ON THE WEAK KÄHLER-RICCI FLOW

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1 ON THE WEAK KÄHLER-RICCI FLOW XIUXIONG CHEN, GANG TIAN, AND ZHOU ZHANG Abstract. I this paper, we defie ad study Kähler-Ricci flow with iitial data ot beig smooth with some atural applicatios. 1. Itroductio Let be a -dimesioal compact (without boudary) Kähler maifold. A Kähler metric g ca be represeted by its Kähler form g o. Cosider the Kähler-Ricci flow o [, T ) g(t) (1.1) = Ric( g(t) ), where g(t) is a family of Kähler metrics ad Ric( g ) deotes the Ricci curvature of g. It is kow that for ay smooth Kähler metric g, there is a uique solutio g(t) of (1.1) for some maximal time T > with g() = g. I geeral, T will deped o the iitial metric g. However, i Kähler maifold case, this oly depeds o the Kähler class ad the first Cher class. This observatio plays a importat role i our itroductio of weak Ricci flow over Kähler maifold. The Ricci flow was itroduced by R. Hamilto i [2]. Extesive research has bee doe i the case of the smooth flow (c.f. [22] [6] [14] [21] [25] [12] [13] [3] or [15] for complete updated refereces). I order to prove the uiqueess of extremal Kähler metrics i full geerality, i [9], the first two amed authors were led to the study of Kähler-Ricci flow i the weak sese. ore precisely, we proved i [9] that for ay iitial L -bouded Kähler metric, there is a uiformly L -bouded solutio of (1.1) with the give iitial metric i a suitable sese. oreover, the volume form of the solutio coverges strogly to the volume form of the iitial Kähler metric i L 2 -topology as t. As oe may expect, such a weak solutio of (1.1) should become smooth immediately after t >. Ideed, we cofirm this i this ote i a more geeral settig. Defiitio 1.1. For ay Kähler curret g with C 1,1 bouded potetial o, there is a uique smooth solutio g(t) (t (, T )) of (1.1) such that lim t + g(t) = g i C 1,α ( α (, 1)) orm at the potetial level. A special case of Riema surfaces was also cosidered i [11] with a completely differet proof. Date: October 9, 28 ad, i revised form, February 13, athematics Subject Classificatio. Primary 53C25; Secodary 53C99, 58J99. Key words ad phrases. Differetial geometry, Geometric evolutio equatio. The first author is supported i part by NSF fuds. The secod author is supported i part by NSF fuds. 1

2 2 XIUXIONG CHEN, GANG TIAN, AND ZHOU ZHANG It was kow that ay Kähler metric with costat scalar curvature is the absolute miimizer of the K-eergy o the space of all Kähler metrics with a fixed Kähler class ([8], [18], [24] ad [9]) 1. From aalytic poit of view, it is a extremely difficult problem to prove the existece of Kähler metric with costat scalar curvature i geeral cases. Oe approach is to costruct weak miimizers of the K-eergy by applyig certai variatioal or cotiuous methods. This seems to be a plausible direct approach but very hard. However, eve if we obtaied a C 1,1 miimizer of the K-eergy fuctioal, we would still face the regularity problem. I [8], the first amed author made the followig cojecture: Ay C 1,1 miimizers of the K-eergy i a give Kähler class must be smooth. As a cosequece of the above theorem, we ca solve this cojecture i caoically polarized cases. Example 1.2. I a Kähler class which is proportioal to the first Cher class, ay L Kähler metric which miimizes the K-eergy fuctioal must be smooth. There is aother motivatio for this short paper. O a geeral Kähler maifold, the Kähler-Ricci flow (1.1) may develop sigularity at fiite time (see [31]). I a o-goig project with his collaborators, the secod amed author proposed some problems of studyig how the Kähler-Ricci flow exteds across the fiite time sigularity (see [28] for more discussios). Oe of them ivolves costructig solutios of the Kähler-Ricci flow with much weaker iitial metrics, possibly o spaces with mild sigularity. Our secod result gives a partial solutio to this problem. First we recall some stadard facts: by Hodge Theorem, the space of all Kähler metrics i a fixed Kähler class give by a smooth Kähler metric is P (, ) = {ϕ C () ϕ = + 1 ϕ > o }. We deote by Cl L P (, ) all bouded fuctios plurisubharmoic with respect to. The for ay ϕ Cl L P (, ), there is a well-defied (Borel) measure ( + 1 ϕ) usig the uderstadig from pluripotetial theory about oge-ampère operator itroduced i [2]. Oe ca also thik of this as a volume form i a weak sese. Now we ca state our secod result. Defiitio 1.3. For ay Kähler potetial ϕ Cl L P (, ) whose volume form is L p (, )(p 3), there is a uique smooth solutio g(t) of (1.1) for t (, T ) such that g(t) coverges to ϕ strogly i the L 2 -topology. The orgaizatio of this ote is as follows: i Sectio 2, we show the existece of the weak solutio for the iduced potetial flow which is equivalet to the Kähler- Ricci flow. C -estimates will be derived for the solutio ad its time derivative. We also examie covergece problem for these weak solutios as t teds to. I Sectio 3, we derive the 2d ad 3rd order estimates for the weak solutios. The Theorem 1.1 ad 1.3 follow. I last sectio, we prove the corollary. 2. Weak solutios of Kähler-Ricci flow First we reduce the Kähler-Ricci flow (1.1) to a scalar flow. To start with, we ote that [ g ] tc 1 () represets a Kähler class wheever t T for a sufficietly small T > depedig oly o [ g ] ad c 1 (). Choose a smooth family of Kähler forms t for t [, T ] such that = ad [ t ] = [ g ] tc 1 (). 1 For Kähler-Eistei metrics, cf. [1], [17] ad [29].

3 ON THE WEAK KÄHLER-RICCI FLOW 3 Write g(t) = t + 1 ϕ(t). The, oe ca show by stadard argumet that g(t) solves (1.1) if ad oly if ϕ(t) solves the followig scalar flow: ϕ (2.1) = log ( t + 1 ϕ) h t, t where h t is defied by Ric( t ) + t = 1 h t, ad X (e h t 1)t =. Clearly such a h( t ) does exist ad is uique. As usual, we will regard either (1.1) or (2.1) as the Kähler-Ricci flow o. For simplicity, we deote by ϕ the Kähler form t + 1 ϕ. To costruct a weak solutio to the Kähler-Ricci flow equatio with o-smooth iitial Kähler potetials, as usual, we use smooth approximatios of the iitial data. Let ϕ be a bouded Kähler potetial such that its volume form is i L p for some p > 1. This actually implies Hölder cotiuity of ϕ from the result i [27]. Let ϕ (s)( s 1) be a 1-parameter Kähler potetials such that the followig properties hold: (1) ϕ () = ϕ ad ϕ (s) P (, ) for < s 1. (2) If ϕ has C 1,1 boud, the ϕ (s) has uiform C 1,1 upper boud ad ϕ (s)(s > ) ϕ strogly i W 2,q (, ) for q sufficietly large. (3) ϕ (s) coverges to ϕ uiformly i L -topology ad the volume form ratio coverges to ϕ strogly i L p (p > 1). Later i Sectio 3, less regularity will be assumed, ad so the approximatio would also be less restrictive. It is kow (cf. [31]) that for ay ϕ i P (, ), there is a uique smooth solutio of (2.1) o [,T] with ϕ as the iitial value. Therefore, we have ϕ(s, t) P (, ) ( < s 1, t [, T ]) satisfyig: (2.2) (2.3) ϕ(s, t) = log ( t + 1 ϕ) h t, t ϕ(s, ) = ϕ (s). Clearly, for each s > fixed, there exists a uiform C 2,α boud for ϕ(s, t) ( t T ). However, the upper boud may well deped o s ad so may blow up whe t, s are both small. If there is a limit ϕ(t) = lim s ϕ(s, t), t [, T ], the we ca regard ϕ(t) as a weak solutio of (2.1) with iitial value ϕ. Lemma 2.1. The solutios ϕ(s, t) coverges uiformly to a family of fuctios ϕ(t) (t [, T ]) as s teds to. Proof. For ay positive s ad s, put ψ(t) = ϕ(s, t) ϕ(s, t), the we have ψ = log ( ϕ(s,t) + 1 ψ) ( t + 1 ϕ(s, t)). The the aximum Priciple implies that sup ψ(t) sup ψ(), that is, sup ϕ(s, t) ϕ(s, t) sup ϕ (s ) ϕ (s). The lemma the follows easily from our choice of the iitial data ϕ (s).

4 4 XIUXIONG CHEN, GANG TIAN, AND ZHOU ZHANG We shall show that ϕ(t) solves the Kähler-Ricci flow ad cosequetly is a weak solutio i a suitable sese. For this purpose, oe eeds to justify that ϕ(t) is a smooth family for t > which solves (2.1) for t > such that lim t + ϕ(t) = ϕ. Of course, this is the core part of this paper. First we derive a few estimates o ϕ(s, t) uiform for s (, 1] ( Sometimes we abbreviate ϕ(s, t) as ϕ for simplicity). These estimates would evetually be passed o to ϕ(t) by takig limit as s teds to i some proper sese. Lemma 2.2. If ϕ is bouded, the there exists a uiform costat C such that for ay s (, 1] ad t [, T ], ϕ(s, t) C. Proof. Choose c such that h t C c for all t [, T ]. Applyig the aximum Priciple to ϕ(s, t) ± c t, we ca obtai mi ϕ (s) c t ϕ(s, t) max ϕ (s) + c t, s >. The the boud o ϕ follows. The ext lemma gives a estimate o the volume form for the Kähler-Ricci flow whe t >. I this ote, ϕ always stads for Laplacia with respect to the flow metric ϕ = t + 1 ϕ. Traditioally, C might stad for differet positive costats at differet places. Lemma 2.3. For t >, the volume form has uiform upper ad positive lower bouds which may deped o t. Proof. Differetiatig the Kähler-Ricci flow equatio we get Similarly, we have ϕ = log ( t + 1 ϕ) t h t, ϕ = ( ϕ ϕ h t tr t ( t) + trϕ ) t ϕ ϕ + C(1 + tr ϕ ). ϕ ϕ ϕ C(1 + tr ϕ ), which will be used for lower boud later i this proof. For the upper boud, let s cosider The t-derivative gives For t small eough, we have F + = ϕ + tϕ. F + = ϕ + ϕ + tϕ t ϕ ϕ + Ct(1 + tr ϕ ) = ϕ ( ϕ + tϕ ) + ϕ ϕ + Ct(1 + tr ϕ ) = ϕ F + + tr ϕ t + Ct(1 + tr ϕ ) ϕ F + + C (C Ct)(1 + tr ϕ ). F + ϕ F + + C.

5 ON THE WEAK KÄHLER-RICCI FLOW 5 Sice F + () is uiformly bouded from above, it follows from the aximum Priciple that F + has a uiform upper boud. For the lower boud, set F = ϕ + tϕ = ϕ th + t log ϕ t. The, usig the same C to deote differet costats at places, F = ϕ + ϕ + tϕ 2ϕ + t( ϕ ϕ C(1 + tr ϕ )) = ϕ (ϕ + tϕ ) ϕ ϕ + 2ϕ Ct(1 + tr ϕ ) = ϕ F + tr ϕ t + 2 log ϕ t ϕ F C + (C Ct)(1 + tr ϕ ) + 2 log ϕ ϕ F C + (C Ct) ( ϕ t ) 1 2h t Ct(1 + tr ϕ ) t + 2 log ϕ. t It s easy to see for x >, with the uderstadig that these two C s are ot the same, x 1 It follows that for t small, we have + C log x C. F ϕ F C. The the aximum Priciple implies that F > C for some uiform costat C, that is, the volume form ϕ has a uiform positive lower boud whe cosidered t uiformly away from iitial time. Next we exam whether itegral estimates. lim ϕ(t) = ϕ. To make sure of this, we derive some t + Lemma 2.4. If ϕ is bouded ad ϕ is i L p (p 1), the for ay s (, 1], there exists a positive fuctio C(t) of t such that ϕ L p C(t). oreover, C(t) is idepedet of s ad is uiformly bouded for t [, T ] with T <. Proof. For λ >, set the modified volume ratio as f ϕ = ϕ t e λϕ. We are cosiderig classic smooth solutio. Let C be a costat satisfyig: t t + Ric( t ) t + Ric() t + h t t C.

6 6 XIUXIONG CHEN, GANG TIAN, AND ZHOU ZHANG Use to deote the gradiet with respect to ϕ. The for λ sufficietly large, log f ϕ = tr ϕ ( t) + ϕ ϕ tr t ( t) λϕ = ϕ log f ϕ + tr ϕ ( t + λ 1 ϕ 1 h t ) + R t tr t h t λϕ ϕ log f ϕ + λ (λ 2 C) tr ϕ t λ log f ϕ + λh t + 2 C ϕfϕ f ϕ Thus, we have fϕ 2 ϕ f 2 ϕ f ϕ For ay p 1, we have d d t fϕ p e λϕ t = p fϕ p 1 f ϕ eλϕ t + = p fϕ p 1 f ϕ eλϕ t + +λ ( f ϕe p λϕ p + 2 C λ log f ϕ. ϕ f ϕ f ϕ 2 ϕ f ϕ + 2 C f ϕ λf ϕ log f ϕ. f p ϕ log ϕ e λϕ t + λ f p ϕ ( R t + tr t h t ) e λϕ t t h t ) t fϕe p λϕ ϕ t fϕ p 1 ( ϕ f ϕ f ϕ 2 ϕ + f ϕ (2 C λ log f ϕ )) e λϕ t + C f ϕ +λ f p ϕe λϕ ( log ϕ t h t ) t f p ϕ e λϕ t fϕ p 2 (p ( ϕ f ϕ f ϕ 2 ϕ ) + f ϕ (2 p C λ(p 1) log f ϕ )) ϕ + C λ f f ϕe p λϕ t ϕ p(p 1) f ϕ p 2 f ϕ 2 ϕe λϕ t + C λ (f ϕ p + fϕ p 1 )e λϕ t. Here we have used the fact that the fuctio xlogx has a lower boud for x >. Sice ϕ is uiformly bouded, it follows that ϕ is uiformly bouded i L p for all t [, T ]. It follows from this lemma that the L p -orm of ϕ(t) is uiformly bouded. By the work of Kolodziej [27], we kow that ϕ(t) C α is uiformly bouded, where α = α(p) > may deped o p > 1. The for ay sequece t i, there is a subsequece, still deoted by t i for simplicity, such that ϕ(t i ) coverges to a C α - fuctio ϕ i the C β -topology for some β (, α). A priori, this limit potetial might well deped o the sequece we choose. Lemma 2.5. If p 3, we have ϕ = ϕ. oreover, ϕ(t) coverges to ϕ strogly i the L 2 -topology, where a smooth volume form is used to defie the L 2 -topology ad the choice clearly does t matter. Proof. First we assume that ϕ is smooth, i. e., oe of ϕ (s)(s > ). I the proof of the precedig lemma, set p = 3 ad we have d f 3 d t ϕe λϕ t 6 f ϕ f ϕ 2 ϕe λϕ t + C fϕe 3 λϕ t + C, which implies (2.4) t f ϕ f ϕ 2 ϕ t d u C.

7 ON THE WEAK KÄHLER-RICCI FLOW 7 Now set f = ϕ. Claim 1: We have for t i some fiite iterval, t t (2.5) f 2 ϕ ϕ d u = f f 2 ϕ t d u C (2.6) t t f 2 ϕ ϕ 2 ϕ ϕ d u C. Proof of Claim 1: Sice ϕ is uiformly bouded, we ca choose a uiform costat c such that ϕ c. The we have f ϕ 2 ϕ 2 ϕ ϕ = 2 f ϕ ( f ϕ ϕ) ϕ (ϕ c) ϕ f ϕ 2 (ϕ c) ϕ ϕ ϕ 1 2 f ϕ 2 ϕ 2 ϕ ϕ + C f ϕ 2 ϕ ϕ f ϕ 2 (ϕ c) ϕ + f ϕ 2 (ϕ c) tr ϕ t ϕ 1 2 f ϕ 2 ϕ 2 ϕ ϕ + C f ϕ 2 ϕ ϕ + C f ϕ 2 ϕ = 1 2 f ϕ 2 ϕ 2 ϕ ϕ + C f ϕ f ϕ 2 ϕt + C f ϕ 3 e λϕ t. It follows by itegratig over t ad (2.4) that t fϕ 2 ϕ 2 ϕ ϕ d u C. To prove the first iequality, we observe The, f = e λϕ f ϕ. f f 2 ϕ t = (f ϕe λϕ ) 2 ϕ ϕ 2 f ϕ 2 e 2λϕ ϕ + 2λ 2 f 2 ϕ ϕ 2 ϕ e 2λϕ ϕ C ( f ϕ 2 ϕ ϕ + f 2 ϕ ϕ 2 ϕ ϕ). Thus the first iequality follows from the secod oe ad (2.4). Claim 1 is proved. Claim 2: For ay smooth o-egative cut-off fuctio χ (fixed), we have, (2.7) f χ 2 ϕ t = χ 2 ϕ ϕ C( χ,l ). Proof of Claim 2: f χ 2 ϕ t = χ 2 ϕ ϕ χ 2 t tr ϕ t ϕ C tr ϕ ( ϕ 1 ϕ) ϕ = C ( ϕ ϕ) ϕ C, where the first iequality follows from a elemetary iequality ad the secod iequality makes use of the positivity of t. I other words, ay smooth cut-off fuctio automatically has uiform W 1,2 orm with respect to ay Kähler metric i ay give Kähler class.

8 8 XIUXIONG CHEN, GANG TIAN, AND ZHOU ZHANG Claim 3: We have (2.8) f ϕ 2 ϕ t = ϕ 2 ϕ ϕ C( ϕ L ). Proof of Claim 3: Let c be the same as i the proof of Claim 1. ϕ 2 ϕ ϕ = (ϕ c) ϕϕ ϕ = (ϕ c)tr ϕ t ϕ (ϕ c) ϕ C. I the last iequality, we have used the fact that ϕ is uiformly bouded. Claim 4: For ay positive χ, the followig iequality holds t (2.9) χ f 2 tr ϕ t t d u C 1 t + C 2 t, where both costats deped oly o χ L ad χ,l. Proof of Claim 4: χ f 2 tr ϕ t t = χ f t ϕ 1 = χ f ϕ χ f 1 ϕ ϕ 1 C + 1χ f ϕ 1 ϕ + ( C C 1 + ( f 2 ϕ ϕ ( 1 + ( f 2 ϕ ϕ ) 1 1f χ ϕ 1 ) 1 ( 2 ϕ 2 ϕ ϕ ) ( f 2 ϕ 2 ϕ ϕ ϕ 2 + ( χ 2 ϕ ϕ ) 1 ) 2. ) 1 2 ( f ) 2 ϕ 2 ϕ ϕ 1 ) 2 We have used Claims 2 ad Claim 3 i derivig the last iequality. The Claim 4 follows from itegratig the above iequality from to t ad usig Claim 1 ad the Schwartz iequality. A straightforward computatio shows log f = tr ϕ ( t) + ϕ ϕ ( tr t t = ϕ log f + tr ϕ ht + t) trt Usig this, we deduce d d t χf 2 t = C Note that ϕf f f 2 ϕ f 2 + Ctr ϕ t + C. ( χ 2f f ) + f 2 tr t t t t χf( ϕ f f 2 ϕ + Cftr ϕ t + Cf) t f ( χ, f) ϕ ϕ + C χf 2 t + C χf 2 tr ϕ t t χ 2 ϕ ϕ f 2 ϕ ϕ + C χf 2 t + C χf 2 tr ϕ t t f 2 ϕ ϕ + C χf 2 t + C χf 2 tr ϕ t t. χ f 2 t C, t [, T ].

9 ON THE WEAK KÄHLER-RICCI FLOW 9 Itegratig the above iequality from t = to t = t i ad usig Claim 4, we have χf 2 t ti χf 2 t + C(t i + t i ). I derivig the above iequality, we used the fact that ϕ is smooth. For geeral ϕ as give, applyig the above to ϕ(s, t i ) for ay s > ad the takig the limit as s teds to, we get for ay ϕ(t i ), ( ) 2 ϕ(ti) χ t i t i χ ( ϕ ) 2 + C(t i + t i ). Here oe ca use Fatou s Lemma ad the strog covergece for the measure at iitial time. O the other had, usig the assumptio that ϕ(t i ) coverges to ϕ, we ca show that ϕ(t i ) coverges weakly to ñ ϕ t i ca justify the covergece of itegratio i a similar way ad have (2.1) χ ( ñ ϕ ) 2 i L 2 (, ). The by takig t i to, we χ ( ϕ ) 2. Sice this holds for ay o-egative smooth cut-off fuctio χ, we have a. e. i. However, It follows ñ ϕ ϕ ϕ ñ = ϕ = vol()! (2.11) ñ ϕ ϕ i L 2 (, ). The uiqueess of oge-ampère equatio for C solutio by Kolodziej as i [26] implies that ϕ = ϕ. Sice {t i } is ay sequece goig to, we have proved that ϕ(t) coverges to ϕ as t teds to. Furthermore, we have χ ( ñ ϕ ) 2 = lim i χ ( ) 2 ϕi. Together with weak covergece, we coclude ϕ i coverges strogly to ñ ϕ L 2 (, ). Hece Lemma 2.5 is proved. i So far, we have show that with assumptio o the volume form, the solutio of the weak Kähler-Ricci flow really goes (back) to the iitial data as t. 3. Higher order estimates I this sectio, we prove the regularity of the weak Ricci flow uder appropriate assumptios o the iitial Kähler potetial ϕ.

10 1 XIUXIONG CHEN, GANG TIAN, AND ZHOU ZHANG 3.1. The C 1,1 case. I this subsectio, we wat to prove the followig Laplacia estimates. Propositio 3.1. [9] If ϕ Cl L P (, ) is C 1,1 -bouded, the t + 1 ϕ(t) C, t [, T ]. I other words, ϕ(t) is uiformly C 1,1 -bouded. oreover, ϕ (t) coverges to ϕ strogly i L 2 space as t. Proof. The C 1,1 assumptio o ϕ gives a L (upper) boud for the volume form. I sight of Lemma 2.5, it suffices to prove the Laplacia estimate. Let ϕ (s) be the smooth approximatios of ϕ ad ϕ(s, t) be the associated solutios as before. Sice ϕ is i C 1,1, we ca arrage ϕ (s) with uiformly bouded C 1,1 -orms. Thus, we oly eed to boud ϕ(s, t) uiformly i C 1,1. For simplicity, ϕ stads for ϕ(s, ) below. By Lemma 2.2, it s uiformly bouded. We first derive a poitwise uiform upper boud o the volume form for t [, T ] as follows. ( ϕ ) ( ϕ ) + C(1 + tr ϕ t ) ( ϕ Cϕ) + C + C(tr ϕ( 1 ϕ) + tr ϕ t ) ( ϕ Cϕ) + C + ( ϕ Cϕ) + C + Cϕ. It ca be reformulated as ( ϕ Cϕ) ( ϕ Cϕ) + C C( ϕ Cϕ). Applyig the aximum Priciple, oe ca easily derive a uiform upper boud for ϕ ϕ Cϕ. Sice ϕ is uiformly bouded, we get a uiform upper boud o, so the volume form is uiformly bouded from above. By stadard computatio as i [32], we have e Cϕ ( ϕ )(e Cϕ tr t ( ϕ )) C+(C ϕ C)tr t ( ϕ )+Ce 1 ϕ 1 trt ( ϕ ) 1. Suppose that the maximum of e Cϕ tr t ( ϕ ) is attaied at some (p, t) i [, T ]. At that poit, assumig t > without loss of geerality, we have C + (C ϕ C)tr t ( ϕ ) + Ce 1 ϕ 1 trt ( ϕ ) 1. Usig the upper boud for ϕ, we have tr t ( ϕ ) 1 ϕ 1 ϕ 1 Ce 1 + Ce 1 trt ( ϕ ) C ϕ C + Ctr t ( ϕ ). e 1 ϕ 1 trt ( ϕ ) This implies a uiform upper boud of tr t ( ϕ ) at (p, t). Sice ϕ is uiformly bouded, it follows that this trace is uiformly bouded. The propositio is thus proved.

11 ON THE WEAK KÄHLER-RICCI FLOW The L case. We are goig to fiish provig Theorems 1.1 ad 1.3 i this subsectio. The estimates will be local for t > uder the weaker assumptio that the iitial Kähler potetial is merely bouded. Let s poit out that i this case, oe ca still defie weak flow usig smooth decreasig approximatio of the iitial bouded Kähler potetial 2. These approximatio flows have potetials decreasig to the weak flow poitwisely. oreover, for two choices of such approximatios, it s ot hard to see the limits would be the same because oe-sided relatio at the iitial time will be preserved alog the flow. All these ca be justified by applyig aximum Priciple as i Lemma 2.1. Thus we have the uiqueess of the weak flow. This also makes sure what we are discussed below is the same weak flow as before whe there is more regularity assumptio. Recall that t is a smooth family of Kähler metrics with [ t ] = [] tc 1 () ad ϕ = t + 1 ϕ. oreover, the Kähler-Ricci flow is reduced to the followig equatio for ϕ: ϕ = log ( t + 1 ϕ) t h t, ϕ(, ) = ϕ. ϕ(t) is obtaied by takig the limit of ϕ(s, t) as s teds to. I order to prove the theorem, we oly eed to get uiform estimates for higher order derivatives of ϕ(s, t) for all s > small. As before, for simplicity, we ca simply assume that ϕ is smooth (as oe of the ϕ(s, )). Choose ay iterval [t 1, t ] with T > t 1 > t >. By Lemmas 2.2 ad 2.3, we have the uiform boud for ϕ ad ϕ for t [t, t 1 ]. Now we derive the Laplacia ad higher order derivative estimates by the stadard methods (see [32], also see [6] or [34]). The estimates may deped o t. As i the last subsectio, we have e Cϕ ( ϕ )(e Cϕ tr t ( ϕ )) C Ctr t ( ϕ ) + Ctr t ( ϕ ) 1, where the boud o ϕ for t [t, t 1 ] has bee used. The boud for e Cϕ tr t ( ϕ ) at t t = is ot available. So we cosider the followig quatity istead Straightforward computatios the gives (t t ) 1 e Cϕ tr t ( ϕ ). e Cϕ ( ϕ )((t t ) 1 e Cϕ tr t ( ϕ )) (t t ) 1 ( C Ctr t ( ϕ ) + Ctr t ( ϕ ) 1 ) ( 1)(t t ) 2 tr t ( ϕ ). At the maximum value poit (p, t) of (t t ) 1 e Cϕ tr t ( ϕ ) i [t, t 1 ], where clearly t > t, ( t t ) 1 ( C Ctr t ( ϕ ) + Ctr t ( ϕ ) 1 ) ( 1)( t t ) 2 tr t ( ϕ ) = C( t t ) 1 (C t + C)( t t ) 2 tr t ( ϕ ) + C( t t ) 1 tr t ( ϕ ) 1. ultiply t t o both sides ad reformulate the above to be C( t t ) (C t + C)( t t ) 1 tr t ( ϕ ) + C(( t t ) 1 tr t ( ϕ )) 1. 2 This is provided by the result i [3] as a simple versio of the more geeral result i [16] for this case.

12 12 XIUXIONG CHEN, GANG TIAN, AND ZHOU ZHANG So we get Thus at (p, t), usig the boud for ϕ, It follows that for all t [t, t 1 ], ( t t ) 1 tr t ( ϕ ) C. ( t t ) 1 e Cϕ tr t ( ϕ ) C. (t t ) 1 e Cϕ tr t ( ϕ ) C, o which implies C tr t ( ϕ ) (t t ) 1. This together with the volume lower boud gives the uiform metric boud locally away from t =. Next, we ca get the third order estimate i a similar fashio. We have a similar iequality as i [6]. Of course, i order to do this, we still traslate the time to guaratee the uiform metric boud. Let s say the time is traslated so that the origial time t = t > is ow the ew iitial time t =. The iequality is as follows, for S = g i j ϕ gϕ k l gϕ λ η ϕ i lλ ϕ jk η, As i [32], we also have ( ϕ )S C S C. ( ϕ ) t ϕ C S C. From the first oe, as i the Laplacia estimate, we cosider ts istead, ( ϕ )(ts) Ct S Ct S. Choosig A > large eough, oe has, for t [, T ], ( ϕ )(ts + A t ϕ) C S C. The fuctio ts + A t ϕ has uiformly bouded at the (ew) iitial time, t =. By aximum Priciple, if its maximum value is achieved at some poit i ad t >, the S is bouded there. Cosequetly, the whole fuctio ts + A t ϕ is also bouded there. So we fially coclude which implies ts + A t ϕ C, S C t. This provides a local C 2,α boud for ϕ. After this, higher order estimates follow from the stadard argumet. Therefore, by Ascoli-Arzela Theorem, we have proved that the weak flow cosidered so far is actually smooth i (, T ]. Theorem 1.1 follows from this ad the secod order estimate i Subsectio 3.1. Theorem 1.3 follows from this ad Lemma 2.5.

13 ON THE WEAK KÄHLER-RICCI FLOW C 1,1 K-eergy miimizer I a earlier paper [9] where the preset work is iitiated, the first two amed authors proved that Propositio 4.1. I ay Kähler class, the volume form of ay C 1,1 K-eergy miimizer belogs to H 1,2 (, ). By Lemma 2.5 ad Propositio 3.1, we ca prove a stroger result. Defiitio 4.2. The C 1,1 miimizer of the K-eergy fuctioal i ay caoical Kähler class with positive first Cher class is ecessarily smooth. Proof. Let φ be a C 1,1 Kähler potetial which miimizes the K-eergy fuctioal i the caoical Kähler class. Accordig to the results proved i the precedig sectios, there is a uique smooth Kähler-Ricci flow ϕ(t)(t > ) such that ϕ = log ϕ + ϕ h. ϕ oreover, lim t = ϕ strogly i L2 (, g ) ad the Kähler potetial ϕ(t) coverges to ϕ strogly i C 1,α () for ay α (, 1). Sice the K-eergy is decreasig alog the Kähler-Ricci flow, we have lim sup E(ϕ(t)) E(ϕ ) = if E(φ). t + φ H Sice the K-eergy is o-icreasig for ϕ(t)(t > ), the for ay t >, we have if E(φ) E(ϕ(t)) if E(φ). φ H φ H I other words, E(ϕ(t)) = if E(φ), t >. φ H Sice ϕ(t) is a smooth Kähler metric, this meas that the scalar curvature of ϕ(t) (t > ) must be costat. Cosequetly, ϕ(t) is a Kähler-Eistei metric for all t >. This i turs implies that ϕ is a fuctio of t oly. Note that ϕ is the strog C 1,α limit of ϕ(t) as t. Therefore, ϕ ϕ(t) is a costat which depeds oly o t. I other words, ϕ is also a Kähler-Eistei metric ad the theorem is the proved. Refereces 1. S. Bado ad T. abuchi. Uiqueess of Eistei Kähler metrics modulo coected group actios. Algebraic geometry, Sedai, 1985, 11 4, Adv. Stud. Pure ath., 1, North-Hollad, Amsterdam, E. Bedford; B. A. Taylor. A ew capacity for plurisubharmoic fuctios. Acta ath. 149 (1982), o. 1-2, Z. Blocki; S. Kolodziej. O regularizatio of plurisubharmoic fuctios o maifolds. Proc. Amer. ath. Soc. 135 (27), o. 7, (electroic). 4. E. Calabi. Extremal Kähler metrics. I Semiar o Differetial Geometry, volume 16 of 12, pages A. of ath. Studies, Uiversity Press, E. Calabi. Extremal Kähler metrics, II. I Differetial geometry ad Complex aalysis, pages Spriger, H. D. Cao. Deformatio of Kähler metrics to Kähler-Eistei metrics o compact Kähler maifolds. Ivet. ath. 81 (1985), o. 2, X. X. Che. Weak limits of Riemaia metrics i surfaces with itegral curvature boud. Calc. Var. Partial Differetial Equatios 6 (1998), o. 3,

14 14 XIUXIONG CHEN, GANG TIAN, AND ZHOU ZHANG 8. X. X. Che. Space of Kähler metrics. Joural of Differetial Geometry, 56(2): , X. X. Che ad G. Tia. Foliatio by holomorphic discs ad its applicatio i Kähler geometry, math.dg/ Submitted. 1. X. X. Che. Space of Kähler metrics (III) O the greatest lower boud of the Calabi eergy ad the lower boud of geodesic distace. Preprit, X. X. Che ad W. Y. Dig. Ricci flow o surfaces with L iitial metrics. Ricci flow o surfaces with degeerate iitial metrics. J. Partial Differetial Equatios 2 (27), o. 3, X. X. Che ad G. Tia. Ricci flow o Kähler-Eistei surfaces. Ivet. ath. 147 (22), o. 3, X. X. Che ad G. Tia. Ricci flow o Kähler-Eistei maifolds. Duke. ath. J. 131, (26), o. 1, B. Chow. The Ricci flow o the 2-sphere. J. Differetial Geom. 33 (1991), o. 2, B. Chow, P. Lu ad L. Ni. Hamilto s Ricci flow. Graduate Studies i athematics, 77. America athematical Society, Providece, RI; Sciece Press, New York, 26. xxxvi+68 pp. ISBN: ; J. -P. D ly,. Pau. Numerical characterizatio of the Kähler coe of a compact Kähler maifold. A. of ath. 159 (24), W.Y. Dig ad G. Tia. Kähler-Eistei metrics ad the geeralized Futaki ivariat. Ivet. ath. 11 (1992), o. 2, S. K. Doaldso. Scalar curvature ad projective embeddigs, II. Q. J. ath. 56 (25), o. 3, S. K. Doaldso. Lower bouds o the Calabi fuctioal. math.dg/ R. Hamilto. Three-maifolds with positive Ricci curvature. J. Diff. Geom., 17:255 36, R. Hamilto. The formatio of sigularities i the Ricci flow, volume II. Iterat. Press, G. Huiske. Ricci deformatio of the metric o a Riemaia maifold. J. Differetial Geom. 21 (1985), o. 1, T. abuchi. K-eergy maps itegratig Futaki ivariats. Tohoku ath. J., 38 (1986), T. abuchi A eergy-theoretic approach to the Hitchi-Kobayashi correspodece for maifolds. I. Ivet. ath. 159 (25), o. 2, G. Perelma. The etropy formula for the Ricci flow ad its geometric applicatios, S., Kolodziej. The complex oge-ampere equatio ad pluripotetial theory. em.amer. ath. Soc. 178 (25), o. 84, x+64 pp. 27. S., Kolodziej. Hölder cotiuity of solutios to the complex oge-ampère equatio with the right had side i L p. The case of compact Kähler maifolds. ArXiv, math.cv/ J. Sog ad G. Tia. The Kähler-Ricci flow o surfaces of positive Kodaira dimesio. Ivetioes math., 17(27), o. 3, G. Tia. Kähler-Eistei metrics with positive scalar curvature. Ivet. ath. 13 (1997), o. 1, G. Tia ad X. H. Zhu Covergece of Kähler-Ricci flow. J. Amer. ath. Soc. 2 (27), o. 3, G. Tia ad Z. Zhag. O the Kähler-Ricci flow o projective maifolds of geeral type. Chiese Aals of athematics - Series B, Volume 27, Number 2, S. T. Yau. O the Ricci curvature of a compact Kähler maifold ad the complex oge- Ampère equatio, I. Comm. Pure Appl. ath. 31 (1978), o. 3, Z. Zhag. O Degeerate oge-ampère Equatios over Closed Kähler aifolds. It. ath. Res. Not. 26, Art. ID 6364, 18 pp. 34. Z. Zhag. Degeerate oge-ampere Equatios over Projective aifolds. PHD Thesis at IT, 26.

15 ON THE WEAK KÄHLER-RICCI FLOW 15 Departmet of athematics, Uiversity of Wiscosi, adiso, WI address: Departmet of athematics, Priceto Uiversity, Priceto, NJ address: Departmet of athematics, Uiversity of ichiga, at A Arbor, I address: zhagou@umich.edu