Conical Kähler–Ricci flows on Fano manifolds

Advances in Mathematics 307 (2017) 1324–1371

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Advances in Mathematics www.elsevier.com/locate/aim

Conical Kähler–Ricci flows on Fano manifolds Jiawei Liu a , Xi Zhang b,∗ a

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China b Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China

a r t i c l e

i n f o

Article history: Received 9 June 2014 Received in revised form 7 December 2016 Accepted 8 December 2016 Available online 27 December 2016 Communicated by Gang Tian MSC: 53C55 32W20

a b s t r a c t In this paper, we study the long-term behavior of conical Kähler–Ricci flows on Fano manifolds. First, by proving uniform regularities for twisted Kähler–Ricci flows, we prove the existence of conical Kähler–Ricci flows by limiting these twisted flows. Second, we obtain uniform Perelman’s estimates along twisted Kähler–Ricci flows by improving the original proof. After that, we prove that if there exists a conical Kähler–Einstein metric, then conical Kähler–Ricci flow must converge to it. © 2016 Elsevier Inc. All rights reserved.

Keywords: Twisted Kähler–Ricci flow Conical Kähler–Ricci flow Conical Kähler–Einstein metric

1. Introduction In recent years, since conical Kähler–Einstein metrics play an important role in the solution of Yau–Tian–Donaldson’s conjecture by Tian [50] and Chen–Donaldson– Sun [9–11], the existence and geometry of conical Kähler–Einstein metrics have been * Corresponding author. E-mail addresses: [email protected] (J. Liu), [email protected] (X. Zhang). http://dx.doi.org/10.1016/j.aim.2016.12.002 0001-8708/© 2016 Elsevier Inc. All rights reserved.

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widely concerned. This type of metrics were studied on Riemann surfaces by McOwen [34] and Troyanov [55], and the existence of these metrics in higher dimensions was first conjectured and examined by Tian [49]. Tian also anticipated that these conical Kähler–Einstein metrics should converge to the complete Tian–Yau Kähler–Einstein metric on the complement of a divisor when the cone angles tend to zero. In [19], Donaldson proposed using these metrics to construct smooth Kähler–Einstein metrics on Fano manifolds by taking a continuity method to deform cone angles of these metrics. Later, Jeffres–Mazzeo–Rubinstein [23] proved Tian’s conjecture on the existence of these metrics along smooth divisors. Berman [2], Brendle [5], Campana–Guenancia–Păun [6], Eyssidieux–Guedj–Zeriahi [20], Guenancia–Păun [21], Li–Sun [25], Mazzeo [30], Mazzeo– Rubinstein [31,32], Song–Wang [46] and Tian [51] etc. also did a lot of work in such a field. For more details, readers can refer to [40] for a survey of the developments. Inspired by these exciting developments, we study the long-time existence and convergence of conical Kähler–Ricci flows to research the existence of conical Kähler–Einstein metrics in this paper. Let (M, ω0 ) be a Fano manifold of complex dimension n, β ∈ (0, 1), D ∈ |−KM | be a smooth divisor, h be a Hermitian metric on −KM with curvature ω0 and s be the defining section of D. Conical Kähler–Ricci flow takes the following form  ∂ω(t) ∂t

= −Ric(ω(t)) + βω(t) + (1 − β)[D],

ω(t)|t=0 = ω∗

(1.1)

√ ¯ 2β . For where [D] is the current of integration along D and ω∗ := ω0 + k −1∂ ∂|s| h small k ∈ R, ω∗ is a conical Kähler metric with cone angle 2πβ along D (see Lemma 2.2 in [23]). Yin [60,61] and Mazzeo–Rubinstein–Sesum [33] studied Ricci flows with conical singularities on surfaces. In higher dimensions, conical Kähler–Ricci flows were first proposed in Jeffres–Mazzeo–Rubinstein’s paper (see Section 2.5 in [23]). Song–Wang (conjecture 5.2 in [46]) made a conjecture on the relation between the convergence of these flows and the greatest Ricci lower bound of M . Definition 1.1. We call ω(t) a long-time solution to conical Kähler–Ricci flow (1.1) if there   √ ¯ exists a metric potential ϕ(t) ∈ C ∞ (M \ D) × [0, ∞) such that ω(t) = ω0 + −1∂ ∂ϕ(t) and satisfies the following conditions. • For any [0, T ], there exists constant C such that C −1 ω∗ ≤ ω(t) ≤ Cω∗

on (M \ D) × [0, T ];

• on (M \ D) × [0, ∞), ω(t) satisfies smooth Kähler–Ricci flow; • on M × [0, ∞), ω(t) satisfies equation (1.1) in the sense of currents; • on [0, T ], there exist constant α ∈ (0, 1) and C ∗ such that ϕ(t) is C α on M with ∗ ∞ respect to ω0 and  ∂ϕ(t) ∂t L (M \D)  C .

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In [50], Tian approximated conical Kähler–Einstein metrics by smooth twisted Kähler–Einstein metrics. Naturally, we approximate conical Kähler–Ricci flows by smooth twisted Kähler–Ricci flows. More precisely, what we have achieved is listed as follows. Based on uniform estimates for twisted Kähler–Ricci flows (Proposition 3.1 and 3.3), we obtain a long-time solution of conical Kähler–Ricci flow (1.1) by using smooth approximation. Then for any β ∈ (0, 1), we consider uniform Perelman’s estimates and Sobolev inequalities along these twisted flows. By the arguments in [27,41], Perelman’s estimates mainly depend on the bound of initial twisted scalar curvatures. But these curvatures may not be bounded uniformly when β ∈ ( 12 , 1). In order to overcome this, we need a key observation (Proposition 5.1) that for any β ∈ (0, 1) the twisted scalar curvatures are uniformly bounded from below along twisted Kähler–Ricci flows when t ≥ 1. Using this observation, we obtain uniform Perelman’s estimates on [1, +∞), which are sufficient to study the convergence of conical Kähler–Ricci flows. At last, we prove that if there exists a conical Kähler–Einstein metric, then conical Kähler–Ricci flow (1.1) must converge to it. This result extends Tian–Zhu’s convergence theorem [53, 54] to the conical case. In fact, we prove the following theorem. Theorem 1.2. Let (M, ω0 ) be a Fano manifold with a smooth divisor D ∈ |−KM |. For any β ∈ (0, 1), there exists a unique long-time solution to conical Kähler–Ricci flow (1.1). Moreover, if there exists a conical Kähler–Einstein metric ωβ,D with cone angle 2πβ ∞ along D, conical Kähler–Ricci flow (1.1) must converge to ωβ,D in Cloc -topology outside 1 α,β divisor D and globally in C -sense for any α ∈ (0, min{ β − 1, 1}). Remark 1.3. If we assume that D ∈ |−λKM | (λ ∈ Q) is a smooth divisor, h is a smooth Hermitian metric on −λKM with curvature λω0 and s is the defining section of D, then the existence theorem is valid for any λ ∈ Q and the convergence result is valid for λ ≥ 1, only if the coefficient β before ω(t) in (1.1) is replaced by 1 − (1 − β)λ. In the first version of this paper, we only proved the global convergence in the sense of currents. The reviewers indicated that this result can be strengthened by C 2,α,β -estimates. In this version, by using the priori estimates obtained by Chu [15] (see also Chen–Wang’s work [14]) and uniform Perelman’s estimates proved in section 5, we improve the global convergence in C α,β -sense. In [56], Wang also considered the long-time existence of conical Kähler–Ricci flows by using limiting method. Wang proved uniform estimates of twisted Kähler–Ricci flows away from D on any time interval [δ, T ] with δ > 0 by using parabolic Evans–Krylov estimates while we obtain uniform estimates away from D on any time interval [0, T ] (see Proposition 3.3). Our arguments are based on the elliptic Schauder estimates which are superior to the parabolic one here, because the latter can only provide us with uniform estimates depending on δ, on Br × [δ, T ] for δ > 0. In this paper, we also study the convergence of conical Kähler–Ricci flow with positive twisted first Chern class, which has not been considered before. After the first version [29] of this paper appeared on

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arXiv, Chen–Wang [14] proved the convergence result when the twisted first Chern class is negative or zero. In [3], Berman–Boucksom–Eyssidieux–Guedj–Zeriahi studied the convergence of Kähler–Ricci flows on Q-Fano varieties with log terminal singularities. By using variational methods, they proved that if the Mabuchi functional is proper, Kähler–Ricci flow must converge to the unique Kähler–Einstein metric in some weak sense. Although their work is independent of Perelman’s estimates, their weakly convergence can’t yield the convergence in C ∞ -topology even if M is non-singular. Here, our goal is to prove the ∞ Cloc -convergence outside D and global C α,β -convergence of conical Kähler–Ricci flows by ∞ using uniform Perelman’s estimates of twisted flows. By adopting uniform Cloc -estimates in this paper, Shen [42] generalized Song–Tian’s [45] and Tian–Zhang’s [52] existence theorems to unnormalized conical Kähler–Ricci flows. Shen [43] also generalized Tian’s elliptic C 2,α -estimates [51] to parabolic case. The paper is organized as follows. In section 2, we set up basic notations for conical Kähler–Einstein metrics and conical (twisted) Kähler–Ricci flows. In section 3, we prove ∞ uniform Laplacian estimates and Cloc -estimates for twisted Kähler–Ricci flows. Then, we get a long-time solution of conical Kähler–Ricci flow (1.1) by limiting a sequence of twisted Kähler–Ricci flows in section 4. In section 5, for any β ∈ (0, 1), we prove uniform Perelman’s estimates along twisted Kähler–Ricci flows when t ≥ 1. In section 6, by choosing suitable initial values, we obtain uniform C 0 -estimates for metric potentials under the assumptions that the twisted Mabuchi K-energy functionals are uniformly proper. After that, we study the convergence of conical Kähler–Ricci flows in the last section. 2. Preliminaries 2.1. Conical Kähler–Einstein metric Let M be a compact Kähler manifold of complex dimension n, and D be a smooth divisor. By saying that a closed positive (1, 1)-current ω with locally bounded potentials is conical Kähler metric with cone angle 2πβ (0 < β ≤ 1) along D, we mean that ω is smooth Kähler metric on M \ D. And near each point p ∈ D, there exists local holomorphic coordinate (z 1 , · · · , z n ) in a neighborhood U of p such that D = {z n = 0} locally. ω is asymptotically equivalent to model conical metric √

−1

n−1 

dz j ∧ dz j +

√

−1|z n |2β−2 dz n ∧ dz n

on U.

(2.1)

j=1

Let ω0 be a smooth Kähler metric and D ⊂ M be a smooth divisor which satisfies c1 (M ) = μ[ω0 ] + (1 − β)c1 (D) with μ ∈ R.

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Definition 2.1. We call ω a conical Kähler–Einstein metric with conic angle 2πβ along D if it is a conical Kähler metric and satisfies Ric(ω) = μω + (1 − β)[D] on M.

(2.2)

Equation (2.2) is classical outside D and it holds in the sense of currents on M . There are other definitions of metrics with conical singularities (see [19,23], etc.). But for conical Kähler–Einstein metrics, these definitions turn out to be equivalent (see Theorem 2 in [23]). 2.2. Twisted Kähler–Ricci flow In [44], by assuming 2πc1 (M ) − β[ω0 ] = [α] = 0 as well as fixing a closed (1, 1)-form θ ∈ [α], Song–Tian introduced twisted Kähler–Einstein metric Ric(ω) = βω + θ.

(2.3)

For more subsequent work, see [2,47,63]. The twisted Kähler–Ricci flow ∂ω(t) = −Ric(ω(t)) + βω(t) + θ ∂t

(2.4)

was studied by Liu [27,28] and Collins–Székelyhidi [16]. When θ = 0, twisted Kähler– Ricci flow turns into Kähler–Ricci flow which was first introduced by Cao [7] to give a parabolic proof of the Calabi–Yau theorem. There are some interesting results on the convergence of Kähler–Ricci flows, see references [8,12,13,18,35–38,44], etc. In particular, on Fano manifolds, Tian–Zhu [53,54] proved that if there exists a Kähler–Einstein metric, then Kähler–Ricci flow with initial metric in c1 (M ) must converge to a Kähler–Einstein metric in C ∞ -topology. The main result in this paper extends theirs to conical case.  t 2 β 2β Denote χ(ε2 + t) = β1 0 (ε +r)r −ε dr. In accordance with [6], for sufficiently small k, √ ωε = ω0 + −1k∂∂χ(ε2 + |s|2h ) are Kähler forms for each ε > 0, and there exists uniform constant (independent of ε) γ > 0 satisfying ωε ≥ γω0 .

(2.5)

In this paper, we use the following twisted Kähler–Ricci flows  ∂ω

ϕε

∂t

= −Ric(ωϕε ) + βωϕε + (1 − β)(ω0 +

ωϕε |t=0 = ωε

√

−1∂∂ log(ε2 + |s|2h ))

(2.6)

√ to approach conical Kähler–Ricci flow (1.1), where ωϕε = ωε + −1∂∂ϕε (t) and (1 − √ β)(ω0 + −1∂∂ log(ε2 + |s|2h )) are smooth closed semi-positive (1,1)-forms. Since twisted

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Kähler–Ricci flows remain in the same Kähler class, we transcribe (2.6) as parabolic Monge–Ampère equations on potentials. 

∂ϕε ∂t

= log

n ωϕ ε ωεn

+ Fε + β(kχ + ϕε ),

ϕε |t=0 = cε0

(2.7)

where constants cε0 (see (6.9) in section 6) are uniformly bounded for ε, Fε = F0 +  √ ωn log( ωεn · (ε2 + |s|2h )1−β ), F0 satisfies −Ric(ω0 ) + ω0 = −1∂∂F0 and V1 M e−F0 dV0 = 1, 0 and χ denotes function χ(ε2 + |s|2h ). 3. Local estimates for twisted Kähler–Ricci flows ∞ In this section, we will prove uniform Laplacian estimates and Cloc -estimates for parabolic Monge–Ampère equations (2.7). In the following sections, by saying a uniform constant, we mean that it is independent of ε and t. We shall use the letter C for a uniform constant which may differ from line to line. We first follow Guenancia–Paun’s arguments [21] to obtain uniform Laplacian estimates.

Proposition 3.1. Let ϕε be solutions of equations (2.7). Assume that there exists a uniform constant C > 0 such that (1) sup |ϕε | ≤ C; M ×[0,T ]

(2)

sup |ϕ˙ ε | ≤ C.

M ×[0,T ]

Then there exists a constant A only depending on ω0 , n, β and C such that A−1 ωε ≤ ωε +

√

−1∂∂ϕε ≤ Aωε ,

on M × [0, T ].

We notice that these estimates are independent of T , thus the above result holds also for time interval [0, +∞). Let us review the appropriate coordinate system (see Lemma 4.1 in [6]). Lemma 3.2. Let (L, h) be the hermitian line bundle associated to a smooth divisor D, and s be a section of L such that D = {s = 0}. Let p0 ∈ D, then there exists an open set Ω ⊂ M centered at p0 , such that for any point p ∈ Ω there exists a coordinate system z = (z 1 , · · · , z n ) and a trivialization η for L such that:  (1) D Ω = {z n = 0}; (2) With respect to the trivialization η, the metric h has the weight ϕ such that ϕ(p) = 0, dϕ(p) = 0, |

∂ |α|+|β| ϕ (p)| ≤ Cα,β ∂z α ∂ z¯β

for constants Cα,β depending only on the multi indexes α, β.

(3.1)

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Proof of Proposition 3.1. Through direct computations, we have (

ωϕn d 1 − Δωϕε ) log trωε ωϕε = (Δωε (ϕ˙ε − log nε ) + Rωε ) dt trωε ωϕε ωε

(3.2)

¯

−

gϕδkε ∂δ trωε ωϕε ∂k¯ trωε ωϕε gεγ s¯ϕεγ tp ϕε¯stp 1 ¯ (gϕp¯qε gϕε m¯j Rωmj ) + { − }. q ε p¯ trωε ωϕε (trωε ωϕε )2 trωε ωϕε

Then we choose a local coordinate system w = (w1 , . . . , wn ) to make (gεi¯i ) identity and (gϕε i¯i ) a diagonal matrix. Since (gϕε i¯i ) is positive definite, we have gϕε i¯i = 1 + ϕεi¯i > 0. It was shown by Aubin [1] and Yau [58] that ¯

gϕδkε ∂δ trωε ωϕε ∂k¯ trωε ωϕε gεγ s¯ϕεγ tp ϕε¯stp − ≤ 0. (trωε ωϕε )2 trωε ωϕε

(3.3)

By substituting (2.7) and (3.3) into (3.2), we have (

 1+ϕ 1 + ϕεjj d 1 εii − Δωϕε ) log trωε ωϕε ≤ − ( + − 2)Rωε iijj (w) dt trωε ωϕε 1 + ϕεjj 1 + ϕεii i≤j

1 βn + (Δωε Fε ) + + β, trωε ωϕε trωε ωϕε

(3.4)

where we use the inequality n = trωε ω0 + kΔωε χ ≥ kΔωε χ. First of all, we deal with the √ ¯ 0 ≥ −Cω0 . Then by term Δωε F0 . There exists a uniform constant C such that −1∂ ∂F (2.5), we have √ ¯ 0 + Cω0 ) ≤ γ −1 (Cn + Δω F0 ), 0 ≤ trωε ( −1∂ ∂F 0 and thus −Cγ −1 ≤ Δωε F0 ≤ γ −1 (Cn + Δω0 F0 ),

(3.5)

which shows that Δωε F0 is uniformly bounded. Claim. By Guenancia–Păun’s arguments in [21], there exists uniform constant C such that −

 1+ϕ 1 + ϕεjj εii ( + − 2)Rωε iijj (w) − trωε ωϕε Δωϕε Ψε,ρ 1 + ϕεjj 1 + ϕεii i≤j

+ Δωε log( ≤C

ωεn · (ε2 + |s|2h )1−β ) ω0n

 1+ϕ 1 + ϕεjj εii ( + ) + Ctrωϕε ωε · trωε ωϕε + C, 1 + ϕεjj 1 + ϕεii i≤j

(3.6)

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

˜ ρ (ε2 + |s|2 ), χρ (ε2 + |s|2 ) = where Ψε,ρ = Cχ h h be determined later.

1 ρ

 |s|2h 0

(ε2 +r)ρ −ε2ρ dr r

1331

and constant C˜ will

Proof. To reader’s convenience, we give the proof here briefly. We first deal with the terms Rωε iijj (w). At point p, we choose coordinate system z = (z 1 , . . . , z n ) in Lemma 3.2. Then the coefficients of the holomorphic bisectional curvature change as follows, Rωε i¯il¯l (w) = Rωε p¯qr¯s (z)

∂z p ∂z q ∂z r ∂z s . ∂wi ∂wi ∂wl ∂wl

(3.7)

At the point p, we have √ ωε ≥ C

−1dz n ∧ d¯ zn n 2 1−β + |z | )

(3.8)

(ε2

∂ for uniform constant C independent of ε and p. Since ( ∂w k ) is unit at the point p under metric ωε ,

|

∂z n 2 | ≤ C(ε2 + |z n |2 )1−β . ∂wk

(3.9)

From the arguments in [21], at the point p, we have Rωε iijj (w) ≥ −C1 ((I) + (II ) + (III )) − C2 ,

(3.10)

where C1 and C2 are uniform constants independent of ε and the point p, (I) =

 i,j

(II ) =

 i,j

(III ) =

∂z n ∂z n 1 || |, 1 | (ε2 + |z n |2 ) 2 ∂wi ∂wj ∂z n 2 ∂z n 1 | | |, 1 | (ε2 + |z n |2 ) 2 ∂wi ∂wj

 i,j

ε2

∂z n ∂z n 1 | i |2 | j |2 . n 2 + |z | ∂w ∂w

Now we need to deal with (I), (II ) and (III ). Taking the coefficients of as an example, by (3.9), the coefficients can be dominated by (I) ≤

 j

(II ) ≤

 j

(III ) ≤

1 (ε2 + |z n |2 )

|

(ε2

in (3.4)

 ∂z n C ∂z n |≤ | j |2 + C, j 2 n 2 β ∂w (ε + |z | ) ∂w j

 ∂z n ∂z n 2 1 C | ≤ | | + C, 1 | (ε2 + |z n |2 )2β−1 ∂wj (ε2 + |z n |2 )β− 2 ∂wj j

 j

β 2

1+ϕεii 1+ϕεjj

∂z n C | j |2 , n 2 β + |z | ) ∂w

(3.11)

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ωn

where constant C is independent of ε and the point p. At the same time, |Δωε log( ωεn · 0 (ε2 + |s|2h )1−β )| can be dominated by  j

∂z n C | j |2 , ˜ (ε2 + |z n |2 )β ∂w

(3.12)

where β˜ = max(β, 1 − β) and C is independent of ε (see section 5.2 in [21]). The choice of the function χρ above is motivated by the following equality √

¯ ρ (ε2 + |s|2 ) = −1∂ ∂χ h

√

−1

D s, D s 1 − ((ε2 + |s|2h )ρ − ε2ρ )ω0 . (ε2 + |s|2h )1−ρ β

(3.13)

Corresponding to ωϕε , we evaluate the Laplacian of the function Ψε,ρ . ˜ ω ωϕ trω ωε + C˜ trωε ωϕε Δωϕε Ψε,ρ ≥ −Ctr ε ε ϕε

n 

(

j=1

1 ∂z n 2 trωε ωϕε | | ). (ε2 + |s|2h )1−ρ ∂wj 1 + ϕεj¯j

˜ we can cancel the terms ˜ and 1 − ρ > β, After taking sufficiently large uniform constant C in (3.11) and (3.12). We prove the claim. 2 Combining (3.5) with (3.6), we have (

  d C  1 − Δωϕε )(log trωε ωϕε + Ψε,ρ ) ≤ )( (1 + ϕεjj )) + n ( dt trωε ωϕε 1 + ϕεii j i + Ctrωϕε ωε +

C +C trωε ωϕε

≤ Ctrωϕε ωε + C. Here we use the fact n ≤ trωϕε ωε · trωε ωϕε in the last inequality. Hence we get (

d − Δωϕε )(log trωε ωϕε + Ψε,ρ − Bϕε ) ≤ −trωϕε ωε + C, dt

where B = C + 1. By the maximum principle, at maximum point p of log trωε ωϕε + Ψε,ρ − Bϕε , trωϕε ωε (p) ≤ C. Since Fε is uniformly bounded (see (25) in [6]), we obtain trωε ωϕε (p) ≤

ωϕn 1 (trωϕε ωε )n−1 (p) nε (p) ≤ C exp(ϕ˙ε − Fε − βϕε − kβχ)(p) ≤ C. (n − 1)! ωε

Hence we have

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

trωε ωϕε ≤ exp(C + Bϕε − Bϕε (p)) ≤ C.

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(3.14)

On the other hand, considering the assumptions on ϕε and ϕ˙ ε , we have C −1 ≤

(ωε +

√

¯ ε )n −1∂ ∂ϕ = exp(ϕ˙ ε − Fε − βϕε − kβχ) ≤ C. n ωε

(3.15)

By (3.14) and (3.15), there exists a uniform constant A such that A−1 ωε ≤ ωε + for any ε and t.

√

−1∂∂ϕε ≤ Aωε

(3.16)

2

∞ Now we consider local Calabi’s C 3 -estimates and Cloc -estimates for twisted Kähler– Ricci flow

∂ωϕ = −Ric(ωϕ ) + λωϕ + θ, (3.17) ∂t √ where c1 (M ) = λ[ω0 ] + [θ], ωϕ = ω0 + −1∂∂ϕ and θ is a smooth closed (1, 1)-form. The above flow is equivalent to equation ωϕn ∂ϕ = log n + f + λϕ, ∂t ω0 where f is the twisted Ricci potential, that is, ¯

√

(3.18)

−1∂∂f = −Ric(ω0 ) + λω0 + θ. Let

¯

S = |∇0 gϕ |2ωϕ = gϕij gϕkl gϕp¯q ∇0i gϕkq¯∇0¯j gϕp¯l , where ∇0 denotes the covariant derivative with respect to metric ω0 . Defining hi k = g0ij gϕjk and Xilk = (∇i h · h−1 )kl , by direct computations, we have Xilk = Γkϕil − Γk0il , S=

|X|2ωϕ ,

k s ∇ϕm V kl − ∇0m V kl = Xms V sl − Xml V ks .

(3.19) (3.20) (3.21)

Here we let ∇ϕ and Γϕ be covariant derivative and Christoffel symbols respectively under metric ωϕ , and Γ0 be the Christoffel symbol with respect to ω0 . In the following arguments, norms  · C k and  · C k,α are all related to metric ω0 unless there is a special statement. We denote the curvature tensors of ωϕ by Rmϕ . Proposition 3.3. Let ϕ(·, t) be a solution of equation (3.18) and satisfy N −1 ω0 ≤ ωϕ ≤ N ω0

on

Br (p) × [0, T ].

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Then there exist constant C  and C  such that on B r2 (p) × [0, T ], we have C , r2 C  ≤ 4. r

S≤ |Rmϕ |2ωϕ

The constant C  depends only on ω0 , N , λ, ϕ(·, 0)C 3 (Br (p)) and θC 1 (Br (p)) ; constant C  depends only on ω0 , N , λ, ϕ(·, 0)C 4 (Br (p)) and θC 2 (Br (p)) . Furthermore, there exist constants Ck1 , Ck2 and Ck3 such that, for any k ≥ 0, |Dk Rmϕ |2ωϕ ≤ Ck1 , ϕ ˙ C k+1,α ≤ Ck2 , ϕC k+3,α ≤ Ck3 on B r2 (p) × [0, T ]. Here constants Ck1 , Ck2 and Ck3 depend only on ω0 , N , λ, ϕ(·, 0)C k+4 (Br (p)) , θC k+2 (Br (p)) , ϕC 0 (Br (p)×[0,T ]) and f C 0 (Br (p)) . Proof. By direct calculations, we have (

d μ q − Δωϕ )S = gϕmγ gϕμβ gϕlα ((gϕβ¯s ∇ϕm θs¯l − ∇ϕ R0βlqm )Xγα dt β + Xml (gϕμs ∇ϕγ θsα − ∇qϕ R0μαqγ )) β μ − Xml Xγα (θp¯q gϕpγ gϕmq gϕμβ gϕlα − gϕmγ θμβ gϕlα + gϕmγ gϕμβ gϕpα gϕlq θp¯q )

− |∇ϕ X|2ωϕ − |∇ϕ X|2ωϕ − λS, s ∇ϕm θl¯q = ∇0m θl¯q − Xml θs¯q , β s s ∇ϕp R0βlqm = ∇0p R0βlqm + Xps R0slqm − Xpl R0βsqm − Xpm R0βlqs .

Hence, the evolution equation of S can be developed as (

d − Δωϕ )S ≤ C(S + 1) − |∇ϕ X|2ωϕ − |∇ϕ X|2ωϕ , dt

(3.22)

where constant C depends only on N , λ, Rm(ω0 )C 1 (Br (p)) and θC 1 (Br (p)) . Let r = r0 > r1 > 2r and ψ be a nonnegative C ∞ cut-off function that is identically equal to 1 on Br1 (p) and vanishes outside Br (p). We also assume that √ ¯ ω ≤ C. |∂ψ|2ω0 , | −1∂ ∂ψ| 0 r2 Through computations, we obtain

(3.23)

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(

1335

d C B − Δωϕ )(ψ 2 S + Btrh) ≤ ( 2 − )S + (B + 1)C. dt r N

(3.24)

Let (x0 , t0 ) be the maximum point of ψ 2 S + Btrh on Br (p) × [0, T ]. If t0 = 0, then S is bounded by initial data ϕ(·, 0)C 3 (Br (p)) . Then we assume that t0 > 0 and that x0 doesn’t lie in the boundary of Br (p). By the maximum principle, 0≤(

C B − )S(x0 , t0 ) + (B + 1)C. r2 N

(3.25)

Taking B = N (C+1) , then S(x0 , t0 ) ≤ C, where C is independent of T . Since 0 ≤ trh ≤ r2 nN , we have S ≤ C + BnN ≤

C r2

on Br1 (p) × [0, T ],

(3.26)

where constant C depends only on N , λ, ϕ(·, 0)C 3 (Br (p)) , θC 1 (Br (p)) and ω0 . By (3.19) and (3.21), (

d − Δωϕ )Rϕ¯ji¯lk = +Rϕ¯jip¯q Rϕ¯lkq¯p + Rϕ¯lip¯q Rϕ¯jkq¯p − Rϕ¯jp¯lq¯Rϕ pi¯qk − Rϕp¯l Rϕ¯ji pk dt − Rϕ¯jh Rϕ hi¯lk − ∇ϕ¯l ∇ϕk θi¯j + λRϕ¯ji¯lk − θ¯jh Rϕh ik¯l . ¯

s s s θs¯j − Xki ∇0¯l θs¯j + Xki X¯lt¯j θst¯, ∇ϕ¯l ∇ϕk θi¯j = ∇0¯l ∇0k θi¯j − X¯ls¯¯j ∇0k θi¯s − ∇0¯l Xki i i i = ∂k¯ Xjl = −Rϕi lkj ∇0k¯ Xjl ¯ + R0 lkj ¯ .

(3.27)

Combining the above equalities with (3.26), we have (

|Rmϕ |ωϕ d − Δωϕ )|Rmϕ |2ωϕ ≤ C(|Rmϕ |3ωϕ + 1 + ) − |∇ϕ Rmϕ |2ωϕ − |∇ϕ Rmϕ |2ωϕ . dt r2

We fix a smaller radius r2 satisfying r1 > r2 > 2r . Let ρ be a cut-off function identically equal to 1 on Br2 (p) and identically equal to 0 outside Br1 . Let ρ satisfy √ ¯ ω ≤ C |∂ρ|2ω0 , | −1∂ ∂ρ| 0 r2 ˆ C r 2 and 2 2 |Rmϕ |ωϕ

for uniform constant C. Let K =

constant Cˆ be large enough such that

K 2

≤

K−S ≤ K. We consider F = ρ K−S +AS. Alike to the previous part, we only consider an inner point (x0 , t0 ) which is a maximum point of F achieved on Br1 (p) × [0, T ]. Using the fact that ∇F = 0 at this point, then we get 2ρ∇ϕ ρ

|Rmϕ |2ωϕ K −S

+ ρ2

∇ϕ |Rmϕ |2ωϕ K −S

+ ρ2

|Rmϕ |2ωϕ ∇ϕ S (K − S)2

Without loss of generality, we assume that |Rmϕ |3ωϕ ≥ 1 +

+ A∇ϕ S = 0.

|Rmϕ |ωϕ r2

(3.28)

at (x0 , t0 ). Then

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(

d − Δωϕ )|Rmϕ |2ωϕ ≤ C|Rmϕ |3ωϕ − |∇ϕ Rmϕ |2ωϕ − |∇ϕ Rmϕ |2ωϕ , dt d C ( − Δωϕ )S ≤ 2 − |∇ϕ X|2ωϕ − |∇ϕ X|2ωϕ dt r

(3.29) (3.30)

on Br1 (p) × [0, T ]. We also note that |∇ϕ |Rmϕ |2ωϕ |ωϕ ≤ |Rmϕ |ωϕ (|∇ϕ Rmϕ |ωϕ + |∇ϕ Rmϕ |ωϕ ), |∇ϕ S|2ωϕ ≤ 2S(|∇ϕ X|2ωϕ + |∇ϕ X|2ωϕ ).

(3.31) (3.32)

Putting (3.28)–(3.32) into the evolution equation of F , at (x0 , t0 ),

(

C|Rmϕ |2ωϕ Cρ2 |Rmϕ |2ωϕ d AC − Δωϕ )F ≤ −A(|∇ϕ X|2ωϕ + |∇ϕ X|2ωϕ ) + 2 + + dt r Kr2 K 2 r2 − − +

ρ2 |Rmϕ |2ωϕ (|∇ϕ X|2ωϕ + |∇ϕ X|2ωϕ ) K2 ρ2 (|∇ϕ Rmϕ |2ωϕ + |∇ϕ Rmϕ |2ωϕ ) K ρ2 (|∇ϕ Rmϕ |2ωϕ + |∇ϕ Rmϕ |2ωϕ ) K

Let Cˆ be sufficiently large so that |∇ϕ X|2ωϕ . By (3.27), we have Cρ2 |Rmϕ |3ωϕ K

≤

8ASQ K

≤

AQ 2 ,

ρ2 |Rmϕ |2ωϕ Q K2

+

+

+

Cρ2 |Rmϕ |3ωϕ K

(3.33)

C|Rmϕ |2ωϕ Kr2 8AS(|∇ϕ X|2ωϕ + |∇ϕ X|2ωϕ ) K

.

where we denote Q = |∇ϕ X|2ωϕ +

+ Cρ2 |Rmϕ |2ωϕ .

So the evolution equation of F can be controlled as follows, (

d AQ AC ˜ + C. − Δωϕ )F ≤ − + 2 + CQ dt 2 r

(3.34)

Now we choose a sufficiently large A such that A ≥ 2(C˜ + 1) and then obtain Q ≤ rC2 at (x0 , t0 ). This implies that |Rmϕ |2ωϕ ≤ rC2 at this point, where C depends only on N , λ, S, θC 2 (Br (p)) and ω0 . Following that F is bounded by rC2 at (x0 , t0 ), where the constant C is independent of T . Hence on Br2 (p) × [0, T ], |Rmϕ |2ωϕ ≤

C , r4

where C depends only on N , λ, ϕ(·, 0)C 4 (Br (p)) , θC 2 (Br (p)) and ω0 .

(3.35)

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Now, we prove C ∞ -estimates of ϕ on B r2 (p). Here, when we say that ϕ is C k,α , we mean that its C k,α -norm can be controlled by a constant which depends only on ω0 , N , λ, r, θC k−1 (Br (p)) , f C 0 (Br (p)) , ϕ(·, 0)C k+1 (Br (p)) and ϕC 0 (Br (p)×[0,T ]) . Likewise, replacing ϕ by ϕ, ˙ we mean that the C k,α -norm of ϕ˙ can be controlled by a constant that depends only on ω0 , N , λ, r, θC k+1 (Br (p)) , f C 0 (Br (p)) , ϕ(·, 0)C k+3 (Br (p)) and ϕC 0 (Br (p)×[0,T ]) . Since |Rmϕ |ωϕ ≤ C on Br2 (p), ϕ˙ is C 1,α . Differentiating equation (3.18), we get ∂f ∂ϕ d ∂ϕ ∂ϕ ¯ ∂g0i¯ ¯ ∂g0i¯ j j = Δωϕ k + gϕij − g0ij + k +λ k. k k k dt ∂z ∂z ∂z ∂z ∂z ∂z

(3.36)

By Calabi’s C 3 -estimates, ϕ is C 2,α and the coefficients of Δωϕ are C 0,α . Since f is the twisted Ricci potential, then Δω0 f = −R(ω0 ) + λn + trω0 θ.

(3.37)

Hence the C 1,α -norm of f on Br2 (p) only depends on ω0 , θC 0 (Br (p)) and f C 0 (Br (p)) . By the standard elliptic Schauder estimates, ϕ is C 3,α on Br3 (p) × [0, T ], where 2r < r3 < r2 . Using the estimates |∇ϕ θ|ωϕ ≤ C, ¯ ϕ ∇ϕ θ|ω ≤ C(1 + |∇ϕ Rmϕ |ω + |∇ϕ X|ω ), |∇ϕ ∇ ϕ ϕ ϕ

(3.38) (3.39)

we have (

d ¯ ϕ ∇ϕ Rmϕ |2 − Δωϕ )|∇ϕ Rmϕ |2ωϕ ≤ −|∇ϕ ∇ϕ Rmϕ |2ωϕ − |∇ ωϕ dt + C|∇ϕ Rmϕ |2ωϕ + |∇ϕ X|2ωϕ + C,

where C depends only on N , λ, θC 0 (Br (p)) and |Rmϕ |2ωϕ . Let  be a cut-off function, identically equal to 1 on Br3 (p) and identically equal to 0 outside Br2 . As mentioned before, we can assume that √ |∂|2ω0 , | −1∂∂|ω0 ≤ C for uniform constant C depending only on ω0 , N and r. Defining H = 2 |∇ϕ Rmϕ |2ωϕ + S + B|Rmϕ |2ωϕ , where B will be determined later. (

d − Δωϕ )H ≤ (C − 2B)|∇ϕ Rmϕ |2ωϕ + C. dt

Let (x0 , t0 ) be the maximum point of H on Br2 (p) × [0, T ]. We assume that t0 > 0 and that x0 doesn’t lie in the boundary of Br2 (p). Taking 2B = C + 1, by the maximum principle, at this point, we have

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J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

|∇ϕ Rmϕ |2ωϕ ≤ C,

(3.40)

where C depends only on N , λ, r, θC 3 (Br (p)) , |Rmϕ |2ωϕ and ω0 . Then at (x0 , t0 ), H is bounded by C which is independent of T . Hence, on Br3 (p) × [0, T ], we obtain |∇ϕ Rmϕ |2ωϕ ≤ C,

(3.41)

where C depends only on N , λ, r, ϕ(·, 0)C 5 (Br (p)) , θC 3 (Br (p)) and ω0 . Differentiating equation (3.17), √ D −1∂ ∂¯ϕ˙ = DRic(ωϕ ) + Dθ, where D denotes the covariant derivative with respect to the metric ωϕ . Taking trace on both side with the metric ωϕ , we have |Δωϕ Dϕ| ˙ ≤ |Rmϕ |ωϕ |∇ϕ| ˙ ωϕ + |DRmϕ |ωϕ + C|X|ωϕ + C.

(3.42)

Since ϕ˙ is C 1,α , |Rmϕ |ωϕ , |DRmϕ |ωϕ and |X|ωϕ are uniformly bounded, we conclude that Dϕ˙ is C 1,α , and ϕ˙ is thus C 2,α . Differentiating equation (3.18) for two times and using the elliptic Schauder estimates, ϕ is C 4,α on Br4 (p) × [0, T ], where r2 < r4 < r3 . We can iterate this procedure by induction and then obtain the uniform bounds for |Dk Rmϕ |2ωϕ , ϕ ˙ C k+1,α and ϕC k+3,α on Brk+3 (p) × [0, T ] for any k ≥ 2, where rk+2 > rk+3 > r2 and these uniform bounds depend only on N , λ, r, ω0 , ϕ(·, 0)C k+4 (Br (p)) , ϕC 0 (Br (p)×[0,T ]) , θC k+2 (Br (p)) and f C 0 (Br (p)) . For more detailed process, readers can see the online version of [29]. 2 Remark 3.4. Considering only the regularities for a single flow (3.18), we can obtain ∞ uniform Cloc -estimates of ϕ by standard Schauder estimates of parabolic equation (see [26]) after getting Calabi’s C 3 -estimates and curvature estimates. Since we want to get conical Kähler–Ricci flow (1.1) by limiting a sequence of twisted Kähler–Ricci flows (2.7) as ε → 0, we need to get uniform C ∞ -estimates of ϕε (·, t) on Br × [0, T ], where Br ⊂⊂ M \ D. But by applying parabolic Schauder estimates, we can only get uniform C ∞ -estimates of ϕε (·, t) on Br × [δ, T ]. Here δ > 0 and these uniform estimates depend on δ. This is the reason why we apply elliptic estimates in the proof of Proposition 3.3. We can also note that the estimates are independent of T , so the results also hold for intervals [0, +∞). 4. The long-time solution to conical Kähler–Ricci flow In this section, we use the estimates obtained in the preceding section to give a long-time solution to conical Kähler–Ricci flow (1.1). We prove

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Theorem 4.1. Assume β ∈ (0, 1). Then there exists a sequence {εi } satisfying εi → 0 as i → +∞ such that the flows (2.7) converge to 

∂ϕ ∂t

ωn

2(1−β)

= log ωϕn + F0 + β(k|s|2β h + ϕ) + log |s|h

(4.1)

0

ϕ|t=0 = c0

√ ∞ ¯ in Cloc -topology on (M \ D) × [0, ∞). Meanwhile, ωϕ(t) = ω∗ + −1∂ ∂ϕ(t) is a long-time solution to conical Kähler–Ricci flow (1.1). Proof. Differentiating equation (2.7) with respect to t, we have d ϕ˙ ε (t) = Δωϕε (t) ϕ˙ ε (t) + β ϕ˙ ε (t). dt

(4.2)

According to the maximum principle, we have sup |ϕ˙ ε (t)| ≤ sup |eβt ϕ˙ ε (0)|, M

where ϕ˙ ε (0) = log

ωεn (ε2 +|s|2h )1−β ω0n

(4.3)

M

+ F0 + β(kχ + cε0 ). Therefore, sup |ϕ˙ ε (t)| ≤ Ceβt for M

uniform constant C. Then on M ×[0, T ], we have ϕε (t)C 0 ≤ CeβT . By Proposition 3.1, there exists a constant C(T ) satisfying C −1 (T )ωε ≤ ωϕε ≤ C(T )ωε on M × [0, T ].

(4.4)

For any K ⊂⊂ M \ D, we have 1 ω0 ≤ ωϕε ≤ N ω0 , N

(4.5)

where the uniform constant N depends only on K and C(T ). Since initial data kχ + cε0 , twisted Ricci potentials F0 + log(ε2 + |s|2h )1−β of ω0 and twisted forms θε = (1 − β)(ω0 + √ −1∂∂ log(ε2 +|s|2h )) are C ∞ uniformly bounded on K, then by Proposition 3.3, ϕε +kχ are C ∞ bounded uniformly on K×[0, T ]. Let K approximate to M \D and T approximate to ∞. By the diagonal rule, we get a sequence which we denote {εi } such that ϕεi (t) ∞ -topology outside divisor D to a function ϕ(t) that is smooth on M \ D. converge in Cloc From (4.4), ωϕ(t) is conical Kähler metric with cone angle 2πβ along D. Next, we prove that the limit ωϕ(t) satisfies conical Kähler–Ricci flow (1.1) in the sense of currents on M × [0, +∞). Let η = η(x, t) be a smooth (n − 1, n − 1)-form with compact support in M × [0, ∞). Without loss of generality, we assume that its compact n ωϕ

support is included in [0, T ). On M × [0, T ], log εi and ϕ(t) are uniformly bounded. On [0, T ], we have

(ε2i +|s|2h )1−β ω0n

2(1−β)

, log

n ωϕ(t) |s|h ω0n

, ϕεi (t)

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∂ωϕεi ∧η = ∂t

M

 = ε →0

√

−1∂ ∂¯

∂ϕεi (t) ∧η ∂t

M



log

ωϕnε (ε2i + |s|2h )1−β i

ω0n

M



−−i−−→



n ωϕ(t)

(log

ω0n

M

√

+ F0 + βϕεi (t)

2(1−β)

+ F0 + βϕ(t) + log |s|h

¯ −1∂ ∂η

√ ¯ ) −1∂ ∂η



(−Ric(ωϕ(t) ) + βωϕ(t) + (1 − β)[D]) ∧ η.

= M

At the same time, it holds that  ωϕεi ∧

∂η εi →0 −−−−→ ∂t

M

 ωϕ(t) ∧

∂η . ∂t

(4.6)

M ∂ϕ

εi (t) On the other hand, ϕεi (t) and ∂t are uniformly bounded on M × [0, T ] while ϕ(t) ∂ϕ(t) and ∂t are uniformly bounded on (M \ D) × [0, T ]. Therefore,



∂ ∂t

ωϕεi

∂ ∧ η −−−−→ ∂t εi →0

M

 ωϕ(t) ∧ η.

(4.7)

M

On [0, T ], we have ∂ ∂t







ωϕ(t) ∧ η = M

M

 − Ric(ωϕ(t) ) + βωϕ(t) + (1 − β)[D] ∧ η

 ωϕ(t) ∧

+

∂η . ∂t

(4.8)

M

Integrating form 0 to ∞ on both sides, we obtain  M ×[0,∞)



∂ω(t) ∧ η dt = − ∂t

ωϕ(t) ∧

M ×[0,∞)



=



∂η dt − ∂t

 ω(0) ∧ η(x, 0) M

 − Ric(ωϕ(t) ) + βωϕ(t) + (1 − β)[D] ∧ η dt.

M ×(0,∞)

By the arbitrariness of η, we prove that ωϕ(t) satisfies conical Kähler–Ricci flow (1.1) in the sense of currents on M × [0, ∞). 2

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Proposition 4.2. For any t ∈ [0, +∞), there exists a constant α ∈ (0, 1) such that ϕ(t) is α-Hölder continuous with respect to ω0 on M . Moreover, ϕ(t)C α (M ) is bounded by a constant C which depends only on n, β, ω0 , ϕ(t) ˙ C 0 (M \D) and ϕ(t)C 0 (M \D) . Proof. Let φ = ϕ + k|s|2β h . Flow (4.1) can be written as (ω0 +

√

˙

¯ n = eφ−F0 −βφ −1∂ ∂φ)

ω0n 2(1−β) |s|h

on M \ D.

(4.9)

Since β ∈ (0, 1), there exists δ such that 2(1 − β)(1 + δ) < 2.  e

2(1−β)

˙ (φ−F 0 −βφ−log |s|h

M

 )(1+δ)

dV0 ≤ C

1

2(1−β)(1+δ) |s|h M

dV0 ≤ C,

where constant C depends only on n, β, ω0 , ϕ(t) ˙ C 0 (M \D) and ϕ(t)C 0 (M \D) . Then by Kolodziej’s Lp -estimates [24], we prove the proposition. 2 Remark 4.3. From Theorem 4.1 and Proposition 4.2, we have ϕ ˙ C 0 (M \D) ≤ C(T ),

ϕC α, α2 (M ) ≤ C(T ),

C −1 (T )ω∗ ≤ ωϕ ≤ C(T )ω∗

on M \ D × [0, T ]. By the uniqueness theorem of conical Kähler–Ricci flows (see Lemma 3.2 in [56]), ωϕ(t) constructed in Theorem 4.1 is the unique long-time solution to conical Kähler–Ricci flow (1.1). 5. Uniform Perelman’s estimates along twisted Kähler–Ricci flows In this section, we first obtain a uniform lower bound for twisted scalar curvatures R(gε (t)) −trgε (t) θε in time interval [δ, ∞), where δ > 0. This conclusion is very important to obtain uniform Perelman’s estimates along twisted Kähler–Ricci flows (2.6), because we have no uniform lower bounds of initial twisted scalar curvatures R(gε (0)) − trgε (0) θε when β ∈ ( 12 , 1). Proposition 5.1. t2 (R(gε (t)) − trgε (t) θε ) are uniformly bounded from below along twisted Kähler–Ricci flows (2.6), that is, there exists a uniform constant C such that, for any t and ε, t2 (R(gε (t)) − trgε (t) θε ) ≥ −C,

(5.1)

where constant C depends only on β and n. In particular, when t ≥ 1, R(gε (t)) − trgε (t) θε ≥ −C.

(5.2)

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Proof. First, we derive the evolution equation of t2 (R(gε (t)) − trgε (t) θε ). (

d − Δgε (t) )(t2 (R(gε (t)) − trgε (t) θε )) dt

= t2 |Ric(gε (t)) − θε |2gε (t) − βt2 (R(gε (t)) − trgε (t) θε ) + 2t(R(gε (t)) − trgε (t) θε ) Assume that (x0 , t0 ) is the minimum point of t2 (R(gε (t)) − trgε (t) θε ) on M × [0, T ]. Case 1, t0 = 0, then we have t2 (R(gε (t)) − trgε (t) θε ) ≥ 0. Case 2, t0 ≥ β2 , then at (x0 , t0 ), 0 ≥ (2t0 − βt20 )(R(gε (t0 )) − trgε (t0 ) θε (x0 ))). Hence R(gε (t0 )) − trgε (t0 ) θε (x0 ) ≥ 0, and then t2 (R(gε (t)) − trgε (t) θε ) ≥ 0. Case 3, 0 < t0 ≤ β2 , without loss of generality, we can assume R(gε (t0 )) − trgε (t0 ) θε (x0 ) ≤ 0. By inequality |Ric(gε (t)) − θε |2gε (t) ≥

(R(gε (t)) − trgε (t) θε )2 , n

at (x0 , t0 ), we have (R(gε (t0 )) − trgε (t0 ) θε (x0 ))2 + 2t0 (R(gε (t0 )) − trgε (t0 ) θε (x0 )) n R(gε (t0 )) − trgε (t0 ) θε (x0 ) √ 2 √ = (t0 + n) − n. n

0 ≥ t20

4n 2 Then t20 (R(gε (t0 )) − trgε (t0 ) θε (x0 )) > − 4n β . Hence t (R(gε (t)) − trgε (t) θε ) ≥ − β . By the above arguments, there exists a uniform constant only depending on n and β such that t2 (R(gε (t)) − trgε (t) θε ) ≥ −C for any t ≥ 0 and ε. When t ≥ 1, we have R(gε (t)) − trgε (t) θε ≥ −C. 2

Now, we prove uniform Perelman’s estimates along twisted Kähler–Ricci flows (2.6) for t ≥ 1 by following the ideas of Sesum–Tian in [41] (see also the twisted case in [27]). Because we have no uniform bounds on the initial data, we will make some improvements in the arguments. In the following sections, we let ∇ be the (1, 0)-type covariant derivative with respect to metric gε (t). Theorem 5.2. Let gε (t) be solutions of twisted Kähler Ricci flows, that is, the corresponding forms ωε (t) satisfy equations (2.6) with initial metrics ωε . Assume that uε (t) ∈ C ∞ (M ) are the twisted Ricci potentials satisfying −Ric(ωε (t)) + βωε (t) + θε =

√

¯ ε (t) −1∂ ∂u

(5.3)

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 √ and V1 M e−uε (t) dVεt = 1, where θε = (1 − β)(ω0 + −1∂∂ log(ε2 + |s|2h )). Then for any β ∈ (0, 1), there exists a uniform constant C such that |R(gε (t)) − trgε (t) θε | ≤ C, uε (t)C 1 (gε (t)) ≤ C, diam(M, gε (t)) ≤ C hold for any t ≥ 1 and ε, where R(gε (t)) − trgε (t) θε and diam(M, gε (t)) are the twisted scalar curvatures and diameters of the manifold respectively. Now we start to prove Theorem 5.2. Firstly, through differentiating equation (5.3)  and V1 M e−uε (t) dVεt = 1, we have d uε (t) = Δωε (t) uε (t) + βuε (t) − aε (t), dt  β aε (t) = uε (t)e−uε (t) dVεt . V

(5.4) (5.5)

M

It is obvious that aε (t) ≤ 0 by Jensen’s inequality. When β ∈ (0, 12 ], by the analogous arguments in [27] or [41], the lower bound of aε (t) can be derived by functional μθε (gε , 1), because the terms max(R(gε ) −trgε θε )− in the lower bound of μθε (gε , 1) can be uniformly M

bounded. However, it does not work when β ∈ ( 12 , 1). Here, for any β ∈ (0, 1), we use uniform Poincaré inequality to get a uniform lower bound of aε(t). We also note that this lower bound is independent of the lower bound of μθε (gε , 1). Lemma 5.3. Let uε (t) satisfy (5.3). Then for any f ∈ C ∞ (M ), ε and t, 1 V



2 −uε (t)

f e

dVεt

1 ≤ βV

M



|∇f |2gε (t) e−uε (t) dVεt

1 +( V

M



f e−uε (t) dVεt )2 .

M

Proof. It suffices to show the lowest strictly positive eigenvalue μ of operator ¯

¯

Lf = −gεij (t)∇i ∇¯j f + gεij (t)∇i uε (t)∇¯j f

(5.6)

satisfying μ ≥ β. Note that L is self-adjoint with respect to the inner product 1 (f, g) = V



f g¯e−uε (t) dVεt

M

and that Ker L = C. Let f be a eigenfunction of eigenvalue μ, f ≡ Constant. ¯

¯

−gεij (t)∇i ∇¯j f + gεij (t)∇i uε (t)∇¯j f = μf.

(5.7)

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By applying ∇k¯ on both sides and adopting Ricci identity, we have ¯

¯

μ∇k¯ f = −gεij (t)∇i ∇k¯ ∇¯j f − gεij (t)Rgs¯ε (t)¯jik¯ ∇s¯f ¯

¯

+ gεij (t)∇i uε (t)∇k¯ ∇¯j f + gεij (t)∇¯j f ∇k¯ ∇i uε (t). ¯

Integrating after multiplying gεlk (t)∇l f e−uε dVεt on both sides, and then using the facts √ ¯ ε (t) and θε is semi-positive, we get that −Ric(ωε (t)) + βωε (t) + θε = −1∂ ∂u  β

|∇f |2gε (t) e−uε (t) dVεt

M



|∇∇f |2gε (t) e−uε (t) dVεt

≤ M

+

1 2

 =μ

 +β

|∇f |2gε (t) e−uε (t) dVεt

M



θε (grad f, J (grad f ))e−uε (t) dVεt

M

|∇f |2gε (t) e−uε (t) dVεt .

M

Hence μ ≥ β.

2

Lemma 5.4. There exists a uniform constant C such that for any t and ε, |aε (t)| ≤ C.

(5.8)

Proof. We only need to prove that aε (t) can be uniformly bounded from below. β d aε (t) = dt V



|∇uε (t)|2gε (t) e−uε (t) dVεt −

M

β2 V



u2ε (t)e−uε (t) dVεt + a2ε (t).

M

By Lemma 5.3, aε (t) are nondecreasing in time t and then aε (t) ≥ ωεn (ε2 +|s|2h )1−β ω0n

β V

 M

uε e−uε dVε0 .

For any β ∈ (0, 1), log + kβχ + F0 can be uniformly bounded (see section 4.5 in [6]). Hence by adjusting χ with a constant (whose variation with respect to ε is bounded), we can assume that uε = log

ωεn (ε2 + |s|2h )1−β + kβχ + F0 . ω0n

(5.9)

Then there exists a uniform constant C such that (5.8) holds. 2 Proposition 5.5. The twisted Ricci potentials uε (t) are uniformly bounded from below along twisted Kähler–Ricci flows (2.6). Proof. From Proposition 5.1 and Lemma 5.4, when t ≥ 1, there exists a uniform constant C1 satisfying

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

Δgε (t) uε (t) − aε (t) ≤ C1 .

1345

(5.10)

1 We conjecture that uε (t) ≥ − 2C for any t ≥ 1 and ε. If not, then there exists β 2C1 (ε0 , y0 , t0 ) such that uε0 (y0 , t0 ) < − β , where t0 ≥ 1. From equation (5.3) and the flow (2.6), we can assume that

uε0 (t) =

d φε (t) dt 0

with φε0 (0) = 0. By similar arguments in [27] or [41], there exists a neighborhood U of y0 such that for any z ∈ U and sufficiently large t, φε0 (z, t) ≤ −C2 eβt ,

(5.11)

where C2 depends only on C1 , β, t0 and ε0 . Applying the Green formula with respect to metric g0 , for sufficiently large t, we have sup φε0 (·, t) ≤ −C3 eβt + C4 ,

(5.12)

M

where C3 depends only on C1 , β, ε0 , t0 and ω0 while C4 depends only on ω0 . On the other hand, by the normalization of uε (t), we have sup uε0 (t) ≥ 0.

(5.13)

M

Let uε0 (xt , t) = sup uε0 (t). Combine (5.13) with M

d (uε (t) − βφε0 (t)) = Δgε0 (t) uε0 (t) − aε0 (t) ≤ C1 . dt 0 At point xt , integrating from 0 to t on both sides, we have sup φε0 (·, t) ≥ −C5 − C1 t,

(5.14)

M

where C5 depends only on β and ω0 . From (5.12) and (5.14), −C3 eβt + C4 ≥ −C5 − C1 t.

(5.15)

When t is sufficiently large, inequality (5.15) is not correct. Thus uε (t) are bounded uniformly from below along flows (2.6) when t ≥ 1. Since aε (t) ≤ 0, (

d − Δgε (t) )uε (t) ≥ βuε (t). dt

By the maximum principle, we deduce that uε (t) are uniformly bounded from below on M × [0, 2] because uε are uniformly bounded (see (5.9)). 2

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J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

Denoting 2 =

d dt

− Δgε (t) , as the computations in [27], we have

2(Δgε (t) uε (t)) = −|∇∇uε (t)|2gε (t) + βΔgε (t) uε (t),

(5.16)

2(|∇uε (t)|2gε (t) ) = −|∇∇uε (t)|2gε (t) − |∇∇uε (t)|2gε (t) 1 + β|∇uε (t)|2gε (t) − θε (grad uε (t), J (grad uε (t))), 2

(5.17)

where J is the complex structure on M . Lemma 5.6. For any β ∈ (0, 1), there exists a uniform constant C independent of t and ε such that on M × [1, 2]. uε (t)C 1 (gε (t)) ≤ C,

(5.18)

|R(gε (t)) − trgε (t) θε | ≤ C.

(5.19)

Proof. In Proposition 5.1 and Proposition 5.5, we have obtained the lower bounds of uε (t) and R(gε (t)) − trgε (t) θε on M × [1, 2]. Since aε (t) ≥ −C for uniform C, then by equation (5.4), we have (

d C − Δgε (t) )(e−βt uε (t) + e−βt ) ≤ 0. dt β

By the maximum principle and the uniform bound of uε , it follows that on M × [0, 2], uε (t) ≤ eβt (uε +

C ) ≤ C. β

Let Hε (x, t) = t|∇uε (t)|2gε (t) + Au2ε (t) and (x0 , t0 ) be the maximum point of Hε (x, t) on M × [0, 2]. By (5.4) and (5.17), we obtain (

d − Δgε (t) )Hε (x, t) ≤ (2β + 1 − 2A)|∇uε (t)|2gε (t) + C, dt

(5.20)

where constant A will be determined later and C is a uniform constant depending only on uε (t)C 0 (M ×[0,2]) and sup |aε (t)|. Case 1, t0 = 0. Then t|∇uε (t)|2gε (t) ≤ Au2ε (x0 , 0) ≤ C, where C is uniform. Case 2, t0 > 0. Let A = β + 1. By the maximum principle, we have |∇uε (x0 , t0 )|2gε (t0 ) ≤ C. Hence t|∇uε (t)|2gε (t) ≤ C. By the above two cases, t|∇uε (t)|2gε (t) ≤ C on M × [0, 2]. Obviously |∇uε (t)|2gε (t) ≤ C on M × [1, 2]. Next, since Δgε (t) uε (t) = −R(gε (t)) +βn +trgε (t) θε , we only need to prove the uniform upper bound of −Δgε (t) uε (t). We take Gε (x, t) = t2 (−Δgε (t) uε (t)) + 2t2 |∇uε (t)|2gε (t) . According to (5.16), (5.17) and

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

¯ ε (t)|2 |∇∇u gε (t) ≥

1347

(Δgε (t) uε (t))2 , n

the evolution equation of Gε (x, t) can be controlled as (

d t2 − Δgε (t) )Gε (x, t) ≤ (βt2 + 2t)(−Δgε (t) uε (t)) − (Δgε (t) uε (t))2 + C dt n

for uniform constant C depending on

sup (t|∇uε (t)|2gε (t) ). Assuming that (x0 , t0 ) is

M ×[0,2]

the maximum point of Gε (x, t) on M × [0, 2]. Case 1, t0 = 0, then −t2 Δgε (t) uε (t) ≤ 0. Case 2, t0 > 0. We assume −Δgε (t) uε (t) > 0 at (x0 , t0 ). We claim that t20 (−Δgε (t0 ) uε (x0 , t0 )) ≤ Bn, where B is a uniform constant to be determined later. If not, t20 (−Δgε (t0 ,) uε (x0 , t0 )) > Bn. By the maximum principle, we have 0 ≤ (−Δgε (t0 ) uε (x0 , t0 ))((4 + 4β) − <

1 (−t20 Δgε (t0 ) uε (x0 , t0 ))) + C n

Bn (4 + 4β − B) + C. 4

We get a contradiction when we let B = 4(1 + β + C). From these two cases, we conclude that −t2 Δgε (t) uε (t) ≤ C for uniform constant C on M × [0, 2]. Furthermore, −Δgε (t) uε (t) ≤ C on M × [1, 2]. 2 Lemma 5.7. There exists a uniform constant C independent of t and ε such that |∇uε (t)|2gε (t) ≤ C(uε (t) + C),

(5.21)

R(gε (t)) − trgε (t) θε ≤ C(uε (t) + C)

(5.22)

for any t ≥ 1 and ε. Proof. It follows from Proposition 5.5 that there exists a uniform constant B > 1 such that uε (t) > −B. Define Hε (t) =

2Hε (t) ≤

|∇uε (t)|2gε (t) uε (t)+2B .

−|∇∇uε (t)|2gε (t) − |∇∇uε (t)|2gε (t) uε (t) + 2B

As the same arguments in [27], we have

+

|∇uε (t)|2gε (t) (2Bβ + aε (t)) (uε (t) + 2B)2

4 |∇uε (t)|4gε (t) |∇∇uε (t)|2gε (t) + |∇∇uε (t)|2gε (t) δ |∇uε (t)|gε (t) −δ + +δ 2 (uε (t) + 2B)3 uε (t) + 2B (uε (t) + 2B)3

−

1 θε (grad uε (t), J (grad uε (t))) ∇uε (t) · ∇Hε (t) + (2 − δ)Re . 2 uε (t) + 2B uε (t) + 2B

Taking δ < 1 and combining Lemma 5.4 with θ(grad uε (t), J (grad uε (t))) ≥ 0,

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2Hε (t) ≤

|∇uε (t)|2gε (t) (2Bβ + C1 )

+ (2 − δ)Re

(uε (t) + 2B)2

4 ∇uε (t) · ∇Hε (t) δ |∇uε (t)|gε (t) − . uε (t) + 2B 2 (uε (t) + 2B)3

From Lemma 5.6, we have sup M

|∇uε (1)|2gε (1) uε (1) + 2B

≤ C3

(5.23)

for uniform constant C3 . Then by the maximum principle, we have Hε (t) ≤ max{C3 , 2(2B + C1 )δ −1 } for any t ≥ 1 and ε. Now we prove the second inequality. Since Δgε (t) uε (t) = βn − R(gε (t)) + trgε (t) θε , we only need to prove the existence of uniform constant C such that −Δgε (t) uε (t) can be controlled by C(uε (t) + C). −Δ ε (t) uε (t) Let Gε = uεg(t)+2B + 2Hε . 2Gε =

−2|∇∇uε (t)|2gε (t) − |∇∇uε (t)|2gε (t) uε (t) + 2B

+ 2Re

+

(−Δgε (t) uε (t) + 2|∇uε (t)|2gε (t) )(2Bβ + aε (t)) (uε (t) + 2B)2

1 θε (grad uε (t), J (grad uε (t))) ∇uε (t) · ∇Gε − . uε (t) + 2B 2 uε (t) + 2B

Since θε is semi-positive, 2Gε ≤

−|∇∇uε (t)|2gε (t) uε (t) + 2B + 2Re

+

(−Δgε (t) uε (t) + 2|∇uε (t)|2gε (t) )(2Bβ + aε (t)) (uε (t) + 2B)2

∇uε (t) · ∇Gε . uε (t) + 2B

In local coordinates,   (Δgε (t) uε (t))2 = ( uεi¯i )2 ≤ n u2εi¯i = n|∇∇uε (t)|2gε (t) . i

(5.24)

i

Therefore, we have 2Gε ≤

|∇uε (t)|2gε (t) (2Bβ + aε (t)) −Δgε (t) uε (t) −Δgε (t) uε (t) 2Bβ + aε (t) ( − )+2 uε (t) + 2B uε (t) + 2B n(uε (t) + 2B) (uε (t) + 2B)2 + 2Re

−Δ

∇uε (t) · ∇Gε . uε (t) + 2B

u (1)

ε (1) ε Since uεg(1)+2B are bounded uniformly by Lemma 5.6, by the maximum principle, there exists a uniform constant C > 0 such that Gε ≤ C for any t ≥ 1 and ε. Hence we ε (t) get u−Δu ≤ C. 2 ε (t)+2B

From (5.21) and discussions in [41] (see Claim 8), we have the following lemma.

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Lemma 5.8. There exists a uniform constant C such that uε (y, t) ≤ C dist2εt (x, y) + C,

(5.25)

R(gε (t)) − trgε (t) θε ≤ C dist2εt (x, y) + C,

(5.26)

|∇uε (t)|gε (t) ≤ C distεt (x, y) + C

(5.27)

for any t ≥ 1 and ε, where uε (x, t) = inf uε (y, t). y∈M

By Lemma 5.8, the statements in Theorem 5.2 will be true if the diam(M, gε (t)) are uniformly bounded when t ≥ 1. In order to prove this, we give the proof of uniform Perelman’s noncollapsing theorem. Before that, we review the twisted Wθ functional and μθ functional.  Wθ (g, f, τ ) =

e−f τ −n (τ (R − trg θ + |∇f |2g ) + βf )dVg ,

M

where g is a Kähler metric, f is a smooth function on M , τ is a positive scale parameter and n is the complex dimension of M . Let μθ (g, τ ) = inf{Wθ (g, f, τ )|f ∈ C ∞ (M ),



1 V

e−f τ −n dV = 1}

M

be μθ functional with respect to metric g. From [27], we have the monotonicity of twisted Wθ and μθ functionals along twisted Kähler–Ricci flows. In the process of proving uniform Perelman’s noncollapsing theorem and the fact that diam(M, gε (t)) can be uniformly bounded, we use the uniform lower bound of functionals μθε (gε (1), τ ), which depends on Sobolev constants CS (M, gε (1)) and max(R(gε (1)) − M

trgε (1) θε )− . In the following proposition, we obtain a uniform control of Sobolev constants CS (M, gε (t)) for any t ∈ [0, 2] and ε > 0. We will present the proof of Proposition 5.9 in Appendix. Proposition 5.9. Let gε (t) be solutions of twisted Kähler–Ricci flows (2.6). Then there exists uniform constant C such that  (

v

2n n−1

dVεt )

n−1 n

 ≤ C(

M

M

 |dv|2gε (t) dVεt

|v|2 dVεt )

+ M

holds for any smooth function v on M , t ∈ [0, 2] and ε > 0. In [27] (see Theorem 2.2), we have

(5.28)

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1350

μθε (gε (1), τ ) ≥ −τ V max(R(gε (1)) − trgε (1) θε )− − nβV (log 2 + αV − n − 1) 1

M

+ βnV log α − βV log V − βnV log τ,

(5.29)

where V is the volume of (M, gε (1)) and α satisfies 4τ ≥ βnαCS (M, gε (1)). Since V ol(M, gε (1)) are fixed while max(R(gε (1)) − trgε (1) θε )− and CS (M, gε (1)) are uniformly M

bounded, by choosing a suitable α, there exists a uniform constant C independent of ε such that μθε (gε (1), τ ) ≥ −C.

(5.30)

Next, let us state and prove uniform Perelman’s noncollapsing theorem. Proposition 5.10. Let gε (t) be solutions of twisted Kähler–Ricci flows (2.6). There exists a uniform constant C such that V olgε (t) (Bgε (t) (x, r)) ≥ Cr2n for any gε (t) satisfying R(gε (t)) − trgε (t) θε ≤ ∂Bgε (t) (x, r) = ∅ and 0 < r < 1.

m r2

on Bgε (t) (x, r) when t ≥ 1, where

Proof. We argue it by contradiction, that is, there exist εk , pk , tk ≥ 1, rk such that Rgεk (tk ) − trgεk (tk ) θεk ≤ rm2 on Bk , εk → 0 and V (rk ) · rk−2n → 0 when k → +∞, where k Bk = Bgεk (tk ) (pk , rk ) and V (rk ) = V olgεk (tk ) (Bgεk (tk ) (pk , rk )). Setting τ (tk ) = rk2 , we define functions as uk (x) = eCk φ(rk−1 distgεk (tk ) (x, pk )), where φ is a smooth function on R, equal to 1 on [0, 12 ], decreasing on [ 21 , 1] and equal to 0 on [1, +∞) and Ck are constants to make uk satisfying 1 V



rk−2n u2k dVεk tk = 1.

M

Hence 1 1 = e2Ck rk−2n V

 φ2 dVεk tk ≤

1 2Ck −2n e rk V (rk ). V

Bk

By assumptions, V (rk )rk−2n → 0 when k → +∞, which shows that Ck → +∞ when k → +∞. We claim that (a) V (rk )rk−2n → 0 as k → +∞; (b) (R(gεk (tk )) − trgεk (tk ) θεk )rk2 ≤ m; (c)

V (rk ) r V ( 2k )

are uniformly bounded.

We only need to prove (c). If fied. If not, for some k, we have

V (rk ) r V ( 2k ) V (rk ) r V ( 2k )

< 5n for any k, then the claim (c) is testi≥ 5n . Let rk =

rk 2 .

We have (rk )−2n V (rk ) ≤

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

1351

r2

(rk )−2n 51n V (rk ) = ( 45 )n rk−2n V (rk ), (rk )2 (R(g(εk (tk )) − trgεk (tk ) θεk ) = 4k (R(gεk (tk )) − trg(tk ) θεk ). Combining (a) and (b), we obtain (rk )2 (R(gεk (tk )) − trgεk (tk ) θεk ) ≤ m and (rk )−2n V (rk ) → 0 when k → +∞. Replace rk by rk . If

 V (rk )

V(

r k 2

< 5n , the demonstration

)

will be terminated. If not, the above process will be repeated. By identity limr→0  V (rk ) r V ( 2k )

n

4 proved in [22] (see (6.9)), we should get

 M

rk−2n

=

n

< 5 at some steps. Then we consider

{pk , rk } obtained from the above. Considering function integral average 1 V

V (r) V ( r2 )

1 2 2 +1 (uk rk

+ rk2n+2 ), we have its

1 (u2 + rk2n+2 )dVεk tk = 1. rk2 + 1 k

Computing functional Wθεk (gεk (tk ), − log r21+1 (u2k + rk2n+2 ), rk2 ), we have k

Wθεk (gεk (tk ), − log 1 = 2 rk + 1 +

−

(1) ≤

(2) ≤



β 2 rk + 1

n rk2 + 1

rk−2n (u2k + rk2n+2 )(R(gεk (tk )) − trgεk (tk ) θεk )rk2 dVεk tk





rk−2n (u2k + rk2n+2 )

M



4u2k |∇uk |2gε

k

M

(tk ) 2 rk dVεk tk

(2)

(u2k + rk2n+2 )2

rk−2n (u2k + rk2n+2 )(log

rk2

1 + log(u2k + rk2n+2 ))dVεk tk (3) +1

rk−2n (u2k + rk2n+2 )dVεk tk ≤ mV,

M



rk−2n e2Ck 4|φ |2 dVεk tk ≤ Crk−2n e2Ck (V (rk ) − V (

M

(3) ≤ 2βV log 2 −

≤C−

(1)

M

1 rk2 + 1

m rk2 + 1

1 (u2 + rk2n+2 ), rk2 ) rk2 + 1 k

2βCk rk2 + 1

β 2 rk + 1 

M



rk−2n u2k log u2k dVεk tk −

M

rk−2n u2k dVεk tk −

β rk2 + 1



β(n + 1) rk2 + 1

rk )), 2

 rk2 log rk2 dVεk tk M

rk−2n e2Ck φ2 log φ2 dVεk tk +

M

Cβ(n + 1)V rk2 + 1

rk ≤ −βV Ck + Crk−2n e2Ck (V (rk ) − V ( )) + C, 2 where C are uniform constants. Combining all these inequalities together and making use of condition (c), we have

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Wθεk (gεk (tk ), − log

rk2

1 rk (u2k + rk2n+2 ), rk2 ) ≤ C − βV Ck + Crk−2n e2Ck (V (rk ) − V ( )) +1 2  ≤ C − Ck βV + C rk−2n e2Ck φ2 dVεk tk Bk

≤ C − Ck βV, where C are uniform constants independent of tk and εk . Considering τ = 1 − (1 − rk2 )e−βtk eβt , by the monotonicity of μθ (g, τ ) (see Theorem 2.5 in [27]), μθεk (gεk (1), 1 − (1 − rk2 )e−β(tk −1) ) ≤ Wθεk (gεk (tk ), − log

1 (u2 + rk2n+2 ), rk2 ) rk2 + 1 k

≤ C − 2Ck βV. Since 0 < 1 − (1 − rk2 )e−β(tk −1) < 1, by (5.30), μθεk (gεk (1), 1 − (1 − rk2 )e−β(tk −1) ) ≥ −C,

(5.31)

where C is independent of εk and tk . Then we get −C ≤ C − Ck βV which does not work when k → +∞. Hence the lemma is proved. 2 Denote dεt (z) = distεt (x, z) and Bεt (k1 , k2 ) = {z|2k1 ≤ dεt (z) ≤ 2k2 }, where uε (x, t) = inf uε (y, t). Considering the annuluses Bεt (k, k + 1), then by Lemma 5.8, M

we have R(gε (t)) − trgε (t) θε ≤ C22k on Bεt (k, k + 1) when t ≥ 1. Interval [2k , 2k+1 ] fits 22k balls of radii 21k . By Proposition 5.10, when t ≥ 1, we have V olgε (t) (Bεt (k, k + 1)) ≥



V olgε (t) (Bεt (xi ,

i

1 )) ≥ C22k−2nk . 2k

(5.32)

Lemma 5.11. When t  1, for any δ > 0, there exist Bεt (k1 , k2 ) such that if diam(M, gε (t)) are large enough, then (a)V olgε (t) (Bεt (k1 , k2 )) < δ, (b)V olgε (t) (Bεt (k1 , k2 )) ≤ 220n V olgε (t) (Bεt (k1 + 2, k2 − 2)). Proof. First, we fix any δ > 0. Since V olgε (t) (M ) is a constant V along twisted Kähler– Ricci flows, it can be uniformly bounded. Let k  1, V = V olgε (t) (Bεt (0, k)) + V olgε (t) (Bεt (k, 3k)) + · · · + V olgε (t) (Bεt (3α−1 k, 3α k)) + · · · , α

where α > m[ Vδ ] + 1, m will be determined later and diam(M, gε (t)) > 23 k+1 . We claim that there must exist 0 ≤ i ≤ α − 1 such that V olgε (t) (Bεt (3i k, 3i+1 k)) < δ. If not, then we have

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

V > V olgε (t) (Bεt (k, 3k)) + · · · + V olgε (t) (Bεt (3α−1 k, 3α k)) ≥ αδ > mδ[

1353

V ] + δ. δ

When we take m satisfying mδ[ Vδ ] +δ > V , the above inequality leads to a contradiction. Thus the claim is proved. Then we determine k1 and k2 . If estimate (b) does not hold, then V olgε (t) (Bεt (3i k, 3i+1 k)) > 220n V olgε (t) (Bεt (3i k + 2, 3i+1 k − 2)). We would consider V olgε (t) (Bεt (3i k +2, 3i+1 k −2)) instead and discuss whether (b) holds for that annulus. If for any p at the p-th step, we are still not able to find suitable radii to satisfy (a) and (b). In that case, at the p-th step we would have V olgε (t) (Bεt (3i k, 3i+1 k)) > 220np V olgε (t) (Bεt (3i k + 2p, 3i+1 k − 2p)). In particular, if 3i k + 2p = 32 3i k, then we have 3i+1 k − 2p = 52 3i k. By (5.32), i 3 5 δ > V olgε (t) (Bεt (3i k, 3i+1 k)) > 25n·3 k V olgε (t) (Bεt ( 3i k, 3i k)) 2 2 3 i 3 i 5n·3i k >2 V olgε (t) (Bεt ( 3 k, 3 k + 1)) 2 2 i

≥ C2(2n+3)·3 k . That would lead to a contradiction if we let k  1. Hence there exists 1 ≤ j ≤ p − 1, V olgε (t) (Bεt (3i k + 2j, 3i+1 k − 2j)) ≤ 220n V olgε (t) (Bεt (3i k + 2(j + 1), 3i+1 k − 2(j + 1))). Let k1 = 3i k + 2j, k2 = 3i+1 k − 2j and then we have k2 − k1 = 2 · 3i k − 4j ≥ 3i k  1. Till now, the proof of the lemma is finished. 2 According to the arguments in [41] (see Lemma 11), we have the following lemma. Lemma 5.12. There must exist r1 ∈ [2k1 , 2k1 +1 ], r2 ∈ [2k2 −1 , 2k2 ] and a uniform constant C such that  (R(gε (t)) − trgε (t) θε )dVεt ≤ C Vˆ < Cδ B(r1 ,r2 )

for t ≥ 1, where δ > 0 and Vˆ = V olgε (t) (Bεt (k1 , k2 )) are obtained in Lemma 5.11. Finally, we prove that diam(M, gε (t)) can be uniformly bounded along twisted flows (2.6) when t ≥ 1 by Perelman’s arguments. There exist a few differences between the proof and the original one. Thus we present a proof of Proposition 5.13 in appendix to readers’ convenience.

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Proposition 5.13. diam(M, gε (t)) are uniformly bounded along twisted Kähler–Ricci flows (2.6) for any t ≥ 1 and ε. Proof of Theorem 5.2. By Lemma 5.8 and Proposition 5.13, we obtain that R(gε (t)) − trgε (t) θε and uε (t) are uniformly bounded from above and |∇uε |gε (t) are uniformly bounded. Combining Proposition 5.1 with 5.5, we prove the theorem. 2 6. Uniform C 0 -estimates for metric potentials ϕε (t) In this section, for any β ∈ (0, 1), we prove uniform Sobolev inequality along twisted Kähler–Ricci flows (2.6). In finite interval, we have proved it in Proposition 5.9. When t ≥ 1, from [28] (see also [59] or [62]), the Sobolev constants along twisted Kähler–Ricci flows depend only on n, max(R(gε (1)) − trgε (1) θε )− and CS (M, gε (1)), where the latter M

two can be uniformly bounded by Theorem 5.2 and Proposition 5.9. We prove the following uniform Sobolev inequality (when t ≥ 1) by Zhang’s arguments [62]. To readers’ convenience, we give its proof in the appendix. Theorem 6.1. Let M be a compact Kähler manifold with complex dimension n ≥ 2 and gε (t) be solutions of twisted Kähler–Ricci flows (2.6). Then there exist uniform constants A and B such that for all v ∈ W 1,2 (M, gε (t)), ε > 0 and t ≥ 1,  (

2n

v n−1 dVεt )

M

n−1 n

 ≤B

 | ∇v |2gε (t) dVεt

v 2 dVεt + A M

+

A 4



(6.1)

M

R(gε (t)) − trgε (t) θε )v 2 dVεt . M

Then by uniform Perelman’s estimates along flows (2.6) and Proposition 5.9, there exists uniform constant C such that for any t ≥ 0 and ε > 0,    n−1 2n 2 n−1 n ( v dVεt ) ≤C | ∇v |gε (t) dVεt + C v 2 dVεt . (6.2) M

M

M

Next, we argue the uniform C 0 -estimates for metric potentials ϕε (t). We will denote φε (t) = ϕε (t) + kχ(ε2 + |s|2h ) and discuss the C 0 -estimates for φε (t). First, we review Aubin’s functionals, Ding’s functional and twisted Mabuchi K-energy functional. Let φt be a path with φ0 = c and φ1 = φ, then  n! Iω0 (φ) = φ(dV0 − dVφ ), V M

n! Jω0 (φ) = V

1  φ˙ t (dV0 − dVφt )dt, 0 M

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

Fω00 (φ) = Jω0 (φ) −

n! V

 φdV0 , M

Fω0 (φ) = Jω0 (φ) −

n! V



φdV0 −



1 1 log( β V

M

Mω0 ,

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n! θ (φ) = −β(Iω0 (φ) − Jω0 (φ)) − V



e−uω0 −βφ dV0 ),

M

n! uω0 (dV0 − dVφ ) + V

M

 log M

ωφn dVφ , ω0n

√ ¯ ω where uω0 is the twisted Ricci potential of ω0 , that is, −Ric(ω0 ) + βω0 + θ = −1∂ ∂u 0  1 −uω0 and V M e dVω0 = 1. The time derivatives of Iω0 , Jω0 and Mω0 , θ along the path φt can be written as follows, n! ∂ Iω (φt ) = ∂t 0 V



n! φ˙t (dV0 − dVφt ) − V

M

n! ∂ Jω (φt ) = ∂t 0 V

 φt Δφ˙t dVφt , M



φ˙ t (dV0 − dVφt ), M

∂ n! Mω0 , θ (φt ) = − ∂t V

 φ˙ t (R(ωφt ) − βn − trωφt θ)dVφt . M

Proposition 6.2. The integrals

 +∞ 0

e−βt ∇uε (t)2L2 dt are uniformly bounded.

Proof. When t ≥ 1, by Theorem 5.2, e−βt ∇uε (t)2L2 ≤ Ce−βt for uniform con +∞ stant C. Thus 1 e−βt ∇uε (t)2L2 dt are uniformly bounded. Then we claim that 1 ∇uε (t)2L2 dt are uniformly bounded. 0 d (Mω0 , dt

0 θε (φε (t)) − βFω0 (φε (t)) −



n! V

φ˙ ε (t)dVεt ) = 0.

(6.3)

M

Hence we have Mω0 , t =β

θε (φε (t))

− Mω0 ,

θε (φε (0))

n! d 0 Fω0 (φε (s))ds + ds V

0



n! φ˙ ε (t)dVεt − V

M

n! =− β V

t 

n! (φ˙ ε (s) − eβs βϕε (0))dVεs ds + V

0 M

−

n! V



(φ˙ ε (0) − βϕε (0))dVε . M

 φ˙ ε (0)dVε0 M

 (φ˙ ε (t) − eβt βϕε (0))dVεt M

(6.4)

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J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

The evolution equation of e−βt (φ˙ ε (t) − eβt βϕε (0)) satisfies (

d − Δgε (t) )(e−βt (φ˙ ε (t) − eβt βϕε (0))) = 0. dt

(6.5)

By the maximum principle, ω n (ε2 + |s|2h )1−β sup |e−βt (φ˙ ε (t) − eβt βϕε (0))| ≤ sup | log ε + kβχ(ε2 + |s|2h ) + F0 | ω0n M M for any t ≥ 0. Hence there exists a uniform constant C such that sup |(φ˙ ε (t) − eβt βϕε (0))| ≤ CeβT .

(6.6)

[0,T ]×M

On the other hand, d Mω0 , dt

θε (φε (t)) = −

n! V

 |∇uε (t)|2gε (t) dVεt .

(6.7)

M

Integrating from 0 to 1 on both sides, we obtain 1 ∇uε (t)2L2 dt = Mω0 ,

θε (φε (0))

− Mω0 ,

θε (φε (1)).

(6.8)

0

Then we prove the claim by using (6.4) and (6.6). 2 Now we consider equations  ∂φ

ε (t) ∂t

= log

n ωφ ε (t) ωεn

+ Fε + βφε (t),

(6.9)

φε (t)|t=0 = cε0 + kχ(ε2 + |s|2h ) where cε0 =

 +∞ −βt 1 e ∇uε (t)2L2 dt β( 0

−

1 V

 M

Fε dVε −

1 V

 M

kβχ(ε2 + |s|2h )dVε ).

Proposition 6.3. There exists a uniform constant C such that φ˙ ε (t)C 0 ≤ C for any ε and t. Proof. Alike to [36], we let 1 αε (t) = V

 M

1 φ˙ ε (t)dVεt = V

 uε (t)dVεt − cε (t). M

(6.10)

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Through computing, we have

e

−βt

t αε (t) = αε (0) −

e−βs ∇φ˙ ε 2L2 ds

0

1 = V

t

 uε dVε − cε (0) −

e−βs ∇φ˙ ε 2L2 ds.

(6.11)

0

M

Substituting uε = Fε + kβχ and −cε (0) = βϕε (0) into (6.11),

e

−βt

+∞  αε (t) = e−βs ∇φ˙ ε 2L2 ds.

(6.12)

t

When t ≥ 1, by Theorem 5.2, +∞  0 ≤ αε (t) = eβ(t−s) ∇φ˙ ε 2L2 ds ≤ C.

(6.13)

t

Then φ˙ ε (t) are bounded uniformly when t ≥ 1. Since φ˙ ε (0) are uniformly bounded, by (4.3), it is easy to see that φ˙ ε (t)C 0 ([0,1]×M ) ≤ C for uniform constant C. 2 Now we establish the relationship among the above functionals. Proposition 6.4. There exists a uniform constant C such that φε (t) satisfies (i)

Mω0 ,

θε (φε (t))

−

βFω00 (φε (t))

n! − V

 φ˙ ε (t)dVεt = Cε , M

|βFω0 (φε (t)) − Mω0 , θε (φε (t))| + |βFω00 (φε (t)) − Mω0 , θε (φε (t))| ≤ C,   (n − 1)! n! (iii) (−φε (t))dVεt − C ≤ Jω0 (φε (t)) ≤ φε (t)dV0 + C, V V (ii)

(iv)

n! V



M

φε (t)dV0 ≤ M

n · n! V

M



(−φε (t))dVεt − (n + 1)Mω0 , M

where Cε in (i) can be bounded by a uniform constant C.

θε (φε (t))

+ C,

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Proof. Following the arguments in [39], we only need to prove the two facts, (1) the constant Cε in (i) can be bounded by a uniform constant C; (2) Mω0 ,

θε (φε (0))

are uniformly bounded.

We note that φ˙ ε are uniformly bounded. (i), (ii), (iii) and (iv) can be easily deduced from the above two facts. Since  n! d (Mω0 , θε (φε (t)) − βFω00 (φε (t)) − (6.14) φ˙ ε (t)dVεt ) = 0, dt V M

we obtain that Mω0 ,

0 θε (φε (t)) − βFω0 (φε (t)) −



n! V

φ˙ ε (t)dVεt M

= Mω0 ,

n! 0 θε (φε (0)) − βFω0 (φε (0)) − V

 φ˙ ε (0)dVε

M

 ωεn (|s|2h + ε2 )1−β βn! log dVε + φε (0)dVε e−F0 ω0n V M M   n! n! − F0 + log(|s|2h + ε2 )1−β dV0 − φ˙ ε (0)dVε , V V

n! = V



M

M

where the last equality can be bounded by a uniform constant. Then we prove fact (2). By the definition of Mω0 , θε , we have Mω0 ,

n! θε (φε (0)) = V

 log M

n! − V

ωεn (|s|2h + ε2 )1−β dVε − βIω0 (φε (0)) + βJω0 (φε (0)) e−F0 ω0n

 F0 + log(|s|2h + ε2 )1−β dV0 . M

Since Iω0 (φε (0)) are uniformly bounded and

1 n Jω0

≤

1 n+1 Iω0

≤ Jω0 , we prove (2). 2

Since uε (t)C 0 can be uniformly bounded, by using uniform Sobolev inequality (6.2) and Poincaré inequality, we obtain the following lemma by the arguments in [39] (see Lemma 10). The proof is completely similar, thus we omit it. Proposition 6.5. We have the following estimates along equations (6.9)  A osc(φε (t)) ≤ φε (t)dV0 + B, V M

where constants A and B are independent of ε and t.

(6.15)

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We define the space of smooth Kähler potentials as H(ω0 ) = {φ ∈ C ∞ (M )| ω0 +

√

¯ > 0}. −1∂ ∂φ

(6.16)

Theorem 6.6. Let φε (t) be solutions of equations (6.9) and θε = (1 − β)(ω0 + √ −1∂∂ log(ε2 + |s|2h )) be smooth closed semi-positive (1, 1)-forms, where s is the defining section of divisor D and h is a smooth Hermitian metric on the line bundle associated to D. If twisted Mabuchi K-energy functionals Mω0 , θε are uniformly proper on H(ω0 ), that is, there exists a uniform function f such that Mω0 ,

θε (φ)

≥ f (Jω0 (φ))

(6.17)

for any ε and φ ∈ H(ω0 ), where f (t) : R+ → R is a monotone increasing function satisfying lim f (t) = +∞. Then there exists a uniform constant C such that t→+∞

φε (t)C 0 ≤ C.

(6.18)

Proof. Since Mω0 , θε (φε (t)) decrease along equations (6.9) and Mω0 , θε (φε (0)) are uniformly bounded, which are proved in Proposition 6.4. It follows that Jω0 (φε (t)) are uniformly bounded from above. Thus by Proposition 6.4 (iii), we have  (−φε (t))dVεt ≤ C.

(6.19)

M

Since Jω0 ≥ 0, applying (6.17), Mω0 ,θε (φε (t)) are uniformly bounded from below. By Proposition 6.4 (iv), we have  φε (t)dV0 ≤ C,

(6.20)

M

where C is a uniform constant. By this inequality and Green’s formula with respect to metric g0 , we get a uniform upper bound of sup φε (t). By the normalization M

1=

1 V

 dVφε (t) = M

n! V



˙

eφε (t)−βφε (t)−Fε dVε M

and the fact that φ˙ ε (t)C 0 are uniformly bounded along equations (6.9), we have  0 < C1 ≤

e−βφε (t) dVε ≤ C2 ,

M

where C1 and C2 are uniform constants. This inequality easily implies a uniform lower bound for sup φε (t). Combining with (6.15) and (6.20), we obtain a uniform bound for M

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φε (t)C 0 . We also conclude that ϕε (t)C 0 ≤ C for uniform constant C because χ are uniformly bounded. 2 7. The convergence of conical Kähler–Ricci flow In this section, we consider the convergence of conical Kähler–Ricci flows. The most important step in the convergence is to obtain uniform C 0 -estimates for φε (t), thus we only need to prove the uniform properness of twisted Mabuchi K-energy functionals Mω0 , θε . On the other hand, we notice that Mω0 , θε are associated with Log Mabuchi K-energy functional Mω0 , (1−β)D . Therefore, we first review some contents of Log Mabuchi K-energy functional which is first introduced by Li–Sun [25]. For any φ ∈ H(ω0 ), Mω0 ,

(1−β)D (φ) = −

n! V

 Hω0 ,(1−β)D (dV0 − dVφ ) M

+

n! V

 log

M

ωφn dVφ − β(Iω0 (φ) − Jω0 (φ)), ω0n

(7.1)

√ ¯ ω ,(1−β)D and where Hω0 ,(1−β)D satisfies −Ric(ω0 ) + βω0 + (1 − β)[D] = −1∂ ∂H 0  1 −Hω0 ,(1−β)D dV0 = 1. It is easy to see that up to a constant Hω0 ,(1−β)D = V M e F0 + (1 − β) log |s|2h . The Log Mabuchi K-energy functional Mω0 , (1−β)D : H(ω0 ) → R is called proper if there is an inequality of the type Mω0 ,

(1−β)D (φ)

≥ f (Jω0 (φ))

(7.2)

for any φ ∈ H(ω0 ), where f (t) : R+ → R is a monotone increasing function satisfying lim f (t) = +∞. By using Donaldson’s openness theorem [19] and the linear property of t→+∞

Log Mabuchi K-energy functional Mω0 ,

(1−β)D ,

Li–Sun [25] proved the following lemma.

Lemma 7.1. (Corollary 1.4 in [25]) If there is a conical Kähler–Einstein metric for β ∈ (0, 1), then the Log Mabuchi K-energy functional Mω0 , (1−β)D is proper. Song–Wang proved a similar result in [46]. In both Li–Sun and Song–Wang’s arguments, Donaldson’s openness theorem plays a key role. Recently, Yao provided an alternative proof of Donaldson’s openness theorem in [57]. Here, we give a remark to Yao’s paper. Let’s review Yao’s idea. Suppose that ωϕβ is a weak conical Kähler–Einstein metric (see Definition 2.1 in [57]). Yao considered the following two parameter continuity paths βε,t with ε ∈ (0, 1] and t ∈ [0, β] to deform Kähler metrics ωψε,β .

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

βε,t :

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⎧ ⎨ Ric(ωφβ ) = tωφβ + (β − t)ωϕε + (1 − β)χε , ε,t

⎩

φβε,0

ε,t

= ψε,β

where {ωϕε } is a sequence of smooth Kähler forms such that ωϕnε approximate ωϕnβ in 1 Lp -sense (p ∈ (1, 1−β )), ψε,β are solutions to βε,0 obtained by using Yau’s results [58], √ and χε = ω0 + −1∂ ∂¯ log(ε2 + |s|2h ). By using Berndtsson’s uniqueness theorem in [4], Yao proved that there exists ε0 > 0 such that the continuity paths βε,t are solvable up to t = β for all ε ∈ (0, ε0 ]. He also got a uniform bound of φβε,β L∞ (M ) for ε ∈ (0, ε0 ] (Proposition 3.11 in [57]). Then he considered the new two parameter continuity paths

ε,t :

⎧ ⎨ Ric(ωuε,t ) = tωuε,t + (1 − t)χε ⎩

uε,β = φβε,β

and proved that there exist δ > 0 and ε˜ > 0 such that uε,t are uniformly bounded for (ε, t) ∈ (0, ε˜] × (β − δ, β + δ) (Proposition 4.1 in [57]). Using these uniform C 0 -estimates of uε,t for any β  ∈ (β − δ, β + δ) and ε ∈ (0, ε˜], he proved that ωuε,β must converge to a weak conical Kähler–Einstein metric ωϕβ with angle 2πβ  along D as ε → 0. This gives another proof of Donaldson’s openness theorem. Remark 7.2. In Yao’s arguments of uniform C 0 -estimates for uε,t , we should find a fixed δ˜ ˜ and ε ∈ (0, ε0 ] beforehand. Since χε and prove that ε,t can be solved for any t ∈ (β, β+ δ) are strictly positive (1, 1)-forms, the linearized operators at t = β, which equal Δφβ +β, ε,β are invertible for some standard Banach spaces. Yao used the standard implicit function theorem to perturb t a little bit in both directions on ε,t for ε ∈ (0, ε0 ]. But in general, the perturbation (β − δ(ε), β + δ(ε)) of t depends on ε, that is, we are not sure whether δ(ε) converge to 0 as ε → 0. To deal with this problem, we can use Székelyhidi’s results in [48] where he proved that for any ω0 ∈ c1 (M ), if there exists a metric ω ˜ ∈ c1 (M ) such that Ric(˜ ω) > kω ˜ , then equation Ric(ω) = kω + (1 − k)ω0

(7.3)

is solvable. Since ε,t can be solved at t = β for ε ∈ (0, ε0 ] while χε ∈ c1 (M ) are Kähler forms, we have Ric(ωuε0 ,β ) > βωuε0 ,β . Obviously, there exists δ˜ such that ˜ u . Ric(ωuε0 ,β ) > (β + δ)ω ε0 ,β

(7.4)

Replacing ω0 and k in (7.3) with χε (any ε ∈ (0, ε0 ]) and t respectively, ε,t can be solved ˜ and ε ∈ (0, ε0 ] by Székelyhidi’s results. for any t ∈ [0, β + δ] Next, we connect the uniform properness of twisted Mabuchi K-energy functionals Mω0 , θε with the properness of Log Mabuchi K-energy functional Mω0 , (1−β){D} .

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Lemma 7.3. If the Log Mabuchi K-energy functional Mω0 , then Mω0 , θε are uniformly proper on H(ω0 ).

(1−β){D}

is proper on H(ω0 ),

Proof. By assumption, we have Mω0 ,

≥ C f˜(Jω0 (φ)) − C.

(1−β)D (φ)

From their definitions, we have Mω0 ,

θε (φ)

≥ Mω0 ,

(1−β)D (φ)

−C

≥ C f˜(Jω0 (φ)) − C, where C is independent of ε. By setting f = C f˜ − C, we get the uniform properness of Mω0 , θε on H(ω0 ). 2 Now we prove the convergence of conical Kähler–Ricci flows. Theorem 7.4. Assume that there exists a conical Kähler–Einstein metric ωβ,D , then flow ∞ (1.1) converges to conical Kähler–Einstein metric ωβ,D in Cloc -topology outside D and 1 α,β globally in C -sense for any α ∈ (0, min{ β − 1, 1}). Proof. First, by computing, we have d Mω0 , dt

n! θε (φε ) = − V

 |∂ φ˙ ε |2gε (t) dVεt .

(7.5)

M

 ˙ 2 Let Yε (t) = n! V M |∂ φε |gε (t) dVεt . By Lemma 7.1 and Lemma 7.3, Mω0 , proper. Hence they can be bounded from below uniformly. For any T , T

are uniformly

T Yε (t)dt ≤

1

θε

Yε (t)dt = Mω0 ,

θε (φε (0))

− Mω0 ,

θε (φε (T ))

≤ C,

(7.6)

0

where C is a uniform constant. Define Y (t) =

n! V

 ˙ 2 dVt . |∂ φ| g(t)

(7.7)

M

From Theorem 5.2, |∂ φ˙ ε |2gε (t) ≤ C when t ≥ 1, then T

εi →0

T

Yεi (t)dt − −−− → 1

Y (t)dt, 1

(7.8)

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where {εi } is obtained in Theorem 4.1. By (7.6), when T → +∞, we get +∞  Y (t)dt < ∞.

(7.9)

1

Hence there exists time sequence tm ∈ [m, m + 1) such that Y (tm ) → 0 as m → +∞. Next, Yε (t) satisfies the following differential identity, n! Y˙ ε (t) = β(n + 1)Yε (t) − V −

n! V

 |∇φ˙ ε |2gε (t) (R(gε (t)) − trgε (t) θε )dVεt − M



|∇∇φ˙ ε |2gε (t) dVεt − M

n! 2V

n! V

 |∇∇φ˙ ε |2gε (t) dVεt M

 θ(∇φ˙ ε , J ∇φ˙ ε )dVεt . M

By Theorem 5.2, we have |R(gε (t)) − trgε (t) θε | ≤ C for a uniform constant C when t  1. Hence Y˙ ε (t) ≤ CYε (t).

(7.10)

Then Yε (t) ≤ eC(t−s) Yε (s) for any t > s. Let εi → 0, we have Y (t) ≤ eC(t−s) Y (s)

(7.11)

when s, t  1. In particular, Y (t) ≤ e2C Y (tm ) for all t ∈ [m + 1, m + 2), hence Y (t) → 0 when t → +∞. Since Mω0 , θε are uniformly proper, ϕε C 0 are uniformly bounded. From Proposition 6.3, ϕ˙ ε C 0 are also uniformly bounded. By Theorem 4.1, we have ϕC 0 ≤ C,

ϕ ˙ C 0 ≤ C,

C −1 ω∗  ωϕ  Cω∗

(7.12)

for uniform constant C on M \D×[0, +∞). Then for any K ⊂⊂ M \D, by Proposition 3.3, there exists a time sequence {ti } such that ϕ(ti ) converge in C ∞ -topology to a smooth function ϕ∞ on K.  |∂(log K

n ωϕ(t i)

ω0n



≤C

|∂ ϕ(t ˙ i )|2g(ti ) dVti → 0. M

2(1−β)

+ F0 + β(k|s|2h + ϕ(ti )) + log |s|h

)|2g0 dV0

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On the other hand, we have  |∂(log K



→

|∂(log K

n ωϕ(t i)

ω0n

2(1−β)

+ F0 + β(k|s|2h + ϕ(ti )) + log |s|h

)|2g0 dV0

ωϕn∞ 2(1−β) 2 + F0 + β(k|s|2h + ϕ∞ ) + log |s|h )|g0 dV0 . ω0n

By the uniqueness of the limit,  |∂(log K

ωϕn∞ 2(1−β) 2 + F0 + β(k|s|2h + ϕ∞ ) + log |s|h )|g0 dV0 = 0. ω0n

Hence Ric(ωϕ∞ ) = βωϕ∞ on K. At the same time, there exists a time subsequence denoted also by {ti } such that ∞ ϕ(ti ) converge in Cloc -topology outside D to a function ϕ∞ which is smooth on M \ D. ∞ We also have ϕ(t ˙ i ) converge to some constant C in Cloc -topology outside D. For any (n − 1, n − 1)-form η, since log 

√

2(1−β)

n ωϕ |s|h ω0n

∂ϕ(ti ) ti →+∞ ∧ η −− −1∂ ∂¯ −−−→ ∂t

M

 (−Ric(ωϕ∞ ) + βωϕ∞ + (1 − β)[D]) ∧ η. M

Let K ⊂⊂ M \D be a compact subset, Then  |

and ϕC 0 are uniformly bounded,

(

√ ∂ϕ(ti ) ¯ ≤| − C) −1∂ ∂η| ∂t

M



√

M \K

 (

¯ = δ, and δ → 0 when K → M \D. −1∂ ∂η

√ ∂ϕ(ti ) i →+∞ ¯ + Cδ ˜ −t− ˜ − C) −1∂ ∂η| −−−→ Cδ. ∂t

K

When letting K → M \ D, we have 

√

−1∂ ∂¯

M

∂ϕ(ti ) ∧η = ∂t



∂ϕ(ti ) √ i →+∞ ¯ −t− −1∂ ∂η −−−→ 0. ∂t

M

Hence, we obtain that Ric(ωϕ∞ ) = βωϕ∞ + (1 − β)[D]

(7.13)

in the sense of currents. Since C −1 ω∗ ≤ ωϕ ≤ Cω∗ , we also have C −1 ω∗ ≤ ωϕ∞ ≤ Cω∗ .

(7.14)

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By the similar arguments as that in the proof of Proposition 4.2, ϕ∞ is Hölder continuous on M . On the basis of the properness of Log Mabuchi K-energy functional, there must exist ωϕ∞ = ωβ,D according to the uniqueness of conical Kähler–Einstein metrics with bound potentials proved by Berman [2]. At last, we claim that ωϕ(t) must converge to ωϕ∞ in C α,β -sense for any α ∈ (0, min{1, β1 −1}). Combining uniform Perelman’s estimates and Proposition 3.1, |dϕ˙ ε |2ωε are uniformly bounded for t  1. We consider the map Ψ : (z 1 , z 2 , · · · , z n−1 , ξ) −→ (z 1 , z 2 , · · · , z n−1 , z n )

(7.15)

near divisor D, where (z 1 , · · · , z n ) is a local holomorphic coordinate, D = {z n = 0} 1 and z n = |ξ| β −1 ξ. By the similar arguments as (A.5) in Appendix, we have Ψ∗ (g ) ≤ C(dρ2 + ρ2 dθ2 ), C

−1

∗

(7.16)

(dρ + ρ dθ ) ≤ Ψ (g∗ ) ≤ C(dρ + ρ dθ ) 2

2

2

2

2

2

(7.17)

for uniform constant C. Hence there exists uniform constant C such that |ϕ˙ ε (x) − ϕ˙ ε (y)| |ϕ˙ ε (x) − ϕ˙ ε (y)| |ϕ˙ ε (x) − ϕ˙ ε (y)| ≤C =C ≤ C. dΨ∗ g∗ (x, y) dΨ∗ gε (x, y) dgε (x, y) Let ε → 0, ϕ ˙ C α,β ˜ ∈ (0, 1). By the uniqueness ˜ (M ) are uniformly bounded for any α theorem of conical Kähler–Ricci flows (see Lemma 3.2 [56]), the flows in section 4 coincide  with the strong one obtained by Chen–Wang [14], hence ϕ are C 2,α ,β for any α ∈ (0, min{1, β1 − 1}). From the priori estimates proved by Chu [15] (see also Chen–Wang’s work [14]) of equation (ω∗ +

√

˙ ¯ n = eϕ−F −1∂ ∂ϕ)

∗

−βϕ

ω∗n ,

(7.18)

ϕC 2,α ,β (M ) are uniformly bounded by n, α , β, ϕ ˙ C α ,β (M ) , ϕC α ,β (M ) , F ∗ C α ,β (M ) ω n |s|

2(1−β)

γ,β and constant C in (7.12), where F ∗ = log ∗ ωhn + F0 + kβ|s|2β with γ = h is C 0 2 min{ β − 2, 1}. Applying this priori estimates and the arguments in Proposition 4.2, there exists uniform α ∈ (0, 1) such that ϕC 2,α,β (M ) are uniformly bounded. Hence ϕC α ,β (M ) are uniformly bounded. ϕC 2,α ,β (M ) are uniformly bounded by using the priori estimate again. If the claim is not true, then there exist α0 ∈ (0, min{1, β1 − 1}), 0 > 0 and a sequence {ti } such that

√  ¯  −1∂ ∂(ϕ(t i ) − ϕ∞ )C α0 ,β (M ) ≥ 0 .

(7.19)

On the other hand, since ϕ(ti )C 2,α ,β (M ) are uniformly bounded for some α > α0 , there exists a subsequence which we also denote it by {ti } such that ϕ(ti ) converge in C 2,α0 ,β -sense to a function ϕ˜∞ and

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J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

√ ¯ ϕ˜∞ − ϕ∞ )C α0 ,β (M ) ≥ 0 .  −1∂ ∂(

(7.20)

ωϕ˜∞ is also a conical Kähler–Einstein metric with Hölder continuous potential. However, ωϕ˜∞ ≡ ωϕ∞ by (7.20), which is impossible by Berman’s uniqueness results. Hence we ∞ prove the claim. Since we have uniform Cloc -estimates for ωϕ , we can also prove that ωϕ ∞ converge to ωϕ∞ in Cloc -topology outside D by the similar arguments as above. 2 Acknowledgments We would like to thank Professor Jiayu Li and Professor Xiaohua Zhu for their useful conversations and suggestions. We also would like to express our gratitude towards the reviewers for their careful reading and valuable suggestions. The authors are supported in part by NSF in China No. 11625106, 11571332 and 11131007. Appendix A In the appendix, we first prove Proposition 5.9 and Theorem 6.1. Proof of Proposition 5.9. Under the appropriate coordinate system (see Lemma 3.2), metrics ωε can be written as follows. √ ωε = ω0 + ke−ϕ (ε2 + |z n |2 e−ϕ )β−1 −1dz n ∧ d¯ zn √ ∂ϕ 2 (ε + |z n |2 e−ϕ )β−1 −1dz α ∧ d¯ zn α ∂z √ ∂ϕ − ke−ϕ z¯n γ (ε2 + |z n |2 e−ϕ )β−1 −1dz n ∧ d¯ zγ ∂ z¯ √ ∂ϕ ∂ϕ zγ + ke−ϕ |z n |2 α γ (ε2 + |z n |2 e−ϕ )β−1 −1dz α ∧ d¯ ∂z ∂ z¯ k ∂2ϕ √ − ((ε2 + |z n |2 e−ϕ )β − ε2β ) α γ −1dz α ∧ d¯ zγ . β ∂z ∂ z¯ − ke−ϕ z n

(A.1)

We consider the maps Ψε : (z 1 , z 2 , · · · , z n−1 , ξ) −→ (z 1 , z 2 , · · · , z n−1 , z n ),

(A.2)

where z n = (ε2β +|ξ|2 ) 2β − 2 ξ. Now, we want to show that Ψ∗ε (gε ) are uniformly equivalent to the Euclidean metric in a small neighborhood of the divisor D. By direct calculations, we only need to deal with the following terms 1

1

Ψ∗ε (ke−ϕ (ε2 + |z n |2 e−ϕ )β−1 dz n · d¯ z n ).

(A.3)

We will show that (A.3) are uniformly equivalent to the Euclidean metric on C. Let √ z n = x + −1y, x = r cos θ and y = r sin θ, then

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

dz n · d¯ z n = dz n ⊗ d¯ z n + d¯ z n ⊗ dz n = 2(dr2 + r2 dθ2 ).

1367

(A.4)

√ We let ξ = u + −1v, u = ρ cos θ1 and v = ρ sin θ1 . By the definition of Ψε , θ1 = θ and 1 1 r = (ε2β + ρ2 ) 2β − 2 ρ. Hence we have Ψ∗ε (ke−ϕ (ε2 + |z n |2 e−ϕ )β−1 dz n · d¯ zn) 1 1 = 2ke−ϕ◦Ψε (ε2 + (ε2β + ρ2 ) β −1 ρ2 e−ϕ◦Ψε )β−1 (ε2β + ρ2 ) β −1 · ((1 + ( β1 − 1)(ε2β + ρ2 )−1 ρ2 )2 dρ2 + ρ2 dθ12 ). Because 1 ≤ (1 + ( β1 − 1)(ε2β + ρ2 )−1 ρ2 )2 ≤

1 β2 ,

(A.5)

we only need to prove that

(ε2 + (ε2β + ρ2 ) β −1 ρ2 e−ϕ◦Ψε )β−1 (ε2β + ρ2 ) β −1 1

1

(A.6)

can be uniformly bounded and the uniform lower bound is away from 0. Firstly, (ε2 + (ε2β + ρ2 ) β −1 ρ2 e−ϕ◦Ψε )β−1 (ε2β + ρ2 ) β −1 1

1

≥ (1 + e−ϕ◦Ψε )β−1 ≥ c > 0, where c is independent of ε. Secondly, we prove that the term (A.6) can be bounded from above. Let εβ = l cos ϑ and ρ = l sin ϑ, where ϑ ∈ [0, π2 ], then (ε2 + (ε2β + ρ2 ) β −1 ρ2 e−ϕ◦Ψε )β−1 (ε2β + ρ2 ) β −1 1

=(

1

1

)1−β ≤ 2( β −1)(1−β) ec(1−β) . 1

2 β

cos ϑ + sin2 ϑe−ϕ◦Ψε

In conclusion, it shows that C1 (dρ2 + ρ2 dθ12 ) ≤ Ψ∗ε (ke−ϕ (ε2 + |z n |2 e−ϕ )β−1 dz n · d¯ z n ) ≤ C2 (dρ2 + ρ2 dθ12 ) for uniform constants C1 and C2 both independent of ε. It is easy to see that the pull-back metric Ψ∗ε (gε ) are uniformly equivalent to the Euclidean metric in a small neighborhood of the divisor D. Therefore, the Sobolev inequality holds if the function v is supported in the above coordinate charts. The global case follows in the standard way by using a partition of unity. Following these arguments,    n−1 2n 2 n−1 n ( v dVε ) ≤ C( |dv|gε dVε + |v|2 dVε ). (A.7) M

M

M

To prove (5.28), we only need to prove that ωε and ωε (t) are uniformly equivalent when t ∈ [0, 2]. Noting that the metric ωε (t) are independent of the choice of the initial constant ϕε (0), without loss of generality, we assume ϕε (0) = 0. By (4.2), we have ϕ˙ ε (t)C 0 (M ×[0,2]) ≤ C and ϕε (t)C 0 (M ×[0,2]) ≤ C for uniform constant C which only

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1368

ω n (ε2 +|s|2 )1−β

depends on log ε ωn h +kβχ(ε2 +|s|2h ) +F0 . Then the uniform equivalence between 0 these metrics follows from Proposition 3.1. 2 Proof of Theorem 6.1. In the proof, we only need to consider the Sobolev inequality along twisted Kähler–Ricci flow (2.6) for t ≥ 1. The proof is almost the same as that in [62] with the only differences that we require the constants independent of ε in addition, thus we give the proof briefly here. Step 1. By using the monotonicity of the functional μθε (gε (s), τ (s)) (Theorem 2.5 in [27]) and taking τ (s) = 1 − e−βt (1 − δ 2 )eβs with δ ∈ (0, 1), we conclude that   2 2 v log v dVεt ≤ δ 2 ((R(gε (t)) − trgε (t) θε )v 2 + 4|∇v|2gε (t) )dVεt M

M

− 2βn log δ + L1 + max(R(gε (1)) − trgε (1) θε )− , M

(A.8)

where L1 = βn log(βnCS (M, gε (1))2 ) − βn log 2 − βn + CS (M,g4 ε (1))2 V − n . Step 2. Fixing a time t0 ≥ 1, we show the upper bound of the short time heat kernel for the fundamental solution of equation 1

1 ∂ Δgε (t0 ) u(x, t) − (R(gε (t0 )) − trgε (t0 ) θε )u(x, t) − u(x, t) = 0. 4 ∂t

(A.9)

Let u be a positive solution of equation (A.9). For the given T ∈ (0, 1] and t ∈ (0, T ], we take p(t) = TT−t . Differentiating up(t) and substituting (A.8) into it, then integrating from t = 0 to t = T on both sides, we have log

u(·, T )∞ ≤ −n log T + L1 + 2βn + 2 max(R(gε (1)) − trgε (1) θε )− . M u(·, 0)1

Since u(x, T ) =

 M

Pε (x, y, T ) ≤

(A.10)

Pε (x, y, T )u(y, 0)dVεt0 , where Pε is the heat kernel of (A.9), exp(L1 + 2βn + 2 max(R(gε (1)) − trgε (1) θε )− ) M

:=

Tn

Step 3. Let Fε = max(R(gε (1)) − trgε (1) θε )− , Ψε,t0 = M

Λ . Tn

1 4 (R(gε (t0 ))

(A.11)

+ trgε (t0 ) θε ) +

Fε + 1  1 and PFε be the heat kernel of Δgε (t0 ) − Ψε,t0 . For t > 0 and y ∈ M , (

d − Δgε (t0 ) )PFε (x, y, t + 1) = −Ψε,t0 PFε (x, y, t + 1). dt

(A.12)

By the maximum principle and (A.11), PFε obeys the global upper bound ˜ −n , PFε (x, y, t) ≤ Ct

t > 0,

where C˜ depends only on Λ and n. Moreover, for any f ∈ L2 (M ), we have

(A.13)

J. Liu, X. Zhang / Advances in Mathematics 307 (2017) 1324–1371

 |

 PFε (x, y, t)f (y)dVεt0 | ≤ (

M

1369

PF2ε (x, y, t)dVεt0 ) 2 f L2 ≤ C˜ 2 t− 2 f L2 1

1

n

M

by Hölder inequality. Then the Sobolev inequality follows Theorem 2.4.2 in [17] and the ˜ 2n and 1 . By the expression of Λ, constants inside the inequality depend only on C, 2n−2 Lemma 5.6 and Proposition 5.9, the constants are independent of ε and t. 2 Proof of Proposition 5.13. If diam(M, gε (t)) are not uniformly bounded, there exist {ti } ⊂ [1, +∞) and εi → 0 such that diam(M, gεi (ti )) → +∞. Let δi → 0 be a sequence of positive numbers, which corresponds to {ti } and {εi }. By Lemma 5.11, we can find sequences {k1i } and {k2i } such that V olgεi (ti ) (Bεi ti (k1i , k2i )) < δi ,

(A.14)

V olgεi (ti ) (Bεi ti (k1i , k2i )) ≤ 220n V olgεi (ti ) (Bεi ti (k1i + 2, k2i − 2)).

(A.15)

For each i, r1i ∈ [2k1 , 2k1 +1 ] and r2i ∈ [2k2 −1 , 2k2 ] are given by Lemma 5.12, φi are cut off i i functions such that φi = 1 on [2k1 +2 , 2k2 −2 ] and φi = 0 on (−∞, r1i ] [r2i , +∞). Define ui (x) = eCi φi (distεi ti (x, pi )), where uεi (pi , ti ) = inf M uεi (y, ti ) and Ci are constants  such that V1 M u2i dVεi ti = 1. i

1 1= V

i

i

 e2Ci φ2i dVεi ti ≤

i

1 2Ci 1 e V ol(Bεi ti (k1i , k2i )) ≤ e2Ci δi . V V

M

Let i → +∞. Since δi → 0, we conclude that Ci → +∞. We consider the functions  1 2 1 1 2 2 (ui + 1) whose integral averages are V M 2 (ui + 1)dVεi ti = 1. By computing, 1 Wθεi (gεi (ti ), − log (u2i + 1), 1) 2   4u2i |∇ui |2gε (ti ) 1 1 2 2 i (ui + 1)(R(gεi (ti )) − trgεi (ti ) θεi + L)dVεi ti + (ui + 1) dVεi ti = 2 2 (u2i + 1)2 M

β + log 2 2

 (u2i + 1)dVεi ti

β − 2

M



M

(u2i + 1) log(u2i + 1)dVεi ti − LV M

1 + (βn + L)V + Ce2Ci V olgεi (ti ) (Bεi ti (k1i , k2i )) 2 2Ci − βV Ci + Ce V olgεi (ti ) (Bεi ti (k1i , k2i )) + C

≤ Ce

2Ci

V

olgεi (ti ) (Bεi ti (k1i , k2i ))

≤ C − Ci βV, where L and C are constants independent of ti and εi and L also satisfies R(gεi (ti )) − trgεi (ti ) θεi + L ≥ 0 for any i. Therefore, by (5.30), we have −C ≤ μθεi (gεi (1), 1) ≤ C − βV Ci ,

(A.16)

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