φ ti sup φ ti. Notice that sup φ ti is bounded from below by zero from (.6) and the fact that I(φ t )=0. Hence ψ ti C 0. The following proposition is the key result of this section. Proposition 4.. Let be a compact complex surface with two Kähler metrics χ 0 and ω. Suppose that ψ C () satisfies the conditions 1 χ ψ = χ 0 + ψ > 0, sup ψ =0, and Λ ω χ ψ Ce A(ψ inf ψ). Then there exists a constant C depending only on, ω, χ 0 and the constants A and C such that ψ C 0 C. We apply this proposition to ψ = ψ ti and obtain a contradiction since Λ ω χ ψti = Λ ω χ φti Ce A(φt i inf t [0,t i ] inf φ t) = Ce A(ψt i inf ψ ti ), where we have used Theorem 3.1 and condition (i) above. It remains to prove the proposition. ProofofProposition4. Let δ be a small positive constant, to be determined later. Set B = A/(1 δ) andletu = e Bψ. Now, for β = n/(n 1) =, the Sobolev inequality for functions f on (, ω) is f β C ( f + f ), for C depending on ω. We will apply this to u p/ for p 1. This gives ( ) 1/β ( ) e Bpβψω C e Bpψ/ ω + e Bpψω. (4.1)

12 960 Ben Weinkove Now calculate e Bpψ/ ω = 1 e Bpψ/ e Bpψ/ ω = B p 1 e Bpψ ψ ψ ω 4 = Bp 1 (e Bpψ ) ψ ω 4 = Bp = Bp = Bp CBp where we have used the estimate e Bpψ 1 ψ ω e Bpψ (χ ψ χ 0 ) ω e Bpψ (Λ ω χ ψ Λ ω χ 0 ) ω = CBp e A inf ψ Λ ω χ ψ Ce A(ψ inf ψ). e Bpψ e A(ψ inf ψ) ω e (p (1 δ))bψω, Then in (4.1), ( ) 1/β pβ ω u C 3 pe A inf ψ p (1 δ) ω u. Raising to the power 1/p and writing γ =1 δ gives u pβ C 1/p 3 p 1/p e (A/p)inf ψ u (p γ)/p p γ. Take the logarithm of both sides to get log u pβ 1 p log C p log p + 1 p sup (p γ) ( Aψ)+ log u p γ. p We now apply the iteration. First, replace p with pβ + γ to get log u pβ +γβ 1+β pβ + γ log C (β log p + log(pβ + γ)) pβ + γ + 1+β pβ + γ sup β(p γ) ( Aψ)+ pβ + γ log u p γ.

13 Convergence of the J-flow on Kähler Surfaces 961 Repeat this procedure, replacing p with pβ + γ to obtain for any positive integer k, log u pβ k+1 +γ(β+β +...+β k ) 1+β + β β k pβ k + γ(1 + β + β β k 1 ) log C pβ k + γ(1 + β β k 1 ) ( βk log p + β k 1 log(pβ + γ) log(pβ k + γ(1 + β β k 1 )) 1+β + β β k + pβ k + γ(1 + β + β β k 1 ) sup ( Aψ) β k (p γ) + pβ k + γ(1 + β + β β k 1 ) log u p γ. (4.) Now set p =1+δ. Then, since β =wehave pβ k + γ(1 + β + β β k 1 )=1+β + β β k + δ. Notice that the second term on the right hand side of (4.) is bounded by Then log p + 1 β log β β k log(βk+1 ) log p +logβ( C 4. k i=1 i +1 β i ) log u pβ k+1 +γ(β+β +...+β k ) log C 3 + C 4 +sup( Aψ)+δmax(log u δ, 0). Using the fact that A =(1 δ)b and Bψ =logu, and letting k tend to infinity, log u C0 C 5 +max(log u δ, 0). Hence we get the following inequality for ψ, ( ( ) 1/δ ψ C 0 C 6 + C 7 max log e δbψω, 0). (4.3) We can now finish the estimate. First, define 1 P (, χ 0 )={Φ C () χ 0 + Φ 0, sup Φ=0}.

14 96 Ben Weinkove Then Proposition.1 of [T1] (see section 4.4, [Ho]) states that there exist constants α>0andc 8 depending only on (, χ 0 ) such that e αφ χn 0 n! C 8 for all Φ P (, χ 0 ). Define δ to be δ =min{ α 4A, 1 } > 0. Then the required estimate follows from (4.3), since ψ belongs to P (, χ 0 ). 5. Convergence of the flow. In this section we complete the proof of the main theorem. We assume, using the result of [C], that a solution φ = φ t for the J-flow exists for all time. From Theorem 3.1 and Theorem 4.1 we have uniform estimates on φ and the derivatives i j φ, using the fact that χ ij = χ 0 ij + i j φ>0. Since the operator 1 n (1 Λ χω), is concave in the χ ij, it is well known that, by the work of Evans [E1, E] and Krylov [Kr] (see also [Tr]), one can deduce a uniform Hölder estimate on the second derivatives i j φ. By differentiating the equation (.1) and applying standard Schauder estimates for parabolic equations (see [LSU] for example), one can obtain uniform estimates on all of the derivatives of φ. It then follows that there is a sequence of times t j such that φ tj converges in C to some smooth function φ. In order to show that we have convergence without having to pass to a subsequence, we will use a modification of the argument in [Ca]. Notice that φ/ satisfies the heat equation ( ) φ = ( ) φ. Since we have uniform bounds for χ ij from above and away from zero, and bounds on χ ij and all the covariant derivatives of χ ij and χ ij, it follows from the Harnack inequality of Li and Yau [LY] and the argument in [Ca]

15 Convergence of the J-flow on Kähler Surfaces 963 that there exist positive constants C 0 and η, which are independent of t, such that ( ) ( ) φ φ sup inf C 0 e ηt. Since φ χ =0, φ/ must take on the value zero somewhere on for each t, andso φ C 0e ηt. Hence for any 0 <s<s,andanyx, s φ(x, s ) φ(x, s) = s s s φ (x, t)dt φ (x, t) dt s C 0 e ηt dt s = C 0 1 η (e ηs e ηs ), which tends to zero as s and s tend to infinity. Hence φ t converges in the C 0 norm to φ. It must converge also in the C topology, since otherwise there would exist an integer N, anɛ>0 and a sequence t j with φ tj φ C N ɛ. Since φ is bounded in all the C k norms, one could pass to a subsequence of the φ tj which would converge to some φ φ, giving the contradiction. This completes the proof. Acknowledgements. This work was completed while the author was a graduate student at Columbia University, and these results form part of his PhD thesis [We]. The author is very grateful to his advisor D.H. Phong for his constant support and advice. He also thanks Jacob Sturm and Jian Song for some helpful conversations, and the referee for some constructive comments.

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