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THE CONICAL KA¨HLER-RICCI FLOW WITH WEAK INITIAL DATA ON FANO MANIFOLD 6 1 0 JIAWEILIUANDXIZHANG 2 y a Abstract. Inthispaper,weprovethelong-timeexistenceanduniquenessof M the conical Kähler-Ricciflow withweak initialdata whichadmits Lp density for some p > 1 on Fano manifold. Furthermore, we study the convergence 7 behavior ofthisflow. 2 ] G 1. Introduction D . ConicalKähler-EinsteinmetricplaysanimportantroleinsolvingtheYau-Tian- h Donaldson’sconjecture(see[6,7,8,45]). Therehasbeenrenewedinterestinconical t a Kähler-Einsteinmetric in recentyears,seereferences [1, 3, 4, 18, 24,26, 30,43]etc. m On the other hand, the conical Kähler-Ricci flow was introduced to attack the [ existence problem of conical Kähler-Einstein metric. The long-time existence and limit behaviour of the conical Kähler-Ricci flow has been widely studied. Chen- 2 v Wang[11]studiedthestrongconicalKähler-Ricciflowandobtainedtheshort-time 0 existence, Y.Q. Wang [48] and the authors [34] got the long-time existence of the 6 conical Kähler-Ricci flow respectively. In [34], the authors also considered the 0 convergence of this flow on Fano manifold with positive twisted first Chern class, 0 they proved that, for any cone angle 0 < 2πβ < 2π, the conical Kähler-Ricci flow 0 . converges to a conical Kähler-Einstein metric if there exists one. Chen-Wang [12] 1 obtained the convergence result of this flow when the twisted first Chern class 0 is negative or zero. Later, by adopting the smooth approximation used by the 6 1 authors in [34], L.M. Shen [38][39] generalized Song-Tian’s result [41] and Tian- : Zhang’s result [46] to the unnormalizedconical Kähler-Ricciflow, and G. Edwards v i [17]obtainedtheuniformscalarcurvatureboundwhenthetwistedfirstChernclass X is negative on the foundation of Song-Tian’s result [42]. r In [34], the authors studied the conical Kähler-Ricci flow which starts with a a model conical Kähler metric (1.1) ω =ω +√ 1k∂∂¯s2β β 0 − | |h on Fano manifold, where ω c (M) is a smooth Kähler metric, s is the defining 0 1 ∈ section of a smooth divisor D λK and h is a smooth Hermitian metric on M ∈|− | λK with curvature λω . In [11, 12, 48], Chen-Wang studied the existence of M 0 − the conical Kähler-Ricci flow from initial (α,β) metric or weak (α,β) metric with other assumptions. In this paper, we mainly study the long-time existence, uniqueness and conver- genceofthe conicalKähler-Ricciflowwithsomeweakinitialdatawhichadmits Lp Keywordsandphrases. twistedKähler-Ricciflow,conicalKähler-Ricciflow,weakinitialdata. AMSMathematics SubjectClassification. 53C55,32W20. 1 2 JIAWEILIUANDXIZHANG densitywith p>1onFanomanifold. We considerthe conicalKähler-Ricciflowby using smooth approximation of the twisted Kähler-Ricci flows as that in [34]. Let M be a Fano manifold with complex dimension n, ω c (M) be a smooth 0 1 ∈ Kähler metric. For any p (0, ], we define the class ∈ ∞ (ω +√ 1∂∂¯ϕ)n (1.2) (M,ω )= ϕ (M,ω ) 0 − Lp(M,ωn) , Ep 0 ∈E 0 | ωn ∈ 0 (cid:8) 0 (cid:9) where the class (1.3) (M,ω )= ϕ PSH(M,ω ) (ω +√ 1∂∂¯ϕ)n = ωn E 0 ∈ 0 |Z 0 − Z 0 (cid:8) M M (cid:9) defined by Guedj-Zeriahi in [21] is the largest subclass of PSH(M,ω ) on which 0 the operator(ω +√ 1∂∂¯)n is well defined and the comparisonprinciple is valid. 0 − · Whenp>1,byS.Kolodziej’sLp-estimate[29]andS.Dinew’s uniquenesstheorem [14] (see also Theorem B in [21]), we know that the functions in (M,ω ) are p 0 E Ho¨lder continuous with respect to ω on M. 0 LetD beadivisoronM. Bysayingaclosedpositive(1,1)-currentω 2πc (M) 1 ∈ with locally bounded potential is a conical Kähler metric with cone angle 2πβ (0 < β < 1 ) along D, we mean that ω is a smooth Kähler metric on M D, and \ near each point p D, there exists local holomorphic coordinate (z1, ,zn) in ∈ ··· a neighborhood U of p such that locally D = zn = 0 , and ω is asymptotically { } equivalent to the model conical metric n 1 − (1.4) √ 1zn 2β 2dzn dzn+√ 1 dzj dzj on U. − − | | ∧ − ∧ X j=1 Assume that D λK (λ Q), µ = 1 (1 β)λ, ωˆ c (M) is a M 1 ∈ | − | ∈ − − ∈ Kähler current which admits Lp density with respect to ωn for some p > 1 and 0 ωˆn = ωn. We study the long-time existence, uniqueness and convergence of M M 0 tRhe followRing conical Kähler-Ricci flow with weak initial data ωˆ ∂ω(t) = Ric(ω(t))+µω(t)+(1 β)[D]. ∂t − −  (1.5)  ω(t) =ωˆ t=0 | From now on, we denote the Kähler current ωˆ = ω +√ 1∂∂¯ϕ := ω with 0 − 0 ϕ0 ϕ (M,ω ) for some p>1. 0 p 0 ∈E Definition 1.1. We call ω(t) a long-time solution to the conical Kähler-Ricci flow (1.5) if it satisfies the following conditions. For any [δ,T] (δ,T >0), there exist constant C such that • C−1ωβ ω(t) Cωβ on [δ,T] (M D); ≤ ≤ × \ On (0, ) (M D), ω(t) satisfies the smooth Kähler-Ricci flow; • ∞ × \ On (0, ) M, ω(t) satisfies equation (1.5) in the sense of currents; • ∞ × Thereexistsmetricpotentialϕ(t) C0 [0, ) M C (0, ) (M D) ∞ • ∈ ∞ × ∩ ∞ × \ such that ω(t)=ω0+√ 1∂∂¯ϕ(t) and(cid:0)lim ϕ(t)(cid:1) ϕ0 L(cid:0)∞(M) =0; (cid:1) − t 0+k − k On [δ,T], there exist constant α (0,1)→and C such that the above metric ∗ • ∈ petential ϕ(t) is Cα on M with respect to ω0 and k∂ϕ∂(tt)kL∞(M\D) 6C∗. THE CONICAL KA¨HLER-RICCI FLOW WITH WEAK INITIAL DATA 3 In Definition 1.1, by saying ω(t) satisfies equation (1.5) in the sense of currents on (0, ) M :=M , we mean that for any smooth (n 1,n 1)-formη(t) with compa∞ct s×upport in (∞0, ) M, we have − − ∞ × ∂ω(t) η(t,x)dt= ( Ric(ω(t))+µω(t)+(1 β)[D]) η(t,x)dt, Z ∂t ∧ Z − − ∧ M∞ M∞ where the integral on the left side can be written as ∂ω(t) ∂η(t,x) η(t,x)dt= ω(t) dt Z ∂t ∧ −Z ∧ ∂t M∞ M∞ it the sense of currents. We study the conical Kähler-Ricci flow (1.5) by using the following twisted Kähler-Ricci flow with weak initial data ω . ϕ0 ∂ωε(t) = Ric(ω (t))+µω (t)+θ , ∂t − ε ε ε  (1.6)  ω (t) =ω , ε |t=0 ϕ0  where θ =(1 β)(λω +√ 1∂∂log(ε2+ s2)) is a smooth closed positive (1,1)- ε − 0 − | |h form,sisthedefinitionsectionofD andhisasmoothHermitianmetricon λK M − with curvature λω . The smooth case of the twisted Kähler-Ricciflow was studied 0 in [13, 17, 19, 20, 32, 33, 34, 38, 48], etc. TherearesomeimportantresultsontheKähler-Ricciflow(aswellasitstwisted versionswith smoothtwisting form)fromweakinitial data,suchasChen-Ding [5], Chen-Tian [9], Chen-Tian-Zhang [10], Guedj-Zeriahi [23], Lott-Zhang [35], Nezza- Lu[36], Song-Tian[41], Székelyhidi-Tosatti[44], Z.Zhang [49]. Here, we study the Kähler-Ricciflowwhichis twistedby non-smoothtwisting form,i.e. the flow(1.5). We first obtain the long-time existence, uniqueness and regularityof the flow (1.6) byfollowingSong-Tian’sargumentsin[41]. Thenwe study the long-timeexistence ofthe conicalKähler-Ricciflow(1.5) by approximatingmethod. Inthis process,in addition to getting the locally uniform regularity of the twisted Kähler-Ricci flow (1.6),the mostimportantstepistoprovethatϕ(t)convergestoϕ inL -normas 0 ∞ t 0+ ( i.e the 4th property in Definition 1.1), where ϕ(t) is a metric potential of → ω(t) with respect to metric ω . Here we need a new idear because of Song-Tian’s 0 method in [41] is invalid. At the same time, we provethe uniqueness ofthe conical Kähler-Ricci flow by Jeffres’ trick [25] and an improvement of the arguments in [48]. In fact, we obtain the following theorem. Theorem 1.2. Let M be a Fano manifold with complex dimension n, ω c (M) 0 1 ∈ be a smooth Kähler metric on M, divisor D λK (λ Q) and ωˆ c (M) be M 1 ∈|− | ∈ ∈ a Kähler current which admits Lp density with respect to ωn for some p > 1 and 0 ωˆn = ωn. Then for any β (0,1), there exists a unique solution ω(t, ) to M M 0 ∈ · tRhe conicaRl Kähler-Ricci flow (1.5) with weak initial data ωˆ. Then we consider the convergence of the conical Kähler-Ricci flow (1.5). When λ>0 and there is no nontrivial holomorphic field which is tangent to D along D, Tian-Zhu[47]provedaMoser-TrudingertypeinequalityforconicalKähler-Einstein manifold and gave a new proof of Donaldson’s openness theorem [16]. Using the Moser-Trudinger type inequality in [47] and following the arguments in [34], we obtain the following convergence result of the conical Kähler-Ricci flow (1.5). 4 JIAWEILIUANDXIZHANG Theorem 1.3. Assume that λ>0 and there is no nontrivial holomorphic field on M tangent to D, if there exists a conical Kähler-Einstein metric with cone angle 2πβ (0<β <1)alongD,thentheconicalKähler-Ricciflow(1.5)mustconvergeto this conical Kähler-Einstein metric in C topology outside divisor D and globally l∞oc in the sense of currents on M. Remark 1.4. In this paper, we only study the convergence with positive twisted first Chern class, i.e. µ = 1 (1 β)λ > 0. When µ 0, one can also get the − − ≤ convergence of the conical Kähler-Ricci flow by following Chen-Wang’s argument in [12]. The paper is organized as follows. In section 2, we prove the long-time exis- tenceanduniquenessofthetwistedKähler-Ricciflow(1.6)byadoptingSong-Tian’s methods in [41]. In section 3, we obtain the existence of a long-time solution to the conical Kähler-Ricci flow (1.5) by limiting the twisted Kähler-Ricci flows, and prove that ϕ(t) converges to ϕ in L -norm as t 0+, where ϕ(t) is a metric 0 ∞ → potential of ω(t) with respect to the metric ω . We also prove the uniqueness of 0 the conicalKähler-Ricciflow with weakinitial data ω . In section4, by using the ϕ0 uniformPerelman’sestimatesalongthetwistedKähler-Ricciflowsobtainedin[34], we prove the convergence theorem under the assumptions in Theorem 1.3. Acknowledgement: The authors would like to thank Professor Jiayu Li for providing many suggestions and encouragements. The first author also would like to thank Professor Xiaohua Zhu for his constant help and encouragements. The secondauthor is partially supported by the NSFC Grants 11131007and 11571332. 2. The long-time existence of the twisted Ka¨hler-Ricci flow with weak initial data In this section, we prove the long-time existence and uniqueness of the twisted Kähler-Ricci flow (1.6) by following Song-Tian’s arguments in [41]. For further consideration in the next section, we shall pay attention to the estimates which are independent of ε. In the following arguments, for the sake of brevity, we only consider the flow (1.6) in the case of λ = 1 (i.e. µ = β), where β (0,1). Our ∈ arguments are also valid for any λ, only if the coefficient β before ω (t) in the case ε of λ=1 is replaced by µ=1 (1 β)λ. We denote − − (ω +√ 1∂∂¯ϕ )n F = 0 − 0 Lp(M,ωn) for p>1. ωn ∈ 0 0 Recall that C (M) is dense in Lp(M,ωn). Therefore there exists a sequence of ∞ 0 positive functions F C (M) such that F ωn = ωn and j ∈ ∞ M j 0 M 0 R R lim F F =0. j Lp(M) j k − k →∞ By considering the complex Monge-Ampère equation (2.1) (ω +√ 1∂∂¯ϕ )n =F ωn 0 − 0,j j 0 and using the stability theorem in [28] (see also [15] or [22]), we have (2.2) lim ϕ0,j ϕ0 L∞(M) =0, j k − k →∞ where ϕ PSH(M,ω ) C (M) satisfy sup(ϕ ϕ )=sup(ϕ ϕ ). 0,j 0 ∞ 0 0,j 0,j 0 ∈ ∩ − − M M THE CONICAL KA¨HLER-RICCI FLOW WITH WEAK INITIAL DATA 5 Let ω = ω +√ 1∂∂ϕ . We prove the long-time existence of the twisted ϕ0,j 0 − 0,j Kähler-Ricci flow (1.6) by using a sequence of smooth twisted Kähler-Ricci flows ∂ωε,j(t) = Ric(ω (t))+βω (t)+θ .  ∂t − ε,j ε,j ε (2.3)  ω (t) =ω ε,j |t=0 ϕ0,j Since the twisted Kähler-Ricci flow preserves the Kähler class, we can write the flow (2.3) as the parabolic Monge-Ampére equation on potentials, ∂ϕε∂,tj(t) =log(ω0+√−1ω∂n∂¯ϕε,j(t))n +F0  0 (2.4) +βϕ (t)+log(ε2+ s2)1 β,  ε,j | |h − ϕ (0)=ϕ ε,j 0,j where F0 satisfies−Ric(ω0)+ω0 = √−1∂∂F0, V1 Me−F0dV0 = 1 and dV0 = ωn0n!. By using the function R (2.5) χ(ε2+ s2)= 1 |s|2h (ε2+r)β −ε2βdr | |h β Z r 0 whichwas givenby Campana-Guenancia-Pa˘un in [4], we canrewrite the flow (2.4) as ∂φε∂,tj(t) =log(ωε+√−1ω∂εn∂¯φε,j(t))n +Fε+β(φε,j(t)+kχ(ε2+|s|2h)), (2.6)  φ (0)=ϕ kχ(ε2+ s2):=φ ε,j 0,j − | |h ε,0,j  where φ (t) = ϕ (t) kχ(ε2 + s2), ω = ω + √ 1k∂∂χ(ε2 + s2), F = F0 +logε(,jωωε0nn · (ε2 +ε,j|s|2h)−1−β). We |kn|how thεat χ(ε02 +|s−|2h) and Fε are| |uhniforεmly bounded (see (15) and (25) in [4]). Proposition 2.1. For any T > 0, there exists constant C depending only on ϕ0 L∞(M), β, n, ω0 and T such that for any t [0,T], ε>0 and j N+, k k ∈ ∈ (2.7) φε,j(t) L∞(M) C. k k ≤ Furthermore, for any j,l, we have (2.8) φε,j(t) φε,l(t) L∞([0,T] M) eβT ϕ0,j ϕ0,l L∞(M). k − k × ≤ k − k In particular, ϕ (t) satisfies ε,j { } (2.9) lim ϕε,j(t) ϕε,l(t) L∞([0,T] M) =0. j.l k − k × →∞ Proof: From equation (2.6), we have ∂e βtφ (t) (e βtω +√ 1∂∂¯e βtφ (t))n − ∂tε,j = e−βtlog − ε (e−βtω )−n ε,j − ε +e βt(F +kβχ(ε2+ s2)) − ε | |h (e βtω +√ 1∂∂¯e βtφ (t))n ≤ e−βtlog − ε (e−βtω )−n ε,j +Ce−βt, − ε which is equivalent to ∂∂t e−βt(φε,j(t)+ Cβ) ≤e−βtlog(cid:0)e−βtωε+√−(1e∂∂¯βet−ωβ)tn(φε,j(t)+ Cβ)(cid:1)n, (cid:0) (cid:1) − ε 6 JIAWEILIUANDXIZHANG where constant C depends only on ϕ0 L∞(M), β, n and ω0. k k For any δ > 0, we denote φ˜ (t) = e βt(φ (t)+ C) δt. Let (t ,x ) be the ε,j − ε,j β − 0 0 maximumpoint of φ˜ (t) on [0,T] M. If t >0,by maximum principle, we have ε,j 0 × ∂ C 0 e βt(φ (t)+ ) (t ,x ) δ ≤ ∂t − ε,j β 0 0 − (cid:0) (cid:1) e βtω +√ 1∂∂¯φ˜ (t) n e βtlog − ε − ε,j (t ,x ) δ ≤ − (cid:0) (e βtω )n (cid:1) 0 0 − − ε δ, ≤ − which is impossible. Hence t =0, then 0 C φ (t) eβtsupφ (0)+δTeβT + (eβT 1). ε,j ε,j ≤ β − M Let δ 0, we obtain → C (2.10) φ (t) eβtsupφ (0)+ (eβT 1). ε,j ε,j ≤ β − M By the same arguments, we can get the lower bound of φ (t) ε,j C (2.11) φ (t) eβtinfφ (0) (eβT 1). ε,j ε,j ≥ M − β − Combining (2.10) and (2.11), we have C kφε,j(t)kL∞(M) ≤eβTkφε,j(0)kL∞(M)+ β(eβT −1)≤C, where constant C depends only on ϕ0 L∞(M), β, n, ω0 and T. k k Let ψ (t)=φ (t) φ (t), then ψ satisfies the following equation ε,j,l ε,j ε,l ε,j,l − (2.12) ∂ψε∂,jt,l(t) =log(cid:0)ωε+√−(1ω∂ε∂¯+φ√ε,l−(t1)∂+∂¯√φε−,l1(∂t)∂¯)ψnε,j,l(t)(cid:1)n +βψε,j,l(t). ψ (0)=ϕ ϕ ε,j,l 0,j 0,l  − By the same arguments as that in the first part, we have ψε,j,l(t) L∞([0,T] M) eβT ϕ0,j ϕ0,l L∞(M). k k × ≤ k − k Since ϕ is a Cauchy sequence in L -norm, we conclude the limit (2.9). (cid:3) 0,j ∞ { } We now prove the uniform equivalence of the volume forms along the complex Monge-Ampère flow (2.6). Lemma2.2. ForanyT >0,thereexistsconstantC dependingonlyon ϕ0 L∞(M), k k n, β, ω and T such that for any t (0,T], ε>0 and j N+, 0 ∈ ∈ tn (ω +√ 1∂∂¯φ (t))n (2.13) ε − ε,j eCt. C ≤ ωn ≤ ε Proof: Let ∆ be the Laplacian operator associated to the Kähler form ε,j ω (t)=ω +√ 1∂∂¯φ (t). Straightforwardcalculations show that ε,j ε ε,j − ∂ (2.14) ( ∆ )φ˙ (t)=βφ˙ (t). ε,j ε,j ε,j ∂t − THE CONICAL KA¨HLER-RICCI FLOW WITH WEAK INITIAL DATA 7 Let H+ (t) = tφ˙ (t) Aφ (t), where A is a sufficiently large number (for ε,j ε,j − ε,j example A = βT + 2). Then H+ (0) = Aφ (0) is uniformly bounded by a ε,j − ε,j constant C depending only on ϕ0 L∞(M), β, n, ω0 and T. k k ∂ ( ∆ )H+ (t) = (1+βt A)φ˙ (t)+A∆ φ (t) ∂t − ε,j ε,j − ε,j ε,j ε,j (2.15) 6 (1+βt A)φ˙ (t)+An. ε,j − Bythemaximumprinciple, H+ (t)isuniformlyboundedfromabovebyaconstant ε,j C depending only on ϕ0 L∞(M), n, β, ω0 and T. k k LetHε−,j(t)=φ˙ε,j(t)+φε,j(t)−nlogt. ThenH−(t) tends to+∞ast→0+ and ∂ n (2.16) (∂t −∆ε,j)Hε−,j(t)=(β+1)φ˙ε,j(t)+trωε,j(t)ωε−n− t. Assumethat(t0,x0)istheminimumpointofHε−,j(t)on[0,T]×M. Weconclude that t >0 and there exists constant C , C and C such that 0 1 2 3 ∂ ωn ωn (t) C (∂t −∆ε,j)Hε−,j(t)|(t0,x0) ≥ C1(ωnε(t))n1 +C2log εω,jn − t3 |(t0,x0) (cid:0) ε,j ε (cid:1) C ωn C (2.17) ≥ 21(ωnε(t))n1 − t3 |(t0,x0), (cid:0) ε,j (cid:1) where constant C depends only on n, C depends only on β and C depends only 1 2 3 on n, ω0, ϕ0 L∞(M), β and T. In inequality (2.17), without loss of generality, we k k assume that ωεnω,jεn(t) > 1 and C21(ωεnω,jεn(t))n1 +C2logωεnω,jεn(t) ≥ 0 at (t0,x0). By the maximum principle, we have (2.18) ωn (t ,x ) C tnωn(x ), ε,j 0 0 ≥ 4 ε 0 where C4 independent of ε and j. Then it easily follows that Hε−,j(t) is bounded from below by a constant C depending only on ϕ0 L∞(M), n, β, ω0 and T. (cid:3) k k In the following lemma, we prove the uniform equivalence of the metrics along the twisted Kähler-Ricci flow (2.3). Lemma2.3. ForanyT >0,thereexistsconstantC dependingonlyon ϕ0 L∞(M), k k n, β, ω and T such that for any t (0,T], ε>0 and j N+, 0 ∈ ∈ (2.19) e−Ctωε ωε,j(t) eCtωε. ≤ ≤ Proof: Let 1 |s|2h (ε2+r)ρ ε2ρ (2.20) Ψ =B − dr ε,ρ ρZ r 0 betheuniformboundfunctionintroducedbyGuenancia-Pa˘unin[24]. Bychoosing suitable B and ρ, and following the arguments in section 2 of [34], we have ∂ ( ∆ )(tlogtr ω (t)+tΨ ) ∂t − ε,j ωε ε,j ε,ρ (2.21) logtr ω (t)+Ctr ω +C, ≤ ωε ε,j ωε,j(t) ε where constant C depends only on n, β, ω and T. 0 8 JIAWEILIUANDXIZHANG LetH (t)=tlogtr ω (t)+tΨ Aφ (t),Abeasufficientlylargeconstant ε,j ωε ε,j ε,ρ− ε,j and(t ,x )be themaximumpointofH (t)on[0,T] M. We needonly consider 0 0 ε,j × t >0. By the inequality 0 1 ωn (t) (2.22) tr ω (t) (tr ω )n 1 ε,j , ωε ε,j ≤ (n 1)! ωε,j(t) ε − ωn − ε we conclude that ∂ ( ∆ )H (t) logtr ω (t)+Ctr ω Aφ˙ (t) Atr ω +C ∂t − ε,j ε,j ≤ ωε ε,j ωε,j(t) ε− ε,j − ωε,j(t) ε A ωn (t) (n 1)logtr ω tr ω (A 1)log ε,j +C, ≤ − ωε,j(t) ε− 2 ωε,j(t) ε− − ωn ε where constant C depends only on ϕ0 L∞(M), n, β, ω0 and T. k k Withoutlossofgenerality,weassumethat Atr ω +(n 1)logtr ω −4 ωε,j(t) ε − ωε,j(t) ε ≤ 0 at (t ,x ). Then at (t ,x ), by Lemma 2.2, we have 0 0 0 0 ∂ A (2.23) ( ∆ )H (t) tr ω Clogt+C. ∂t − ε,j ε,j ≤−4 ωε,j(t) ε− By the maximum principle, at (t ,x ), 0 0 1 (2.24) tr ω Clog +C. ωε,j(t) ε ≤ t By using inequality (2.22), at (t ,x ), 0 0 1 (2.25) trωεωε,j(t)≤C(log t +1)n−1eCt ≤e2tC, where constant C depends only on ϕ0 L∞(M), n, β, ω0 and T. Hence we have k k (2.26) trωεωε,j(t)≤eCt for some constant C depending only on ϕ0 L∞(M), n, β, ω0 and T. k k Furthermore, by inequality (2.22) again, we know (2.27) trωε,j(t)ωε ≤eCt, where constant C depends only on ϕ0 L∞(M), n, β, ω0 and T. From (2.26) and k k (2.27), we prove the lemma. (cid:3) By Lemma 2.3 and the fact that ω > γω for some uniform constant γ (see ε 0 inequality (24) in [4]), we have (2.28) e−Ctω0 ωε,j(t) CεeCtω0, ≤ ≤ on(0,T] M, where C is a uniformconstantandC depends onε. We nextprove ε × the Calabi’s C3-estimates. Denote (2.29) Sε,j =|∇ω0ωε,j(t)|2ωε,j(t) =gεim,¯jgεk,¯ljgεp,q¯j∇0i(gε,j)kq¯∇0m(gε,j)p¯l. Lemma 2.4. For any T > 0 and ε >0, there exist constants C and C such that ε for any t (0,T] and j N+, ∈ ∈ (2.30) Sε,j CεeCt, ≤ where constant C depends only on ϕ0 L∞(M), n, β, ω0 and T, and constant Cε k k depends in addition on ε. THE CONICAL KA¨HLER-RICCI FLOW WITH WEAK INITIAL DATA 9 Proof: By the similar arguments in [33] or [34] and choosing sufficiently large α and β, we have ∂ (2.31) ( ∆ε,j)(e−2tαSε,j) Cεe−αtSε,j +Cε, ∂t − ≤ ∂ (2.32) (∂t −∆ε,j)(e−2tγtrω0ωε,j(t)) ≤ Cε−Cε−1e−3tγSε,j. By choosing A =C (C +1) and α=3γ, ε ε ε ∂ (2.33) (∂t −∆ε,j)(e−2tαSε,j +Aεe−2tγtrω0ωε,j(t))≤−e−3tγSε,j +Cε. By the maximum principle, we have (2.34) Sε,j CεeCt on (0,T] M ≤ × for some constant C depending only on ϕ0 L∞(M), n, β, ω0 and T, and constant k k C depending in addition on ε. (cid:3) ε By using the Schauder regularity theory and equation (2.4), we get the high order estimates of ϕ (t). ε,j Proposition 2.5. For any 0<δ <T < , ε>0 and k 0, there exists constant ∞ ≥ Cε,δ,T,k depending only on δ, T, ε, k, n, β, ω0 and ϕ0 L∞(M), such that for any k k j N+, ∈ (2.35) ϕ (t) C . ε,j ε,δ,T,k k kCk [δ,T] M ≤ × (cid:0) (cid:1) By (2.9), for any T > 0, ϕ (t) converges to ϕ (t) L ([0,T] M) uniformly ε,j ε ∞ ∈ × in L ([0,T] M). For any 0 < δ < T < and ε > 0, ϕ (t) is uniformly ∞ ε,j × ∞ bounded(depends onε)in C ([δ,T] M). Thereforeϕ (t) convergesto ϕ (t)in ∞ ε,j ε × C ([δ,T] M). Hence for any ε>0, ϕ (t) C ((0, ) M). ∞ ε ∞ × ∈ ∞ × Proposition 2.6. For any ε>0, ϕ (t) C0([0, ) M) and ε ∈ ∞ × (2.36) lim ϕε(t) ϕ0 L∞(M) =0. t 0+k − k → Proof: For any (t,z) (0,T] M, ∈ × ϕ (t,z) ϕ (z) ϕ (t,z) ϕ (t,z) + ϕ (t,z) ϕ (z) ε 0 ε ε,j ε,j 0,j | − | ≤ | − | | − | (2.37) +ϕ (z) ϕ (z). 0,j 0 | − | Since ϕ (t) is a Cauchy sequence in L ([0,T] M), ε,j ∞ × (2.38) lim ϕε(t,z) ϕε,j(t,z) L∞([0,T] M) =0. j k − k × →∞ From (2.2), we have (2.39) lim ϕ0,j(z) ϕ0(z) L∞(M) =0, j k − k →∞ For any ǫ>0, there exists N such that for any j >N, ǫ sup ϕ (t,z) ϕ (t,z) < , ε ε,j | − | 3 [0,T] M × ǫ sup ϕ (z) ϕ (z) < . 0,j 0 | − | 3 M 10 JIAWEILIUANDXIZHANG On the other hand, fix such j, there exists 0<δ <T such that ǫ (2.40) sup ϕ (t,z) ϕ < . ε,j 0,j | − | 3 [0,δ] M × Combining the above estimates together, for any t [0,δ] and z M, ∈ ∈ (2.41) ϕ (t,z) ϕ (z) <ǫ. ε 0 | − | This completes the proof of the lemma. (cid:3) Proposition2.7. ϕ (t)istheuniquesolutiontotheparabolicMonge-Ampèreequa- ε tion ∂ϕ∂εt(t) =log(ω0+√−ω1∂n∂¯ϕε(t))n +F0  0 (2.42) +βϕ (t)+log(ε2+ s2)1 β, (0, ) M  ε | |h − ∞ × ϕ (0)=ϕ ε 0  in the space of C0 [0, ) M C (0, ) M . ∞ ∞ × ∩ ∞ × (cid:0) (cid:1) (cid:0) (cid:1) Proof: By proposition 2.6, we only need to prove the uniqueness. Suppose there exists another solution ϕ˜ (t) C0 [0, ) M C (0, ) M to the ε ∞ ∈ ∞ × ∩ ∞ × Monge-Ampère equation (2.42). (cid:0) (cid:1) (cid:0) (cid:1) Let ψ (t)=ϕ˜ (t) ϕ (t). Then ε ε ε − (2.43) ∂ψ∂εt(t) =log(cid:0)ω0+√−(ω10∂+∂¯√ϕε−(1t)∂+∂¯√ϕε−(1t)∂)∂n¯ψε(t)(cid:1)n +βψε(t). ψ (0)=0 ε  For any T >0, by the same arguments as that in the proof of Proposition 2.1, we have ψε(t) L∞([0,T] M) eβT ψε(0) L∞(M) =0. k k × ≤ k k Hence ψ (t)=0, that is ϕ˜ (t)=ϕ (t). (cid:3) ε ε ε By the similar arguments as that in [41], we prove the uniqueness theorems of the twisted Kähler-Ricci flow. Theorem 2.8. Let M be a Fano manifold with complex dimension n, ω c (M) 0 1 ∈ be a smooth Kähler metric on M and ωˆ c (M) be a Kähler current which admits 1 ∈ Lp density with respect to ωn for some p > 1 and ωˆn = ωn. Then there 0 M M 0 exists a unique solution ω (t) C (0, ) M toRthe twisteRd Kähler-Ricci flow ε ∞ ∈ ∞ × (1.6) with initial data ωˆ in the follow(cid:0)ing sense. (cid:1) (1) ∂ωε(t) = Ric(ω (t))+βω (t)+θ on (0, ) M; ∂t − ε ε ε ∞ × (2) There exists ϕ (t) C0 [0, ) M C (0, ) M such that ω (t) = ε ∞ ε ∈ ∞ × ∩ ∞ × ω +√ 1∂∂¯ϕ (t) and (cid:0) (cid:1) (cid:0) (cid:1) 0 ε − (2.44) lim ϕε(t) ϕ0 L∞(M) =0, t 0+k − k → where ϕ (M,ω ) is a metric potential of ωˆ with respect to ω . In particular, 0 p 0 0 ∈ E ω (t) converges in the sense of distribution to ωˆ as t 0. ε →