The interior regularity of the Calabi flow on a toric surface
aa r X i v : . [ m a t h . DG ] F e b THE INTERIOR REGULARITY OF THE CALABI FLOWON A TORIC SURFACE
XIUXIONG CHEN, HONGNIAN HUANG AND LI SHENG
Abstract.
Let X be a toric surface with Delzant polygon P and u ( t )be a solution of the Calabi flow equation on P . Suppose the Calabi flowexists in [0 , T ). By studying local estimates of the Riemann curvatureand the geodesic distance under the Calabi flow, we prove a uniforminterior estimate of u ( t ) for t < T . Contents
1. Introduction 12. Notations and Setup 53. Distance Control 64. Curvature Control 134.1. Evolution equation of the Calabi flow with a cut-off function 175. Main Theorem 28References 281.
Introduction
According to Calabi [3, 4], a K¨ahler metric is called extremal if the gra-dient vector field of its scalar curvature is a holomorphic vector field. Whenthis vector field vanishes, it is a constant scalar curvature K¨ahler (cscK)metric. One of the core problems in K¨ahler geometry is to establish theexistence of cscK metrics with appropriate algebraic conditions (c.f. Yau-Tian-Donaldson conjecture). It is now known that the existence of cscKmetrics implies K -stability on a polarized K¨ahler manifold X [30, 26, 18, 11].But the existence problem for general cscK metrics in a non-canonical classis largely open.In 1982, Calabi proposed to deform any K¨ahler potential in the directionof its scalar curvature:(1) ∂ϕ∂t = R ϕ − R, where R is the average of R ϕ . This is a parabolic flow of 4th order andone can establish short time existence for smooth initial data [7]. The firstnamed author conjectured that the Calabi flow exists for all time (c.f. [9]). Date : 11/25/2012.The second named author is financially supported by the FMJH (Fondationmath´ematique Jacques Hadamard).
In a joint work with W. He, the first named author proved that the Calabiflow can be extended indefinitely if the
Ric curvature is bounded [7]. Morerecently, joint with R. Feng, the second named author proved the globalexistence of the Calabi flow in a special K¨ahler manifold X = C / ( Z + i Z )[19]. Since the work of [7], there are extensive work in the subject of theCalabi flow, c.f., [36, 20, 21, 22, 25, 32, 23, 33] etc.In a remarkable series of work [14, 15, 16, 17], Donaldson proved the ex-istence of constant scalar curvature K¨ahler metric in a polarized K -stabletoric surface. In [37], X. Zhu and B. Zhou studied the weak solution of theAbreu’s equation. In [6], B. Chen, A.-M. Li and the third named authorextended Donaldson’s work to the case of extremal K¨ahler metric in toricsurface.Inspired by Donaldson’s program on toric surfaces, the first and secondnamed author studied the long time existence of the Calabi flow [10] in toricK¨ahler surface. In our current paper, let X be a toric surface with Delzantpolygon P and u be a sympletic potential in P. By Abreu’s work, we canreduce the scalar curvature equation to(2) A u = − U ij (cid:18) D u ) (cid:19) ij where U ij is the cofactor matrix of u ij . Following the tradition, we denotethe average of scalar curvature R as A in toric settings. Suppose that u ( t )is a one parameter family of symplectic potentials satisfying the Calabi flowequation. Then,(3) ∂u∂t = A + U ij (cid:18) D u ) (cid:19) ij . Suppose the Calabi flow exists in the time interval [0 , T ). We then havethe following theorem.
Theorem 1.1.
For any constant ǫ > , let P ǫ be the subset containing allthe points in P whose Euclidean distance to ∂P is greater than ǫ . Thenthere exist constant C ( ǫ ) and C ( k, ǫ ) for all k ∈ Z + such that for any t < T and any point x ∈ P ǫ , we have ( D u )( t, x ) > C ( ǫ ) , | u | C k < C ( k, ǫ ) . This is a parabolic version of interior estimates of Donaldson [15] wherehe proved the same estimates for constant scalar curvature K¨ahler toricmetric. In Donaldson’s proof, it is crucial that the scalar curvature is uni-form bounded. Viewing equation (2) as a second order elliptic equationon D u ) , Donaldson controlled the lower bound of det( D u ) in P ǫ bymaximum principle via some cleverly constructed barrier functions. He ob-tained the upper bound by adopting the techniques of Trudinger and Wang[34, 35] which also relies on the construction of the barrier functions and themaximum principle. Then appealing to the real linearized Monge-Amp`eretheory [1, 2] and Schauder’s estimates, he obtained( D u )( x ) > C ( ǫ ) , | u | C k < C ( k, ǫ ) . HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 3
In our case, however, we only have L control of A t along the Calabi flow. Apointwise bound on the det( D u ) seems to be beyond reach due to the lackof maximum principle. Thus, we are not able to apply Donaldson’s methodin our paper. A crucial step is to prove the following intermediate theorem. Theorem 1.2.
For every small constant ǫ > , there exists C ( ǫ ) > suchthat for any t < T and x ∈ P ǫ , we have Q ( t, x ) d u ( t ) ( x, ∂P ǫ ) < C ( ǫ ) , where Q ( t, x ) = ( | Rm | + |∇ Rm | / + |∇ Rm | / )( t, x ) . One immediate remark is that, even if this theorem holds, we do not havecontrol on curvature in the interior, unless we can prove that the geometricdistance from interior to the boundary is non-trivial. As a first step in ourpaper, we prove the following:
Theorem 1.3.
For any small ǫ > , there exists a constant C ( ǫ ) suchthat for any t < T , we have d u ( t ) ( P ǫ , P ǫ ) > C ( ǫ ) . In other words, the interior of the polygon contains some geometric ball offixed size during the flow at finite times.
Now, we discuss the main ideas in this paper.We use a blow-up analysis method adopted from [8] [9]. However, thereare substantial new difficulties arising in our settings:(1) Unlike [8], we do not have a uniform Sobolev constant bound here.This causes troubles in both ends: to obtain convergence of the re-scaling sequence, to classify the “bubble” in the limit as well as tocontrol the higher derivatives (so that we can “bootstrap” conver-gence to draw contradictions). The lack of injectivity radius controlcan be overcomed, if we know that the scalar curvature is a constant.Then, we can study equation (2) in the tangential space and obtainthe higher derivatives of the curvature there. This method is notavailable to us since we only have integral estimates of of curvaturefunctions. To overcome the difficulty of collapsing, we needs to uni-formly control |∇ Rm | after blowing up as in [19].(2) In both [8] and [9], they blow up at the global maximum of the curva-ture function | Rm | at each time slice. Therefore, one has curvaturecontrol globally (over both forward time and space) after blowing up.For Ricci flow, W. Shi [29] can obtain higher derivatives control viamaximum principle. With similar ideas in mind, but using integralestimate, W. He and the first named author [8] control Z X |∇ k Rm | dg instead. Here curvature uniform bound is crucial in the long calcu-lations. Then the Sobolev constant bound enables them to obtain XIUXIONG CHEN, HONGNIAN HUANG AND LI SHENG higher regularities control of Rm , i.e., |∇ k Rm | .There are two immediate difficulties in estimating the evolutionof the above integral in our setting. First, the geometrical quantity Q ( t, x ) is not known to be bounded in any time slice t < Z X f |∇ k Rm | dg. Upon controlling this quantity, by using M-condition and uniformcurvature bound, we obtain that ( c.f. [16])( D u )( x ) < C, where x is around the blowup point. The higher regularities followfrom here. We remark that this is one of the main reasons why wehave to restrict ourselves to an interior estimate at this stage. Herewe adopt similar arguments in the appendix of [19] to give us thehigher regularities control of Rm .(3) To deal with the cut off function f , some of the technical difficultiescome from integration by parts, for example, R X f dg is not bounded.The more substantial difficulty is that we need to choose some par-abolic box in the polygon in the sense of geodesic distance. Then,the issue of how the geodesic distance changes over time is crucial.In fact, one of the main challenges in a geometric flow (Ricci flow,Calabi flow) is to compare the distance function in different timeslices, for example t = 0 and t = −
1. Note that the evolution ofmetric is controlled by the Hessian of the scalar curvature which, atthe present stage, is precisely what we hope to control.A key step is that we divide the geodesic segment which realizesthe distance at time t = − | Rm | is large, the set where the concentration is controlledand the oscillation of metric, comparing to t = 0, is large and the setwhere the oscillation is bounded and the concentration is bounded . For the first set, it is easy to show that its Lebesgue measure iscontrolled. For the second set, we prove that its Lebesgue measureis also controlled because the evolution of metric is controlled by theHessian of the scalar curvature, hence by the difference of the Calabienergy at t = − t = 0 which is bounded. Therefore, we areable to prove that, if the distance between x and ∂P ǫ is L at t = 0, Roughly speaking, a point contains a nontrivial concentration energy if there is certainline segment (in Euclidean sense, passing through this point) where the integration of | Rm | is nontrivial over this line segment. HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 5 then the distance between x and ∂P ǫ is at least cL at t = − c .Finally, to prove Theorem (1.1) using Theorem (1.2), one needs to showthat the geodesic distance and Euclidean distance are somehow equivalentin P ǫ . This is proved in Theorem (1.3). Notice that the length of a curve is Z s p D ( u )( γ ′ ( s ) , γ ′ ( s )) ds. (4)The key observation is that (4) is bounded from below because Z P T race ( u ij ( t )) < C uniformly.With all the results obtained so far, one can control Rm and its covariantderivatives in P ǫ . Then one can get Theorem (1.1) using Krylov-Safonovand Schauder’s estimates. Remark 1.4.
After the celebrated work of G. Perelman [28] , it has nowbecome a powerful tool in the Ricci flow to select a local maximum of cer-tain geometry quantity (like curvature) and to apply careful analysis in localparabolic box (c.f. [12, 13] ). While we clearly draw inspirations from Perel-man’s work, it seems still a novelty to localize estimates in the Calabi flow.There are obvious, substantial new difficulties arising because of its higherorder. Some of these new difficulties are fundamentals and require thoroughnew investigation and invention. However, the first named author suspectsthat it will be a constant theme in the study of geometry flow to deal withthe oscillations of the distance functions over time parameter.
Acknowledgment:
The second and third named author would liketo express their gratitude to Professor Paul Gauduchon and Frank Pacardfor their support. The second named author would like to thank ProfessorPengfei Guan and Vestislav Apostolov for stimulating discussions. The thirdnamed author would like to thank Professor Anmin Li for his support.2.
Notations and Setup
Let X be a K¨ahler manifold with a complex structure J and a K¨ahlerclass [ ω ]. There is a one to one correspondence between the sets of allK¨ahler metrics and a set of relative K¨ahler potentials H , where H = { ϕ ∈ C ∞ ( M ) | ω + i∂ ¯ ∂ϕ > } / R . The Calabi flow equation is ∂ϕ∂t = R ϕ − R, XIUXIONG CHEN, HONGNIAN HUANG AND LI SHENG where R ϕ is the scalar curvature and R is its average. The Calabi flowdecreases the Calabi energy which is Z X ( R ϕ − R ) dω nϕ . The evolution equation of the bisectional curvature of the Calabi flow is ∂Rm∂t = −△ Rm + ∇ Rm ∗ Rm + ∇ Rm ∗ ∇ Rm.
Suppose X is a toric manifold with a Delzant polytope P . Then the toricinvariant K¨ahler metric is one to one corresponding to the set of symplecticpotentials u satisfying the Guillemin boundary conditions up to an affinefunction. The Calabi flow equation in the sympletic side is ∂u∂t = A − A u , where A u = − P i,j u ijij by Abreu’s work and A is its average. The distancebetween any two symplectic potentials u and u is sZ P ( u − u ) dµ. In the calculations of later sections, we also adopt the following notations: • | · | E : Euclidean metric. • | · | u or | · | : Riemannian metric. • D : Euclidean derivative. • ∇ : Covariant derivative.At the end of this section, we would like to introduce the M -conditionfrom Donaldson’s paper [16]. Let l be a line interval of P parameterizingby p + sν, s ∈ [ − R, R ], where ν is a unit vector. We say u satisfying the M -condition on l if | ( Du ( p − Rν ) − Du ( p + Rν )) · ν | E < M. We say u satisfying the M -condition on P if for any line interval l ⊂ P , u satisfies the M -condition on l .3. Distance Control
Our goal in this section is to prove Theorem (1.3). Before going into thedetails of the proof, we would like to explain the ideas of the proof to ourreaders. Our first observation is Corollary (3.2), i.e., Z P T race ( u ij ( t )) < C, where C is a positive constant independent of t . This is a consequence ofthe facts that the Calabi flow decreases the Calabi energy and the geodesicdistance in the space of relative K¨ahler potentials. Our second observationis Proposition (3.6) where we show that if T race ( u ij ) is large at one point x , then we can show that along any interval l with fixed diameter passingthrough x in P ǫ , R l T race ( u ij ) is also large, provided that the L norm of HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 7 Rm on l is bounded. With these two observations and the fact that theRiemann length of a curve is Z s q u ij ( γ ′ ( s ) , γ ′ ( s )) ds, we can prove Theorem (1.3).Now we proceed to give some lemmas in order to prove Corollary (3.2). ByTheorem 1.5 of Calabi and Chen [5], the Calabi flow decreases the distance,we obtain that Z P ( u (0) − u ( t/ dµ ≥ Z P ( u ( t/ − u ( t )) dµ. It implies that there exists a constant
C > t < T , wehave Z P u ( t ) dµ < C. Since the Calabi energy decreases under the Calabi flow, we have for any t < T , Z P A ( t ) dµ < C. Thus we obtain that for any t < T , (cid:12)(cid:12)(cid:12)(cid:12)Z P u ( t ) A ( t ) dµ (cid:12)(cid:12)(cid:12)(cid:12) < C. Also (cid:12)(cid:12)(cid:12)(cid:12)Z P u (0) A ( t ) dµ (cid:12)(cid:12)(cid:12)(cid:12) < C. Then we have (cid:12)(cid:12)(cid:12)(cid:12)Z P ( u ( t ) − u (0)) A ( t ) dµ (cid:12)(cid:12)(cid:12)(cid:12) < C. Integration by parts as Lemma 3.3.5 in [14], we have (cid:12)(cid:12)(cid:12)(cid:12)Z P ( u ( t ) − u (0)) ij u ij ( t ) dµ (cid:12)(cid:12)(cid:12)(cid:12) < C + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z ∂P ( u ( t ) − u (0)) dσ (cid:12)(cid:12)(cid:12)(cid:12) . Thus we have (cid:12)(cid:12)(cid:12)(cid:12)Z P u ij (0) u ij ( t ) dµ (cid:12)(cid:12)(cid:12)(cid:12) < C + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z ∂P ( u ( t ) − u (0)) dσ (cid:12)(cid:12)(cid:12)(cid:12) . Similar calculations show (cid:12)(cid:12)(cid:12)(cid:12)Z P u ij ( t ) u ij (0) dµ (cid:12)(cid:12)(cid:12)(cid:12) < C + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z ∂P ( u ( t ) − u (0)) dσ (cid:12)(cid:12)(cid:12)(cid:12) . We also observe the following lemma.
Lemma 3.1.
There exists a constant C such that for any t < T , we have (cid:12)(cid:12)(cid:12)(cid:12)Z ∂P ( u ( t ) − u (0)) dσ (cid:12)(cid:12)(cid:12)(cid:12) < C. XIUXIONG CHEN, HONGNIAN HUANG AND LI SHENG
Proof.
Notice that for any t < T , by the calculations of section 5 in [22], wehave Z t Z P u ij ( s ) A ik ( s ) A jl ( s ) u kl ( s ) dµds = Z P A (0) − A ( t ) dµ. Thus we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z t Z P u ij ( s ) A ij ( s ) dµdt (cid:12)(cid:12)(cid:12)(cid:12) < C. Integration by parts, we get (cid:12)(cid:12)(cid:12)(cid:12)Z t Z ∂P A ( s ) dσdt (cid:12)(cid:12)(cid:12)(cid:12) < C. Since Z ∂P ( u ( t ) − u (0)) dσ = Z t Z ∂P A − A ( s ) dσdt, we obtain the result. (cid:3) As a corollary, we have
Corollary 3.2.
There exists a constant C such that for any t < T , we have (cid:12)(cid:12)(cid:12)(cid:12)Z P u ij ( t ) u ij (0) dµ (cid:12)(cid:12)(cid:12)(cid:12) < C, (cid:12)(cid:12)(cid:12)(cid:12)Z P u ij (0) u ij ( t ) dµ (cid:12)(cid:12)(cid:12)(cid:12) < C. To continue, we prove the following lemma first which is an extension ofLemma 4 in [16].
Lemma 3.3.
Suppose l is a line interval inside P such that R l | Rm | isbounded by a constant C . We parametrize l = { p + sν : − R ≤ s ≤ R } , where p ∈ l is the middle point of l and ν is a unit vector. Suppose u satisfiesthe M -condition on l , then • R > , we have u ij (0) ν i ν j ≤ e M − C . • R ≤ , we have u ij (0) ν i ν j ≤ e M/ − CR .
Proof.
We can suppose that ν is the unit vector in the x direction and p is the origin point. Let H ( s ) = u ( s, . We apply the definition of the M -condition to obtain Z R − R H ( s ) ds ≤ M. Since d ds H ( s ) − ≤ | Rm | ( s ) , for any s >
0, we obtain (cid:18) H (cid:19) ′ ( s ) − (cid:18) H (cid:19) ′ (0) ≤ Z s | Rm | ( s ) ds ≤ C √ s . HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 9
Suppose H (0) − = ǫ , (cid:0) H (cid:1) ′ (0) = ǫ . Then H ( s ) − − H (0) − = Z s (cid:18) H (cid:19) ′ ( s ) ds ≤ Z s C √ s + ǫ ds = ǫ s + C ( s ) / . Thus H ( s ) + H ( − s ) ≥ ǫ + ǫ s + Cs / + 1 ǫ − ǫ s + Cs / ≥ ǫ + Cs / . This gives M ≥ Z R − R H ( s ) ds ≥ Z R dsǫ + Cs / If R >
1, then we have M ≥ Z dsǫ + Cs + C = C + 2 ln( ǫ + C ) − ǫ , which implies ǫ ≥ e C − M . If R ≤
1, then we have M ≥ Z R dsǫ + Cs = 2 ln CR + ǫ ǫ , which implies ǫ ≥ CRe M/ − . Thus we obtain the results. (cid:3)
Now let x be the axis along l and z be another axis. We have the followingobservation due to Donaldson [16]: Lemma 3.4. | u zzxx | ( s ) ≤ | Rm | ( s ) u zz ( s ) u xx ( s ) and | u xxxx | ( s ) ≤ | Rm | ( s ) u xx ( s ) u xx ( s ) . Proof.
Let s be the origin point. Set v ( x, z ) = u ( x + az, z ) , where a is some constant to be determined. Then( D v )( x, z ) = (cid:18) a (cid:19) ( D u )( x + az, z ) (cid:18) a (cid:19) . It shows that( D u ) − ( x + az, z ) = (cid:18) a (cid:19) ( D v ) − ( x, z ) (cid:18) a (cid:19) . It means that u xx ( x + az, z ) = v xx ( x, z ) and u zz ( x + az, x ) = v zz ( x, z ). Thuswe have u zzxx ( x + az, z ) = v zzxx ( x, z ) . Since v xz ( x, z ) = au xx ( x + az, z ) + u xz ( x + az, z ), we can choose an appro-priate constant a such that v xz (0 ,
0) = 0. Thus calculations in Lemma 4.3of [22] show that | v zzxx (0 , v zz (0 , v xx (0 , |≤ vuut X i,j,k,l ( v ijkl ) (0 , v kk (0 , v ll (0 , v ii (0 , v jj (0 , s X i,j,k,l v ijkl (0 , v klij (0 , | Rm | (0 , . Thus | u zzxx (0 , | = | v zzxx (0 , | ≤ u zz (0 , u xx (0 , | Rm | (0 , . Notice that u xx ( x + az, z ) = v xx ( x, z ) + 2 av xz ( x, z ) + a v zz ( x, z ) . Thus u xxxx ( x + az, z ) = v xxxx ( x, z ) + 2 av xzxx ( x, z ) + a v zzxx ( x, z ) . Notice that | v xxxx | (0 , ≤ | Rm | (0 , | v xzxx | (0 , ≤ p v zz (0 , v xx (0 , | Rm | (0 , . Thus we obtain | u xxxx | (0 , ≤ | Rm | (0 , a p v zz (0 , v xx (0 ,
0) + a v zz (0 , v xx (0 , ≤ | Rm | (0 , a v zz (0 , v xx (0 , ≤ | Rm | (0 , v xx (0 ,
0) + a v zz (0 , v xx (0 , ≤ | Rm | (0 , u xx (0 , u xx (0 , . (cid:3) Lemma 3.5.
Let l be a line interval in P parameterizing by l = p + sν, s ∈ [ − R, R ] and ν is a unit vector parallel to the x -axis. Suppose R R − R | Rm | ( s ) ds < C and u satisfies the M -condition on l . Then thereexists s > depending on M, C such that one of the following case occurs: • For any s ∈ [0 , s ] , u zz ( s ) ≥ u zz (0) / . • For any s ∈ [ − s , , u zz ( s ) ≥ u zz (0) / . HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 11
Also we have either for any s ∈ [0 , s ] , u xx ( s ) ≥ u xx (0) / . or for any s ∈ [ − s , , u xx ( s ) ≥ u xx (0) / . Proof.
By Lemma (3.3), we know that u xx ( s ) , s ∈ [ − R, R ] is bounded by aconstant C . Thus we have for s ∈ [ − R, R ], − C | Rm | ( s ) u zz ( s ) ≤ u zzxx ( s ) ≤ C | Rm | ( s ) u zz ( s ) . Let f ( s ) = ln u zz ( s ). We have f ′′ ( s ) + f ′ ( s ) = u zzxx u zz ( s ) . Thus for s ∈ [ − R, R ], we obtain | f ′′ + f ′ | ( s ) ≤ C | Rm | ( s ) . Dividing both sides by 1 + f ′ , we obtain (cid:12)(cid:12)(cid:12)(cid:12) f ′′ f ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | Rm | ( s ) . Integrating both sides, we have | arctan( f ′ ( s )) − arctan( f ′ (0)) | ≤ C p | s | Without loss of generality, we can assume f ′ (0) ≥
0. Then for s ∈ [0 , R ], wehave arctan( f ′ ( s )) ≥ − C √ s which is f ′ ( s ) ≥ tan( − C √ s ) . Thus by choose s appropriately, we conclude that for any s ∈ [0 , s ], f ( s ) ≥ f (0) − Cs / and u zz ( s ) ≥ u zz (0) / . Similar calculations show u xx ( s ) ≥ u xx (0) / s ∈ [0 , s ]. (cid:3) Our discussions lead to the following proposition.
Proposition 3.6.
Suppose there exists a unit vector ν such that ( D u )(0)( ν, ν ) < ǫ, then Z R − R trace ( u ij ( s )) ds ≥ Cǫ , where C only depends on M, R and R R − R | Rm | ( s ) ds . Proof.
Since ( D u )(0)( ν, ν ) < ǫ , we conclude that the smallest eigenvalueof ( D u )(0) must be less than ǫ . Thus T r (( D u ) − )(0) > /ǫ . Withoutloss of generality, we can assume that u zz (0) > /ǫ . Then our previousdiscussions show that there exists s > s ∈ [0 , s ] (or forall s ∈ [ − s , u zz ( s ) ≥ u zz (0)2 . Then Z R − R trace ( u ij ( s )) ds ≥ Z R u zz ( s ) ds ≥ Z min { s ,R } ǫ ds = min { s , R } ǫ . Thus our conclusion holds. (cid:3)
Now we give a proof of Theorem (1.3).
Proof of Theorem (1.3).
Recall that for any t < T we have Z P ( u ( t )) dµ < C. Then it is easy to see that there exists a constant
M > t < T, x ∈ P ǫ / , we have | u ( t, x ) | < M, | Du ( t, x ) | < M. It shows that for any t < T and any line interval l ⊂ P ǫ / , u ( t ) satisfies the M -condition on l .For any t < T , let c be a geodesic realizing the minimum distance between ∂P ǫ and ∂P ǫ . Denote ǫ to be the geodesic length of c . We want to showthat ǫ cannot be too small.The first case is that c is far away from the vertex. To simply our nota-tions, we can assume the boundary of P, P ǫ , ∂P ǫ are x = 0 , x = ǫ , x = 2 ǫ respectively. We will also suppress t in the following calculations. We let x, y be two axises. Let C be a constant such that Z P | Rm | dx < C . Also let G = { x ∈ [ ǫ , ǫ ] | Z x = x | Rm | < C ǫ } . Then the Lebesgue measure of G is greater than ǫ /
2. So G represents the“good lines”. Let the curve c be parametrized by its Euclidean length: c = { ( x ( s ) , y ( s )) | s ∈ [0 , s ] } HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 13
It is easy to see that s ≥ ǫ . Since Z s p ( D u )( c ′ ( s ) , c ′ ( s )) ds = ǫ, we conclude that the Lebesgue measure of the following set B = { x ∈ G | ∃ s ∈ [0 , s ] , x ( c ( s )) = x , ( D u )( c ′ ( s ) , c ′ ( s )) ≤ ǫ ǫ } is greater than ǫ /
4. In fact, B are the bad points in G ∩ c .On one hand, after choosing R appropriately, Proposition (3.6) tells usthat Z P ǫ T r ( u ij ( x )) dµ ≥ Z x ∈B Z x = x T r ( u ij ) dµ ≥ ǫ Cǫ ǫ = Cǫ , where C depending only on ǫ , M, R and R P | Rm | dµ .On the other hand, from Corollary (3.2), we know that there exists aconstant C > t < T , we have Z P ǫ T r ( u ij ( t, x )) dµ < C . Thus we conclude that ǫ is bounded from below uniformly.The remaining case we should consider is that the curve c is close to avertex. Then without loss of generality, we can let the boundary of P to be x = 0 & y = 0 and the boundary of P ǫ to be x = ǫ & y = ǫ . Withoutloss of generality, we can assume that half of the curve c , in the sense ofEuclidean distance, will touch the set { ( x, y ) ∈ P ǫ \ P ǫ | x ∈ [ ǫ, ǫ ] } . Then our previous arguments provide the lower bound of ǫ . (cid:3) Curvature Control
In this section, we will use the blow-up analysis to prove Theorem (1.2)which is the key step in the proof of Theorem (1.1).Suppose that the conclusion is not true, then we can find a sequence ofpoints ( x i , t i ) , x i ∈ P ǫ , t i < T such that Q ( t i , x i ) d u ( t i ) ( x i , ∂P ǫ ) = max t ≤ t i , x ∈ P ǫ Q ( t, x ) d u ( t ) ( x, ∂P ǫ )and lim i →∞ Q ( t i , x i ) d g ( t i ) ( x i , ∂P ǫ ) = ∞ . Let λ i = Q ( t i , x i ), we define a sequence of the Calabi flow u i ( t ) , t ≤ u i ( t, x ) = λ i u (cid:18) t + t i λ i , x + x i λ i (cid:19) . Let Q i ( t, x ) = ( | Rm | + |∇ Rm | / + |∇ Rm | / ) u i ( t, x ) , then Q i (0 ,
0) = 1. We denote P ( i ) = λ i P, P ( i ) ǫ to be λ i P ǫ . The followinglemma shows that we can pick a backward parabolic neighborhood in thesymplectic side for sufficiently large i . Note that the Calabi energy and the M -condition are scaling invariant. Proposition 4.1.
For sufficiently large i and for any t ∈ [ − , , we have d u i ( t ) (0 , ∂P ( i ) ǫ ) ≥ cd u i (0) (0 , ∂P ( i ) ǫ ) for some uniform constant c .Proof. Without loss of generality, we can just prove for the case t = −
1. Wesuppress i in the following calculations. Notice that Z P A ( − dµ − Z P A (0) dµ = Z − Z P u ij ( t ) A ik ( t ) A jl ( t ) u kl ( t ) dµdt. Notice that we can assume Z P A ( − dµ − Z P A (0) dµ < . Thus Z − Z P u ij ( t ) A ik ( t ) A jl ( t ) u kl ( t ) dµdt < . For any unit vector ν , we write u and A as the second derivative of u and A in the direction of ν . We have | log u (0 , x ) − log u ( − , x ) | = | Z − A u dt |≤ sZ − A u dt ≤ sZ − u ij ( t ) A ik ( t ) A jl ( t ) u kl ( t ) dt. In the last inequality, we use the calculation method in Lemma (3.4). Weconclude that the Lebesgue measure of the set { x ∈ P | ∃ unit vector ν, s . t . | log(u ij ν i ν j )(0 , x) − log(u ij ν i ν j )( − , x) | > } is less than 1.Let L be the geodesic distance between (0 ,
0) and ∂P ǫ at t = 0. Suppose γ is the curve realizing the minimum distance between (0 ,
0) and ∂P ǫ at t = −
1. Then the geodesic length of γ at t = 0 is greater than L . Since u ( t )satisfies the M -condition, we conclude that the Euclidean distance betweenthe end points of γ is greater than ( √ − L M . HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 15
The first case is that γ is a straight line. We can assume that γ lies inthe x -axis. Notice that R P | Rm | dµ < C . Then the Lebesgue measure ofthe following set B = { x ∈ γ | Z x = x | Rm | dy > C } is less than 1.Notice that for any x ∈ P whose geodesic distance to (0 ,
0) at t = 0 isless than L/
2, we have Q (0 , x ) ≤
4. Let u be the second derivative of u along the x -axis. Since u ( t ) satisfies the M -condition, we conclude that u (0 , x ) < C . Let ( x ,
0) be the point in γ such that the geodesic distancebetween (0 ,
0) and ( x ,
0) is L/ t = 0. Thus we conclude that Z ( x, ∈B , x ≤ x p u ( x ) dx < C. Since L ≫
1, after removing B from γ , the geodesic length of γ is still greaterthan L/ t = 0. Our goal is to prove that the geodesic length of γ \B isgreater than cL at t = − c . Thus without lossof generality, we can assume that B is an empty set.Now set˜ B = γ \{ x ∈ [0 , x ] | ( D u )(0 , ( x, ≤ D u )( − , ( x, } = { x ∈ [0 , x ] | ∃ ν s.t. ( u ij )(0 , ( x, ν, ν ) < u ij )( − , ( x, ν, ν ) } . We want to show that the Lebesgue measure of ˜ B is well controlled. Oncewe know this, we can argue that Z x ∈ [0 ,x ] , ( x, ∈ ˜ B √ u (0 , ( x, dx < C. Thus Z x ∈ [0 ,x ] , ( x, / ∈ ˜ B √ u (0 , ( x, dx > L/ . Hence Z x ∈ [0 ,x ] , ( x, / ∈ ˜ B √ u ( − , ( x, dx > L/ . Next we prove that the Lebesgue measure of ˜ B is well controlled. Noticethat for every point x ∈ ˜ B , we obtain that there exists a unit vector ν suchthat ( u ij ) t =0 ( x, ν, ν ) < u ij ) t = − ( x, ν, ν ) . Applying Lemma (3.5), we conclude that there exists a uniform constant y such that for each point y ∈ [0 , y ]( u ij ) t = − ( x, y )( ν, ν ) ≥
12 ( u ij ) t = − ( x, ν, ν ) ≥ u ij ) t =0 ( x, ν, ν ) . Since u ( t ) satisfies the M -condition, the geodesic distance is controlled bythe Euclidean distance. Applying Lemma 7 of [16], we can choose y appro-priately such that for any y ∈ [0 , y ]2( u ij ) t =0 ( x, ν, ν ) ≥ ( u ij ) t =0 ( x, y )( ν, ν ) . Thus ( u ij ) t = − ( x, y )( ν, ν ) ≥ u ij ) t =0 ( x, y )( ν, ν ) . Then there is a unit vector ˜ ν such thatln( u ij ) t = − ( x, y )(˜ ν, ˜ ν ) ≤ ln( u ij ) t =0 ( x, y )(˜ ν, ˜ ν ) − ln 25 . It shows that | ln( u ij ) t = − ( x, y )(˜ ν, ˜ ν ) − ln( u ij ) t =0 ( x, y )(˜ ν, ˜ ν ) | ≥ ln 25 . Thus the Lebesgue measure of ˜
B × [0 , y ] is well controlled.The remaining case is that γ is not a straight line. Let x -axis and y -axisbe perpendicular to each other. The following calculations will be done at t = 0. Let p be the first point in γ such that the geodesic distance between(0 ,
0) and p is L/
2. We replace γ by the points in γ connecting (0 ,
0) and p . Let B x , B y be any union of disjoint open intervals in the x -axis and y -axis respectively. We assume that the Lebesgue measure of B x and B y areless than ǫ which is a positive constant to be determined later. Notice thatthe Riemann length of γ , i.e., ¯ L , is greater than L . We want to show thatone of the following is true: • For any B x , after removing the points of γ whose x -coordinate liesin B x , the length of γ is larger than c L . • For any B y , after removing the points of γ whose y -coordinate liesin B y , the length of γ is larger than c L .The above constant c is to be determined. Suppose not, then we concludethat the sum of length of intervals in γ whose x -coordinate in B x and y -coordinate in B y is greater than ¯ L − c L . We want to show that this wouldlead to a contraction for some ǫ and c . Notice that ( D u ) < C I for someuniform constant C . We have the following observation. Claim 4.2.
Let γ ⊂ γ be a curve such that the geodesic distance betweenthe ends point of γ is √ C . Let | γ | u be the geodesic length of γ . Thenthere exists C depending on M, ǫ such that | γ \B x × B y | u ≥ C p C . Let us assume that our claim holds. Then we can pick a successive se-quence of points p i ∈ γ, i = 1 , . . . , N such that for any i = 1 , . . . , N − p i and p i +1 is 20 √ C . Moreover, the geodesicdistance between p and p N is greater than L/
3. Let γ N be the curve in γ connecting p and p N . Then our previous claim shows that | γ N \B x × B y | u ≥ C L/ . Notice that | γ \B x × B y | u < c L . We derive a contradiction by setting6 c < C .Now we turn to give a proof of our previous claim. Without loss ofgenerality, we assume that the two end points of γ are (0 ,
0) and p . Wecan further assume that for any point p ∈ γ , the geodesic distance between(0 ,
0) and p is less than or equal to 20 √ C .The simple case is that h , i and h , i are the eigenvectors of ( D u )(0 , B u ((0 , , √ C ) is almost an ellipsoid. HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 17
Moreover, for any point x ∈ B u ((0 , , √ C ), we have a uniform constant C such that 1 C ( D u )(0 , ≤ ( D u )( x ) ≤ C ( D u )(0 , . Let ( x ,
0) be at the boundary of B u ((0 , , √ C ). Notice that x ≥ x -coordinate of p isgreater than C x for some uniform constant C . We have the followinginequalities: | γ | u ≥ √ C √ u (0 , C x − ǫ ) ≥ C p C . The more complicated case is that we need to rotate the coordinate systemsto get x ′ -axis and y ′ -axis so that the h , i and h , i axis are eigenvector of( D u )(0 , B ′ x ′ and B ′ y ′ with Lebesguemeasure less than 2 ǫ such that B x × B y ⊂ B ′ x ′ × B ′ y ′ . Then our earlier arguments apply. So without loss of generality, we havethat for any B x with Lebesgue measure less than ǫ , after removing the set ofpoints in γ whose x -coordinate is in B x , the length of γ is still greater than c L . Then we can apply the arguments in the case where γ is a straight lineto obtain the conclusion as follows: • By choosing C > x -axis is less than ǫ/ B = { x | ∃ y s.t. ( x , y ) ∈ γ & Z x = x | Rm | ( x , y ) dy ≥ C } . • Let γ be the sets of points in γ whose geodesic distance to (0 , t = 0 is less than L/
2. By choosing C > x -axis is less than ǫ/ B = { x / ∈ B | ∃ y s.t. ( x , y ) ∈ γ and ∃ ν s.t. ( u ij ) t =0 ( x , y )( ν, ν ) < C ( u ij ) t = − ( x , y )( ν, ν ) } . Then after removing ( B ∪ B ) × y from γ , we know that the length of γ at t = 0 is greater than c L for some uniform constant c >
0, then the lengthof γ at t = − cL for some uniform constant c . (cid:3) Evolution equation of the Calabi flow with a cut-off function.
By the previous discussions, we can conclude that • For any point ( t, p ) ∈ [ − , × P ǫ , Q ( t, p ) × d t ( p, ∂P ǫ ) ≤ L. • For any t ∈ [ − , , d t ((0 , , ∂P ǫ ) ≥ c √ L. • d ((0 , , ∂P ǫ ) = √ L. Thus without loss of generality, we can assume that | Rm | ( t, x ) ≤ B u t ((0 , ,
2) and | Rm | (0 ,
0) = 1 at t = 0.Now let us consider a C cutoff function ψ : [0 , → [0 ,
1] such that • ψ ( t ) = 1 , ≤ t ≤ / • ψ ( t ) = 0 , t ≥ • | ψ ′ ( t ) | ≤ Cψ ( t ) a − a • | ψ ′′ ( t ) | ≤ Cψ ( t ) a − a where a ∈ Z + to be determined.Let r t ( · ) = d t ( · , (0 , P with induced metric g ( t ) = u ( t ) ij dx i dx j . It is naturally to extend r to be a function in the whole manifold such thatit is invariant under the T action. In this case, r is the distance functionto a submanifold T and we define f t = ψ ( r t ) to be a cutoff function. Wesuppress t in the following calculations.It is easy to see that away from { the cut locus of (0 , ⊂ P } × T , |∇ f | = | ψ ′ ||∇ r | ≤ Cψ ( t ) a − a and △ f = ψ ′′ |∇ r | + ψ ′ △ r. Since the curvature is bounded, we can show that |△ r | is also bounded for0 ≤ r ≤ △ r is bounded by an universal constant C away from { the cut locus of (0 , } × T . Notice that we only need topay attention to the case where 1 / ≤ r ≤ △ r , we apply the Weitzenb¨ock formula, i.e. | Hess r | + ∂∂r ( △ r ) + Ric( ∂∂r , ∂∂r ) = 0Let λ , . . . , λ be the eigenvalues of Hess r . Then | Hess r | = λ + · · · + λ n ≥ ( λ + · · · + λ ) r )) △ r ) △ r ) ∂∂r ( △ r ) + C ≤ φ = △ r . Then φ ′ Cφ ≥ , (5)where C < △ r at the point x and γ is the unique minimiz-ing geodesic connecting p = (0 , , x ∈ P . It is also easy to see that γ is theminimizing geodesic in the whole manifold. Taking x , x , . . . approaching p along γ , and rescaling the metric by r ( x i ) , we getting a sequence of manifoldconverging to standard R in Cheeger-Gromov sense. Using the geodesic HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 19 spherical coordinate in P , we can write the metric in the whole manifold(locally) as g = r ⊗ r + h ij dθ i ⊗ dθ j + g ij dη i ⊗ dη j Hence in the limiting process, r is always fixed in the coordinate system.Since the metric converges in C ∞ , we conclude that r ( x i ) △ r ( x i ) convergesto 1. Thus φ ( x i ) ∼ r ( x i ), plugging into inequality (5) and integrating itgives △ r ≤ √− C coth( √− Cr )For the lower bound of △ r , if △ r ( x ) is very negative, then φ is very closeto 0, hence 1 + Cφ > C >
0. Let γ ( s ) be a geodesic connecting γ (0) = p and γ ( s ) = x where s is the arc-length parameter of γ . Then φ ( s + s )approaches to 0 as s > φ will reach to0 for a small s >
0. Thus we obtain a contradiction if s is not close to aconjugate point in γ .Now we have the following lemma: Lemma 4.3. If | Rm | ≤ in B u ( p, , then we can construct a cutoff func-tion f such that |∇ f | ≤ Cf a − a , |△ f | ≤ Cf a − a , where C is a universal constant. Next, we are going to derive a sequence of integral inequalities which willbe useful in calculating the evolution equation of the Calabi flow with acut-off function. We will do the calculations in the complex side. Pleasekeep in mind that our cut-off function only define in the polygon and we donot have any control in the torus direction. For example, the integral of f is not necessary bounded. In Proposition (4.6), we bypass this difficulty byusing that the integral of f | Rm | is bounded. To simplify the notations, wewrite R X f dg as R f . Lemma 4.4.
For every k ∈ Z + , there is a ( k ) ∈ Z + , < b ( k ) < such thatfor all a > a ( k ) , > b > b ( k ) , Z f b |∇ k Rm | ≤ ǫ Z f |∇ k +1 Rm | + C ( k, ǫ ) Proof.
We derive the inequality by induction. For k = 0, it is obvious. For k >
0, we have Z f b |∇ k Rm | = Z ∇ f b ∗ ∇ k − Rm ∗ ∇ k Rm + Z f b ∇ k +1 Rm ∗ ∇ k − Rm ≤ ǫ Z f b |∇ k Rm | + C ( ǫ ) Z f b − a |∇ k − Rm | + ǫ Z f |∇ k +1 Rm | + C ( ǫ ) Z f b − |∇ k − Rm | . Choose b ( k ) , a ( k ) such that2 b ( k ) − > b ( k − , b ( k ) − a ( k ) > b ( k − , a ( k ) > a ( k − , then by induction, we have C ( ǫ ) Z f b − a |∇ k − Rm | ≤ ǫ Z |∇ k Rm | + C ( k, ǫ )and C ( ǫ ) Z f b − |∇ k − Rm | ≤ ǫ Z |∇ k Rm | + C ( k, ǫ ) . Thus we obtain the desired inequality. (cid:3)
Corollary 4.5.
For every k ∈ Z + and ǫ > , there are ¯ a ( k ) , C ( k, ǫ ) suchthat for all a ≥ ¯ a ( k ) Z f |∇ i Rm | < ǫ Z f |∇ k Rm | + C ( k, ǫ ) , for < i < k. (6) Proof.
We derive the inequality by induction. For k = 1, it is obvious. For k >
1, we have Z f |∇ k − Rm | = Z f ∇ k Rm ∗ ∇ k − Rm + Z ∇ f ∗ ∇ k − Rm ∗ ∇ k − Rm ≤ ǫ Z f |∇ k Rm | + C ( ǫ ) Z f |∇ k − Rm | + ǫ Z f |∇ k − Rm | + C ( ǫ ) Z f a − a |∇ k − Rm | . Hence Z f |∇ k − Rm | ≤ ǫ Z f |∇ k Rm | + C ( k, ǫ ) + C ( ǫ ) Z f a − a |∇ k − Rm | . We choose ¯ a ( k ) such that¯ a ( k ) − a ( k ) > b ( k − , ¯ a ( k ) ≥ a ( k ) . Applying Proposition (4.4), we obtain the result. (cid:3)
Proposition 4.6.
For every positive integer k ≥ and every p ≥ , thereexists constants c ( k, p ) ∈ Z + , C ( k, p ) > depending only on k, p such thatfor a > c ( k, p ) and < i < k, Z f |∇ i Rm | pki ≤ C ( k, p ) (cid:18) Z f |∇ k Rm | p + Z f |∇ k Rm | (cid:19) . HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 21
Proof.
We will derive this inequality by induction on k . Notice that | Rm | isbounded in the support of f . For k = 2, we have Z f |∇ Rm | p ≤ Z |∇ f || Rm ||∇ Rm | p − + C Z f |∇ Rm || Rm ||∇ Rm | p − ≤ Z (cid:18) f |∇ Rm | p + Cf a − pa | Rm | p (cid:19) + Z (cid:18) f |∇ Rm | p + Cf |∇ Rm | p (cid:19) . Since f a − pa | Rm | p ≤ C | Rm | , we have Z f |∇ Rm | p ≤ C (1 + Z f |∇ Rm | p ) . Now let us assume that the inequality holds up to k −
1. Let c ( k, p ) > max(¯ a ( k ) , c ( k − , pk/ ( k − i < k −
1, by induction andCorollary (4.5), we have Z f |∇ i Rm | pki ≤ C (cid:18) Z f |∇ k − Rm | pkk − + Z f |∇ k Rm | (cid:19) . Thus we only need to show that Z f |∇ k − Rm | pkk − ≤ C (cid:18) Z f |∇ k Rm | p + Z f |∇ k Rm | (cid:19) . Using integration by parts, we get Z f |∇ k − Rm | pkk − ≤ Z |∇ f ||∇ k − Rm ||∇ k − Rm | pkk − − + C Z f |∇ k Rm ||∇ k − Rm ||∇ k − Rm | pkk − − ≤ Z (cid:18) f |∇ k − Rm | pkk − + Cf − pka ( k − |∇ k − Rm | pkk − (cid:19) + Z f (cid:18) C |∇ k Rm | p + 14 |∇ k − Rm | pkk − + ǫ ( p, k ) |∇ k − Rm | pkk − (cid:19) , where ǫ ( p, k ) is to be determined. The only term we need to worry about is Z f − pka ( k − |∇ k − Rm | pkk − . Now let 1 s = pkk − − pkk − − , r = 1 − s . Using Young’s inequality xy ≤ x r r + y s s , we obtain f − pka ( k − |∇ k − Rm | pkk − ≤ f − r pka ( k − |∇ k − Rm | r + f |∇ k − Rm | pkk − s . It is easy to see that by further increasing c ( p, k ) then b = 1 − r pka ( k − Z f − pka ( k − |∇ k − Rm | pkk − ≤ C Z f − r pka ( k − |∇ k − Rm | + ǫ ( p, k ) Z f |∇ k − Rm | pkk − ≤ C (cid:18) Z f |∇ k − Rm | (cid:19) + Cǫ ( p, k ) Z f |∇ k − Rm | pkk − . We obtain the conclusion by choosing ǫ ( p, k ) sufficiently small. (cid:3) Theorem 4.7.
Under our settings, we get the following evolution inequality ∂∂t Z X f |∇ k Rm | dg ≤ − Z X f |∇ k +2 Rm | dg + C in the sense of distribution, where C is a constant depending only on k and the Calabi energy. Before we get into the proof, let us derive the evolution formula for (cid:18) ∂∂t + △ (cid:19) |∇ k Rm | . Let us recall that ∂Rm∂t = −△ Rm + ∇ Rm ∗ ∇ Rm + ∇ Rm ∗ Rm, where ∇ = ∂ + ¯ ∂ and △ = ∂ ¯ ∂ = ¯ ∂∂ . Then ∂ ∇ k Rm∂t = −△ ∇ k Rm + X i + j = k +2 ∇ i Rm ∗ ∇ j Rm.
So, ∂∂t |∇ k Rm | = h ∂∂t ∇ k Rm, ∇ k Rm i + h∇ k Rm, ∂∂t ∇ k Rm i + ∇ Rm ∗ ∇ k Rm ∗ ∇ k Rm = −h△ ∇ k Rm, ∇ k Rm i − h∇ k Rm, △ ∇ k Rm i + X i + j = k +2 ∇ i Rm ∗ ∇ j Rm ∗ ∇ k Rm.
Moreover,
HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 23 △ |∇ k Rm | = △h△∇ k Rm, ∇ k Rm i + △h∇ k Rm, △∇ k Rm i + △ (cid:16) h∇ i ∇ k Rm, ∇ ¯ i ∇ k Rm i + h∇ ¯ i ∇ k Rm, ∇ i ∇ k Rm i (cid:17) = h△ ∇ k Rm, ∇ k Rm i + h∇ i △∇ k Rm, ∇ ¯ i ∇ k Rm i + h∇ ¯ i △∇ k Rm, ∇ i ∇ k Rm i + h△∇ k Rm, △∇ k Rm i + h△∇ k Rm, △∇ k Rm i + h∇ i ∇ k Rm, ∇ ¯ i △∇ k Rm i + h∇ ¯ i ∇ k Rm, ∇ i △∇ k Rm i + h∇ k Rm, △ ∇ k Rm i + △ (cid:16) |∇ k +1 Rm | (cid:17) . Since h∇ i △∇ k Rm, ∇ ¯ i ∇ k Rm i + h∇ i ∇ k Rm, ∇ ¯ i △∇ k Rm i = △| ( ∇ k Rm ) ,i | − | ( ∇ k Rm ) ,ij | − | ( ∇ k Rm ) ,i ¯ j | + Rm ∗ ∇ k +1 Rm ∗ ∇ k +1 Rm + ∇ Rm ∗ ∇ k Rm ∗ ∇ k +1 Rm and h∇ i ∇ k Rm, ∇ ¯ i △∇ k Rm i + h∇ ¯ i ∇ k Rm, ∇ i △∇ k Rm i = △| ( ∇ k Rm ) , ¯ i | − | ( ∇ k Rm ) , ¯ ij | − | ( ∇ k Rm ) , ¯ i ¯ j | + Rm ∗ ∇ k +1 Rm ∗ ∇ k +1 Rm + ∇ Rm ∗ ∇ k Rm ∗ ∇ k +1 Rm.
We get (cid:18) ∂∂t + △ (cid:19) |∇ k Rm | = −| ( ∇ k Rm ) ,ij | − | ( ∇ k Rm ) ,i ¯ j | − | ( ∇ k Rm ) , ¯ ij | − | ( ∇ k Rm ) , ¯ i ¯ j | + 2 |△∇ k Rm | +2 △ (cid:16) |∇ k +1 Rm | (cid:17) + X i + j = k +2 ∇ i Rm ∗ ∇ j Rm ∗ ∇ k Rm + Rm ∗ ∇ k +1 Rm ∗ ∇ k +1 Rm.
Now we are ready to prove Theorem (4.7)
Proof. ∂∂t Z X f |∇ k Rm | dg = Z X ∂∂t (cid:16) f |∇ k Rm | (cid:17) + f |∇ k Rm | △ R dg = Z X (cid:18) ∂∂t + △ (cid:19) (cid:16) f |∇ k Rm | (cid:17) + f |∇ k Rm | △ R dg = Z X f (cid:18) ∂∂t + △ (cid:19) (cid:16) |∇ k Rm | (cid:17) + ∂f∂t |∇ k Rm | + f ∇ k Rm ∗ ∇ k Rm ∗ ∇ Rm − ( △ f )( △|∇ k Rm | ) dg = Z X f (cid:16) −| ( ∇ k Rm ) ,ij | − | ( ∇ k Rm ) ,i ¯ j | − | ( ∇ k Rm ) , ¯ ij | − | ( ∇ k Rm ) , ¯ i ¯ j | + 2 |△∇ k Rm | (cid:17) dg + Z X f X i + j = k +2 ∇ i Rm ∗ ∇ j Rm ∗ ∇ k Rm + Rm ∗ ∇ k +1 Rm ∗ ∇ k +1 Rm dg + Z X ∂f∂t |∇ k Rm | + ( △ f )(2 |∇ k +1 Rm | − △|∇ k Rm | ) dg Now let us deal with the above three integral one by one. For the firstintergral, since − Z X f | ( ∇ k Rm ) ,ij | = Z X f ¯ i ( ∇ k Rm ) ,ij ( ∇ k Rm ) , ¯ j + f ( ∇ k Rm ) ,ij ¯ i ( ∇ k Rm ) , ¯ j = Z X ∇ f ∗ ∇ k +2 Rm ∗ ∇ k +1 Rm + f ( ∇ k Rm ) ,i ¯ ij ( ∇ k Rm ) , ¯ j + f Rm ∗ ∇ k +1 Rm ∗ ∇ k +1 Rm = Z X ∇ f ∗ ∇ k +2 Rm ∗ ∇ k +1 Rm − f ( ∇ k Rm ) ,i ¯ i ( ∇ k Rm ) , ¯ jj + f Rm ∗ ∇ k +1 Rm ∗ ∇ k +1 Rm = Z X ∇ f ∗ ∇ k +2 Rm ∗ ∇ k +1 Rm + f ∇ k +2 Rm ∗ ∇ k Rm ∗ Rm − f ( ∇ k Rm ) ,i ¯ i ( ∇ k Rm ) ,j ¯ j + f Rm ∗ ∇ k +1 Rm ∗ ∇ k +1 Rm ≤ Z X −|△∇ k Rm | + ǫf |∇ k +2 Rm | + Cf a − a |∇ k +1 Rm | + Cf |∇ k Rm | + Cf |∇ k +1 Rm | By Lemma (4.4) and Corollary (4.5), we get − Z X f | ( ∇ k Rm ) ,ij | ≤ C + Z X −|△∇ k Rm | + ǫ Z X f |∇ k +2 Rm | Also − Z X f | ( ∇ k Rm ) ,i ¯ j | = − Z X f ( ∇ k Rm ) ,i ¯ j ( ∇ k Rm ) ,j ¯ i + f ∇ k +2 Rm ∗ ∇ k Rm ∗ Rm = Z X f ¯ j ( ∇ k Rm ) ,i ( ∇ k Rm ) , ¯ ij + f ( ∇ k Rm ) ,i ( ∇ k Rm ) ,j ¯ i ¯ j + f ∇ k +2 Rm ∗ ∇ k Rm ∗ Rm = Z X ∇ f ∗ ∇ k +2 Rm ∗ ∇ k +1 Rm + f ( ∇ k Rm ) ,i ( ∇ k Rm ) ,j ¯ j ¯ i + f ∇ k +2 Rm ∗ ∇ k Rm ∗ Rm = Z X ∇ f ∗ ∇ k +2 Rm ∗ ∇ k +1 Rm − f ( ∇ k Rm ) ,i ¯ i ( ∇ k Rm ) ,j ¯ j + f ∇ k +2 Rm ∗ ∇ k Rm ∗ Rm ≤ Z X −|△∇ k Rm | + ǫf |∇ k +2 Rm | + Cf a − a |∇ k +1 Rm | + Cf |∇ k Rm | . By Lemma (4.4) and Corollary (4.5), we get − Z X f | ( ∇ k Rm ) ,i ¯ j | ≤ C + Z X −|△∇ k Rm | + ǫ Z X f |∇ k +2 Rm | Hence Z X f (cid:16) −| ( ∇ k Rm ) ,ij | − | ( ∇ k Rm ) ,i ¯ j | − | ( ∇ k Rm ) , ¯ ij | − | ( ∇ k Rm ) , ¯ i ¯ j | + 2 |△∇ k Rm | (cid:17) ≤ C − ( 12 − ǫ ) Z X f |∇ k +2 Rm | Now we estimate the second integral. For i > , j >
0, by Proposition(4.6), we have (cid:12)(cid:12)(cid:12)(cid:12)Z X f ∇ i Rm ∗ ∇ j Rm ∗ ∇ k Rm (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z X f |∇ i Rm | k +4 i (cid:19) i k +4 (cid:18)Z X f |∇ j Rm | k +4 j (cid:19) j k +4 (cid:18)Z X f |∇ k Rm | (cid:19) ≤ (cid:18) C + C Z X f |∇ k +2 Rm | (cid:19) i k +4 (cid:18) C + C Z X f |∇ k +2 Rm | (cid:19) j k +4 (cid:18) C + ǫ Z X f |∇ k +2 Rm | (cid:19) ≤ C + ǫ Z X f |∇ k +2 Rm | . Also we have (cid:12)(cid:12)(cid:12)(cid:12)Z X f Rm ∗ ∇ k Rm ∗ ∇ k +2 Rm (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + ǫ Z X f |∇ k +2 Rm | , (cid:12)(cid:12)(cid:12)(cid:12)Z X f Rm ∗ ∇ k +1 Rm ∗ ∇ k +1 Rm (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + ǫ Z X f |∇ k +2 Rm | . Now we try to control the third integral. Notice that we pick (0 ,
0) inthe symplectic side. So it will move in the complex side as t changes. ByLegendre transformation, the coordinate of (0 ,
0) in the complex side is ξ t = Du t (0 , c ( s ), parameterizing by s = t . Forany point ξ in the complex side, let γ t be the minimizing geodesic connecting ξ t and ξ at time t . We first control the upper bound of ∂d t ∂t ( ξ, ξ t ) at t = t . Let t → t +0 . We can replace γ t by γ t plus the part of ˜ c connecting ξ t and ξ t . Let ¯ γ t be the part of ˜ c connecting ξ t and ξ t . Let ˜ d t be its Riemannlength at t . Then ∂ ˜ d t ∂t ( t ) = |∇ A t | u t (0 , . By our definition of Q , we know that |∇ A t | u t (0 ,
0) is bounded. Let | γ | u be the Riemannian length of γ . Then in the sense of distribution, we have ∂d t ( ξ, ξ t ) ∂t | t = t ≤ ∂ | γ t | u t ∂t ( t ) + C. Notice that if the distance between ξ and ξ is less than 2, we have ∂ | γ t | u t ∂t ( t ) ≤ Z γ t |∇ A | u < C. So we have obtained the upper bound of ∂d t ∂t ( ξ, ξ t ) at t = t in the sense ofdistribution. Next we want to get the lower bound of ∂d t ∂t ( ξ, ξ t ) at t = t inthe sense of distribution. Notice that d t ( ξ, ξ t ) ≥ d t ( ξ, ξ ) − | ¯ γ t | u t . So we only need to get a lower bound of ∂d t ( ξ,ξ ) ∂t | at t = t . It is well knownthat ∂d t ( ξ, ξ ) ∂t ( ξ, ξ t ) ≥ ∂ | γ t | u t ∂t ( t ) ≥ − Z γ t |∇ A | u > − C. So we conclude that ∂f∂t is bounded in the sense of distribution.Together with △|∇ k Rm | = ∇ k +2 Rm ∗ ∇ k Rm + ∇ k +1 Rm ∗ ∇ k +1 Rm, we have (cid:12)(cid:12)(cid:12)(cid:12)Z X ∂f∂t |∇ k Rm | + ( △ f )(2 |∇ k +1 Rm | − △|∇ k Rm | ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z X Cf a − a |∇ k Rm | + ǫf |∇ k +2 Rm | + Cf a − a |∇ k Rm | + Cf a − a |∇ k +1 Rm | ≤ C + ǫ Z X f |∇ k +2 Rm | . Combining the above results together, we get the conclusion. (cid:3)
Now let us define F k ( t ) = k X i =0 t i Z X f |∇ i Rm | dg ( t − , HE INTERIOR REGULARITY OF THE CALABI FLOW ON A TORIC SURFACE 27 where t ∈ [0 , ∂F k ∂t = k X i =1 it i − Z X f |∇ i Rm | + k X i =0 t i ∂∂t Z X f |∇ i Rm | ≤ k X i =1 it i − Z X f |∇ i Rm | + k X i =0 t i (cid:18) − Z X f |∇ i +2 Rm | + C (cid:19) ≤ k − X i =0 t i (cid:18) − Z X f |∇ i +2 Rm | + ( i + 1) Z X f |∇ i +1 Rm | (cid:19) + C ≤ C Hence F (1) − F (0) ≤ C . Then we have the following result: Corollary 4.8.
At time t = 0 , for every k ∈ Z + , there is a constant C ( k ) depending only on k , such that Z X f |∇ k Rm | dg < C ( k ) Especially, Z B ( p , ) |∇ k Rm | dµ < C ( k ) where p = (0 , , B ( p , ) is a geodesic ball in P and dµ is the standardEuclidean measure on P . Now applying the arguments in the appendix of [19], we have
Corollary 4.9.
For any p ∈ B ( p , ) . |∇ k Rm | ( p ) < C ( k ) . Moreover we have
Corollary 4.10.
For any
C > , there exists an i ( C ) ∈ Z + such that forany i > i ( C ) , p ∈ B ( p , C ) , we have |∇ k Rm ( i ) | ( p ) < C ( k ) . Now we are ready to prove Theorem (1.2).
Proof of Theorem (1.2).
We are essentially dealing two cases. The first caseis that for sufficiently large i , | Rm ( i ) (0 , p ) | has a uniform lower bound. Thearguments of [19] can be applied to rule out this case. The second case isthat there is a sequence of i such that | Rm ( i ) (0 , p ) | →
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