Cusp Kähler–Ricci flow on compact Kähler manifolds

Abstract

In this paper, we study the limiting flow of conical Kähler–Ricci flows when the cone angles tend to 0. We prove the existence and uniqueness of this limiting flow with cusp singularity on compact Kähler manifold M which carries a smooth hypersurface D such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this limiting flow converge to a unique cusp Kähler–Einstein metric.

1 Introduction

In this paper, we study the limiting flow of (twisted) conical Kähler–Ricci flows when the cone angles tend to 0. Our motivation for considering this limiting flow is to study the existence of singular Kähler–Einstein metric when the cone angle is 0. In [39], Tian anticipated that the complete Tian–Yau Kähler–Einstein metric on the complement of a divisor should be the limit of conical Kähler–Einstein metrics when the cone angles tend to 0.

Let M be a compact Kähler manifold with complex dimension n and $D\subset M$ be a smooth hypersurface. Here, by supposing that the twisted canonical bundle $K_M+D$ is ample, we prove the long-time existence, uniqueness and convergence of the limiting flow of twisted conical Kähler–Ricci flows when the cone angles tend to 0. Since this limiting flow admits cusp singularity along D, we call it cusp Kähler–Ricci flow. As an application, we show the existence of cusp Kähler–Einstein metric [22, 40] by using cusp Kähler–Ricci flow.

The conical Kähler–Ricci flow was introduced to attack the existence problem of conical Kähler–Einstein metric. This equation was first proposed in Jeffres–Mazzeo–Rubinstein’s paper (Section 2.5 in [19]). Song–Wang made some conjectures on the relation between the convergence of conical Kähler–Ricci flow and the greatest Ricci lower bound of M (conjecture 5.2 in [37]). The long-time existence, regularity and limit behavior of conical Kähler–Ricci flow have been widely studied, see the works of Liu–Zhang [27, 28], Chen-Wang [7, 8], Wang [45], Shen [34, 35], Edwards [11], Nomura [33], Liu–Zhang [26] and Zhang [47].

By saying a closed positive (1, 1)-current $\omega$ is conical Kähler metric with cone angle $2\pi \beta$ ($0<\beta \le 1$ ) along D, we mean that D is locally given by $\{z^{n}=0\}$ and $\omega$ is asymptotically equivalent to model conical metric

$$\begin{aligned} \sqrt{-1} \sum _{j=1}^{n-1} \mathrm{d}z^{j}\wedge \mathrm{d}\overline{z}^{j}+\frac{\sqrt{-1}\mathrm{d}z^{n}\wedge \mathrm{d}\overline{z}^{n}}{|z^{n}|^{2(1-\beta )}}. \end{aligned}$$

(1.1)

And by saying a closed positive (1, 1)-current $\omega$ is cusp Kähler metric along D, we mean that D is locally given by $\{z^{n}=0\}$ and $\omega$ is asymptotically equivalent to model cusp metric

$$\begin{aligned} \sqrt{-1} \sum _{j=1}^{n-1} \mathrm{d}z^{j}\wedge \mathrm{d}\overline{z}^{j}+\frac{\sqrt{-1}\mathrm{d}z^{n}\wedge \mathrm{d}\overline{z}^{n}}{|z^{n}|^{2}\log ^2|z^{n}|^{2}}. \end{aligned}$$

(1.2)

For more about cusp Kähler metrics, please see Auvray’s works [1, 2].

Let $\omega _0$ be a smooth Kähler metric on M and satisfy $c_1(K_M)+c_1(D)=[\omega _0]$. We denote $D=\{s=0\}$, where s is a holomorphic section of the line bundle $L_D$ associated to D. In [28], we proved the long-time existence, uniqueness, regularity and convergence of conical Kähler–Ricci flow with weak initial data $\omega _{\varphi _0}\in \mathcal {E}_{p}(M,\omega _{0})$ when $p>1$, where

$$\begin{aligned} \mathcal {E}_{p}(M,\omega _{0})= & {} \left\{ \varphi \in \mathcal {E}(M,\omega _{0})\ |\ \frac{(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi )^{n}}{\omega _{0}^{n}}\in L^{p}(M,\omega _{0}^{n}) \right\} ,\\ \mathcal {E}(M,\omega _{0})= & {} \left\{ \varphi \in PSH(M,\omega _{0})\ |\ \int _{M} (\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi )^{n}= \int _{M} \omega ^{n}_{0}\right\} . \end{aligned}$$

Thanks to Kołodziej’s theorem (Theorem 2.4.2 in [24]), potentials in the class $\mathcal {E}_{p}(M,\omega _{0})$ with $p>1$ are continuous. Furthermore, by Kołodziej’s $L^{p}$-estimate (Theorem 2.1 in [23]) and Dinew’s uniqueness theorem (Theorem 1.2 in [10], see also Theorem B in [14]), we know that the potentials in $\mathcal {E}_{p}(M,\omega _{0})$ with $p>1$ are Hölder continuous with respect to $\omega _{0}$ on M.

Let $\rho$ be a smooth closed (1, 1)-form and $\hat{\omega }_\beta =\omega _0+\sqrt{-1}\tau \partial \bar{\partial }|s|_h^{2\beta }$, where h is a smooth hermitian metric on $L_D$ and $\tau$ is a small constant. When $c_1(M)=\mu [\omega _0]+(1-\beta )c_1(D)+[\rho ]$$(\mu \in \mathbb {R})$, by our arguments in [28], there exists a unique long-time solution of twisted conical Kähler–Ricci flow

$$\begin{aligned} \frac{\partial \omega _{\beta }(t)}{\partial t}= & {} -Ric(\omega _{\beta }(t))+\mu \omega _{\beta }(t)+(1-\beta )[D]+\rho . \nonumber \\ \omega _{\beta }(t)|_{t=0}= & {} \omega _{\varphi _0} \end{aligned}$$

(1.3)

Definition 1.1

We call $\omega _\beta (t)$ a long-time solution to twisted conical Kähler–Ricci flow (1.3) if it satisfies the following conditions.

(1):

For any $[\delta , T]$ ($\delta , T>0$), there exists constant C such that

$$\begin{aligned} C^{-1}\hat{\omega }_\beta \le \omega _{\beta }(t)\le C\hat{\omega }_\beta \quad \mathrm{on}\quad [\delta ,T]\times (M{\setminus } D); \end{aligned}$$

(2):

on $(0,\infty )\times (M{\setminus } D)$, $\omega _{\beta }(t)$ satisfies smooth twisted Kähler–Ricci flow;

(3):

on $(0,\infty )\times M$, $\omega _{\beta }(t)$ satisfies Eq. (1.3) in the sense of currents;

(4):

there exists metric potential $\varphi _{\beta }(t)\in C^{0}\left( [0,\infty )\times M\right) \cap C^{\infty }\left( (0,\infty )\times (M{\setminus } D)\right)$ such that $\omega _{\beta }(t)=\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi _{\beta }(t)$ and $\lim \limits _{t\rightarrow 0^{+}}\Vert \varphi _{\beta }(t)-\varphi _{0}\Vert _{L^{\infty }(M)}=0$;

(5):

on $[\delta , T]$, there exist constant $\alpha \in (0,1)$ and $C^{*}$ such that the above metric potential $\varphi _{\beta }(t)$ is $C^{\alpha }$ on M with respect to $\omega _{0}$ and $\Vert \frac{\partial \varphi _{\beta }(t)}{\partial t}\Vert _{L^{\infty }(M{\setminus } D)}\leqslant C^{*}$.

From Guenancia’s result (Lemma 3.1 in [16]),

$$\begin{aligned} \omega _{\beta }=\omega _{0}-\sqrt{-1}\partial \bar{\partial }\log \left( \frac{1-|s|_{h}^{2\beta }}{\beta }\right) ^{2}:=\omega _{0}+\sqrt{-1}\partial \bar{\partial }\psi _{\beta } \end{aligned}$$

(1.4)

is a conical Kähler metric with cone angle $2\pi \beta$ along D. Hence, $\omega _\beta \in \mathcal {E}_{p}(M,\omega _{0})$ for $p\in (1,\frac{1}{1-\beta })$. By direct calculations, it is obvious that $\omega _\beta \ge \frac{1}{2}\omega _0$ for choosing suitable hermitian metric h. Since $\psi _\beta$ converge to $\psi _0$ in $C_{loc}^{\infty }$-sense outside D and globally in $L^1$-sense on M as $\beta \rightarrow 0$, $\omega _\beta$ converge to cusp Kähler metric

$$\begin{aligned} \omega _{cusp}=\omega _{0}-\sqrt{-1}\partial \bar{\partial }\log \log ^2|s|_h^2:=\omega _{0}+\sqrt{-1}\partial \bar{\partial }\psi _{0} \end{aligned}$$

(1.5)

in $C_{loc}^{\infty }$-sense outside D and globally in the sense of currents. Here we remark that $\frac{\omega _{\psi _0}^n}{\omega _{0}^{n}}\in L^1(M)$ but $\frac{\omega _{\psi _0}^n}{\omega _{0}^{n}}\notin L^p(M)$ for $p>1$. We pick $\theta \in c_1(D)$ a smooth real closed (1, 1)-form such that $[D]=\theta +\sqrt{-1}\partial \bar{\partial }\log |s|_h^2$, then we consider twisted conical Kähler–Ricci flow

$$\begin{aligned} \frac{\partial \omega _{\beta }(t)}{\partial t}= & {} -Ric(\omega _{\beta }(t))-\omega _{\beta }(t)+(1-\beta )[D]+\beta \theta .\nonumber \\ \omega _{\beta }(t)|_{t=0}= & {} \omega _{\beta } \end{aligned}$$

(1.6)

Since $c_1(K_M)+c_1(D)=[\omega _0]$, flow (1.6) preserves the Kähler class, that is, $[\omega _\beta (t)]=[\omega _0]$. We write (1.6) as parabolic complex Monge–Ampère equation on potentials,

$$\begin{aligned} \frac{\partial \varphi _{\beta }(t)}{\partial t}= & {} \log \frac{(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi _{\beta }(t))^{n}}{\omega _{0}^{n}}-\varphi _{\beta }(t)+h_{0}+(1-\beta )\log |s|_{h}^{2}\nonumber \\ \varphi _{\beta }(0)= & {} \psi _\beta \end{aligned}$$

(1.7)

on $(0,\infty )\times (M{\setminus } D)$, where $h_{0}\in C^\infty (M)$ satisfies $-Ric(\omega _{0})+\theta -\omega _{0}=\sqrt{-1}\partial \overline{\partial }h_{0}$. By proving uniform estimates (independent of $\beta$) for twisted conical Kähler–Ricci flows (1.6), we obtain the limiting flow which is called cusp Kähler–Ricci flow

$$\begin{aligned} \frac{\partial \omega (t)}{\partial t}= & {} -Ric(\omega (t))-\omega (t)+[D].\nonumber \\ \omega (t)|_{t=0}= & {} \omega _{cusp} \end{aligned}$$

(1.8)

Definition 1.2

We call $\omega (t)$ a long-time solution to cusp Kähler–Ricci flow (1.8) if it satisfies the following conditions.

(1):

For any $[\delta , T]$ ($\delta , T>0$), there exists constant C such that

$$\begin{aligned} C^{-1}\omega _{cusp}\le \omega (t)\le C\omega _{cusp}\quad \mathrm{on}\quad [\delta ,T]\times (M{\setminus } D); \end{aligned}$$

(2):

on $(0,\infty )\times (M{\setminus } D)$, $\omega (t)$ satisfies smooth Kähler–Ricci flow;

(3):

on $(0,\infty )\times M$, $\omega (t)$ satisfies Eq. (1.8) in the sense of currents;

(4):

there exists $\varphi (t)\in C^{0}\left( [0,\infty )\times (M{\setminus } D)\right) \cap C^{\infty }\left( (0,\infty )\times (M{\setminus } D)\right)$ such that

$$\begin{aligned} \omega (t)=\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi (t)\ \ \mathrm{and}\ \ \lim \limits _{t\rightarrow 0^{+}}\Vert \varphi (t)-\psi _{0}\Vert _{L^{1}(M)}=0; \end{aligned}$$

(5):

on (0, T], $\Vert \varphi (t)-\psi _0\Vert _{L^{\infty }(M{\setminus } D)}\leqslant C$;

(6):

on $[\delta , T]$, there exist constant C such that $\Vert \frac{\partial \varphi (t)}{\partial t}\Vert _{L^{\infty }(M{\setminus } D)}\leqslant C$.

There are some important results on Kähler–Ricci flows (as well as its twisted versions with smooth twisting forms) from weak initial data, such as Chen–Ding [5], Chen–Tian–Zhang [6], Guedj–Zeriahi [15], Di Nezza–Lu [9], Song-Tian [36], Székelyhidi–Tosatti [38] and Zhang [48] etc. In [29, 30], Lott–Zhang proved some significant results on various Kähler–Ricci flows with singularities. In particular in [29], in $M{\setminus } D$, they thoroughly studied the existence and convergence of Kähler–Ricci flow whose initial metric is finite volume Kähler metric with “superstandard spatial asymptotics” (Definition 8.10 in [29]) and gave some interesting examples. Their flow keeps “superstandard spatial asymptotics” and this type of metrics contain cusp Kähler metrics. Here we study the limiting behavior of conical Kähler–Ricci flows when the cone angles tend to 0 on M. The limiting flow admits non-smooth twisting form globally, cusp singularity along D and weak initial data, it is a solution of cusp Kähler–Ricci flow (1.8) and can be seen as Lott–Zhang’s case when we restrict it in $M{\setminus } D$. In this limiting process, we need prove uniform estimates (independent of $\beta$) for a sequence of conical flows and consider the asymptotic behavior of the weak solution to the limiting flow when t tend to $0^+$. There are some other interesting results on singular Ricci flows, see Ji–Mazzeo–Sesum [20], Kleiner–Lott [21], Mazzeo–Rubinstein–Sesum [32], Topping [43] and Topping–Yin [44].

In [28], we studied conical Kähler–Ricci flows which are twisted by non-smooth twisting forms and start from weak initial data with $L^p$-density for $p>1$. Here, by limiting conical flows (1.6), we prove that the limiting flow with weak initial data $\omega _{cusp}$ is a long-time solution to cusp Kähler–Ricci flow (1.8). This initial metric only admits $L^1$-density. For obtaining this limiting flow, in addition to getting uniform estimates ( independent of $\beta$) of flows (1.6), it is important to prove that $\varphi (t)$ converge to $\psi _{0}$ globally in $L^{1}$-sense and locally in $L^{\infty }$-sense outside D as $t\rightarrow 0^+$. In this process, we need to construct auxiliary function, and we also need a key observation (Proposition 2.8 and 2.9) that both $\psi _\beta$ and $\varphi _\beta (t)$ are monotone decreasing as $\beta \searrow 0$. Then we obtain a uniqueness result of cusp Kähler–Ricci flow. In fact, we obtain the following theorem.

Theorem 1.3

Let M be a compact Kähler manifold and $\omega _{0}$ be a smooth Kähler metric. Assume that $D\subset M$ is a smooth hypersurface which satisfies $c_1(K_M)+c_1(D)=[\omega _0]$. Then the twisted conical Kähler–Ricci flows (1.6) converge to a unique long-time solution $\omega (t)=\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi (t)$ of cusp Kähler–Ricci flow (1.8) in $C_{loc}^{\infty }$-sense in $(0,\infty )\times (M{\setminus } D)$ and globally in the sense of currents.

Remark 1.4

The uniqueness in Theorem 1.3 need to be understood in this sense: if $\phi (t)\in C^{0}\left( [0,\infty )\times (M{\setminus } D)\right) \cap C^{\infty }\left( (0,\infty )\times (M{\setminus } D)\right)$ is a solution to equation

$$\begin{aligned} \frac{\partial \phi (t)}{\partial t}= & {} \log \frac{(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\phi (t))^{n}}{\omega _{0}^{n}}-\phi (t)+h_{0}+\log |s|_{h}^{2}\nonumber \\ \phi (0)= & {} \psi _0 \end{aligned}$$

(1.9)

on $(0,\infty )\times (M{\setminus } D)$ and satisfies (1), (4), (5) and (6) in Definition 1.2, then $\phi (t)$ lies below $\varphi (t)$ which is obtained by limiting twisted conical Kähler–Ricci flows (1.6) in Theorem 1.3. When $n=1$, this uniqueness property is called “maximally stretched” in Topping’s (Remark 1.9 in [42]) and Giesen-Topping’s (Theorem 1.2 in [13]) works.

Remark 1.5

Since $K_M+D$ is ample, $K_{M}+(1-\beta )D$ is also ample for sufficiently small $\beta$. Guenancia [16] proved that cusp Kähler–Einstein metric is the limit of conical Kähler–Einstein metrics with background metrics $\omega _{0}-\beta \theta$ as $\beta \rightarrow 0$. The cohomology classes are changing in this process. But in the flow case, we cannot obtain above uniqueness of the limiting flow if we choose the approximating flows that are conical Kähler–Ricci flows with background metrics $\omega _{0}-\beta \theta$. In fact, if we choose the approximating flows that are conical Kähler–Ricci flows

$$\begin{aligned} \frac{\partial \tilde{\omega }_{\beta }(t)}{\partial t}= & {} -Ric(\tilde{\omega }_{\beta }(t))-\tilde{\omega }_{\beta }(t)+(1-\beta )[D]\nonumber \\ \tilde{\omega }_{\beta }(t)|_{t=0}= & {} \omega _{0}-\beta \theta +\sqrt{-1}\partial \bar{\partial }\psi _\beta \end{aligned}$$

(1.10)

with background metrics $\omega _{0}-\beta \theta$, that is, $\tilde{\omega }_{\beta }(t)=\omega _{0}-\beta \theta +\sqrt{-1}\partial \bar{\partial }\tilde{\varphi }_{\beta }(t)$, we can also get a long-time solution $\tilde{\omega }(t)=\omega _{0}+\sqrt{-1}\partial \bar{\partial }\tilde{\varphi }(t)$ to Eq. (1.8). But we do not know whether $\tilde{\varphi }(t)$ is unique or maximal. We can only prove $\tilde{\varphi }_\beta (t)+\beta \log |s|_h^2\nearrow \tilde{\varphi }(t)$ outside D as $\beta \searrow 0$. However, by the uniqueness result in Theorem 1.3, $\tilde{\varphi }(t)$ must lie below $\varphi (t)$. Therefore, we set the background metric to $\omega _{0}$ in this paper.

At last, we prove the convergence of cusp Kähler–Ricci flow (1.8).

Theorem 1.6

Cusp Kähler–Ricci flow (1.8) converges to a Kähler–Einstein metric with cusp singularity along D in $C_{loc}^{\infty }$-topology outside hypersurface D and globally in the sense of currents.

Kobayashi [22] and Tian–Yau [40] asserted that if the twisted canonical bundle $K_M+D$ is ample, then there is a unique (up to constant multiple) complete cusp Kähler–Einstein metric with negative Ricci curvature in $M{\setminus } D$. The above convergence result recovers the existence of this cusp Kähler–Einstein metric.

The paper is organized as follows. In Sect. 2, we prove the long-time existence and uniqueness of cusp Kähler–Ricci flow (1.8) by limiting twisted conical Kähler–Ricci flows (1.6) and constructing auxiliary function. In Sect. 3, we prove the convergence theorem.

2 The long-time existence of cusp Kähler–Ricci flow

In this section, we prove the long-time existence of cusp Kähler–Ricci flow by limiting twisted conical Kähler–Ricci flows (1.6), and we also prove the uniqueness theorem. For further consideration in the following arguments, we shall pay attention to the estimates which are independent of $\beta$.

From our arguments (Sections 2 and 3 in [28]), we know that there exists a unique long-time solution $\varphi _\beta (t)\in C^0\left( [0,\infty )\times M\right) \bigcap C^\infty \left( (0,\infty )\times (M{\setminus } D)\right)$ to Eq. (1.7). Let $\phi _{\beta }(t)=\varphi _{\beta }(t)-\psi _{\beta }$, we write the Eq. (1.7) as

$$\begin{aligned} \frac{\partial \phi _{\beta }(t)}{\partial t}= & {} \log \frac{(\omega _{\beta }+\sqrt{-1}\partial \bar{\partial }\phi _{\beta }(t))^{n}}{\omega _{\beta }^{n}}-\phi _{\beta }(t)+h_{\beta }\nonumber \\ \phi _{\beta }(0)= & {} 0 \end{aligned}$$

(2.1)

in $(0,\infty )\times (M{\setminus } D)$, where $h_{\beta }=-\psi _{\beta }+h_{0}+\log \frac{|s|_{h}^{2(1-\beta )}\omega _{\beta }^{n}}{\omega _{0}^{n}}$ is uniformly bounded by constant C independent of $\beta$.

Lemma 2.1

There exists constant C independent of $\beta$ and t such that

$$\begin{aligned} \Vert \phi _{\beta }(t)\Vert _{L^{\infty }(M)}\leqslant C. \end{aligned}$$

(2.2)

Proof

Fix $T>0$. For any $\varepsilon >0$, we let $\chi _{\beta ,\varepsilon }(t)=\phi _{\beta }(t)+\varepsilon \log |s|_{h}^{2}$. Since $\chi _{\beta ,\varepsilon }(t)$ is smooth in $M{\setminus } D$, bounded from above and goes to $-\infty$ near D, it achieves its maximum in $M{\setminus } D$. Let $(t_{0}, x_{0})$ be the maximum point of $\chi _{\beta ,\varepsilon }(t)$ on $[0,T]\times M$ with $x_{0}\in M{\setminus } D$. If $t_{0}=0$, then we have

$$\begin{aligned} \phi _{\beta }(t)\leqslant -\varepsilon \log |s|_{h}^{2}. \end{aligned}$$

(2.3)

If $t_{0}\ne 0$. At $(t_{0}, x_{0})$, we have

$$\begin{aligned} 0\leqslant \frac{\partial \chi _{\beta ,\varepsilon }(t)}{\partial t}= & {} \log \frac{(\omega _{\beta }+\sqrt{-1}\partial \bar{\partial }\phi _{\beta }(t))^{n}}{\omega _{\beta }^{n}}-\phi _{\beta }(t)+h_{\beta }\\= & {} \log \frac{(\omega _{\beta }+\sqrt{-1}\partial \bar{\partial }\chi _{\beta ,\varepsilon }(t)+\varepsilon \theta )^{n}}{\omega _{\beta }^{n}}-\phi _{\beta }(t)+h_{\beta }\\\le & {} n\log 2-\phi _{\beta }(t)+C. \end{aligned}$$

Hence, $\phi _{\beta }(t_{0},x_{0})\leqslant C$ and

$$\begin{aligned} \phi _{\beta }(t)\leqslant C-\varepsilon \log |s|_{h}^{2}, \end{aligned}$$

(2.4)

where constant C independent of $\beta$, t and $\varepsilon$. Let $\varepsilon \rightarrow 0$, we have $\phi _{\beta }(t)\leqslant C$ in $M{\setminus } D$. Since $\phi _{\beta }(t)$ is continuous, $\phi _{\beta }(t)\leqslant C$ on M.

For the lower bound, we can reproduce the same arguments with $\tilde{\chi }_{\beta ,\varepsilon }(t)=\phi _{\beta }(t)-\varepsilon \log |s|_{h}^{2}$, and get $\phi _{\beta }(t)\ge C$ on M. $\square$

We now prove the uniform equivalence of volume forms along complex Monge–Ampère Eq. (2.1). We first recall the following lemma.

Lemma 2.2

If $\omega _1$ and $\omega _2$ are positive (1, 1)-forms, then

$$\begin{aligned} n\left( \frac{\omega _1^n}{\omega _2^n}\right) ^{\frac{1}{n}}\leqslant tr_{\omega _2}\omega _1\leqslant n\left( \frac{\omega _1^n}{\omega _2^n}\right) (tr_{\omega _1}\omega _2)^{n-1}. \end{aligned}$$

(2.5)

The proof of Lemma 2.2 follows from eigenvalue considerations (section 2 [46]).

Lemma 2.3

For any $T>0$, there exists constant C independent of $\beta$ such that for any $t\in (0,T]$,

$$\begin{aligned} \frac{t^n}{C}\le \frac{(\omega _{\beta }+\sqrt{-1}\partial \bar{\partial }\phi _{\beta }(t))^{n}}{\omega _{\beta }^n}\le e^{\frac{C}{t}}\ \ \ \ \ in\ \ M{\setminus } D. \end{aligned}$$

(2.6)

Proof

For any $t>0$, we assume that $t\in [\delta ,T]$ with $\delta >0$. Let $\Delta _{\beta ,t}$ be the Laplacian operator associated to $\omega _\beta (t)=\omega _{\beta }+\sqrt{-1}\partial \bar{\partial }\phi _{\beta }(t)$. Straightforward calculations show that

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta _{\beta ,t}\right) \dot{\phi }_{\beta }(t)=-\dot{\phi }_{\beta }(t). \end{aligned}$$

(2.7)

Let $H_{\beta ,\varepsilon }^+(t)=(t-\delta )\dot{\phi }_{\beta }(t)-\phi _{\beta }(t)+\varepsilon \log |s|_h^2$. Since $H_{\beta ,\varepsilon }^+(t)$ is smooth in $M{\setminus } D$, bounded from above and goes to $-\infty$ near D, it achieves its maximum in $M{\setminus } D$. Let $(t_{0}, x_{0})$ be the maximum point of $H_{\beta ,\varepsilon }^+(t)$ on $[\delta ,T]\times M$ with $x_{0}\in M{\setminus } D$. If $t_{0}=\delta$, then

$$\begin{aligned} (t-\delta )\dot{\phi }_{\beta }(t)\le C-\varepsilon \log |s|_h^2, \end{aligned}$$

(2.8)

where constant C independent of $\beta$, $\delta$, t and $\varepsilon$. If $t_0\ne \delta$, then at $(t_0,x_0)$, we have

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta _{\beta ,t}\right) H_{\beta ,\varepsilon }^+(t)= & {} -(t-\delta )\dot{\phi }_{\beta }(t)+n+tr_{\omega _\beta (t)}(-\omega _\beta +\varepsilon \theta )\nonumber \\\leqslant & {} -(t-\delta )\dot{\phi }_{\beta }(t)+n \end{aligned}$$

(2.9)

for sufficiently small $\varepsilon$. By the maximum principle, we have

$$\begin{aligned} (t-\delta )\dot{\phi }_{\beta }(t)\le C-\varepsilon \log |s|_h^2, \end{aligned}$$

(2.10)

where constant C independent of $\beta$, $\delta$, t and $\varepsilon$. Let $\varepsilon \rightarrow 0$ and then $\delta \rightarrow 0$, we have

$$\begin{aligned} \dot{\phi }_{\beta }(t)\le \frac{C}{t}\ \ \ \ \ on\ \ (0,T]\times (M{\setminus } D), \end{aligned}$$

(2.11)

where constant C independent of $\beta$ and t.

Let $H_{\beta ,\varepsilon }^-(t)=\dot{\phi }_{\beta }(t)+2\phi _{\beta }(t)-n\log (t-\delta )-\varepsilon \log |s|_h^2$. Then $H_{\beta ,\varepsilon }^-(t)$ tend to $+\infty$ either $t\rightarrow \delta ^+$ or $x\rightarrow D$. By computing, we also have

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta _{\beta ,t}\right) H_{\beta ,\varepsilon }^-(t) \ge \dot{\phi }_{\beta }(t)-2n-\frac{n}{t-\delta }+tr_{\omega _{\beta }(t)}\omega _\beta . \end{aligned}$$

(2.12)

Assume that $(t_0,x_0)$ is the minimum point of $H_{\beta ,\varepsilon }^-(t)$ on $[\delta ,T]\times M$ with $t_0>\delta$ and $x_0\in M{\setminus } D$. Thanks to Lemmas 2.1 and 2.2, there exists constant $C_1$ and $C_2$ such that

$$\begin{aligned} 0\geqslant \left( \frac{\partial }{\partial t}-\Delta _{\beta ,t}\right) H_{\beta ,\varepsilon }^-(t)|_{(t_0,x_0)}\ge & {} \left( C_1\left( \frac{\omega _{\beta }^n}{\omega ^{n}_{\beta }(t)}\right) ^{\frac{1}{n}}+\log \frac{\omega ^n_{\beta }(t)}{\omega _{\beta }^n} -\frac{C_2}{t-\delta }\right) |_{(t_0,x_0)}\nonumber \\\ge & {} \Big (\frac{C_1}{2}\Big (\frac{\omega _{\beta }^n}{\omega ^{n}_{\beta }(t)}\Big )^{\frac{1}{n}} -\frac{C_2}{t-\delta }\Big )|_{(t_0,x_0)}, \end{aligned}$$

(2.13)

where constant $C_1$ depends only on n, $C_2$ depends only on n, $\omega _{0}$ and T. In inequality (2.13), we assume $\frac{\omega _{\beta }^n}{\omega ^{n}_{\beta }(t)}>1$ and $\frac{C_1}{2}(\frac{\omega _{\beta }^n}{\omega ^{n}_{\beta }(t)})^{\frac{1}{n}}+\log \frac{\omega ^n_{\beta }(t)}{\omega _{\beta }^n}\ge 0$ at $(t_0,x_0)$. Other cases are easy to get the lower bound (2.16) for $\dot{\phi }_\beta$. By the maximum principle, we have

$$\begin{aligned} \omega ^{n}_{\beta }(t_0,x_0)\ge C_4(t_0-\delta )^n \omega _{\beta }^n(x_0), \end{aligned}$$

(2.14)

where $C_4$ independent of $\beta$, $\varepsilon$ and $\delta$. Then it easily follows that

$$\begin{aligned} \dot{\phi }_\beta (t)\ge -C+n\log (t-\delta )+\varepsilon \log |s|_h^2, \end{aligned}$$

(2.15)

where constant C independent of $\beta$, $\varepsilon$ and $\delta$. Let $\varepsilon \rightarrow 0$ and then $\delta \rightarrow 0$, we have

$$\begin{aligned} \dot{\phi }_{\beta }(t)\ge -C+n\log t\ \ \ \ \ \mathrm{on}\ \ (0,T]\times (M{\setminus } D), \end{aligned}$$

(2.16)

where constant C independent of $\beta$. By (2.11) and (2.16), we obtain (2.6). $\square$

We first recall Guenancia’s results about the curvature of $\omega _\beta$ ( Theorem 3.2 [16]).

Lemma 2.4

There exists a constant C depending only on M such that for all $\beta \in (0,\frac{1}{2}]$, the holomorphic bisectional curvature of $\omega _{\beta }$ is bounded by C.

Next, we prove the uniform equivalence of metrics along twisted conical Kähler–Ricci flows (1.6) by Chern–Lu inequality.

Lemma 2.5

For any $T>0$, there exists constant C independent of $\beta$ such that for any $t\in (0,T]$ and $\beta \in (0,\frac{1}{2}]$,

$$\begin{aligned} e^{-\frac{C}{t}}\omega _\beta \le \omega _{\beta }(t)\le e^{\frac{C}{t}}\omega _\beta \ \ \ \ \ in\ \ M{\setminus } D. \end{aligned}$$

(2.17)

Proof

By Chern–Lu inequality [4, 31] (see also Proposition 7.1 in [19]), in $M{\setminus } D$, we have

$$\begin{aligned} \Delta _{\beta ,t}\log tr_{\omega _{\beta }(t)}\omega _{\beta }\geqslant \frac{\big (Ric(\omega _\beta (t)),\omega _\beta \big )_{\omega _\beta (t)}}{tr_{\omega _{\beta }(t)}\omega _{\beta }}-C tr_{\omega _{\beta }(t)}\omega _{\beta }, \end{aligned}$$

(2.18)

where $(\ ,\ )_{\omega _\beta (t)}$ is the inner product with respect to $\omega _\beta (t)$ and constant C depends on the upper bound for the holomorphic bisectional curvature of $\omega _\beta$. In $M{\setminus } D$, we also have

$$\begin{aligned} \frac{\partial }{\partial t}\log tr_{\omega _{\beta }(t)}\omega _{\beta }=\frac{\big (Ric(\omega _\beta (t))+\omega _\beta (t)-\beta \theta ,\omega _\beta \big )_{\omega _\beta (t)}}{tr_{\omega _{\beta }(t)}\omega _{\beta }}. \end{aligned}$$

(2.19)

By using (2.18) and (2.19), we have

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta _{\beta ,t}\right) \log tr_{\omega _{\beta }(t)}\omega _{\beta }\le C tr_{\omega _{\beta }(t)}\omega _{\beta }+1, \end{aligned}$$

(2.20)

where constant C independent of $\beta$.

Let $H_{\beta ,\varepsilon }(t)=(t-\delta )\log tr_{\omega _{\beta }(t)}\omega _{\beta }-A\phi _{\beta }(t)+\varepsilon \log |s|_h^2$, A be a sufficiently large constant and $(t_0,x_0)$ be the maximum point of $H_{\beta ,\varepsilon }(t)$ on $[\delta ,T]\times (M{\setminus } D)$. We know that $x_0 \in M{\setminus } D$ and we need only consider $t_0>\delta$. By direct calculations,

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta _{\beta ,t}\right) H_{\beta ,\varepsilon }(t)\le & {} \log tr_{\omega _{\beta }(t)}\omega _{\beta }+Ctr_{\omega _{\beta }(t)}\omega _{\beta }-A\dot{\phi }_{\beta }(t)-Atr_{\omega _{\beta }(t)}\omega _{\beta }+\varepsilon tr_{\omega _{\beta }(t)}\theta +C\\\le & {} -\frac{A}{2}tr_{\omega _{\beta }(t)}\omega _{\beta }+\log tr_{\omega _{\beta }(t)}\omega _{\beta }-A\log \frac{\omega _{\beta }^n(t)}{\omega _{\beta }^n}+C, \end{aligned}$$

where constant C independent of $\beta$ and $\delta$.

Without loss of generality, we assume that $-\frac{A}{4}tr_{\omega _{\beta }(t)}\omega _{\beta }+\log tr_{\omega _{\beta }(t)}\omega _{\beta }\le 0$ at $(t_0,x_0)$. Then at $(t_0,x_0)$, by Lemma 2.3, we have

$$\begin{aligned} \left( \frac{\partial }{\partial t}-\Delta _{\beta ,t}\right) H_{\beta ,\varepsilon }(t)\le -\frac{A}{4}tr_{\omega _{\beta }(t)}\omega _{\beta }-An\log (t-\delta )+C. \end{aligned}$$

(2.21)

By the maximum principle, at $(t_0,x_0)$,

$$\begin{aligned} tr_{\omega _{\beta }(t)}\omega _{\beta }\le C\log \frac{1}{t-\delta }+C, \end{aligned}$$

(2.22)

which implies that

$$\begin{aligned} (t-\delta )\log tr_{\omega _{\beta }(t)}\omega _{\beta }\le (t_0-\delta ) \log \left( C\log \frac{1}{t_0-\delta }+C\right) +C-\varepsilon \log |s|_h^2. \end{aligned}$$

(2.23)

Let $\varepsilon \rightarrow 0$ and then $\delta \rightarrow 0$, on $(0,T]\times (M{\setminus } D)$,

$$\begin{aligned} tr_{\omega _{\beta }(t)}\omega _{\beta }\le e^\frac{C}{t}. \end{aligned}$$

(2.24)

By using Lemmas 2.2 and 2.3, we have

$$\begin{aligned} tr_{\omega _{\beta }}\omega _{\beta }(t)\le e^{\frac{C}{t}}, \end{aligned}$$

(2.25)

where C independent of $\beta$. From (2.24) and (2.25), we prove the lemma. $\square$

By the argument as that in Lemma 3.1 of [28], we get the following local Calabi’s $C^3$-estimates and curvature estimates.

Lemma 2.6

For any $T>0$ and $B_r(p)\subset \subset M{\setminus } D$, there exist constants C, $C'$ and $C''$ depend only on n, T, $\omega _0$ and $dist_{\omega _0}(B_r(p),D)$ such that

$$\begin{aligned} S_{\omega _{\beta }(t)}\le & {} \frac{C'}{r^{2}}e^{\frac{C}{t}},\\ |Rm_{\omega _{\beta }(t)}|_{\omega _{\beta }(t)}^{2}\le & {} \frac{C''}{r^{4}}e^{\frac{C}{t}} \end{aligned}$$

on $(0,T]\times B_{\frac{r}{2}}(p)$.

Since $\phi _\beta (t)=\varphi _\beta (t)-\psi _\beta$ and $\psi _\beta \in C^0(M)\cap C^\infty (M{\setminus } D)$, establishing local uniform estimates for $\phi _\beta (t)$ in $M{\setminus } D$ is equivalent to establish the estimates for $\varphi _\beta (t)$. By using the standard parabolic Schauder regularity theory (Theorem 4.9 in [25]), we obtain the following proposition.

Proposition 2.7

For any $0<\delta<T<\infty$, $k\in \mathbb {N}^{+}$ and $B_r(p)\subset \subset M{\setminus } D$, there exists constant $C_{\delta ,T,k,p,r}$ depends only on n, $\delta$, k, T, $\omega _0$ and $dist_{\omega _{0}}(B_r(p),D)$ such that for $\beta \in (0,\frac{1}{2}]$,

$$\begin{aligned} \Vert \varphi _{\beta }(t)\Vert _{C^{k}\left( [\delta ,T]\times B_r(p)\right) }\le C_{\delta ,T,k,p,r}. \end{aligned}$$

(2.26)

Through a further observation to $\psi _\beta$ and Eq. (1.7), we prove the monotonicity of $\psi _\beta$ and $\varphi _\beta (t)$ with respect to $\beta$.

Proposition 2.8

For any $x\in M$, $\psi _\beta (x)$ is monotone decreasing as $\beta \searrow 0$.

Proof

By direct computations, for any $x\in M{\setminus } D$, we have

$$\begin{aligned} \frac{\mathrm{d}\psi _\beta }{\mathrm{d}\beta }=2\frac{\beta |s|_h^{2\beta } \log |s|_h^2+1-|s|_h^{2\beta }}{\beta (1-|s|_h^{2\beta })}. \end{aligned}$$

(2.27)

Denote $f_\beta (a)=\beta a^\beta \log a+1-a^\beta$ for $\beta >0$ and $a\in [0,1]$. By computing, we get

$$\begin{aligned} f'_{\beta }(a)=\beta ^2a^{\beta -1}\log a\le 0. \end{aligned}$$

(2.28)

Hence $f_\beta (a)\ge f_\beta (1)=0$ and we have $\frac{\mathrm{d}\psi _\beta }{\mathrm{d}\beta }\ge 0$. $\square$

Proposition 2.9

For any $(t,x)\in (0,\infty )\times M$, $\varphi _\beta (t,x)$ is monotone decreasing as $\beta \searrow 0$.

Proof

By the arguments in section 3 of [28], we obtain Eq. (1.7) by approximating equations

$$\begin{aligned} \frac{\partial \varphi _{\beta ,\varepsilon }(t)}{\partial t}= & {} \log \frac{(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi _{\beta ,\varepsilon }(t))^{n}}{\omega _{0}^{n}}-\varphi _{\beta ,\varepsilon }(t)+h_{0}+\log (\varepsilon ^2+|s|_{h}^{2})^{1-\beta }\nonumber \\ \varphi _{\beta ,\varepsilon }(0)= & {} \psi _\beta \end{aligned}$$

(2.29)

For $\beta _1<\beta _2$, let $\psi _{1,2}(t)=\varphi _{\beta _1,\varepsilon }(t)-\varphi _{\beta _2,\varepsilon }(t)$. On $[\eta ,T]\times M$ with $\eta >0$ and $T<\infty$,

$$\begin{aligned}&\ \frac{\partial }{\partial t}(e^{t-\eta }\psi _{1,2}(t))\nonumber \\&\quad \le e^{t-\eta }\log \frac{\left( e^{t-\eta }\omega _{0}+\sqrt{-1}\partial \bar{\partial }e^{t-\eta }\varphi _{\beta _2,\varepsilon }(t)+\sqrt{-1}\partial \bar{\partial }e^{t-\eta }\psi _{1,2}(t)\right) ^{n}}{(e^{t-\eta }\omega _{0}+\sqrt{-1}\partial \bar{\partial }e^{t-\eta }\varphi _{\beta _2,\varepsilon }(t))^{n}}. \end{aligned}$$

(2.30)

Let $\tilde{\psi }_{1,2}(t)=e^{t-\eta }\psi _{1,2}(t)-\delta (t-\eta )$ with $\delta >0$ and $(t_0, x_0)$ be the maximum point of $\tilde{\psi }_{1,2}(t)$ on $[\eta ,T]\times M$. If $t_0>\eta$, by the maximum principle, at this point,

$$\begin{aligned} 0\le \frac{\partial }{\partial t}\tilde{\psi }_{1,2}(t)=\frac{\partial }{\partial t}\left( e^{t-\eta }\psi _{1,2}(t)\right) -\delta \le -\delta \end{aligned}$$

(2.31)

which is impossible, hence $t_0=\eta$. So for any $(t,x)\in [\eta ,T]\times M$,

$$\begin{aligned} \psi _{1,2}(t,x)\le e^{-t+\eta }\sup \limits _{M}\psi _{1,2}(\eta ,x)+T\delta . \end{aligned}$$

(2.32)

Since $\lim \limits _{t\rightarrow 0^+}\parallel \varphi _{\beta ,\varepsilon }(t)-\psi _\beta \parallel _{L^\infty (M)}=0$, let $\eta \rightarrow 0$, we get

$$\begin{aligned} \psi _{1,2}(t,x)\le e^{-t}\sup \limits _{M}(\psi _{\beta _1}-\psi _{\beta _2})+T\delta \le T\delta . \end{aligned}$$

(2.33)

Here we use Proposition 2.8 in the last inequality. Let $\delta \rightarrow 0$ and then $\varepsilon \rightarrow 0$, we conclude that $\varphi _{\beta _1}(t,x)\le \varphi _{\beta _2}(t,x)$. $\square$

For any $[\delta , T]\times K\subset \subset (0,\infty )\times M{\setminus } D$ and $k\ge 0$, $\Vert \varphi _{\beta }(t)\Vert _{C^{k}([\delta ,T]\times K)}$ is uniformly bounded by Proposition 2.7. Let $\delta$ approximate to 0, T approximate to $\infty$ and K approximate to $M{\setminus } D$, by diagonal rule, we get a sequence $\{\beta _i\}$, such that $\varphi _{\beta _i}(t)$ converge in $C^\infty _{loc}$-topology in $(0,\infty )\times (M{\setminus } D)$ to a function $\varphi (t)$ that is smooth on $C^\infty \left( (0,\infty )\times (M{\setminus } D)\right)$ and satisfies equation

$$\begin{aligned} \frac{\partial \varphi (t)}{\partial t}=\log \frac{(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi (t))^{n}}{\omega _{0}^{n}}-\varphi (t)+h_0 +\log |s|_{h}^{2} \end{aligned}$$

(2.34)

in $(0,\infty )\times (M{\setminus } D)$. Since $\varphi _\beta (t)$ is monotone decreasing as $\beta \rightarrow 0$, $\varphi _{\beta }(t)$ converge in $C^\infty _{loc}$-topology in $(0,\infty )\times (M{\setminus } D)$ to $\varphi (t)$. For any $T>0$,

$$\begin{aligned} e^{-\frac{C}{t}}\omega _{cusp}\le \omega (t)\le e^{\frac{C}{t}}\omega _{cusp}\ \ on\ (0,T]\times (M{\setminus } D), \end{aligned}$$

(2.35)

where $\omega (t)=\omega _0+\sqrt{-1}\partial \overline{\partial }\varphi (t)$, constants C depend only on n, $\omega _{0}$ and T.

Next, by using the monotonicity of $\varphi _{\beta }(t)$ with respect to $\beta$ and constructing auxiliary function, we prove the $L^1$-convergence of $\varphi (t)$ as $t\rightarrow 0^{+}$ as well as $\varphi (t)$ converge to $\psi _{0}$ in $L^{\infty }$-norm as $t\rightarrow 0^{+}$ on any compact subset $K\subset \subset M{\setminus } D$.

Lemma 2.10

There exists a unique $v_\beta \in PSH(M,\omega _0)\bigcap L^\infty (M)$ to equation

$$\begin{aligned} (\omega _0+\sqrt{-1}\partial \bar{\partial }v_\beta )^n=e^{v_\beta -h_0}\frac{\omega _0^n}{|s|_h^{2(1-\beta )}}. \end{aligned}$$

(2.36)

Furthermore, $v_\beta \in C^{2,\alpha ,\beta }(M)$ and $\parallel v_\beta -\psi _\beta \parallel _{L^\infty (M)}$ can be uniformly bounded by constant C independent of $\beta$.

Proof

By Kołodziej’s theorem (Theorem 2.4.2 in [23], see also Theorem 4.1 in [12]), there exists a unique continuous solution $v_\beta$ to Eq. (2.36). Then by Guenancia–P$\breve{a}$un’s regularity estimates ( Theorem B in [17], see also Theorem 1.4 in [26]), $v_\beta \in C^{2,\alpha ,\beta }(M)$ (readers can refer to page 5731 in [3] for more details about the space $C^{2,\alpha ,\beta }(M)$). Next, we prove $\parallel v_\beta -\psi _\beta \parallel _{L^\infty (M)}$ can be uniformly bounded. Let $u_\beta =v_\beta -\psi _\beta$, we write Eq. (2.36) as

$$\begin{aligned} (\omega _\beta +\sqrt{-1}\partial \bar{\partial }u_\beta )^n=e^{u_\beta +h_\beta }\omega _\beta ^n, \end{aligned}$$

(2.37)

where $h_{\beta }=\psi _{\beta }-h_{0}+\log \frac{\omega _{0}^{n}}{|s|_{h}^{2(1-\beta )}\omega _{\beta }^{n}}$ is uniformly bounded independent of $\beta$. Define $\chi _{\beta ,\varepsilon }=u_\beta +\varepsilon \log |s|_h^2$. Then $\sqrt{-1}\partial \bar{\partial }\chi _{\beta ,\varepsilon }=\sqrt{-1}\partial \bar{\partial }u_\beta -\varepsilon \theta$ in $M{\setminus } D$. Since $\chi _{\beta ,\varepsilon }$ is smooth in $M{\setminus } D$, bounded from above and goes to $-\infty$ near D, it achieves its maximum in $M{\setminus } D$. Let $x_{0}$ be the maximum point of $\chi _{\beta ,\varepsilon }$ on M with $x_{0}\in M{\setminus } D$. By the maximum principle,

$$\begin{aligned} e^{u_\beta +h_\beta }\omega _\beta ^n(x_0)=(\omega _\beta +\sqrt{-1} \partial \bar{\partial }u_\beta )^n(x_0)=(\omega _\beta +\sqrt{-1} \partial \bar{\partial }\chi _{\beta ,\varepsilon }+\varepsilon \theta )^n(x_0) \le 2^n\omega _\beta ^n(x_0). \end{aligned}$$

Hence, $u_\beta \le C-\varepsilon \log |s|_h^2$, where constant C independent of $\beta$ and $\varepsilon$. Let $\varepsilon \rightarrow 0$, we get the uniform upper bound of $u_\beta$. By the similar arguments, we can obtain the uniform lower bound of $u_\beta$. $\square$

Proposition 2.11

$\varphi (t)\in C^0\left( [0,\infty )\times (M{\setminus } D)\right)$ and

$$\begin{aligned} \lim \limits _{t\rightarrow 0^+}\Vert \varphi (t)-\psi _{0}\Vert _{L^{1}(M)}=0. \end{aligned}$$

(2.38)

Proof

By the monotonicity of $\varphi _{\beta }(t)$ with respect to $\beta$, for any $(t,z)\in (0,T]\times (M{\setminus } D)$, we have

$$\begin{aligned} \varphi (t,z)-\psi _0(z)\le & {} \varphi _{\beta }(t,z)-\psi _0(z)\nonumber \\\le & {} |\varphi _{\beta }(t,z)-\psi _{\beta }(z)|+|\psi _{\beta }(z)-\psi _{0}(z)|. \end{aligned}$$

(2.39)

Since $\psi _\beta$ converge to $\psi _0$ in $C^\infty _{loc}$-sense outside D as $\beta \rightarrow 0$, we can insure that for any $\epsilon >0$ and $K\subset \subset M{\setminus } D$, there exists N such that for $\beta _1<\frac{1}{N}$,

$$\begin{aligned} \Vert \psi _{\beta _1}(z)-\psi _{0}(z)\Vert _{L^{\infty }(K)}<\frac{\epsilon }{2}. \end{aligned}$$

(2.40)

Fix such $\beta _{1}$. Since by the definition of the flow (1.6)

$$\begin{aligned} \lim \limits _{t\rightarrow 0^+}\Vert \varphi _{\beta }(t,z)-\psi _\beta \Vert _{L^{\infty }(M)}=0, \end{aligned}$$

(2.41)

there exists $0<\delta _1<T$ such that

$$\begin{aligned} \sup \limits _{[0,\delta _1]\times M}|\varphi _{\beta _1}(t,z)-\psi _{\beta _1}| <\frac{\epsilon }{2}. \end{aligned}$$

(2.42)

Combining the above inequalities together, for any $t\in (0,\delta _1]$ and $z\in K$

$$\begin{aligned} \sup \limits _{[0,\delta _1]\times K}(\varphi (t,z)-\psi _0(z))<\epsilon . \end{aligned}$$

(2.43)

We define function

$$\begin{aligned} H_{\beta }(t)=(1-te^{-t})\psi _\beta +te^{-t}v_{\beta }+h(t)e^{-t}, \end{aligned}$$

(2.44)

where $v_\beta$ and $u_\beta =v_\beta -\psi _\beta$ are obtained in Lemma 2.10, and

$$\begin{aligned} h(t)=(1-e^t-t)\Vert u_{\beta }\Vert _{L^\infty (M)}+n(t\log t-t)e^{t}-n\int _0^te^{s}s\log s \mathrm{d}s. \end{aligned}$$

Straightforward calculations show that

$$\begin{aligned} \frac{\partial }{\partial t}H_{\beta }(t)+H_{\beta }(t)= & {} \psi _\beta +e^{-t}u_\beta -e^{-t}\Vert u_{\beta }\Vert _{L^\infty (M)}-\Vert u_{\beta }\Vert _{L^\infty (M)}+n\log t-nt\\\le & {} \psi _\beta +u_\beta +n\log t-nt\\= & {} v_\beta +n\log t-nt. \end{aligned}$$

Therefore, we have

$$\begin{aligned} e^{\frac{\partial }{\partial t}H_{\beta }(t)+H_{\beta }(t)}\omega _0^n\le t^ne^{-nt}e^{v_{\beta }}\omega _0^n. \end{aligned}$$

Note that $te^{-t}<1$, we have

$$\begin{aligned} \omega _0+\sqrt{-1}\partial \overline{\partial }H_{\beta }(t)= & {} (1-te^{-t})(\omega _0+\sqrt{-1}\partial \overline{\partial }\psi _\beta )+te^{-t}(\omega _0+\sqrt{-1}\partial \overline{\partial }v_{\beta })\\\ge & {} te^{-t}(\omega _0+\sqrt{-1}\partial \overline{\partial }v_\beta ). \end{aligned}$$

Combining the above inequalities,

$$\begin{aligned} (\omega _0+\sqrt{-1}\partial \overline{\partial }H_{\beta }(t))^n&\ge t^ne^{-n t}(\omega _0+\sqrt{-1}\partial \overline{\partial }\varphi _\beta )^n\\&\ge e^{-h_{0}+\frac{\partial }{\partial t}H_{\beta }(t)+H_{\beta }(t)}\frac{\omega _{0}^{n}}{|s|_{h}^{2(1-\beta )}}, \end{aligned}$$

which is equivalent to

$$\begin{aligned} \frac{\partial }{\partial t}H_{\beta }(t)\le & {} \log \frac{(\omega _0+\sqrt{-1}\partial \overline{\partial }H_{\beta }(t))^n}{\omega _{0}^{n}}-H_{\beta }(t)+h_{0}+\log |s|_{h}^{2(1-\beta )}.\nonumber \\ H_{\beta }(0)= & {} \psi _\beta \end{aligned}$$

(2.45)

Next, we prove $H_\beta (t)\le \varphi _\beta (t)$ by using Jeffres’ trick [18]. For any $0<t_1<T<\infty$ and $a>0$.

Denote $\Psi (t)=H_\beta (t)+a|s|_h^{2q}-\varphi _\beta (t)$ and $\hat{\Delta }=\int _0^1 g_{sH_\beta (t)+(1-s)\varphi _\beta (t)}^{i\bar{j}}\frac{\partial ^2}{\partial z^i\partial \bar{z}^j}\mathrm{d}s$, where $0<q<1$ is determined later. $\Psi (t)$ evolves along the following equation

$$\begin{aligned} \frac{\partial \Psi (t)}{\partial t}\leqslant \hat{\Delta }\Psi (t)-a\hat{\Delta }|s|_h^{2q}-\Psi (t)+a|s|_h^{2q}. \end{aligned}$$

Since

$$\begin{aligned} \omega _0+\sqrt{-1}\partial \overline{\partial }H_{\beta }(t)&\ge (1-te^{-t})(\omega _0+\sqrt{-1}\partial \overline{\partial }\psi _\beta )\ge \frac{1}{4}\omega _0,\\ \omega _0+\sqrt{-1}\partial \overline{\partial }\varphi _{\beta }(t)&\ge e^{-\frac{C(T)}{t_1}}\omega _\beta \ge \frac{1}{2}e^{-\frac{C(T)}{t_1}}\omega _0,\\ \sqrt{-1}\partial \overline{\partial }|s|_h^{2q}&=q^2|s|_h^{2q}\sqrt{-1}\partial \log |s|_h^{2}\wedge \overline{\partial }\log |s|_h^{2} +q|s|_h^{2q}\sqrt{-1}\partial \overline{\partial }\log |s|_h^{2}, \end{aligned}$$

we obtain the following inequalities

$$\begin{aligned} \frac{1}{2}\min \left( \frac{1}{2},e^{-\frac{C(T)}{t_1}}\right) \omega _0 \le \frac{1}{2}s\omega _{\beta }+(1-s)e^{-\frac{C(T)}{t_1}}\omega _\beta \le s\omega _{H_\beta (t)}+(1-s)\omega _{\varphi _\beta (t)}. \end{aligned}$$

Hence, we have

$$\begin{aligned} \hat{\Delta }|s|_h^{2q}&\ge q|s|_h^{2q}\int _0^1g_{sH_\beta (t)+(1-s) \varphi _\beta (t)}\left( \frac{\partial ^2}{\partial z^i\partial \bar{z}^j}\log |s|_h^{2}\right) \mathrm{d}s\\&=-q|s|_h^{2q}\int _0^1g_{sH_\beta (t)+(1-s)\varphi _\beta (t)}^{i\bar{j}}\theta _{i\bar{j}}\mathrm{d}s\\&\ge -\frac{1}{2}C(t_1,T)q|s|_h^{2q}g_{\beta }^{i\bar{j}}g_{0,i\bar{j}}\ge -C(t_1,T) \end{aligned}$$

in $M{\setminus } D$, where constant $C(t_1,T)=2K\max (2,e^{\frac{C(T)}{t_1}})$ independent of a and $\theta \leqslant K\omega _0$. Then we have

$$\begin{aligned} \frac{\partial \Psi (t)}{\partial t}\le \hat{\Delta }\Psi (t)-\Psi (t)+aC(t_1,T). \end{aligned}$$

Let $\tilde{\Psi }=e^{(t-t_1)}\Psi -aC(t_1,T)e^{(t-t_1)}-\varepsilon (t-t_1)$. By choosing suitable $0<q<1$, we can assume that the space maximum of $\tilde{\Psi }$ on $[t_1,T]\times M$ is attained away from D. Let $(t_0,x_0)$ be the maximum point. If $t_0>t_1$, by the maximum principle, at $(t_0,x_0)$, we have

$$\begin{aligned} 0\le \left( \frac{\partial }{\partial t}-\hat{\Delta }\right) \tilde{\Psi }(t)\le -\varepsilon , \end{aligned}$$

which is impossible, hence $t_0=t_1$. Then for $(t,x)\in [t_1,T]\times M$, we obtain

$$\begin{aligned} H_\beta (t)-\varphi _\beta (t) \le \Vert H_\beta (t_1,x)-\varphi _\beta (t_1,x)\Vert _{L^\infty (M)}+aC(t_1,T)+\varepsilon T \end{aligned}$$

Since $\lim \limits _{t\rightarrow 0^+}\Vert H_{\beta }(t,z)-\psi _\beta \Vert _{L^{\infty }(M)}=0$ and (2.41), let $a\rightarrow 0$ and then $t_1\rightarrow 0^+$,

$$\begin{aligned} H_\beta (t)-\varphi _\beta (t)\le \varepsilon T. \end{aligned}$$

It shows that $H_\beta (t)\le \varphi _\beta (t)$ after we let $\varepsilon \rightarrow 0$. For any $(t,z)\in (0,T]\times (M{\setminus } D)$

$$\begin{aligned} \varphi _{\beta }(t,z)-\psi _0(z)&\ge te^{- t}u_{\beta }+h(t)e^{-t}+\psi _\beta -\psi _0\nonumber \\&\ge -Ct-C(1-e^{-t})+h_{1}(t)e^{-t} , \end{aligned}$$

(2.46)

where $h_{1}(t)=n(t\log t-t)e^{t}-n\int _0^te^{s}s\log s\ \mathrm{d}s$, constant C is independent of $\beta$ thanks to Lemma 2.10. Letting $\beta \rightarrow 0$, we have

$$\begin{aligned} \varphi (t,z)-\psi _0(z)\ge -Ct-C(1-e^{-t})+h_{1}(t)e^{-t}. \end{aligned}$$

(2.47)

There exists $\delta _2$ such that for any $t\in [0,\delta _2]$ and $z\in M{\setminus } D$,

$$\begin{aligned} \varphi (t,z)-\psi _0(z)>-\frac{\epsilon }{2}. \end{aligned}$$

(2.48)

Let $\delta =\min (\delta _1,\delta _2)$, then for any $t\in (0,\delta ]$ and $z\in K$,

$$\begin{aligned} -\epsilon<\varphi (t,z)-\psi _{0}(z)<\epsilon . \end{aligned}$$

(2.49)

This, together with Proposition 2.7, insures that $\varphi (t)\in C^0\left( [0,\infty )\times (M{\setminus } D)\right)$. Since $\psi _\beta$ converge to $\psi _0$ in $L^1$-sense on M, for sufficiently small $\beta _2$, we have

$$\begin{aligned} \int _M|\psi _{\beta _2}(z)-\psi _{0}(z)|\ \omega _0^n<\frac{\epsilon }{2}. \end{aligned}$$

(2.50)

By (2.39), (2.41) and (2.48), there exists $\delta$ such that for any $t\in (0,\delta )$,

$$\begin{aligned} \int _M|\varphi (t)-\psi _{0}(z)|\ \omega _0^n<\epsilon , \end{aligned}$$

(2.51)

which implies (2.38). $\square$

Theorem 2.12

$\omega (t)=\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi (t)$ is a long-time solution to cusp Kähler–Ricci flow (1.8).

Proof

We should only prove that $\omega (t)$ satisfies Eq. (1.8) in the sense of currents on $(0,\infty )\times M$.

Let $\eta =\eta (t,x)$ be a smooth $(n-1,n-1)$-form with compact support in $(0,\infty )\times M$. Without loss of generality, we assume that its compact support is included in $(\delta ,T)\times M$ ($0<\delta<T<\infty$). On $[\delta ,T]\times (M{\setminus } D)$, $\log \frac{\omega _{\beta }^{n}(t)|s|_{h}^{2(1-\beta )}}{\omega _{0}^{n}}-\psi _\beta$, $\log \frac{\omega ^{n}(t)|s|_{h}^{2}}{\omega _{0}^{n}}-\psi _0$, $\varphi _{\beta }(t)-\psi _\beta$ and $\varphi (t)-\psi _0$ are uniformly bounded. On $[\delta ,T]$, we have

$$\begin{aligned}&\ \int _{M}\frac{\partial \omega _{\beta }(t)}{\partial t}\wedge \eta =\int _{M}\sqrt{-1}\partial \bar{\partial }\frac{\partial \varphi _{\beta }(t)}{\partial t}\wedge \eta \\&\quad =\int _{M}\left( \log \frac{\omega _{\beta }^{n}(t)|s|_{h}^{2(1-\beta )}}{\omega _{0}^{n}}-\psi _\beta -(\varphi _{\beta }(t)-\psi _\beta )+h_0\right) \sqrt{-1}\partial \bar{\partial }\eta \\&\quad \xrightarrow {\beta \rightarrow 0}\int _{M}\left( \log \frac{\omega ^{n}(t)|s|_{h}^{2}}{\omega _{0}^{n}}-\psi _0-(\varphi (t)-\psi _0)+h_0\right) \sqrt{-1}\partial \bar{\partial }\eta \\&\quad =\int _{M} (-Ric (\omega (t))- \omega (t) +[D])\wedge \eta . \end{aligned}$$

In the above limit process, we make use of the uniform convergence theorem. At the same time, there also holds

$$\begin{aligned} \int _M \omega _{\beta }(t)\wedge \frac{\partial \eta }{\partial t}&=\int _M \omega _{0}\wedge \frac{\partial \eta }{\partial t}+\int _M \varphi _{\beta }(t)\sqrt{-1}\partial \bar{\partial }\frac{\partial \eta }{\partial t}\nonumber \\&\xrightarrow {\beta \rightarrow 0}\int _M \omega _0\wedge \frac{\partial \eta }{\partial t}+\int _M \varphi (t)\sqrt{-1}\partial \bar{\partial }\frac{\partial \eta }{\partial t}\nonumber \\&=\int _M \omega (t)\wedge \frac{\partial \eta }{\partial t}. \end{aligned}$$

(2.52)

On the other hand,

$$\begin{aligned} \frac{\partial }{\partial t}\int _M \omega _{\beta }(t)\wedge \eta&=\int _M \varphi _{\beta }(t)\sqrt{-1}\partial \bar{\partial }\frac{\partial \eta }{\partial t}+\int _M \omega _0\wedge \frac{\partial \eta }{\partial t}+\int _M \frac{\partial \varphi _{\beta }(t)}{\partial t} \sqrt{-1}\partial \bar{\partial }\eta \nonumber \\&\xrightarrow {\beta \rightarrow 0} \int _M \varphi (t)\sqrt{-1}\partial \bar{\partial }\frac{\partial \eta }{\partial t}+\int _M \omega _0\wedge \frac{\partial \eta }{\partial t}+\int _M \frac{\partial \varphi }{\partial t} \sqrt{-1}\partial \bar{\partial }\eta \nonumber \\&=\frac{\partial }{\partial t}\int _M \omega (t)\wedge \eta . \end{aligned}$$

(2.53)

Combining equality

$$\begin{aligned} \frac{\partial }{\partial t}\int _M \omega _{\beta }(t)\wedge \eta =\int _{M}\frac{\partial \omega _{\beta }(t)}{\partial t}\wedge \eta +\int _M \omega _{\beta }(t)\wedge \frac{\partial \eta }{\partial t} \end{aligned}$$

with equalities (2.52)–(2.53), on $[\delta ,T]$, we have

$$\begin{aligned} \frac{\partial }{\partial t}\int _M \omega (t)\wedge \eta= & {} \int _{M} \left( -Ric (\omega (t))- \omega (t)+[D]\right) \wedge \eta \nonumber \\&+\int _M \omega (t)\wedge \frac{\partial \eta }{\partial t}. \end{aligned}$$

(2.54)

Since $Supp\ \eta \subset (\delta ,T)\times M$, we have

$$\begin{aligned} \int _0^{+\infty }\frac{d}{d t}\int _M \omega (t)\wedge \eta \ \mathrm{d}t=0. \end{aligned}$$

(2.55)

Integrating form 0 to $\infty$ on both sides of Eq. (2.54),

$$\begin{aligned} \int _{(0,\infty )\times M} \frac{\partial \omega (t)}{\partial t}\wedge \eta ~\mathrm{d}t&=-\int _{(0,\infty )\times M} \omega (t)\wedge \frac{\partial \eta }{\partial t}~\mathrm{d}t=-\int _{0}^{\infty }\int _M \omega (t)\wedge \frac{\partial \eta }{\partial t}~\mathrm{d}t\\&=\int _{0}^{\infty }\int _{M} \left( -Ric (\omega (t))- \omega (t) + [D]\right) \wedge \eta ~\mathrm{d}t\\&=\int _{(0,\infty )\times M} \left( -Ric (\omega (t))- \omega (t) +[D]\right) \wedge \eta ~\mathrm{d}t. \end{aligned}$$

By the arbitrariness of $\eta$, we prove that $\omega (t)$ satisfies cusp Kähler–Ricci flow (1.8) in the sense of currents on $(0,\infty )\times M$. $\square$

Now we prove the uniqueness theorem.

Theorem 2.13

Let $\tilde{\varphi }(t)\in C^0\left( [0,\infty )\times (M{\setminus } D)\right) \bigcap C^\infty \big ((0,\infty )\times (M{\setminus } D)\big )$ be a long-time solutions to parabolic Monge–Ampère equation

$$\begin{aligned} \frac{\partial \varphi (t)}{\partial t}=\log \frac{(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi (t))^{n}}{\omega _{0}^{n}}-\varphi (t)+h_{0} +\log |s|_{h}^{2} \end{aligned}$$

(2.56)

in $(0,\infty )\times (M{\setminus } D)$. If $\tilde{\varphi }$ satisfies

• For any $0<\delta<T<\infty$, there exists uniform constant C such that

  $$\begin{aligned} C^{-1}\omega _{cusp}\le \omega _{0}+\sqrt{-1}\partial \bar{\partial } \tilde{\varphi }(t)\le C\omega _{cusp}\ \ \ on\ \ [\delta ,T]\times (M{\setminus } D); \end{aligned}$$

• on (0, T], $\Vert \tilde{\varphi }(t)-\psi _0\Vert _{L^{\infty }(M{\setminus } D)}\leqslant C$;

• on $[\delta , T]$, there exist constant $C^{*}$ such that $\Vert \frac{\partial \tilde{\varphi }(t)}{\partial t}\Vert _{L^{\infty }(M{\setminus } D)}\leqslant C^{*}$;

• $\lim \limits _{t\rightarrow 0^{+}}\Vert \tilde{\varphi }(t)-\psi _{0}\Vert _{L^{1}(M)}=0$.

Then $\tilde{\varphi }(t)\le \varphi (t)$.

Proof

For any $0<t_1<T<\infty$ and $a>0$. Denote $\Psi (t)=\tilde{\varphi }(t)+a\log |s|_h^2-\varphi _\beta (t)$ and $\hat{\Delta }=\int _0^1 g_{s\tilde{\varphi }(t)+(1-s)\varphi _\beta (t)}^{i\bar{j}}\frac{\partial ^2}{\partial z^i\partial \bar{z}^j}\mathrm{d}s$. We note that $\tilde{\varphi }(t)$ is bounded from above because it is a $\omega _0$-psh function. $\Psi (t)$ evolves along the following equation

$$\begin{aligned} \frac{\partial \Psi (t)}{\partial t}=\hat{\Delta }\Psi (t)-a\hat{\Delta }\log |s|_h^{2}-\Psi (t)+(a+\beta )\log |s|_h^2. \end{aligned}$$

Since $-\sqrt{-1}\partial \bar{\partial }\log |s|_h^2=\theta$, we obtain

$$\begin{aligned} -\hat{\Delta }\log |s|_h^2=\int _0^1g_{s\tilde{\varphi }(t)+(1-s)\varphi _\beta (t)}^{i\bar{j}}\theta _{i\bar{j}}\mathrm{d}s\le C(t_1,T) \end{aligned}$$

in $M{\setminus } D$. Then we obtain

$$\begin{aligned} \frac{\partial \Psi (t)}{\partial t}\le \hat{\Delta }\Psi (t)-\Psi (t)+aC(t_1,T). \end{aligned}$$

Then by the arguments as that in Proposition 2.11, on $[t_1,T]\times (M{\setminus } D)$,

$$\begin{aligned} \tilde{\varphi }(t)-\varphi _\beta (t)\le e^{-(t-t_1)}\sup _M\left( \tilde{\varphi }(t_1)-\varphi _\beta (t_1)\right) . \end{aligned}$$

Since $\tilde{\varphi }(t_1)$ converge to $\psi _0$ in $L^1$-sense and $\varphi _\beta (t)$ converge to $\psi _\beta$ in $L^\infty$-sense as $t_{1}\rightarrow 0^+$, by Hartogs Lemma, we have

$$\begin{aligned} \tilde{\varphi }(t)-\varphi _\beta (t)\le e^{-t}\sup _M(\psi _0-\psi _\beta )\le 0, \end{aligned}$$

after we let $t_1\rightarrow 0$. Hence $\tilde{\varphi }(t)\le \varphi (t)$ in $(0,\infty )\times (M{\setminus } D)$. $\square$

Remark 2.14

If M is a compact Kähler manifold with smooth hypersurface D. We can also consider unnormalized cusp Kähler–Ricci flow

$$\begin{aligned} \frac{\partial \hat{\omega }(t)}{\partial t}= & {} -Ric(\hat{\omega }(t))+[D].\nonumber \\ \hat{\omega }(t)|_{t=0}= & {} \omega _{cusp} \end{aligned}$$

(2.57)

If we define $\omega (t)=e^{-t}\hat{\omega }(e^t-1)$, then flow (2.57) is actually the same as normalized cusp Kähler–Ricci flow (1.8) only modulo a scaling. Let

$$\begin{aligned} T_0=\sup \{\ t\ |\ [\omega _0]-t(c_1(M)-c_1(D))>0\}. \end{aligned}$$

(2.58)

Combining the arguments of Tian–Zhang [41] and Liu–Zhang [28] with the arguments in this paper, there exists a unique solution to flow (2.57) on $[0,T_0)$ in some weak sense which is similar as Definition 1.2.

3 The convergence of cusp Kähler–Ricci flow

In this section, we prove the convergence theorem of cusp Kähler–Ricci flow (1.8).

Proof of Theorem 1.6:

Differentiating Eq. (2.56) in time t, we have

$$\begin{aligned} \left( \frac{d}{\mathrm{d}t}-\Delta _t\right) \frac{\partial \varphi }{\partial t}=-\frac{\partial \varphi }{\partial t} \end{aligned}$$

(3.1)

on $[\delta ,T]\times (M{\setminus } D)$ with $\delta >0$. For any $\varepsilon >0$,

$$\begin{aligned} \left( \frac{d}{\mathrm{d}t}-\Delta _t\right) \left( \frac{\partial \varphi }{\partial t}+\varepsilon \log |s|_h^2\right)&=-\frac{\partial \varphi }{\partial t}+\varepsilon tr_{\omega (t)}\theta \nonumber \\&\le -\left( \frac{\partial \varphi }{\partial t}+\varepsilon \log |s|_h^2\right) +\varepsilon C(\delta ,T), \end{aligned}$$

(3.2)

where constant $C(\delta ,T)$ independent of $\varepsilon$. For any $\eta >0$, let $H=e^{t-\delta }(\frac{\partial \varphi }{\partial t}+\varepsilon \log |s|_h^2)-\varepsilon e^{t-\delta } C(\delta ,T)-\eta (t-\delta )$. Since $\frac{\partial \varphi }{\partial t}$ is bounded on $[\delta ,T]\times (M{\setminus } D)$, the maximum point $(t_0,x_0)$ of H satisfies $x_0\in M{\setminus } D$. If $t_0>\delta$, by the maximum principle, we get a contradiction. Hence, $t_0=\delta$. Then we have

$$\begin{aligned} \frac{\partial \varphi }{\partial t}\le C(\delta )e^{-t}-\varepsilon \log |s|_h^2+\varepsilon C(\delta ,T)+\eta T. \end{aligned}$$

(3.3)

Let $\varepsilon \rightarrow 0$, $\eta \rightarrow 0$ and then $T\rightarrow \infty$, we obtain

$$\begin{aligned} \frac{\partial \varphi }{\partial t}\le C(\delta )e^{-t}\ \ \ in\ \ [\delta ,\infty )\times (M{\setminus } D). \end{aligned}$$

(3.4)

By the same arguments, we can get the lower bound of $\frac{\partial \varphi }{\partial t}$. In fact, we obtain

$$\begin{aligned} |\frac{\partial \varphi }{\partial t}|\le C(\delta )e^{-t}\ \ \ in\ \ [\delta ,\infty )\times (M{\setminus } D). \end{aligned}$$

(3.5)

For $\delta<t<s$,

$$\begin{aligned} |\varphi (t)-\varphi (s)|\le C(\delta )(e^{-t}-e^{-s})\ \ \ in\ \ [\delta ,\infty )\times (M{\setminus } D). \end{aligned}$$

(3.6)

Therefore, $\varphi (t)$ converge exponentially fast in $L^\infty$-sense to $\varphi _\infty$ in $M{\setminus } D$. Then making use of the arguments in the proofs of Lemma 2.5, Lemma 2.6 and Proposition 2.7 for flow (1.8), we can prove that $\varphi (t)$ are locally $C^k$-bounded (independent of t) for any $k\in \mathbb {N}^+$ outside D. Since $\varphi (t)$ converge in $L^\infty$-sense to $\varphi _\infty$, $\varphi (t)$ must converge to $\varphi _\infty$ in $C^\infty _{loc}$-sense in $M{\setminus } D$. At the same time, For any smooth $(n-1,n-1)$-form $\eta$,

$$\begin{aligned} \int _M\frac{\partial \omega (t)}{\partial t}\wedge \eta =\int _M\frac{\partial \varphi (t)}{\partial t}\sqrt{-1}\partial \bar{\partial }\eta \xrightarrow {t\rightarrow \infty }0 \end{aligned}$$

(3.7)

while

$$\begin{aligned} \int _M\frac{\partial \omega (t)}{\partial t}\wedge \eta&=\int _M\sqrt{-1}\partial \bar{\partial }(\log \frac{|s|_h^2(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi (t))^{n}}{\omega _{0}^{n}}-\varphi (t)+h_{0})\wedge \eta \\&=\int _M\left( \log \frac{|s|_h^2(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi (t))^{n}}{\omega _{0}^{n}}-\psi _0-(\varphi (t)-\psi _0)+h_{0}\right) \sqrt{-1}\partial \bar{\partial }\eta \\&\xrightarrow {t\rightarrow \infty }\int _M\left( \log \frac{|s|_h^2(\omega _{0}+\sqrt{-1}\partial \bar{\partial }\varphi _\infty )^{n}}{\omega _{0}^{n}}-\psi _0-(\varphi _\infty -\psi _0)+h_{0}\right) \sqrt{-1}\partial \bar{\partial }\eta \\&=\int _M(-Ric(\omega _\infty )-\omega _\infty +[D])\wedge \eta . \end{aligned}$$

which implies the convergence in the sense of currents. $\square$