Stability of the conical K\"ahler-Ricci flows on Fano manifolds

TL;DR

This paper proves the stability of conical K"ahler-Ricci flows on Fano manifolds near a conical K"ahler-Einstein metric, under a weaker energy condition, and provides new proofs of key existence and openness results.

Contribution

It establishes stability of conical K"ahler-Ricci flows under a bounded below Log Mabuchi energy, relaxing previous properness assumptions.

Findings

Flow converges to conical K"ahler-Einstein metrics for nearby cone angles.

Provides parabolic proofs of Donaldson's openness theorem.

Confirms existence conjecture for positive Ricci curvature cases.

Abstract

In this paper, we study the stability of the conical K\"ahler-Ricci flows on Fano manifolds. That is, if there exists a conical K\"ahler-Einstein metric with cone angle $2\pi\beta$ along the divisor, then for any $\beta'$ sufficiently close to $\beta$, the corresponding conical K\"ahler-Ricci flow converges to a conical K\"ahler-Einstein metric with cone angle $2\pi\beta'$ along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence before. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical K\"ahler-Einstein metrics with positive Ricci curvatures.

Equations

$$(CKR{F}_{{\mu }_{\gamma }}):\{\begin{array}{rl}& \frac{\partial {\omega }_{\gamma }(t)}{\partial t}=-Ric({\omega }_{\gamma }(t))+{\mu }_{\gamma }{\omega }_{\gamma }(t)+(1-\gamma )[D],\\ & \\ & {\omega }_{\gamma }(t){|}_{t=0}=\hat{\omega }\end{array}$$

$$(TKR{F}_{{\mu }_{\gamma },\epsilon }):\{\begin{array}{rl}& \frac{\partial {\omega }_{\gamma ,\epsilon }(t)}{\partial t}=-Ric({\omega }_{\gamma ,\epsilon }(t))+{\mu }_{\gamma }{\omega }_{\gamma ,\epsilon }(t)+(1-\gamma ){\theta }_{\epsilon }\\ & \\ & {\omega }_{\gamma ,\epsilon }(t){|}_{t=0}=\hat{\omega }\end{array}$$

$$\begin{array}{rl}{A}_{{\mu }_{\gamma },\epsilon }(t)& =\frac{1}{V}{\int }_M{u}_{\gamma ,\epsilon }(t)e^{-u_{\gamma ,\epsilon }(t)}d{V}_{\gamma ,\epsilon }(t),\\ A_{{\mu }_{\gamma }}(t)& =\frac{1}{V}{\int }_M{u}_{\gamma }(t)e^{-u_{\gamma }(t)}d{V}_{\gamma }(t).\end{array}$$

$$\Vert u_{\beta ,\epsilon }(t){\Vert }_{C^0(M)}+os{c}_M{\phi }_{\beta ,\epsilon }(t)+\underset{M}{\sup }t{r}_{{\omega }_{\epsilon }^{\beta }}{\omega }_{\beta ,\epsilon }(t)⩽D_{\beta }$$

$$\Vert u_{\gamma ,\epsilon }(t){\Vert }_{C^0(M)}+os{c}_M{\phi }_{\gamma ,\epsilon }(t)+\underset{M}{\sup }t{r}_{{\omega }_{\epsilon }^{\gamma }}{\omega }_{\gamma ,\epsilon }(t)⩽L_{\beta }+1$$

$$\begin{array}{rl}|R({\omega }_{\gamma ,\epsilon }(t))-(1-\gamma )t{r}_{{\omega }_{\gamma ,\epsilon }(t)}{\theta }_{\epsilon }|& ⩽C,\\ \Vert u_{\gamma ,\epsilon }(t){\Vert }_{C^1({\omega }_{\gamma ,\epsilon }(t))}& ⩽C,\\ diam(M,{\omega }_{\gamma ,\epsilon }(t))& ⩽C\end{array}$$

$$(TKR{F}_{{\mu }_{\gamma },\epsilon }^{\beta }):\{\begin{array}{rl}& \frac{\partial {\omega }_{\gamma ,\epsilon }^{\beta }(t)}{\partial t}=-Ric({\omega }_{\gamma ,\epsilon }^{\beta }(t))+{\mu }_{\gamma }{\omega }_{\gamma ,\epsilon }^{\beta }(t)+({\mu }_{\beta }-{\mu }_{\gamma }){\omega }_{{\phi }_{\epsilon }}+(1-\beta ){\theta }_{\epsilon }\\ & \\ & {\omega }_{\gamma ,\epsilon }^{\beta }(t){|}_{t=0}=\hat{\omega }\end{array}$$

$$(CKR{F}_{{\mu }_{\gamma }}^{\beta }):\{\begin{array}{rl}& \frac{\partial {\omega }_{\gamma }^{\beta }(t)}{\partial t}=-Ric({\omega }_{\gamma }^{\beta }(t))+{\mu }_{\gamma }{\omega }_{\gamma }^{\beta }(t)+({\mu }_{\beta }-{\mu }_{\gamma }){\omega }_{{\phi }_{\beta }}+(1-\beta )[D].\\ & \\ & {\omega }_{\gamma }^{\beta }(t){|}_{t=0}=\hat{\omega }\end{array}$$

$$\begin{array}{rl}{A}_{{\mu }_{\gamma },\epsilon }^{\beta }(t)& =\frac{1}{V}{\int }_M{u}_{\gamma ,\epsilon }^{\beta }(t)e^{-u_{\gamma ,\epsilon }^{\beta }(t)}d{V}_{\gamma ,\epsilon }^{\beta }(t),\\ A_{{\mu }_{\gamma }}^{\beta }(t)& =\frac{1}{V}{\int }_M{u}_{\gamma }^{\beta }(t)e^{-u_{\gamma }^{\beta }(t)}d{V}_{\gamma }^{\beta }(t).\end{array}$$

$$\Vert {\psi }_{{\beta }_0,\epsilon }^{\beta }(t){\Vert }_{C^0(M)}⩽C_{{\beta }_0}fort\in [0,+\infty ).$$

$$\Vert {\dot{\psi }}_{\gamma ,\epsilon }^{\beta }(t){\Vert }_{C^0(M)}⩽{\hat{C}}_{{\beta }_0}$$

$$\Vert {\psi }_{\gamma ,\epsilon }^{\beta }(t){\Vert }_{C^0(M)}⩽\max (\Vert {\phi }_{\beta }{\Vert }_{C^0(M)}+\frac{{\hat{C}}_{{\beta }_0}+|{\xi }_{\beta }|}{{\mu }_{{\beta }_0}},C_{{\beta }_0})+1$$

$$\Vert {\psi }_{\beta ,\epsilon }(t){\Vert }_{C^0(M)}⩽B_{\beta }on[0,+\infty ).$$

$$\Vert {\psi }_{\gamma ,\epsilon }(t){\Vert }_{C^0(M)}⩽\max (\Vert {\phi }_{\beta }{\Vert }_{C^0(M)}+\frac{C_{\beta }+|{\xi }_{\beta }|}{{\mu }_{\beta }},B_{\beta })+1$$

$$\begin{split}\Omega_{\lambda}=\bigg{\{}s\in(\beta,1]\bigg{|}&\ (CKRF_{\mu_{\gamma}})\ converges\ to\ a\ conical\ K\ddot{a}hler-Einstein\ metric\ with\\ &\ Ricci\ curvature\ \mu_{\gamma}\ and\ cone\ angle\ 2\pi\gamma\ along\ D\ for\ \gamma\in(\beta,s]\bigg{\}}.\end{split}$$

$$(TCKR{F}_{{\nu }_{\gamma }}):\{\begin{array}{rl}& \frac{\partial {\omega }_{\gamma }(t)}{\partial t}=-Ric({\omega }_{\gamma }(t))+{\nu }_{\gamma }{\omega }_{\gamma }(t)+(1-\gamma )\kappa {\omega }_0+(1-\gamma )[D].\\ & \\ & {\omega }_{\gamma }(t){|}_{t=0}=\hat{\omega }\end{array}$$

$$\begin{split}\Omega_{\kappa}=\bigg{\{}s\in(\hat{\beta},1]\bigg{|}&\ (TCKRF_{\nu_{\gamma}})\ converges\ to\ a\ twisted\ conical\ K\ddot{a}hler-Einstein\ metric\\ &\ \ with\ twisted\ form\ (1-\gamma)\kappa\omega_{0}\ ,\ twisted\ Ricci\ curvature\ \nu_{\gamma}\ and\ cone\\ &\ \ angle\ 2\pi\gamma\ along\ D\ for\ \gamma\in(\hat{\beta},s]\bigg{\}}.\end{split}$$

$$\sqrt{-1}\sum _{j=1}^{n-1}d{z}_j\wedge d{\overline{z}}_j+\sqrt{-1}|z_n{|}^{2\beta -2}d{z}_n\wedge d{\overline{z}}_{n}onU.$$

$$Ric(\omega )=\mu \omega +(1-\beta )[D]onM.$$

$$C^{-1}{\omega }_{\gamma }⩽{\omega }_{\gamma }(t)⩽C{\omega }_{\gamma }on[\delta ,T]\times (M\setminus D);$$

$${\int }_{M_{\infty }}\frac{\partial {\omega }_{\gamma }(t)}{\partial t}\wedge \eta (t,x)dt={\int }_{M_{\infty }}(-Ric({\omega }_{\gamma }(t))+{\mu }_{\gamma }{\omega }_{\gamma }(t)+(1-\gamma )[D])\wedge \eta (t,x)dt,$$

$${\int }_{M_{\infty }}\frac{\partial {\omega }_{\gamma }(t)}{\partial t}\wedge \eta (t,x)dt=-{\int }_{M_{\infty }}{\omega }_{\gamma }(t)\wedge \frac{\partial \eta (t,x)}{\partial t}dt.$$

$$C^{-1}{\omega }_{\beta }⩽{\omega }_{\gamma }^{\beta }(t)⩽C{\omega }_{\beta }on[\delta ,T]\times (M\setminus D);$$

$$({\int }_M{v}^{\frac{2n}{n-1}}d{V}_{\epsilon }^{\gamma }{)}^{\frac{n-1}{n}}⩽C({\int }_M|dv{|}_{{\omega }_{\epsilon }^{\gamma }}^{2}d{V}_{\epsilon }^{\gamma }+{\int }_M|v{|}^{2}d{V}_{\epsilon }^{\gamma })$$

$$({\int }_M{v}^{2}d{V}_{\epsilon }^{\gamma }{)}^{\frac{1}{2}}⩽C({\int }_M|dv{|}_{{\omega }_{\epsilon }^{\gamma }}d{V}_{\epsilon }^{\gamma }+{\int }_M|v|d{V}_{\epsilon }^{\gamma })$$

$$\begin{array}{rl}|R({\omega }_{\gamma ,\epsilon }(t))-(1-\gamma )t{r}_{{\omega }_{\gamma ,\epsilon }(t)}{\theta }_{\epsilon }|& ⩽C,\\ \Vert u_{\gamma ,\epsilon }(t){\Vert }_{C^1({\omega }_{\gamma ,\epsilon }(t))}& ⩽C,\\ diam(M,{\omega }_{\gamma ,\epsilon }(t))& ⩽C\end{array}$$

$$\begin{array}{rl}{I}_{{\omega }_0}(\varphi )& =\frac{1}{V}{\int }_M\varphi (d{V}_0-d{V}_{\varphi }),\\ J_{{\omega }_0}(\varphi )& =\frac{1}{V}{\int }_0^1{\int }_M{\dot{\varphi }}_t(d{V}_0-d{V}_{{\varphi }_t})dt,\end{array}$$

$$\begin{array}{rl}{\mathcal{M}}_{k,\theta }(\varphi )& =-k(I_{{\omega }_0}(\varphi )-J_{{\omega }_0}(\varphi ))-\frac{1}{V}{\int }_M{u}_{{\omega }_0}(d{V}_0-d{V}_{\varphi })\\ & +\frac{1}{V}{\int }_M\log \frac{{\omega }_{\varphi }^n}{{\omega }_0^n}d{V}_{\varphi },\end{array}$$

$$\begin{split}\mathcal{M}^{\beta}_{\mu_{\gamma},\varepsilon}(\phi)&=-\mu_{\gamma}(I_{\omega_{0}}(\phi)-J_{\omega_{0}}(\phi))+\frac{1}{V}\int_{M}\log\frac{\omega_{\phi}^{n}}{\omega_{0}^{n}}dV_{\phi}\\ &\ -\frac{1}{V}\int_{M}\Big{(}F_{0}+(\mu_{\beta}-\mu_{\gamma})\varphi_{\varepsilon}+(1-\beta)\log(\varepsilon^{2}+|s|^{2}_{h})\Big{)}(dV_{0}-dV_{\phi}),\\ \mathcal{M}_{\mu_{\gamma},\varepsilon}(\phi)&=-\mu_{\gamma}(I_{\omega_{0}}(\phi)-J_{\omega_{0}}(\phi))+\frac{1}{V}\int_{M}\log\frac{\omega_{\phi}^{n}}{\omega_{0}^{n}}dV_{\phi}\\ &\ -\frac{1}{V}\int_{M}\Big{(}F_{0}+(1-\gamma)\log(\varepsilon^{2}+|s|^{2}_{h})\Big{)}(dV_{0}-dV_{\phi}).\end{split}$$

$$\begin{split}\mathcal{M}_{\mu_{\gamma}}(\phi)&=-\mu_{\gamma}(I_{\omega_{0}}(\phi)-J_{\omega_{0}}(\phi))+\frac{1}{V}\int_{M}\log\frac{\omega_{\phi}^{n}}{\omega_{0}^{n}}dV_{\phi}\\ &\ -\frac{1}{V}\int_{M}\Big{(}F_{0}+(1-\gamma)\log|s|^{2}_{h}\Big{)}(dV_{0}-dV_{\phi}),\end{split}$$

Full text

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LiuF. LastnameMagdeburg ZhangF. LastnameHefei \contactInstitut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, universitätsplatz 2, 39106, Magdeburg, Germanyjiawei.liu@ovgu.de \contactKey Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, and School of Mathematical Sciences, University of Science and Technology of China, JinZhai Road, 230026, Hefei, Chinamathzx@ustc.edu.cn \researchsupportedThe first author is supported by the Special Priority Program SPP 2026 “Geometry at Infinity" from the German Research Foundation (DFG), and the second author is supported by NSF in China No.11625106, 11571332 and 11721101.

1 Introduction

Since the conical Kähler-Einstein metrics play an important role in the solution of the Yau-Tian-Donaldson’s conjecture which has been proved by Chen-Donaldson-Sun [6, 7, 8] and Tian [51], the existence and geometry of the conical Kähler-Einstein metrics have been widely concerned. The conical Kähler-Einstein metrics have been studied by Berman [1], Brendle [3], Campana-Guenancia-P$\breve{a}$un [4], Donaldson [17], Guenancia-P$\breve{a}$un [21], Guo-Song [22, 23], Jeffres [25], Jeffres-Mazzeo-Rubinstein [26], Li-Sun [32], Mazzeo [38], Song-Wang [48], Tian-Wang [52] and Yao [58] etc. For more details, readers can refer to Rubinstein’s article [44].

The conical Kähler-Ricci flows were introduced to attack the existence of the conical Kähler-Einstein metrics. These flows were first proposed in Jeffres-Mazzeo-Rubinstein’s paper (see Section $2.5$ in [26]), then Song-Wang (conjecture $5.2$ in [48]) made a conjecture on the relation between the convergence of these flows and the greatest Ricci lower bounds of the manifolds. Then the existence, regularity and limit behavior of the conical Kähler-Ricci flows have been studied by Chen-Wang [9, 10], Edwards [19, 20], Liu-Zhang [34], Liu-Zhang [36, 37], Nomura [39], Shen [45, 46], Wang [57] and Zhang [62, 63] etc.

Let $M$ be a Fano manifold with complex dimension $n$, $\omega_{0}\in c_{1}(M)$ be a smooth Kähler metric and $D\in|-\lambda K_{M}|$ ($0<\lambda\in\mathbb{Q}$) be a smooth divisor. Assume that the Kähler current $\hat{\omega}\in c_{1}(M)$ admits $L^{p}$-density with respect to $\omega_{0}^{n}$ for some $p>1$ and satisfies $\int_{M}\hat{\omega}^{n}=\int_{M}\omega_{0}^{n}$. Let $\gamma\in(0,1)$ and $\mu_{\gamma}=1-(1-\gamma)\lambda$. The conical Kähler-Ricci flows take the following form.

[TABLE]

where $[D]$ is the current of integration along $D$. In [37], we proved the existence of these flows by using smooth approximation (see also [36, 57] for the stronger initial metrics). Let $c_{1,\gamma}(M)=c_{1}(M)-(1-\gamma)[D]$ be the twisted first Chern class. Then $c_{1,\gamma}(M)=\mu_{\gamma}[\omega_{0}]$ in the Fano case. When $\mu_{\gamma}$ is negative or zero, Chen-Wang [10] proved that the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with Ricci curvature $\mu_{\gamma}$ and cone angle $2\pi\gamma$ along $D$. When $\mu_{\gamma}>0$ is sufficiently small and $\lambda\geqslant 1$, Li-Sun (see section $2.3$ in [32], when $\lambda=1$, see also Berman’s work [1] and Jeffres-Mazzeo-Rubinstein’s work [26]) proved that the Log Mabuchi energy $\mathcal{M}_{\mu_{\gamma}}$ is proper by using its definition and the property that the Log $\alpha$-invariant is positive. Then the convergence of the conical Kähler-Ricci flows $(CKRF_{\mu_{\gamma}})$ follows from the arguments in [37]. In other $\mu_{\gamma}>0$ cases, there are obstacles. In [37], under the assumptions that there exists a conical Kähler-Einstein metric with Ricci curvature $\mu_{\gamma}$ and cone angle $2\pi\gamma$ along $D$ and no nontrivial holomorphic vector fields tangent to $D$, we deduced their convergence.

If there exists a conical Kähler-Einstein metric $\omega_{\varphi_{\beta}}$ with cone angle $2\pi\beta$ along $D$, then $\omega_{\varphi_{\beta}}$ obtains the minimum of the Log Mabuchi energy $\mathcal{M}_{\mu_{\beta}}$ (Corollary $2.10$ in [32]), and thus this energy is bounded from blow. Furthermore, when $\lambda\geqslant 1$, Li-Sun (Corollary $1.7$ in [32]) proved that $\mathcal{M}_{\mu_{\beta}}$ is proper by using Berman’s properness theorem (Theorem $1.5$ in [1]) and Donaldson’s openness theorem (Theorem $2$ in [17]). For $\lambda>0$, Tian-Zhu (Theorem $0.1$ in [56]) proved this property by assuming in addition that there is no nontrivial holomorphic vector fields tangent to $D$. Darvas-Rubinstein solved Tian’s properness conjectures and gave more general properness theorems (Theorem $2.12$ in [14]). Under the above assumptions, by using the properness of $\mathcal{M}_{\mu_{\beta}}$ and the uniform Perelman’s estimates, we [36, 37] proved that the conical Kähler-Ricci flows $(CKRF_{\mu_{\beta}})$ with $\mu_{\beta}>0$ converge to $\omega_{\varphi_{\beta}}$. It is worth noting that the properness of $\mathcal{M}_{\mu_{\beta}}$ is a stronger condition than that $\mathcal{M}_{\mu_{\beta}}$ is bounded from below.

In this paper, we study the stability of the conical Kähler-Ricci flows by only using the weaker condition that the Log Mabuchi energy is bounded form below. We hope that this method can play a positive role in researching the relation between the limit behavior of the (conical) Kähler-Ricci flows and the stability of the manifolds, and studying the existence of the Kähler-Einstein metrics with cone angle zero by using parabolic method. The first problem is related to a parabolic type of Yau-Tian-Donaldson’s conjecture, and the latter one is related to a Tian’s conjecture [50] that the complete Tian-Yau Kähler-Einstein metric on the complement of $D$ should be the limit of the conical Kähler-Einstein metrics as the cone angles tend to zero.

Theorem 1.1**.**

Assume that $\lambda>0$ and there is no nontrivial holomorphic vector fields on $M$ tangent to $D$. If $\mu_{\beta}$ $(0<\beta<1)$ is positive, we further assume that there is a conical Kähler-Einstein metric with Ricci curvature $\mu_{\beta}$ and cone angle $2\pi\beta$ along $D$. Then for any $\beta^{\prime}$ sufficiently close to $\beta$, the conical Kähler-Ricci flow $(CKRF_{\mu_{\beta^{\prime}}})$ converges to a conical Kähler-Einstein metric with Ricci curvature $\mu_{\beta^{\prime}}$ and cone angle $2\pi\beta^{\prime}$ along $D$ in $C_{loc}^{\infty}$-topology outside $D$ and globally in $C^{\alpha,\beta^{\prime}}$-sense for any $\alpha\in(0,\min\{1,\frac{1}{\beta^{\prime}}-1\})$.

Remark 1.2**.**

When $\lambda\geqslant 1$, the assumption that there is no nontrivial holomorphic vector fields tangent to $D$ can be removed according to Theorem $1.5$ of Berman [1] (see also Corollary $2.21$ in [32]) or Theorem $2.8$ of Song-Wang [48].

As in [37], we still use the twisted Kähler-Ricci flows

[TABLE]

to study the conical Kähler-Ricci flows $(CKRF_{\mu_{\gamma}})$, where $\theta_{\varepsilon}=\lambda\omega_{0}+\sqrt{-1}\partial\overline{\partial}\log(\varepsilon^{2}+|s|_{h}^{2})$ are smooth closed positive $(1,1)$-forms, $s$ is the definition section of $D$ and $h$ is a smooth Hermitian metric on $-\lambda K_{M}$ with curvature $\lambda\omega_{0}$. We assume that $|s|_{h}^{2}<\frac{1}{2}$ by rescaling $h$.

Let $u_{\gamma,\varepsilon}(t)$ and $u_{\gamma}(t)$ be the twisted Ricci potentials of $\omega_{\gamma,\varepsilon}(t)$ and $\omega_{\gamma}(t)$ with normalization $\frac{1}{V}\int_{M}e^{-u_{\gamma,\varepsilon}(t)}dV_{\gamma,\varepsilon}(t)=1$ and $\frac{1}{V}\int_{M}e^{-u_{\gamma}(t)}dV_{\gamma}(t)=1$ respectively. We denote $A_{\mu_{\gamma},\varepsilon}(t)$ and $A_{\mu_{\gamma}}(t)$ be the functionals

[TABLE]

When $\mu_{\gamma}=0$, the functional $A_{\mu_{\gamma}}(t)$ is well-defined (Theorem 3.6) and converges to zero as $t$ tend to $\infty$ (Theorem 3.7). When $\mu_{\gamma}>0$, from the uniform Perelman’s estimates (Theorem 2.10), the functional $A_{\mu_{\gamma}}(t)$ is well-defined. Furthermore, $A_{\mu_{\gamma},\varepsilon}(t)$ is nondecreasing along $(TKRF_{\mu_{\gamma},\varepsilon})$ (Theorem 2.12), and $A_{\mu_{\gamma}}(t)$ converges to [math] as $t$ tend to $\infty$ when the Log Mabuchi energy $\mathcal{M}_{\mu_{\gamma}}$ is bounded from below (Theorem 2.14).

Remark 1.3**.**

In this paper, when considering the convergence for a sequence of twisted Kähler-Ricci flows as $t$ tend to $\infty$, we use the functionals $A_{\mu_{\gamma},\varepsilon}(t)$ and $A_{\mu_{\gamma}}(t)$. For a single (twisted) Kähler-Ricci flow, after obtaining the uniform $C^{\infty}$-estimates, we usually use the energy $\|\nabla u(t)\|_{L^{2}(M)}$ (see Liu-Wang [35], Phong-Sturm et al.[41, 42, 43] and Tian-Zhu [54]) or the (twisted) Perelman’s entropy (see Collins-Sz$\acute{e}$kelyhidi [11] and Tian-Zhu et al.[55, 53]) to prove the convergence. But in our case, we need consider the convergence for a specific choice sequence of twisted Kähler-Ricci flows. We do not use $\|\nabla u_{\gamma,\varepsilon}(t)\|_{L^{2}(M)}$ or the twisted Perelman’s entropy because we can not prove that these functionals converge to [math] uniformly (that is, the convergence depends on $\mu_{\gamma}$ and $\varepsilon$). In order to overcome these difficulties, we use the functionals $A_{\mu_{\gamma},\varepsilon}(t)$ and $A_{\mu_{\gamma}}(t)$, especially the convergence of $A_{\mu_{\gamma}}(t)$ and the monotonicity of $A_{\mu_{\gamma},\varepsilon}(t)$, to get the convergence we need. Although the twisted Perelman’s entropies are also monotonous along the twisted Kähler-Ricci flow (see Liu [33] and Collins-Sz$\acute{e}$kelyhidi [11]), we still do not use these entropies because there is no details about them in this conical case. On the manifolds with isolated conical singularity, Dai-Wang [12, 13] and Ozuch [40] developed Perelman’s entropies, and recently Kröncke-Vertman [30] improved the regularities for the minimizers of these entropies and studied their monotonicity along the Ricci de Turck flow.

There are three cases in Theorem 1.1, that is, $\mu_{\beta}$ is negative, zero and positive. The last two cases are difficult.

Case $1$. If $\mu_{\beta}$ is negative, $\mu_{\gamma}$ is also negative when $\gamma$ sufficiently close to $\beta$. Then we can prove Theorem 1.1 without any assumptions because the uniform estimates along the twisted Kähler-Ricci flows can be deduced directly (see also Chen-Wang’s work [10]).

Case $2$. If $\mu_{\beta}$ is zero (that is, $\lambda=\frac{1}{1-\beta}>1$ ), then $\mu_{\gamma}$ is a negative or sufficiently small positive number when $\gamma$ close to $\beta$. The case of $\mu_{\gamma}<0$ can be solved as in Case $1$. There are difficulties when $\mu_{\gamma}>0$, because no direct uniform Perelman’s estimates (independent of $\mu_{\gamma}$) can be used (these estimates depend on $\frac{1}{\mu_{\gamma}}$ from the proof of Proposition $5.5$ in [36]). For overcoming these difficulties, we need some new arguments.

First, we prove a uniform lower bound independent of $\mu_{\gamma}$ for the twisted scalar curvature $R(\omega_{\mu_{\gamma},\varepsilon}(t))-(1-\gamma)tr_{\omega_{\mu_{\gamma},\varepsilon}(t)}\theta_{\varepsilon}$ along $(TKRF_{\mu_{\gamma},\varepsilon})$ when $t\geqslant 1$ (Lemma 3.1). This uniform lower bound plays an important role in proving the uniform (Perelman’s) estimates (independent of $\mu_{\gamma}$) along $(TKRF_{\mu_{\gamma},\varepsilon})$ for $\gamma$ sufficiently close to $\beta$.

Next, without any assumptions, we prove that there exists uniform constant $D_{\beta}$ such that along the twisted Kähler-Ricci flows $(TKRF_{0,\varepsilon})$,

[TABLE]

for any $\varepsilon\in(0,\frac{1}{2}]$ and $t\geqslant 1$ (Theorem 3.6). Here we denote $\varphi_{\gamma,\varepsilon}(t)$ is the metric potential of $\omega_{\gamma,\varepsilon}(t)$ with background metric $\omega_{0}$, and $\omega_{\varepsilon}^{\gamma}$ is smooth Kähler metric given in equation $(\ref{TKRF9})$. For a single (twisted) Kähler-Ricci flow in the case of $\mu_{\beta}=0$, the bound on the oscillation of metric potential follows from Yau’s $C^{0}$-estimates for the elliptic complex Monge-Ampère equation. The main tools are the Sobolev inequality, the Poincaré inequality and the Moser iteration (see Lemma $3$ in [5] or Theorem $4.6$ in [49]). Since we consider a sequence of equations and do not have uniform Poincaré inequality, we here need modified the original proof when we adopt this method. We remark that this bound can also be deduced from Kołodeizj’s $L^{p}$-estimates [28, 29]. Then by using estimates $(\ref{20190223})$ and the smooth approximation [36, 37], we prove that the conical Kähler-Ricci flow $(CKRF_{0})$ converges to a Ricci flat conical Kähler-Einstein metric with cone angle $2\pi\beta$ along $D$ (Theorem 3.7).

At last, by using the continuity of $(TKRF_{\mu_{\gamma},\varepsilon})$ with respect to $\mu_{\gamma}$ and $\varepsilon$ (Remark 3.10), we deduce the uniform estimates (independent of $\mu_{\gamma}$) along $(TKRF_{\mu_{\gamma},\varepsilon})$ for $\gamma$ sufficiently close to $\beta$. In this process, we mainly use the uniform lower bound of the twisted scalar curvature $R(\omega_{\mu_{\gamma},\varepsilon}(t))-(1-\gamma)tr_{\omega_{\mu_{\gamma},\varepsilon}(t)}\theta_{\varepsilon}$, the convergence of $A_{0}(t)$, the monotonicity of $A_{\mu_{\gamma},\varepsilon}(t)$ and Dinew’s uniqueness theorem (Theorem $1.2$ in [16], see also Berndtsson’s work [2]). We prove the following lemma.

Lemma 1.4**.**

Assume that $\lambda>1$ and $\beta=1-\frac{1}{\lambda}$ (that is, $\mu_{\beta}=0$). Then there exists a constant $\delta(\lambda)>0$ such that

[TABLE]

for any $\varepsilon\in(0,\delta(\lambda))$, $\gamma\in(\beta,\beta+\delta(\lambda))$ and $t\in[\frac{1}{\delta(\lambda)},+\infty)$, where $L_{\beta}$ comes from $(\ref{20190304})$.

In [36, 37], we prove the uniform Perelman’s estimates along the twisted Kähler-Ricci flows $(TKRF_{\mu_{\gamma},\varepsilon})$ when $\mu_{\gamma}>0$. These estimates are important in studying the convergence of the conical Kähler-Ricci flows $(CKRF_{\mu_{\gamma}})$. From the proof of Proposition $5.5$ in [36], we know that these estimates depend on $\frac{1}{\mu_{\gamma}}$. So they are not uniform for $\mu_{\gamma}\in(0,1)$. In Lemma 1.4, we have obtained the uniform bound for $\|u_{\gamma,\varepsilon}(t)\|_{C^{0}(M)}$ when $\mu_{\gamma}\in(0,\eta(\lambda))$ with $\lambda>1$ and the uniform lower bound for the twisted scalar curvature $R(\omega_{\mu_{\gamma},\varepsilon}(t))-(1-\gamma)tr_{\omega_{\mu_{\gamma},\varepsilon}(t)}\theta_{\varepsilon}$, then by using Lemma 3.11, we get the uniform Perelman’s estimates independent of $\mu_{\gamma}$ along $(TKRF_{\mu_{\gamma},\varepsilon})$ when $\lambda>1$.

Theorem 1.5**.**

Let $\omega_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$. When $\lambda>1$, there exists uniform constant $C$ such that

[TABLE]

hold for any $\mu_{\gamma}\in(0,1]$, $t\geqslant 1$ and $\varepsilon\in(0,\delta(\lambda))$, where $\delta(\lambda)$ is the constant in Lemma 1.4 and $diam(M,\omega_{\gamma,\varepsilon}(t))$ is the diameter of the manifold with respect to $\omega_{\gamma,\varepsilon}(t)$.

Remark 1.6**.**

When $\lambda=1$ (resp. $0<\lambda<1$), we can not get the uniform Perelman’s estimates independent of $\mu_{\gamma}\in(0,1]$ (resp. $\mu_{\gamma}\in(1-\lambda,1]$) by this method. Because there is no uniform estimates as $(\ref{20190223})$ when $\mu_{\gamma}=0$ (resp. $\mu_{\gamma}=1-\lambda$), that is, $\gamma=0$.

By using Lemma 1.4, the convergence of $(CKRF_{\mu_{\gamma}})$ with $\gamma\in(\beta,\beta+\delta(\lambda))$ follows from the arguments in [36, 37]. It should be noted that we still don’t need any assumptions in this case. As a corollary of the first two cases, we get the following existence theorem for the conical Kähler-Einstein metrics by using conical Kähler-Ricci flows. This result implies the existence conjecture for the conical Kähler-Einstein metrics with positive Ricci curvatures (when $\lambda=1$, they correspond to the cone angles), which is given by Donaldson [18] and proved by Berman, Jeffres-Mazzeo-Rubinstein and Li-Sun (see Theorem $1.5$ in [1], Corollary $1$ in [26] and Theorem $1.1$ in [32]).

Theorem 1.7**.**

Assume $\lambda>1$. There exist conical Kähler-Einstein metrics with Ricci curvatures $\mu_{\gamma}$ and cone angles $2\pi\gamma$ along $D$ when $\gamma\in(0,1-\frac{1}{\lambda}+\delta(\lambda))$ for some $\delta(\lambda)>0$.

Case $3$. If $\mu_{\beta}$ is positive, then $\mu_{\gamma}$ is still positive when $\gamma$ sufficiently close to $\beta$. In this case, we can not get the uniform estimates along $(TKRF_{\mu_{\beta},\varepsilon})$ by no assumptions as in Case $2$. We need the assumptions in Theorem 1.1. Let $\omega_{\varphi_{\beta}}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi_{\beta}$ be a conical Kähler-Einstein metric with Ricci curvature $\mu_{\beta}$ and cone angle $2\pi\beta$ along $D$. We approximate $\varphi_{\beta}$ with a decreasing sequence of smooth $\omega_{0}$-psh functions $\varphi_{\varepsilon}$. Now we introduce the twisted Kähler-Ricci flows

[TABLE]

with $\gamma\in[1-\frac{1}{\lambda},\beta]$ (that is, $\mu_{\gamma}\in[0,\mu_{\beta}]$). By the arguments in section $3$ of [37], when $\varepsilon$ tend to [math], the limit flows of $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ are the twisted conical Kähler-Ricci flows

[TABLE]

In the following arguments, we restrict $\gamma\in[1-\frac{1}{\lambda},\beta]$ in the flows $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ and $(CKRF^{\beta}_{\mu_{\gamma}})$. In fact, $\omega^{\beta}_{\beta,\varepsilon}(t)=\omega_{\beta,\varepsilon}(t)$ and $\omega^{\beta}_{\beta}(t)=\omega_{\beta}(t)$ by the uniqueness results (see Proposition $2.7$ and Theorem $3.7$ in [37]). Let $u^{\beta}_{\gamma,\varepsilon}(t)$ and $u^{\beta}_{\gamma}(t)$ be the twisted Ricci potentials of $\omega^{\beta}_{\gamma,\varepsilon}(t)$ and $\omega^{\beta}_{\gamma}(t)$ with normalization $\frac{1}{V}\int_{M}e^{-u^{\beta}_{\gamma,\varepsilon}(t)}dV^{\beta}_{\gamma,\varepsilon}(t)=1$ and $\frac{1}{V}\int_{M}e^{-u^{\beta}_{\gamma}(t)}dV^{\beta}_{\gamma}(t)=1$ respectively. We denote functionals

[TABLE]

When $\mu_{\gamma}=0$, the functional $A^{\beta}_{\mu_{\gamma}}(t)$ is well-defined (Theorem 4.14) and converges to zero as $t$ tend to $\infty$ (Theorem 4.15). When $\mu_{\gamma}\in(0,\mu_{\beta}]$, from the uniform Perelman’s estimates (Theorem 4.7), the functional $A^{\beta}_{\mu_{\gamma}}(t)$ is well-defined. Furthermore, when $\mu_{\gamma}\in(0,\mu_{\beta}]$, $A^{\beta}_{\mu_{\gamma},\varepsilon}(t)$ is nondecreasing along $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ (Theorem 2.12), and $A^{\beta}_{\mu_{\gamma}}(t)$ converges to [math] as $t$ tend to $\infty$ when the Log Mabuchi energy $\mathcal{M}^{\beta}_{\mu_{\gamma}}$ is bounded from below (Theorem 4.8).

First, by similar arguments as in Case $2$, we get the uniform estimates along $(TKRF^{\beta}_{0,\varepsilon})$ (Theorem 4.14) and then obtain the uniform estimates along $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ when $\mu_{\gamma}$ sufficiently close to [math] (Lemma 4.19). We let $\psi^{\beta}_{\gamma,\varepsilon}(t)$ be the metric potential (with specific choice of initial data, see $(\ref{TKRF5})$) of $\omega^{\beta}_{\gamma,\varepsilon}(t)$ with background metric $\omega_{0}$. Fix a $\mu_{\beta_{0}}>0$ obtained in Lemma 4.19, there exists constant $C_{\beta_{0}}$ independent of $\varepsilon\in(0,\delta)$ such that

[TABLE]

At the same time, there exists uniform constant $\hat{C}_{\beta_{0}}$ such that

[TABLE]

for any $\varepsilon>0$, $\gamma\in[\beta_{0},\beta]$ and $\ t\in[1,+\infty)$ (Proposition 4.9).

Then, by using the uniform estimates $(\ref{05})$ and $(\ref{05001})$, the continuity of $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ with respect to $\mu_{\gamma}$ and $\varepsilon$ (Remark 4.13), the convergence of $A^{\beta}_{\mu_{\gamma}}(t)$, the monotonicity of $A^{\beta}_{\mu_{\gamma},\varepsilon}(t)$ and Berndtsson’s uniqueness theorem [2] for the conical Kähler-Einstein metrics with bounded potentials, we deduce the uniform estimates (independent of $\gamma$) of $\psi^{\beta}_{\gamma,\varepsilon}(t)$ for $\gamma\in(\beta_{0},\beta)$. That is, we obtain the following lemma.

Lemma 1.8**.**

Under the same assumptions as in Theorem 1.1. For above $\mu_{\beta_{0}}>0$, there exists a constant $\delta_{\beta_{0}}>0$ depending on $\beta_{0}$ such that

[TABLE]

for any $\varepsilon\in(0,\delta_{\beta_{0}})$, $\gamma\in[\beta_{0},\beta)$ and $t\in[\frac{1}{\delta_{\beta_{0}}},+\infty)$, where $C_{\beta_{0}}$, $\hat{C}_{\beta_{0}}$ and $\xi_{\beta}$ are the constants in $(\ref{05})$, $(\ref{05001})$ and $(\ref{2019022301})$ respectively.

Therefore, there exists a uniform constant $B_{\beta}$ such that for any $\varepsilon\in(0,\delta_{\beta_{0}})$,

[TABLE]

Then by using the arguments in [37], we get the convergence of the conical Kähler-Ricci flows $(CKRF_{\mu_{\beta}})$. Instead of using the properness of $\mathcal{M}_{\mu_{\beta}}$ in [36, 37], here we only use the weaker condition that $\mathcal{M}_{\mu_{\beta}}$ is bounded from below.

Theorem 1.9**.**

Assume that $\lambda>0$ and there is no nontrivial holomorphic fields on $M$ tangent to $D$. If $\mu_{\beta}$ $(0<\beta<1)$ is positive, we further assume that there is a conical Kähler-Einstein metric with Ricci curvature $\mu_{\beta}$ and cone angle $2\pi\beta$ along $D$. Then the conical Kähler-Ricci flow $(CKRF_{\mu_{\beta}})$ converges to a conical Kähler-Einstein metric with Ricci curvature $\mu_{\beta}$ and cone angle $2\pi\beta$ along $D$ in $C_{loc}^{\infty}$-topology outside divisor $D$ and globally in $C^{\alpha,\beta}$-sense for any $\alpha\in(0,\min\{1,\frac{1}{\beta}-1\})$.

We denote $\psi_{\gamma,\varepsilon}(t)$ is the metric potential (with specific choice of initial data, see $(\ref{TKRF6})$) of $\omega_{\gamma,\varepsilon}(t)$ with background metric $\omega_{0}$. At last, by similar arguments as above, we prove

Lemma 1.10**.**

Under the same assumptions as in Theorem 1.1. There exists a constant $\delta>0$ such that

[TABLE]

for any $\varepsilon\in(0,\delta)$, $\gamma\in(\beta-\delta,\beta+\delta)$ and $t\in[\frac{1}{\delta},+\infty)$, where $B_{\beta}$ is the constant in $(\ref{007})$, $\xi_{\beta}$ is the constant in $(\ref{2019022301})$ and $C_{\beta}$ is the constant in Proposition 4.10.

Then the convergence of the conical Kähler-Ricci flows $(CKRF_{\mu_{\gamma}})$ with $\gamma\in(\beta-\delta,\beta+\delta)$ follows from the arguments in [36, 37].

It is generally known that Donaldson’s celebrated openness theorem played an important role in solving the Yau-Tian-Donaldson’s conjecture. It was first proposed by Donaldson [17]. Then Yao [59] (see a remark in [36]) and Tian-Zhu [56] gave different proofs by using elliptic methods. Here, as a corollary of Theorem 1.1, we give a parabolic proof of Donaldson’s openness theorem.

Theorem 1.11**.**

(Donaldson’s openness theorem)   Assume that $\lambda>0$ and there is no nontrivial holomorphic vector fields on $M$ tangent to $D$. If there is a conical Kähler-Einstein metric with Ricci curvature $\mu_{\beta}$ and cone angle $2\pi\beta$ along $D$, then there exist conical Kähler-Einstein metrics with Ricci curvature $\mu_{\beta^{\prime}}$ and cone angles $2\pi\beta^{\prime}$ along $D$ for $\beta^{\prime}$ sufficiently close to $\beta$.

We give a further remark of our project which is related to the parabolic version of the Yau-Tian-Donaldson’s conjecture.

Remark 1.12**.**

When $\lambda>1$. We denote $\mu_{\beta}=0$ (that is, $\beta=1-\frac{1}{\lambda}$) and define a set $\Omega_{\lambda}$ as follows

[TABLE]

From Theorem 1.1, $\Omega_{\lambda}$ contains $(\beta,\beta_{0})$ with some $\beta_{0}\in(\beta,1)$, so it is non-empty. Furthermore, $\Omega_{\lambda}$ is open in $(\beta,1)$. If we can combine the closeness of $\Omega_{\lambda}$ with the stability of the manifolds, then $1\in\Omega_{\lambda}$ and thus the smooth Kähler-Ricci flow converges to a Kähler-Einstein metric.

When $\lambda=1$, we denote $\kappa>0$ sufficiently small and $\nu_{\gamma}=1-(1-\gamma)(1+\kappa)$. We consider the twisted conical Kähler-Ricci flows

[TABLE]

Assume that $\nu_{\hat{\beta}}=0$ (that is, $\hat{\beta}=1-\frac{1}{1+\kappa}>0$). We define set $\Omega_{\kappa}$ as follows

[TABLE]

By similar arguments as Theorem 1.1, we know that $\Omega_{\kappa}$ is open in $(\hat{\beta},1)$ and contains $(\hat{\beta},\beta_{0})$ for some $\beta_{0}\in(\hat{\beta},1)$. If $\Omega_{\kappa}$ is closed under some stability assumption on the manifold, then $1\in\Omega_{\kappa}$ which means that the smooth Kähler-Ricci flow converges to a Kähler-Einstein metric.

The above problem can be seen as a parabolic version of the Yau-Tian-Donaldson’s conjecture. In our subsequent work, we will focus on the relation between the closeness of $\Omega_{\lambda}$ (resp. $\Omega_{\kappa}$) and the K-stability of the manifold.

This paper is organized as follows. In section $2$, we recall some results of the twisted Kähler-Ricci flows and conical Kähler-Ricci flows in the literature. In section $3$, we study the Case 2 of Theorem 1.1. We first give a uniform lower bound independent of $\mu_{\gamma}$ for the twisted scalar curvatures $R(\omega_{\mu_{\gamma},\varepsilon}(t))-(1-\gamma)tr_{\omega_{\mu_{\gamma},\varepsilon}(t)}\theta_{\varepsilon}$. Then after we get some uniform estimates along the twisted Kähler-Ricci flows $(TKRF_{0,\varepsilon})$ and $(TKRF_{\mu_{\gamma},\varepsilon})$, we prove Lemma 1.4. At last, we prove Lemma 1.8 and Lemma 1.10 after we get some uniform regularities for the twisted Kähler-Ricci flows $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ in section $5$.

Acknowledge: The first author would like to thank Professors Jiayu Li, Miles Simon and Xiaohua Zhu for their useful discussions, constant help and encouragement. He also would like to thank Professor Xiangwen Zhang, Doctors Chao Li and Xishen Jin for their several useful comments. Part of this work was carried out while the first author’s visit to the University of Newcastle and the University of Adelaide in Australia. He is grateful to Professors James McCoy and Thomas Leistner for their invitations, and the universities for their hospitality.

2 Preliminaries

In this section, we give some known results about the twisted Kähler-Ricci flows and the (twisted) conical Kähler-Ricci flows. 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 $\omega$ with locally bounded potentials is conical Kähler metric with cone angle $2\pi\beta$ ($0<\beta<1$) along $D$, we mean that $\omega$ is smooth Kähler metric on $M\setminus D$. And near each point $p\in D$, there exists local holomorphic coordinate $(z_{1},\cdots,z_{n})$ in a neighborhood $U$ of $p$ such that $D=\{z_{n}=0\}$ and $\omega$ is asymptotically equivalent to the model conical metric

[TABLE]

Definition 2.1**.**

Let $\omega_{0}$ be a smooth Kähler metric and $D\subset M$ be a smooth divisor which satisfies $c_{1}(M)=\mu[\omega_{0}]+(1-\beta)[D]$ with $\mu\in\mathbb{R}$. We call $\omega$ a conical Kähler-Einstein metric with Ricci curvature $\mu$ and cone angle $2\pi\beta$ along $D$ if it is a conical Kähler metric with cone angle $2\pi\beta$ along $D$ and satisfies

[TABLE]

Equation (10) is classical outside $D$ and it holds in the sense of currents on $M$. There are other definitions of metrics with conical singularities (see [17, 26], etc.). But for conical Kähler-Einstein metrics, these definitions turn out to be equivalent (see Theorem $2$ in [26]).

In this paper, by saying that a conical Kähler-Einstein metric with cone angle $2\pi\beta$ along $D$ we also mean that its Ricci curvature is $\mu_{\beta}$.

We write $\hat{\omega}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi_{0}$. Let $\omega_{\gamma}=\omega_{0}+\frac{k}{\gamma^{2}}\sqrt{-1}\partial\bar{\partial}|s|_{h}^{2\gamma}$ with $\gamma\in(0,1)$ be the conical Kähler metric which is given by Donaldson [17]. Now, in the Fano case, we list some results proved in [37].

Theorem 2.2**.**

(Theorem $2.8$ in [37]) There exists a unique long-time solution $\omega_{\gamma,\varepsilon}(t)$ to the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$ in the following sense.

• •

$\omega_{\gamma,\varepsilon}(t)$* satisfies the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$ on $(0,\infty)\times M$;*

• •

There exists a metric potential \varphi_{\gamma,\varepsilon}(t)\in C^{0}\big{(}[0,\infty)\times M\big{)}\cap C^{\infty}\big{(}(0,\infty)\times M\big{)} such that $\omega_{\gamma,\varepsilon}(t)=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi_{\gamma,\varepsilon}(t)$ and $\lim\limits_{t\rightarrow 0^{+}}\|\varphi_{\gamma,\varepsilon}(t)-\varphi_{0}\|_{L^{\infty}(M)}=0$.

Theorem 2.3**.**

(Theorem $1.2$ in [37]) There exists a unique long-time solution $\omega_{\gamma}(t)$ to the conical Kähler-Ricci flow $(CKRF_{\mu_{\gamma}})$ in the following sense.

• •

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

[TABLE]

• •

On $(0,\infty)\times(M\setminus D)$, $\omega_{\gamma}(t)$ satisfies the smooth Kähler-Ricci flow;

• •

On $(0,\infty)\times M$, $\omega_{\gamma}(t)$ satisfies equation $(CKRF_{\mu_{\gamma}})$ in the sense of currents;

• •

There exists a metric potential \varphi_{\gamma}(t)\in C^{0}\big{(}[0,\infty)\times M\big{)}\cap C^{\infty}\big{(}(0,\infty)\times(M\setminus D)\big{)} such that $\omega_{\gamma}(t)=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi_{\gamma}(t)$ and $\lim\limits_{t\rightarrow 0^{+}}\|\varphi_{\gamma}(t)-\varphi_{0}\|_{L^{\infty}(M)}=0$;

• •

On $[\delta,T]$, there exist constants $\alpha\in(0,1)$ and $C^{\ast}$ such that $\varphi_{\gamma}(t)$ is $C^{\alpha}$ on $M$ with respect to $\omega_{0}$ and $\|\frac{\partial\varphi_{\gamma}(t)}{\partial t}\|_{L^{\infty}(M\setminus D)}\leqslant C^{\ast}$.

Remark 2.4**.**

In Theorem 2.3, by saying that $\omega_{\gamma}(t)$ satisfies equation $(CKRF_{\mu_{\gamma}})$ in the sense of currents on $M_{\infty}:=(0,\infty)\times M$, we mean that for any smooth $(n-1,n-1)$-form $\eta(t)$ with compact support in $(0,\infty)\times M$, we have

[TABLE]

where the integral on the left side can be written as

[TABLE]

By using the similar arguments as that in the sections $2$ and $3$ of [37], we have the following results.

Theorem 2.5**.**

There exists a unique long-time solution $\omega^{\beta}_{\gamma,\varepsilon}(t)$ to the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ in the following sense.

• •

$\omega^{\beta}_{\gamma,\varepsilon}(t)$* satisfies the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ on $(0,\infty)\times M$;*

• •

There exists a metric potential \varphi^{\beta}_{\gamma,\varepsilon}(t)\in C^{0}\big{(}[0,\infty)\times M\big{)}\cap C^{\infty}\big{(}(0,\infty)\times M\big{)} such that $\omega^{\beta}_{\gamma,\varepsilon}(t)=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi^{\beta}_{\gamma,\varepsilon}(t)$ and $\lim\limits_{t\rightarrow 0^{+}}\|\varphi^{\beta}_{\gamma,\varepsilon}(t)-\varphi_{0}\|_{L^{\infty}(M)}=0$.

Theorem 2.6**.**

There exists a unique long-time solution $\omega^{\beta}_{\gamma}(t)$ to the conical Kähler-Ricci flow $(CKRF^{\beta}_{\mu_{\gamma}})$ in the following sense.

• •

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

[TABLE]

• •

On $(0,\infty)\times(M\setminus D)$, $\omega^{\beta}_{\gamma}(t)$ satisfies the smooth twisted Kähler-Ricci flow;

• •

On $(0,\infty)\times M$, $\omega^{\beta}_{\gamma}(t)$ satisfies equation $(CKRF^{\beta}_{\mu_{\gamma}})$ in the sense of currents;

• •

There exists a metric potential \varphi^{\beta}_{\gamma}(t)\in C^{0}\big{(}[0,\infty)\times M\big{)}\cap C^{\infty}\big{(}(0,\infty)\times(M\setminus D)\big{)} such that $\omega^{\beta}_{\gamma}(t)=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi^{\beta}_{\gamma}(t)$ and $\lim\limits_{t\rightarrow 0^{+}}\|\varphi^{\beta}_{\gamma}(t)-\varphi_{0}\|_{L^{\infty}(M)}=0$;

• •

On $[\delta,T]$, there exist constants $\alpha\in(0,1)$ and $C^{\ast}$ such that $\varphi^{\beta}_{\gamma}(t)$ is $C^{\alpha}$ on $M$ with respect to $\omega_{0}$ and $\|\frac{\partial\varphi^{\beta}_{\gamma}(t)}{\partial t}\|_{L^{\infty}(M\setminus D)}\leqslant C^{\ast}$.

Remark 2.7**.**

By Liu-Zhang’s results (Theorem $3.8$ in [34], see also Theorem $3.10$ in [37]). The solutions $\omega_{\gamma,\varepsilon}(t)$ and $\omega^{\beta}_{\gamma,\varepsilon}(t)$ are $C^{\alpha,\beta}$ for any $\alpha\in(0,\min\{1,\frac{1}{\beta}-1\})$ when $t>0$.

From the proof of Proposition $5.9$ in [36], we have the following uniform Sobolev inequalities when the cone angles away from [math].

Theorem 2.8**.**

Assume that $n\geqslant 2$. For any $\beta_{0}\in(0,1)$, there exists uniform constant $C$ such that

[TABLE]

hold for any smooth functions $v$ on $M$, $\gamma\in[\beta_{0},1)$ and $\varepsilon>0$, where $dV_{\varepsilon}^{\gamma}=\frac{(\omega_{\varepsilon}^{\gamma})^{n}}{n!}$.

Theorem 2.9**.**

Assume that $n=1$. Then for any $\beta_{0}\in(0,1)$, there exists uniform constant $C$ such that

[TABLE]

holds for any smooth functions $v$ on $M$, $\gamma\in[\beta_{0},1)$ and $\varepsilon>0$.

In [36, 37], we proved the uniform Perelman’s estimates for $\mu_{\gamma}>0$, which play an important role in studying the convergence of the conical Kähler-Ricci flows. These estimates mainly depend on the uniform lower bounded of the twisted scalar curvature $R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}$, the uniform Sobolev inequality $\mathcal{C}_{S}(M,\omega_{\gamma,\varepsilon}(1))$ and $\frac{1}{\mu_{\gamma}}$. Hence when $\mu_{\gamma}$ away from [math], we have the following uniform Perelman’s estimates by using the arguments in [36].

Theorem 2.10**.**

Let $\omega_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$. Then for any $0<\mu_{\gamma_{1}}<\mu_{\gamma_{2}}\leqslant 1$, there exists uniform constant $C$, such that

[TABLE]

hold for any $\mu_{\gamma}\in[\mu_{\gamma_{1}},\mu_{\gamma_{2}}]$, $t\geqslant 1$ and $\varepsilon>0$.

Remark 2.11**.**

The uniform Perelman’s estimates for $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ can also be established by using the similar arguments as that in [36], but there exist a few details need to be verified in the proof, so we present them in section $4$.

In the proof of uniform Perelman’s estimates, we prove that the functional $A_{\mu_{\gamma},\varepsilon}(t)$ with $\mu_{\gamma}>0$ is nondecreasing by using the uniform Poincaré inequality along along the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$. Since $(\mu_{\beta}-\mu_{\gamma})\omega_{\varphi_{\varepsilon}}+(1-\beta)\theta_{\varepsilon}$ are positive closed $(1,1)$-forms when $\mu_{\gamma}\in(0,\mu_{\beta}]$, there also exists uniform Poincaré inequality along $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$. So $A^{\beta}_{\mu_{\gamma},\varepsilon}(t)$ is nondecreasing along $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ when $\mu_{\gamma}\in(0,\mu_{\beta}]$.

Theorem 2.12**.**

When $\mu_{\gamma}\in(0,1]$, the functional $A_{\mu_{\gamma},\varepsilon}(t)$ is nondecreasing along the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$.

When $\mu_{\gamma}\in(0,\mu_{\beta}]$, the functional $A^{\beta}_{\mu_{\gamma},\varepsilon}(t)$ is nondecreasing along the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$.

Now we recall Aubin’s functionals. Let $\phi_{t}$ be a path with $\phi_{0}=c$ and $\phi_{1}=\phi$, then

[TABLE]

where $dV_{0}=\frac{\omega_{0}^{n}}{n!}$ and $dV_{\phi}=\frac{\omega_{\phi}^{n}}{n!}$. They satisfy $0\leqslant\frac{1}{n}J_{\omega_{0}}\leqslant\frac{1}{n+1}I_{\omega_{0}}\leqslant J_{\omega_{0}}$. The twisted Mabuchi energy is defined as

[TABLE]

where $u_{\omega_{0}}$ satisfies $-Ric(\omega_{0})+k\omega_{0}+\theta=\sqrt{-1}\partial\bar{\partial}u_{\omega_{0}}$ and $\frac{1}{V}\int_{M}e^{-u_{\omega_{0}}}dV_{0}=1$. We denote $\mathcal{M}^{\beta}_{\mu_{\gamma},\varepsilon}$ and $\mathcal{M}_{\mu_{\gamma},\varepsilon}$ are the twisted Mabuchi energies with twisted forms $(\mu_{\beta}-\mu_{\gamma})\omega_{\varphi_{\varepsilon}}+(1-\beta)\theta_{\varepsilon}$ and $(1-\gamma)\theta_{\varepsilon}$ respectively. Then

[TABLE]

The Log Mabuchi energy is denoted by

[TABLE]

and the twisted Log Mabuchi energy is defined as

[TABLE]

By using Berndtsson’s convexity theorem [2], Li-Sun (Corollary $2.10$ in [32]) proved that the conical Kähler-Einstein metric is the minimum point of the Log Mabuchi energy.

Theorem 2.13**.**

(Corollary $2.10$ in [32]) If there exists a conical Kähler-Einstein metric $\omega_{\varphi_{\beta}}$ with cone angle $2\pi\beta$ along $D$, then $\omega_{\varphi_{\beta}}$ obtains the minimum of $\mathcal{M}_{\mu_{\beta}}$.

Combining the uniform Perelman’s estimates, the Perelman’s non-collapsing theorem and the uniform Poincaré inequality along $(TKRF_{\mu_{\gamma},\varepsilon})$, we get the following estimates and convergence for $A_{\mu_{\gamma}}(t)$ by using the arguments in Lemma $4.3$ of [33] or Lemma $3$ of [42].

Theorem 2.14**.**

For $\mu_{\gamma}\in(0,1]$, there exists constant $C$ such that for $t\geqslant 1$,

[TABLE]

Furthermore, if $\mathcal{M}_{\mu_{\gamma}}$ is bounded form below, then $A_{\mu_{\gamma}}(t)$ converge to [math] as $t\rightarrow+\infty$.

Remark 2.15**.**

We can also get a similar result for $A^{\beta}_{\mu_{\gamma}}(t)$ after we prove the uniform Perelman’s estimates along the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$ in section $4$.

3 Proof of the case $\mu_{\beta}=0$

Assume that $\lambda>1$. Then $\mu_{\beta}=0$ when $\beta=1-\frac{1}{\lambda}\in(0,1)$. In this section, we study the convergence of the conical Kähler-Ricci flow $(CKRF_{\mu_{\gamma}})$ when $\mu_{\gamma}$ is positive and sufficiently small. For $\mu_{\gamma}\in(0,1]$ (that is, $\gamma\in(\beta,1]$), there is no direct uniform Perelman’s estimates independent of $\mu_{\gamma}$ along the twisted Kähler-Ricci flows $(TKRF_{\mu_{\gamma},\varepsilon})$ and hence we do not have the uniform estimates for $\dot{\psi}_{\gamma,\varepsilon}(t)$, so we need some new discussions. In this section, we always assume that $\mu_{\beta}=0$ and $\mu_{\gamma}\in[\mu_{\beta},1]$ (that is, $\gamma\in[\beta,1]$) unless otherwise specified, and normalize $\varphi_{0}$ by

[TABLE]

First, by improving the proof of Proposition $5.1$ in [36], we get the uniform lower bound of the twisted scalar curvature $R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}$ when $t\geqslant 1$.

Lemma 3.1**.**

$R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}$* are uniformly bounded from below by $-4n$ along the twisted Kähler-Ricci flows $(TKRF_{\mu_{\gamma},\varepsilon})$, that is, for any $\mu_{\gamma}\in[0,1]$, $\varepsilon>0$ and $t\geqslant 1$, we have*

[TABLE]

Beweis.

First, we derive the evolution equation of (t-\frac{1}{2})^{2}\Big{(}R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}\Big{)}.

[TABLE]

Let $(t_{0},x_{0})$ be the minimum point of (t-\frac{1}{2})^{2}\Big{(}R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}\Big{)} on $[\frac{1}{2},1]\times M$.

$Case\ 1$, $t_{0}=\frac{1}{2}$, then we have (t-\frac{1}{2})^{2}\Big{(}R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}\Big{)}\geqslant 0.

$Case\ 2$, $t_{0}\neq\frac{1}{2}$, without loss of generality, we can assume $R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}\leqslant 0$ at $(t_{0},x_{0})$. By inequality

[TABLE]

we have

[TABLE]

Then (t_{0}-\frac{1}{2})^{2}\Big{(}R(\omega_{\gamma,\varepsilon}(t_{0}))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t_{0})}\theta_{\varepsilon}\Big{)}\geqslant-2n(t_{0}-\frac{1}{2})\geqslant-n. Hence,

[TABLE]

for any $\mu_{\gamma}\in[0,1]$, $\varepsilon>0$ and $t\in[\frac{1}{2},1]$. In particular, when $t=1$, we have

[TABLE]

Then we consider the lower bound of $R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}$ on $[1,\infty)\times M$.

[TABLE]

By maximum principle, on $[1,\infty)\times M$, we have

[TABLE]

for any $\mu_{\gamma}\in[0,1]$, $t\geqslant 1$ and $\varepsilon>0$. We complete the proof this lemma. ∎

We write the flows $(TKRF_{\mu_{\gamma},\varepsilon})$ as parabolic Monge-Ampère equations

[TABLE]

where $F_{0}$ satisfies $-Ric(\omega_{0})+\omega_{0}=\sqrt{-1}\partial\overline{\partial}F_{0}$ and $\frac{1}{V}\int_{M}e^{-F_{0}}dV_{0}=1$. Let $\chi_{\gamma}$ with $\gamma\in(0,1)$ be the function $\chi_{\gamma}(\varepsilon^{2}+|s|_{h}^{2})=\frac{1}{\gamma}\int_{0}^{|s|_{h}^{2}}\frac{(\varepsilon^{2}+r)^{\gamma}-\varepsilon^{2\gamma}}{r}dr$ which is given by Campana-Guenancia-P$\breve{a}$un [4]. Denote F_{\gamma,\varepsilon}=F_{0}+\log\Big{(}\frac{(\omega^{\gamma}_{\varepsilon})^{n}}{\omega_{0}^{n}}\cdot(\varepsilon^{2}+|s|_{h}^{2})^{1-\gamma}\Big{)}, $\phi_{\gamma,\varepsilon}(t)=\varphi_{\gamma,\varepsilon}(t)-k\chi_{\gamma}$ and $\omega^{\gamma}_{\varepsilon}=\omega_{0}+\sqrt{-1}k\partial\overline{\partial}\chi_{\gamma}$. Equation (23) can be written as

[TABLE]

We first prove the uniform $C^{0}$-estimates (independent of $\mu_{\gamma}$) for $\phi_{\gamma,\varepsilon}$.

Lemma 3.2**.**

For any $T>0$, there exists constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$, $\omega_{0}$ and $T$ such that for any $t\in[0,T]$, $\varepsilon>0$ and $\mu_{\gamma}\in[0,1]$ (that is, $\gamma\in[\beta,1]$),

[TABLE]

Beweis.

Since $\beta=1-\frac{1}{\lambda}>0$ and $\gamma\in[\beta,1]$, from Campana-Guenancia-P$\breve{a}$un’s results (see $(15)$ and $(25)$ in [4]), there exist uniform constants $A_{0}$ and $A_{1}$, such that for any $\varepsilon>0$ and $\gamma\in[\beta,1]$,

[TABLE]

Then we have

[TABLE]

which is equivalent to

[TABLE]

For any $\delta>0$, we denote $\tilde{\phi}_{\gamma,\varepsilon}(t)=e^{-\mu_{\gamma}t}\phi_{\gamma,\varepsilon}(t)-A_{0}\int_{0}^{t}e^{-\mu_{\gamma}s}ds-\delta t$. Let $(t_{0},x_{0})$ be the maximum point of $\tilde{\phi}_{\gamma,\varepsilon}(t)$ on $[0,T]\times M$. If $t_{0}>0$, by maximum principle, we have

[TABLE]

which is impossible. Hence $t_{0}=0$, then

[TABLE]

Let $\delta\rightarrow 0$, we obtain

[TABLE]

Hence there exists a constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$, $\omega_{0}$ amd $T$ such that $\phi_{\gamma,\varepsilon}(t)\leqslant C$ for any $t\in[0,T]$, $\varepsilon>0$ and $\mu_{\gamma}\in[0,1]$. By similar arguments we can get the uniform lower bound for $\phi_{\gamma,\varepsilon}(t)$. ∎

Then by using the arguments in section $2$ of [37] (see also Song-Tian’s work [47]), we have the following lemmas.

Lemma 3.3**.**

For any $T>0$, there exists a constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$, $\omega_{0}$ and $T$, such that for any $t\in(0,T]$, $\varepsilon>0$ and $\mu_{\gamma}\in[0,1]$,

[TABLE]

Lemma 3.4**.**

For any $T>0$, there exists a constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$, $\omega_{0}$ and $T$ such that for any $t\in(0,T]$, $\varepsilon>0$ and $\mu_{\gamma}\in[0,1]$,

[TABLE]

Hence away from time [math], on any compact subset in $M\setminus D$, $\omega_{\gamma,\varepsilon}$ are uniformly equivalent to $\omega_{0}$. Then Evans-Krylov-Safonov’s estimates (see [31]) imply the high order regularities.

Lemma 3.5**.**

For any $0<\delta<T<\infty$, $k\in\mathbb{N}^{+}$ and $B_{r}(p)\subset\subset M\setminus D$, there exist constants $C_{\delta,T,k,p,r}$, such that for any $\varepsilon>0$ and $\mu_{\gamma}\in[0,1]$,

[TABLE]

where constants $C_{\delta,T,k,p,r}$ depend on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$, $\delta$, $k$, $T$, $\omega_{0}$ and $dist_{\omega_{0}}(B_{r}(p),D)$.

Next, we prove the uniform estimates along the twisted Kähler-Ricci flows $(TKRF_{0,\varepsilon})$ and the convergence of the conical Kähler-Ricci flows $(CKRF_{0})$.

Theorem 3.6**.**

There exists a uniform constant $D_{\beta}$ such that

[TABLE]

for any $\varepsilon\in(0,\frac{1}{2}]$ and $t\geqslant 1$.

Beweis.

From Lemma 3.3, $\|\dot{\varphi}_{\beta,\varepsilon}(1)\|_{C^{0}(M)}$ are uniformly bounded for any $\varepsilon>0$. Then the maximum principle implies that $\|\dot{\varphi}_{\beta,\varepsilon}(t)\|_{C^{0}(M)}$ are uniformly bounded for any $\varepsilon>0$ and $t\geqslant 1$. Since $u_{\beta,\varepsilon}(t)=\dot{\varphi}_{\beta,\varepsilon}(t)+c_{\beta,\varepsilon}(t)$ and $\frac{1}{V}\int_{M}e^{-u_{\beta,\varepsilon}(t)}dV_{\beta,\varepsilon}(t)=1$, $c_{\beta,\varepsilon}(t)$ and $u_{\beta,\varepsilon}(t)$ are uniformly bounded. Denote $\tilde{\varphi}_{\beta,\varepsilon}(t)=\varphi_{\beta,\varepsilon}(t)-\frac{1}{V}\int_{M}\varphi_{\beta,\varepsilon}(t)dV_{0}$, then $\tilde{\varphi}_{\beta,\varepsilon}(t)$ satisfies

[TABLE]

By using the Green’s formula with respect to $\omega_{0}$ and $-\Delta_{\omega_{0}}\tilde{\varphi}_{\beta,\varepsilon}(t)\leqslant n$, we have

[TABLE]

for any $t\geqslant 0$ and $\varepsilon>0$, where constant $C$ depends only on $n$ and $\omega_{0}$. Therefore, there exists a constant $A$ such that $A-\tilde{\varphi}_{\beta,\varepsilon}(t)\geqslant 1$. Since $\chi_{\beta}$ is positive, using integration by parts, we have

[TABLE]

where constant $C$ depends only on $\beta$, $n$ and $\omega_{0}$. On the other hand, from Campana-Guenancia-P$\breve{a}$un’s work [4], there exist constants $\delta$ and $C$ such that

[TABLE]

Since $A-\tilde{\varphi}_{\beta,\varepsilon}(t)$ is positive, we conclude that

[TABLE]

Using the normalization $(\ref{201903})$, we obtain

[TABLE]

Hence there exists a uniform constant such that for any $\varepsilon>0$ and $t\geqslant 0$,

[TABLE]

Denote $\tilde{\phi}_{\beta,\varepsilon}(t)=\phi_{\beta,\varepsilon}(t)-\frac{1}{V}\int_{M}\phi_{\beta,\varepsilon}(t)dV_{0}$. Since $\chi_{\beta}$ is uniformly bounded, by $(\ref{201911})$ and $(\ref{201912})$, there exist constants $B$ and $C$ such that for any $\varepsilon>0$ and $t\geqslant 0$,

[TABLE]

Let $f_{\varepsilon t}=B-\tilde{\phi}_{\beta,\varepsilon}(t)\geqslant 1$. For any $\alpha\geqslant 0$ and $t\geqslant 1$,

[TABLE]

Hence the $L^{2}$-norm of $\nabla f_{\varepsilon t}^{\frac{\alpha}{2}+1}$ with respect to $\omega_{\varepsilon}^{\beta}$ can be bounded as follows

[TABLE]

When $n\geqslant 2$, we denote $\kappa=\frac{n}{n-1}$ and $p=\alpha+2$. By using the uniform Sobolev inequality $(\ref{1.5.6})$, we have

[TABLE]

Hence there exists uniform constant $C$ such that

[TABLE]

Let $p_{i}=p_{i-1}\kappa$ and $p_{0}=\frac{2n}{n-1}$. By Moser’s iteration, we have

[TABLE]

After letting $i\rightarrow\infty$, we deduce

[TABLE]

On the other hand, taking $\alpha=0$ in $(\ref{20190001})$ and using Hölder inequality, we have

[TABLE]

Combining $(\ref{201913})$ with $(\ref{yewan13})$, for $t\geqslant 1$, we have

[TABLE]

Since $\chi_{\beta}$ is uniformly bounded, and $osc_{M}\varphi_{\beta,\varepsilon}(t)$ are uniformly bounded for $t\in[0,1]$,

[TABLE]

for any $\varepsilon>0$ and $t\geqslant 0$.

Now we denote $\hat{\phi}_{\beta,\varepsilon}(t)=\phi_{\beta,\varepsilon}(t)-\frac{1}{V}\int_{M}\phi_{\beta,\varepsilon}(t)e^{-F_{0}+C_{\beta,\varepsilon}}\frac{dV_{0}}{(\varepsilon^{2}+|s_{h}^{2}|)^{1-\beta}}$, where $C_{\beta,\varepsilon}$ is the normalization constant such that $\frac{1}{V}\int_{M}e^{-F_{0}+C_{\beta,\varepsilon}}\frac{dV_{0}}{(\varepsilon^{2}+|s|_{h}^{2})^{1-\beta}}=1$. This normalization implies that for any $\varepsilon\in(0,\frac{1}{2}]$,

[TABLE]

Hence $C_{\beta,\varepsilon}\leqslant 0$. Since $\frac{1}{V}\int_{M}\hat{\phi}_{\beta,\varepsilon}(t)e^{-F_{0}+C_{\beta,\varepsilon}}\frac{dV_{0}}{(\varepsilon^{2}+|s|_{h}^{2})^{1-\beta}}=0$, we have

[TABLE]

From $(\ref{AAA})$, we have $osc_{M}\hat{\phi}_{\beta,\varepsilon}(t)\leqslant C$. Hence we conclude that

[TABLE]

for any $\varepsilon\in(0,\frac{1}{2}]$ and $t\geqslant 0$. On the other hand, from Lemma 3.4, there exists uniform constant $C$ such that for any $\varepsilon\in(0,\frac{1}{2}]$,

[TABLE]

By using the arguments in the proof of Proposition $3.1$ in [36], for any $t>1$, there exist uniform constant $B$ and $C$ such that

[TABLE]

By Jensen’s inequality, we have

[TABLE]

Hecne $\inf\limits_{[1,t]}\frac{1}{V}\int_{M}\phi_{\beta,\varepsilon}(s)e^{-F_{0}+C_{\beta,\varepsilon}}\frac{dV_{0}}{(\varepsilon^{2}+|s|_{h}^{2})^{1-\beta}}=\frac{1}{V}\int_{M}\phi_{\beta,\varepsilon}(t)e^{-F_{0}+C_{\beta,\varepsilon}}\frac{dV_{0}}{(\varepsilon^{2}+|s|_{h}^{2})^{1-\beta}}$, and then there exists uniform constant $C$ such that

[TABLE]

for any $\varepsilon\in(0,\frac{1}{2}]$.

When $n=1$, we can not use the uniform Sobolev inequality $(\ref{1.5.6})$. The estimates for $osc_{M}\varphi_{\gamma,\varepsilon}(t)$ follows from Kołodziej’s $L^{p}$-estimates [28, 29] directly. The other proofs are similar as above. ∎

Since $\|\dot{\varphi}_{\beta,\varepsilon}(t)\|_{C^{0}(M)}$ are uniformly bounded, we have

[TABLE]

for any $t\geqslant 1$ and $\varepsilon\in(0,\frac{1}{2}]$. Combining this inequality with $(\ref{201903040002})$, we get

[TABLE]

and then obtain the uniform $C^{\infty}_{loc}$-estimates in $M\setminus D$ along $(TKRF_{0,\varepsilon})$ for any $\varepsilon\in(0,\frac{1}{2}]$ and $t\geqslant 1$. At the same time, the twisted Mabuchi energy $\mathcal{M}_{0,\varepsilon}(\varphi_{\beta,\varepsilon}(t))$ are uniformly bounded from below. Then by using the smooth approximation and the arguments in section $7$ of [36], we get the convergence of the conical Kähler-Ricci flows $(CKRF_{0})$. In [10], Chen-Wang obtained this result by considering the conical Kähler-Ricci flow directly. In fact, by using similar arguments as above, we get the following convergence theorem.

Theorem 3.7**.**

Let $M$ be a Kähler manifold, $\omega_{0}$ be a smooth Kähler metric and $D$ be a smooth divisor. Assume that the Kähler current $\hat{\omega}\in c_{1}(M)$ admits $L^{p}$-density with respect to $\omega_{0}^{n}$ for some $p>1$ and satisfies $\int_{M}\hat{\omega}^{n}=\int_{M}\omega_{0}^{n}$. If $c_{1}(M)=(1-\gamma)[D]$ with $\gamma\in(0,1)$, then the conical Kähler-Ricci flow $(CKRF_{0})$ converges to a Ricci flat conical Kähler-Einstein metric with cone angle $2\pi\gamma$ along $D$ in $C_{loc}^{\infty}$-topology outside divisor $D$ and globally in $C^{\alpha,\gamma}$-sense for any $\alpha\in(0,\min\{1,\frac{1}{\gamma}-1\})$.

Proposition 3.8**.**

There exists a constant $\hat{\delta}$, such that for any $t\in[0,\hat{\delta})$, $\varepsilon\in[0,\frac{1}{2}]$ and $z\in M$, $\varphi_{\gamma,\varepsilon}(t)$ is increasing with respect to $\gamma\in[\beta,1]$.

Beweis.

We first consider the case $\varepsilon\in(0,\frac{1}{2}]$. By Kołodziej’s results [28, 29], there exists a Hölder continuous solution $u_{\gamma,\varepsilon}$ to the equation

[TABLE]

and $u_{\gamma,\varepsilon}$ satisfies

[TABLE]

where the normalization constant $\hat{C}$ is uniformly bounded independent of $\varepsilon$ and $\gamma$, constant $C$ depends only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$ and $F_{0}$. We define function

[TABLE]

where

[TABLE]

From the arguments in the proof of Proposition $3.5$ in [37], we know that $\psi_{\gamma,\varepsilon}(t)$ is a subsolution of equation $(\ref{TKRF7})$ and hence

[TABLE]

where $h_{1}(t)$ with $h_{1}(0)=0$ is a continuous function independent of $\gamma$ and $\varepsilon$. Then there exists $\hat{\delta}$, such that $h_{1}(t)>-\frac{1}{2}$ for any $t\in[0,\hat{\delta}]$. By the normalization $(\ref{nom0})$, we have

[TABLE]

for any $\gamma\in[\beta,1]$, $\varepsilon\in(0,1]$, $t\in(0,\hat{\delta})$ and $z\in M$. For $\beta\leqslant\gamma_{1}\leqslant\gamma_{2}\leqslant 1$, on $[0,\hat{\delta}]\times M$,

[TABLE]

Then the maximum principle implies that $\varphi_{\mu_{\gamma_{1}},\varepsilon}(t)\leqslant\varphi_{\mu_{\gamma_{2}},\varepsilon}(t)$ for any $\varepsilon\in(0,\frac{1}{2}]$, $t\in(0,\hat{\delta})$ and $z\in M$. Letting $\varepsilon\rightarrow 0$, we get the $\varepsilon=0$ case. ∎

Proposition 3.9**.**

Let $\{\varepsilon_{i}\}\in[0,\frac{1}{2}]$ and $\{\gamma_{i}\}\in[\beta,1]$ (that is, $\mu_{\gamma_{i}}\in[0,1]$). We assume that $\varepsilon_{i}$ and $\gamma_{i}$ converge to $\varepsilon_{\infty}$ and $\gamma_{\infty}$ respectively. For any $[\delta,T]$ $(0<\delta<T<\infty)$, there exists $\alpha\in(0,1)$ such that $\varphi_{\gamma_{i},\varepsilon_{i}}(t)$ converge to $\varphi_{\gamma_{\infty},\varepsilon_{\infty}}(t)$ in $C^{\alpha^{\prime}}$-sense on $[\delta,T]\times M$ for any $\alpha^{\prime}\in(0,\alpha)$, and on $(0,\infty)\times(M\setminus D)$ the convergence is in $C^{\infty}_{loc}$-sense.

Beweis.

On $[\delta,T]\times M$ with $0<\delta<T<\infty$, by Lemma 3.2 and Lemma 3.3, $\varphi_{\gamma_{i},\varepsilon_{i}}(t)$ and $\dot{\varphi}_{\gamma_{i},\varepsilon_{i}}(t)$ are uniformly bounded. By Kołodziej’s $L^{p}$-estimates [28, 29], there exists $\alpha\in(0,1)$ such that $\|\varphi_{\gamma_{i},\varepsilon_{i}}(t)\|_{C^{\alpha}(M,\omega_{0})}$ are uniformly bounded. Then there exists a subsequence $\varphi_{\gamma_{i_{k}},\varepsilon_{i_{k}}}(t)$ converge to a function $\varphi_{\infty}(t)$ in $C^{\alpha^{\prime}}$-sense on $[\delta,T]\times M$ with $\alpha^{\prime}\in(0,\alpha)$, and in $C^{\infty}_{loc}$-sense on $(0,\infty)\times(M\setminus D)$ from Lemma 3.5. Furthermore, on $(0,\infty)\times M$ (or $(0,\infty)\times(M\setminus D)$ if $\varepsilon_{\infty}=0$), $\varphi_{\infty}(t)$ satisfies equation

[TABLE]

We claim that $\varphi_{\infty}(t)=\varphi_{\gamma_{\infty},\varepsilon_{\infty}}(t)$. Fix $\gamma$, $t$ and $z\in M$, by Proposition $3.3$ in [37], $\varphi_{\gamma,\varepsilon}(t)$ is decreasing as $\varepsilon\searrow 0$. Combining this with Proposition 3.8 and $(\ref{201906002})$, we have

[TABLE]

on $[0,\hat{\delta})\times M$ for any $\gamma_{i_{k}}$ and $\varepsilon_{i_{k}}$. Let $k\rightarrow\infty$, we have

[TABLE]

Since $\varphi_{1,\frac{1}{2}}(t)$ converge to $\varphi_{0}$ in $L^{\infty}$-sense as $t\rightarrow 0$ and $h_{1}(0)=0$, $\varphi_{\infty}(t)$ converge to $\varphi_{0}$ in $L^{\infty}$-sense as $t\rightarrow 0$. By the uniqueness results (see Proposition $2.7$ or Theorem $3.7$ in [37]), we prove the claim. Next, we prove that $\varphi_{\gamma_{i},\varepsilon_{i}}(t)$ converge to $\varphi_{\gamma_{\infty},\varepsilon_{\infty}}(t)$ in $C^{\alpha^{\prime}}$-sense on $[\delta,T]\times M$ for any $\alpha^{\prime}\in(0,\alpha)$. If this is not true, then there exists $\alpha_{0}\in(0,\alpha)$, $\epsilon_{0}>0$ and a sequence $\varphi_{\gamma_{j_{k}},\varepsilon_{j_{k}}}(t)$ such that

[TABLE]

Since $\|\varphi_{\gamma_{j_{k}},\varepsilon_{j_{k}}}(t)\|_{C^{\alpha^{\prime}}([\delta,T]\times M)}$ are uniformly bounded for some $\alpha^{\prime}\in(\alpha_{0},\alpha)$, there exists a subsequence which we also denote it by $\varphi_{\gamma_{j_{k}},\varepsilon_{j_{k}}}(t)$ such that $\varphi_{\gamma_{j_{k}},\varepsilon_{j_{k}}}(t)$ converge in $C^{\alpha_{0}}$-sense to a function $\tilde{\varphi}_{\infty}(t)$ as $k\rightarrow\infty$ and

[TABLE]

By the similar arguments as above, $\tilde{\varphi}_{\infty}(t)$ is also a solution of the equation $(\ref{TKRF7})$ with $\mu_{\gamma_{\infty}}$ and $\varepsilon_{\infty}$. However, $\tilde{\varphi}_{\infty}(t)\not\equiv\varphi_{\gamma_{\infty},\varepsilon_{\infty}}(t)$ by $(\ref{3.22.6116})$, which is impossible by the uniqueness theorem. Hence we prove the $C^{\alpha}$-convergence. From the uniform $C^{\infty}_{loc}$-estimates for $\varphi_{\gamma_{i},\varepsilon_{i}}(t)$, we can also prove that $\varphi_{\gamma_{i},\varepsilon_{i}}(t)$ converge to $\varphi_{\gamma_{\infty},\varepsilon_{\infty}}(t)$ in $C^{\infty}_{loc}$-topology on $(0,\infty)\times(M\setminus D)$ by the similar arguments. ∎

Remark 3.10**.**

As a consequence of Proposition 3.9, on $[\delta,T]\times M$, the $C^{\alpha}$-norm of $\varphi_{\gamma,\varepsilon}(t)$ on $[\delta,T]\times M$ (for some $\alpha\in(0,1)$) is continuous with respect to $\gamma\in[\beta,1]$ and $\varepsilon\in[0,\frac{1}{2}]$. Furthermore, If we fix $\varepsilon\in(0,\frac{1}{2}]$, then the $C^{\infty}$-norm of $\varphi_{\gamma,\varepsilon}(t)$ on $[\delta,T]\times M$ is continuous with respect to $\gamma\in[\beta,1]$.

Straightforward calculation shows that

[TABLE]

where $\mathcal{J}$ is the complex structure on $M$.

Lemma 3.11**.**

If $\|u_{\gamma,\varepsilon}(t)\|_{C^{0}(M)}\leqslant A$ on $[T_{1},T_{2}]$ with ($1<T_{1}<T_{1}+1<T_{2}\leqslant\infty$). Then there exists uniform constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$, $\omega_{0}$ and $A$ such that on $M\times[T_{1}+1,T_{2}]$,

[TABLE]

Beweis.

We first prove that there exists uniform constant $C$ such that

[TABLE]

Calculation shows that $u_{\gamma,\varepsilon}(t)$ can be written as

[TABLE]

where $C_{\gamma,\varepsilon,1}$ is the normalization constant such that $\frac{1}{V}\int_{M}e^{-u_{\gamma,\varepsilon}(1)}dV_{\gamma,\varepsilon}(1)=1$. By Lemma 3.2 and Lemma 3.3, $C_{\gamma,\varepsilon,1}$ and $u_{\gamma,\varepsilon}(1)$ are uniformly bounded by a constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$ and $\omega_{0}$, and hence $A_{\mu_{\gamma},\varepsilon}(1)$ are uniformly bounded. Since $A_{\mu_{\gamma},\varepsilon}(t)$ is increasing along $(TKRF_{\mu_{\gamma},\varepsilon})$ when $\mu_{\gamma}>0$, $A_{\mu_{\gamma},\varepsilon}(t)$ is uniformly bounded from below for any $\varepsilon>0$, $\gamma\in(\beta,1]$ and $t\geqslant 1$. On the other hand, by Jensen’s inequality, $A_{\mu_{\gamma},\varepsilon}(t)\leqslant 0$.

Let $H_{\gamma,\varepsilon}(t)=(t-T_{1})|\nabla u_{\gamma,\varepsilon}(t)|^{2}_{\omega_{\gamma,\varepsilon}(t)}+\frac{3}{2}u^{2}_{\gamma,\varepsilon}(t)$ and $(t_{0},x_{0})$ be the maximum point of $H_{\gamma,\varepsilon}(t)$ on $[T_{1},T_{1}+1]\times M$. Combining $(\ref{3.22.16})$ with

[TABLE]

we obtain

[TABLE]

where constant $C$ depends only on $A$, $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$ and $\omega_{0}$.

$Case\ 1$, $t_{0}=T_{1}$. Then $(t-T_{1})|\nabla u_{\gamma,\varepsilon}(t)|^{2}_{\omega_{\gamma,\varepsilon}(t)}\leqslant\frac{3}{2}u^{2}_{\gamma,\varepsilon}(T_{1})\leqslant\frac{3}{2}A^{2}$.

$Case\ 2$, $t_{0}>T_{1}$. By the maximum principle, we have $|\nabla u_{\gamma,\varepsilon}(t_{0},x_{0})|^{2}_{\omega_{\gamma,\varepsilon}(t_{0})}\leqslant C$. Hence $(t-T_{1})|\nabla u_{\gamma,\varepsilon}(t)|^{2}_{\omega_{\gamma,\varepsilon}(t)}\leqslant C$.

From the above two cases, $(t-T_{1})|\nabla u_{\gamma,\varepsilon}(t)|^{2}_{\omega_{\gamma,\varepsilon}(t)}\leqslant C$ on $[T_{1},T_{1}+1]\times M$. Obviously $|\nabla u_{\gamma,\varepsilon}(T_{1}+1)|^{2}_{\omega_{\gamma,\varepsilon}(T_{1}+1)}$ are uniformly bounded by a constant $C$ which depends only on $A$, $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$ and $\omega_{0}$.

Since $\Delta_{\omega_{\gamma,\varepsilon}(t)}u_{\gamma,\varepsilon}(t)=-R(\omega_{\gamma,\varepsilon}(t))+\mu_{\gamma}n+(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}$, we only need to prove the uniform upper bound of $-\Delta_{\omega_{\gamma,\varepsilon}(t)}u_{\gamma,\varepsilon}(t)$.

We take $G_{\gamma,\varepsilon}(t)=(t-T_{1})^{2}(-\Delta_{\omega_{\gamma,\varepsilon}(t)}u_{\gamma,\varepsilon}(t))+2(t-T_{1})^{2}|\nabla u_{\gamma,\varepsilon}(t)|^{2}_{\omega_{\gamma,\varepsilon}(t)}$. According to (LABEL:3.22.16) and

[TABLE]

the evolution equation of $G_{\gamma,\varepsilon}(t)$ can be controlled as

[TABLE]

where constant $C_{0}$ depends only on $A$, $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$ and $\omega_{0}$. Assuming that $(t_{0},x_{0})$ is the maximum point of $G_{\gamma,\varepsilon}(t)$ on $[T_{1},T_{1}+1]\times M$.

$Case\ 1$, $t_{0}=T_{1}$, then $(t-T_{1})^{2}(-\Delta_{\omega_{\gamma,\varepsilon}(t)}u_{\gamma,\varepsilon}(t))\leqslant 0$.

$Case\ 2$, $t_{0}>T_{1}$. We assume $-\Delta_{\omega_{\gamma,\varepsilon}(t)}u_{\gamma,\varepsilon}(t)>0$ at $(t_{0},x_{0})$ without loss of generality. We claim that $(t_{0}-T_{1})^{2}(-\Delta_{\omega_{\gamma,\varepsilon}(t_{0})}u_{\gamma,\varepsilon}(t_{0},x_{0}))\leqslant n(3+C_{0})$. If not, by the maximum principle, we have

[TABLE]

We get a contradiction. From these two cases, we conclude that $-(t-T_{1})^{2}\Delta_{\omega_{\gamma,\varepsilon}(t)}u_{\gamma,\varepsilon}(t)\leqslant C$ on $[T_{1},T_{1}+1]\times M$ for some constant $C$ depending only on $A$, $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$ and $\omega_{0}$. Furthermore, $R(\omega_{\gamma,\varepsilon}(T_{1}+1))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(T_{1}+1)}\theta_{\varepsilon}=-\Delta_{\omega_{\gamma,\varepsilon}(T_{1}+1)}u_{\gamma,\varepsilon}(T_{1}+1)+\mu_{\gamma}n\leqslant C$.

Since $\|u_{\gamma,\varepsilon}(t)\|_{C^{0}(M)}\leqslant A$ on $[T_{1},T_{2}]$, there exists a uniform constant $B>1$ such that $u_{\gamma,\varepsilon}(t)>-B$ on $[T_{1},T_{2}]$. Define $H_{\gamma,\varepsilon}(t)=\frac{|\nabla u_{\gamma,\varepsilon}(t)|_{\omega_{\gamma,\varepsilon}(t)}^{2}}{u_{\gamma,\varepsilon}(t)+2B}$. By arguments in [36],

[TABLE]

Taking $\delta<1$, since $\theta_{\varepsilon}(grad\ u_{\gamma,\varepsilon}(t),\mathcal{J}(grad\ u_{\gamma,\varepsilon}(t)))\geqslant 0$, we have

[TABLE]

From $(\ref{1.5.2})$, we have

[TABLE]

where $C_{2}$ depends only on $A$, $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$ and $\omega_{0}$. The maximum principle implies that $H_{\gamma,\varepsilon}(t)\leqslant\max(C_{2},\ 2(2B+C_{1})\delta^{-1})$ for $t\in[T_{1}+1,T_{2}]$. We get the first inequality in $(\ref{3.22.18})$.

Now we prove the second inequality. Let $G_{\gamma,\varepsilon}(t)=\frac{-\Delta_{\omega_{\gamma,\varepsilon}(t)}u_{\gamma,\varepsilon}(t)}{u_{\gamma},{\varepsilon}(t)+2B}+2H_{\gamma,\varepsilon}(t)$.

[TABLE]

Since $\theta_{\varepsilon}$ is semi-positive,

[TABLE]

By using inequality

[TABLE]

we have

[TABLE]

Since $\frac{-\Delta_{\omega_{\gamma,\varepsilon}(T_{1}+1)}u_{\gamma,\varepsilon}(T_{1}+1)}{u_{\gamma,\varepsilon}(T_{1}+1)+2B}$ are bounded uniformly, by the maximum principle, there exists a uniform constant $C$ depending only on $A$, $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$ and $\omega_{0}$ such that $G_{\gamma,\varepsilon}(t)\leqslant C$ for any $t\in[T_{1}+1,T_{2}]$. Hence we get $-\Delta_{\omega_{\gamma,\varepsilon}(t)}u_{\gamma,\varepsilon}(t)\leqslant C(u_{\gamma,\varepsilon}(t)+2B)$ and hence $R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}\leqslant C(u_{\gamma,\varepsilon}(t)+C)$ on $[T_{1}+1,T_{2}]\times M$. ∎

Now, we recall some uniform Sobolev inequalities along the twisted Kähler-Ricci flows $(TKRF_{\mu_{\gamma},\varepsilon})$. From the proof of Theorem $6.1$ in [36] (see also [35, 60, 61]), when $t\geqslant 1$, the Sobolev constants along $(TKRF_{\mu_{\gamma},\varepsilon})$ depend only on $n$, $\omega_{0}$, $\max\limits_{M}(R(\omega_{\gamma,\varepsilon}(1))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(1)}\theta_{\varepsilon})^{-}$ and $\mathcal{C}_{S}(M,\omega_{\gamma,\varepsilon}(1))$, where the latter two are uniformly bounded from Theorem 2.8, Lemma 3.1 and Lemma 3.4 . Hence we have the following Sobolev inequality.

Theorem 3.12**.**

Let $M$ be Fano manifold of complex dimension $n\geq 2$ and $\omega_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$. There exist uniform constants A and B depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$ and $\omega_{0}$ such that

[TABLE]

for any $v\in W^{1,2}(M,\omega_{\gamma,\varepsilon}(t))$, $\gamma\in(\beta,1]$, $\varepsilon>0$ and $t\geq 1$.

By using Hus’s work (see Theorem $1$ in [24]) and the arguments in the proof of Theorem $6.1$ in [36] (see also [35, 60, 61]), we have the following uniform Sobolev inequality for $n=1$.

Theorem 3.13**.**

Let $M$ be a Fano manifold of complex dimension $1$ and $\omega_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$. For $n_{0}>1$, there exist uniform constants A and B depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n_{0}$ and $\omega_{0}$ such that

[TABLE]

for any $v\in W^{1,2}(M,\omega_{\gamma,\varepsilon}(t))$, $\gamma\in(\beta,1]$, $\varepsilon>0$ and $t\geq 1$.

Combining these uniform Sobolev inequalitities with Lemma 3.1, we have the following Theorem by following Jiang’s work (Theorem $1.12$ in [27]).

Theorem 3.14**.**

Let $M$ be a Fano manifold of complex dimension $n$ and $\omega_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$. Let $f$ be a non-negative Lipschitz continuous function on $[0,\infty)\times M$ satisfying

[TABLE]

on $[0,\infty)\times M$ in the weak sense, where $a\geqslant 0$. For $p>0$, there exists a constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $a$, $p$, $\omega_{0}$ and $n_{0}$ ($n_{0}=n$ if $n\geqslant 2$, $n_{0}>1$ if $n=1$) such that

[TABLE]

holds for any $T<t<T+1$ with $T\geqslant 1$, $\gamma\in(\beta,1]$ and $\varepsilon>0$.

Let $\omega_{\varphi_{\beta}}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi_{\beta}$ be the Ricci flat conical Kähler-Einstein metric obtained in Theorem 3.7, then $\varphi_{\beta}\in C^{0}(M)\cap C^{\infty}(M\setminus D)$ satisfies

[TABLE]

where $\hat{C}_{\beta}$ is the normalization constant such that $\frac{1}{V}\int_{M}e^{-F_{0}+\hat{C}_{\beta}}\frac{dV_{0}}{|s|_{h}^{2(1-\beta)}}=1$. By using the normalization of $\frac{1}{V}\int_{M}e^{-F_{0}}dV_{0}=1$, we conclude that $\hat{C}_{\beta}=-\log\frac{1}{V}\int_{M}e^{-F_{0}}\frac{dV_{0}}{|s|_{h}^{2(1-\beta)}}\leqslant 0$. By Kołodziej’s $L^{p}$-estimates [28, 29], we have

[TABLE]

At the same time, by Guenancia-P$\breve{a}$un’s results [4, 21] (see also Liu-Zhang [34]), there exists constant $M_{\beta}$ such that

[TABLE]

We denote

[TABLE]

where $D_{\beta}$ is the constant in $(\ref{1904000})$. Next, we prove Lemma 1.4.

Proof of Lemma 1.4.    If this lemma is not true. For $\delta_{1}<\min(\frac{1}{2},1-\beta)$, there exist $\varepsilon_{1}\in(0,\delta_{1})$, $\gamma^{\prime}_{1}\in(\beta,\beta+\delta_{1})$ and $t^{\prime}_{1}\in[\frac{1}{\delta_{1}},\infty)$, such that

[TABLE]

Assume $t^{\prime}_{1}\in[\frac{1}{\delta_{1}},T_{1}]$, then we have

[TABLE]

Theorem 3.6 and Remark 3.10 imply that there exists $\gamma_{1}\in(\beta,\gamma^{\prime}_{1})$, such that

[TABLE]

Hence there exists $t_{1}\in[\frac{1}{\delta_{1}},T_{1}]$ such that

[TABLE]

and then

[TABLE]

For $\delta_{2}=\min(\frac{1}{2},\frac{1}{T_{1}+1},\varepsilon_{1},\gamma_{1}-\beta)$, there exist $\varepsilon_{2}\in(0,\delta_{2})$, $\gamma^{\prime}_{2}\in(\beta,\beta+\delta_{2})$ and $t^{\prime}_{2}\in[\frac{1}{\delta_{2}},\infty)$, such that

[TABLE]

Assume $t^{\prime}_{2}\in[T_{1}+1,T_{2}]$, we have

[TABLE]

Theorem 3.6 and Remark 3.10 imply that there exists $\gamma_{2}\in(\beta,\gamma^{\prime}_{2})$, such that

[TABLE]

Hence there exists $t_{2}\in[T_{1}+1,T_{2}]$ such that

[TABLE]

and then

[TABLE]

After repeating above process, we get a subsequence $\varphi_{\gamma_{i},\varepsilon_{i}}(t_{i})$ with $\varepsilon_{i}\searrow 0$, $\gamma_{i}\searrow\beta$, $t_{i}\nearrow\infty$ and $t_{i}\in[T_{i-1}+1,T_{i}]$ satisfying

[TABLE]

and

[TABLE]

We claim that there exists a uniform constant $A$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $\omega_{0}$ and $n$, such that

[TABLE]

Combining the arguments below $(\ref{201903120015})$ with Lemma 3.1, we have

[TABLE]

Then for the above sequence, we have

[TABLE]

Integrating form $t$ to $T_{i-1}+1$ on both sides, where $t\in[T_{i-1},T_{i-1}+1]$, we obtain

[TABLE]

Hence there exists a uniform constant $C$ such that

[TABLE]

for any $t\in[T_{i-1},T_{i-1}+1]$, $\varepsilon_{i}$ and $\gamma_{i}$. By using Lemma 3.1 again, we have

[TABLE]

for any $\gamma\in[\beta,1]$, $\varepsilon>0$ and $t\geqslant 1$. Since

[TABLE]

the twisted Mabuchi energy $\mathcal{M}_{\mu_{\gamma},\varepsilon}$ is decreasing along the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$. By Lemma 3.2 and Lemma 3.3, we have

[TABLE]

for any $\gamma\in[\beta,1]$, $\varepsilon>0$ and $t\geqslant 1$. On the other hand, we write equation $(\ref{TKRF7})$ as

[TABLE]

Integrating above equation on both sides, there exists uniform constant such that for any $t\geqslant 1$, $\gamma\in[\beta,1]$ and $\varepsilon>0$,

[TABLE]

By using $(\ref{1915})$, we have

[TABLE]

for any $t\in[T_{i-1}+1,T_{i}]$, $\gamma_{i}$ and $\varepsilon_{i}$. Hence $\|\dot{\varphi}_{\gamma_{i},\varepsilon_{i}}(t)-\mu_{i}\varphi_{\gamma_{i},\varepsilon_{i}}(t)\|_{C^{0}(M)}$ are uniformly bounded on $[T_{i-1}+1,T_{i}]$. Combining this with $(\ref{1915})$, we have

[TABLE]

Integrating $(\ref{1916001})$ from $T_{i-1}$ to $T_{i-1}+1$ on both sides, we have

[TABLE]

for any $\gamma_{i}$ and $\varepsilon_{i}$. Straightforward calculation shows that

[TABLE]

Applying Theorem 3.14 to $\varphi_{\gamma_{i},\varepsilon_{i}}(t)$, we have

[TABLE]

for any $t\in(T_{i-1},T_{i-1}+1]$, $\gamma_{i}$ and $\varepsilon_{i}$ by using $(\ref{201903060000001})$ and $(\ref{201903060000002})$. Hence there exists uniform constant $C$ such that for any $\gamma_{i}$, $\varepsilon_{i}$ and $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$,

[TABLE]

By Jensen’s inequality and normalization $\frac{1}{V}\int_{M}e^{-u_{\gamma,\varepsilon}(t)}dV_{\gamma,\varepsilon}(t)=1$, we have

[TABLE]

When $\mu_{\gamma}\geqslant 0$, combining this with $(\ref{2019030701})$, we control the evolution of $u_{\gamma_{i},\varepsilon_{i}}(t)$ as

[TABLE]

Then we have

[TABLE]

Integrating form $t$ to $T_{i-1}+1$ on both sides, where $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$, we obtain

[TABLE]

Hence there exists a uniform constant $C$ such that

[TABLE]

for any $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$, $\varepsilon_{i}$ and $\gamma_{i}$. Since $u_{\gamma,\varepsilon}(t)=\dot{\varphi}_{\gamma,\varepsilon}(t)+\hat{c}_{\gamma,\varepsilon}(t)$, where $\hat{c}_{\gamma,\varepsilon}(t)$ is only a function of time $t$, from $(\ref{191911})$ and $(\ref{190312005})$, we have

[TABLE]

for any $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$, $\varepsilon_{i}$ and $\gamma_{i}$. By using $(\ref{2019030701})$ again, we have

[TABLE]

Hence

[TABLE]

Integrating form $t$ to $T_{i-1}+1$ on both sides, where $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$, we obtain

[TABLE]

and then

[TABLE]

By using Lemma 3.1, we have

[TABLE]

Hence

[TABLE]

Integrating form $t$ to $T_{i-1}+1$ on both sides, where $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$, we obtain

[TABLE]

Hence

[TABLE]

Integrating form $t$ to $T_{i-1}+1$ on both sides of $(\ref{1903120011})$ and $(\ref{1903120012})$, where $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$,

[TABLE]

From above arguments, for any $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$, $\varepsilon_{i}$ and $\gamma_{i}$, we get

[TABLE]

Let $\psi_{\gamma_{i},\varepsilon_{i}}(t)=\varphi_{\gamma_{i},\varepsilon_{i}}(t)+\tilde{C}_{\gamma_{i},\varepsilon_{i}}e^{\mu_{\gamma_{i}}t}$ and denote

[TABLE]

where F_{\gamma_{i},\varepsilon_{i}}(1)=F_{0}+\log\Big{(}\frac{\omega_{\gamma_{i},\varepsilon_{i}}^{n}(1)(\varepsilon^{2}+|s|_{h}^{2})^{1-\gamma_{i}}}{\omega_{0}^{n}}\Big{)}. We know that $\tilde{C}_{\gamma_{i},\varepsilon_{i}}$ is well-defined from the uniform Perelman’s estimates (here the estimates depend on $\mu_{\gamma_{i}}$) and that $\psi_{\gamma_{i},\varepsilon_{i}}(t)$ is a solution of equation $(\ref{TKRF7})$ with initial data $\varphi_{0}+\tilde{C}_{\gamma_{i},\varepsilon_{i}}$. From $(\ref{201903120015})$, $u_{\gamma_{i},\varepsilon_{i}}(t)$ can be written as

[TABLE]

where $C_{\gamma_{i},\varepsilon_{i},1}$ is the normalization constant such that $\frac{1}{V}\int_{M}e^{-u_{\gamma_{i},\varepsilon_{i}}(1)}dV_{\gamma_{i},\varepsilon_{i}}(1)=1$. By Lemma 3.2 and Lemma 3.3, $C_{\gamma_{i},\varepsilon_{i},1}$ and $u_{\gamma_{i},\varepsilon_{i}}(1)$ are uniformly bounded independent of $\varepsilon_{i}$ and $\gamma_{i}$. Let $u_{\gamma_{i},\varepsilon_{i}}(t)=\dot{\psi}_{\gamma_{i},\varepsilon_{i}}(t)+c_{\gamma_{i},\varepsilon_{i}}(t)$. Then

[TABLE]

Next, we prove that there exists a uniform constant $C$ such that for any $\varepsilon_{i}$ and $\gamma_{i}$,

[TABLE]

We let

[TABLE]

Through computing, we have

[TABLE]

Integrating this equality form $1$ to $t$ on both sides, we have

[TABLE]

Putting $(\ref{201903120016})$ and $(\ref{333})$ into above equality, we have

[TABLE]

Therefore, by using $(\ref{2019030701})$, there exists uniform constant $C$ such that for any $\gamma_{i}$, $\varepsilon_{i}$ and $t\in[T_{i-1}+\frac{1}{4},T_{i-1}+1]$,

[TABLE]

Then we conclude that $\|\dot{\psi}_{\gamma_{i},\varepsilon_{i}}(t)\|_{C^{0}(M)}$ is uniformly bounded on $[T_{i-1}+\frac{1}{4},T_{i-1}+1]$ by using $(\ref{190312006})$. Since $osc_{M}\psi_{\gamma_{i},\varepsilon_{i}}(t)=osc_{M}\varphi_{\gamma_{i},\varepsilon_{i}}(t)$, $osc_{M}\psi_{\gamma_{i},\varepsilon_{i}}(t)$ are uniform bounded on $[T_{i-1}+\frac{1}{4},T_{i-1}+1]$. We have proved that there exists constant $C$ depending only on $L_{\beta}$, $n$, $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$ and $\omega_{0}$ such that

[TABLE]

From above estimates, we also have

[TABLE]

which implies that $\|\mu_{\gamma_{i}}\psi_{\gamma_{i},\varepsilon_{i}}(t)\|_{C^{0}(M)}$ are uniformly bounded on $[T_{i-1}+\frac{1}{4},T_{i-1}+1]$. From the arguments in Lemma $2.3$ of [37], there exist uniform constant $B$ and $C$ such that, on $(T_{i-1}+\frac{1}{4},T_{i-1}+1]\times M$, we have

[TABLE]

where $(t_{1},z_{1})$ is the minimum point of $\psi_{\gamma_{i},\varepsilon_{i}}(t)$ on $[T_{i-1}+\frac{1}{4},T_{i-1}+1]\times M$, and we use that $\chi_{\gamma}$ with $\gamma\in[\beta,1)$ can be uniformly bounded by a constant $C$ depending only on $\beta$ in the second inequality. So on $t\in[T_{i-1}+\frac{1}{2},T_{i-1}+1]\times M$, we have

[TABLE]

Hence we prove the claim $(\ref{1916})$.

From $(\ref{1916})$, for $t\in[T_{i-1}+\frac{1}{2},T_{i}]$, $\gamma_{i}$ and $\varepsilon_{i}$, we have

[TABLE]

Combining this with $(\ref{192202})$, we conclude that the $C^{0}$-norms of $\dot{\varphi}_{\gamma_{i},\varepsilon_{i}}(t)-\mu_{\gamma_{i}}\varphi_{\gamma_{i},\varepsilon_{i}}(t)$ and then $\dot{\phi}_{\gamma_{i},\varepsilon_{i}}(t)-\mu_{\gamma_{i}}\phi_{\gamma_{i},\varepsilon_{i}}(t)$ are uniformly bounded on $[T_{i-1}+\frac{1}{2},T_{i}]$. So we obtain

[TABLE]

where we use that $F_{\gamma,\varepsilon}$ with $\gamma\in[\beta,1)$ can be uniformly bounded by a constant $C$ depending only on $\beta$, $n$ and $\omega_{0}$. Hence for any $t\in[T_{i-1}+\frac{1}{2},T_{i}]$, $\gamma_{i}$ and $\varepsilon_{i}$, we have

[TABLE]

By Evans-Krylov-Safonov’s estimates, for any $B_{r}(p)\subset\subset M\setminus D$, there is uniform constant $C_{k,p,r}$ such that

[TABLE]

for any $t\in[T_{i-1}+\frac{3}{4},T_{i}]$, $\gamma_{i}$ and $\varepsilon_{i}$. In particular, for any $K\subset\subset M\setminus D$, we have

[TABLE]

for any $i$ and $k\in\mathbb{N}^{+}$. Then $\varphi_{\gamma_{i},\varepsilon_{i}}(t_{i})$ (by taking a subsequence if necessary) converge to a function $\varphi_{\beta}(\infty)\in C^{\infty}(M\setminus D)$ in $C^{\infty}_{loc}$-topology in $M\setminus D$. By estimates $(\ref{192400010101})$, the Lebesgue Dominated Convergence theorem implies that $\int_{M}(\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi_{\beta}(\infty))^{n}=\int_{M}\omega_{0}^{n}$.

Let $u_{\beta}(\infty)$ be the twisted Ricci potential of $\omega_{\varphi_{\beta}(\infty)}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi_{\beta}(\infty)$. For any $K\subset\subset M\setminus D$, since $\omega_{\gamma_{i},\varepsilon_{i}}(t_{i})$ are uniformly equivalent to $\omega_{0}$ on $K$ by $(\ref{192400010101})$ and $\mu_{\gamma_{i}}$ tend to [math], we conclude that $u_{\beta}(\infty)$ satisfies

[TABLE]

By the Lebesgue Dominated Convergence theorem again, we get that $\frac{1}{V}\int_{M}e^{-u_{\beta}(\infty)}dV_{\beta}(\infty)=1$. Since $A_{\mu_{\gamma},\varepsilon}(t)$ is increasing along $(TKRF_{\mu_{\gamma},\varepsilon})$ with $\mu_{\gamma}>0$, for any $j<i$,

[TABLE]

the second inequality due to Jensen’s inequality. Let $i\rightarrow\infty$, we have

[TABLE]

Since $\mu_{\beta}=0$, from Theorem 3.7, the conical Kähler-Ricci flow $(CKRF_{\mu_{\beta}})$ converge to a conical Kähler-Einstein metric. By using the normalization of the twisted Ricci potential $u_{\beta}(t)$, we conclude that $u_{\beta}(t)$ converge to [math] in $C^{\infty}_{loc}$-topology on $M\setminus D$, and hence $A_{\mu_{\beta}}(t)$ converge to [math]. This implies that

[TABLE]

after letting $j\rightarrow\infty$. Since $\log$ is strictly concave, we conclude $u_{\beta}(\infty)=0$ by using Jensen’s inequality and its normalization. This means that $\omega_{\varphi_{\beta}(\infty)}$ is a Ricci flat conical Kähler-Einstein metric with cone angle $2\pi\beta$ along $D$, hence $\varphi_{\beta}(\infty)$ satisfies equation $(\ref{201902230101})$. Then Dinew’s uniqueness theorem (Theorem $1.2$ in [16]) implies that $\varphi_{\beta}(\infty)=\varphi_{\beta}+C$. Letting $i\rightarrow\infty$ in $(\ref{2000019})$, we get

[TABLE]

This leads to a contradiction. Thus Lemma 1.4 is proved.

We denote constant

[TABLE]

where F_{\gamma,\varepsilon}(1)=F_{0}+\log\Big{(}\frac{\omega_{\gamma,\varepsilon}^{n}(1)(\varepsilon^{2}+|s|_{h}^{2})^{1-\gamma}}{\omega_{0}^{n}}\Big{)}. Since $\mu_{\gamma}>0$, $\tilde{C}_{\gamma,\varepsilon}$ are well-defined and uniformly bounded by the uniform Perelman’s estimates (independent of $\varepsilon$ but depend on $\mu_{\gamma}$). We denote $\psi_{\gamma,\varepsilon}(t)=\varphi_{\gamma,\varepsilon}(t)+\tilde{C}_{\gamma,\varepsilon}e^{\gamma t}$, which is a solution to the equation

[TABLE]

From the Proposition $4.5$ in [37], we know that $\|\dot{\psi}_{\gamma,\varepsilon}(t)\|_{C^{0}(M)}$ are uniform bounded for any $\varepsilon>0$ and $t\geqslant 1$. For any $\gamma\in(\beta,\beta+\delta(\lambda))$ obtained in Lemma 1.4, we can get the uniform $C^{0}$-estimates and Laplacian $C^{2}$-estimates of $\psi_{\gamma,\varepsilon}(t)$ for any $\varepsilon\in(0,\delta(\lambda))$ and $t\geqslant 1$. Hence the twisted Mabuchi energy $\mathcal{M}_{\mu_{\gamma},\varepsilon}$ are uniformly bounded from below along the twisted Kähler-Ricci flow $(TKRF_{\mu_{\gamma},\varepsilon})$, that is, there exists uniform constant $C$ such that for $\varepsilon\in(0,\delta(\lambda))$ and $t\geqslant 1$,

[TABLE]

Then by using the arguments in [36, 37], we deduce the convergence of the conical Kähler-Ricci flows $(CKRF_{\mu_{\gamma}})$ with $\gamma\in(\beta,\beta+\delta(\lambda))$.

Proof of Theorem 1.5.  Combining Lemma 1.4, Lemma 3.1 with Lemma 3.11, we obtain that $|R(\omega_{\gamma,\varepsilon}(t))-(1-\gamma)tr_{\omega_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}|$ and $\|u_{\gamma,\varepsilon}(t)\|_{C^{1}(\omega_{\gamma,\varepsilon}(t))}$ are uniform bounded for any $\varepsilon\in(0,\delta(\lambda))$, $\gamma\in(\beta,\beta+\delta(\lambda))$ and $t\in[\frac{1}{\delta(\lambda)},+\infty)$. From Lemma 1.4, we also know that there exists uniform constant $C$ such that

[TABLE]

hold for any $\varepsilon\in(0,\delta(\lambda))$, $\gamma\in(\beta,\beta+\delta(\lambda))$ and $t\in[\frac{1}{\delta(\lambda)},+\infty)$. Combining this with $(\ref{192202})$, we conclude that $\|\dot{\varphi}_{\gamma,\varepsilon}(t)-\mu_{\gamma}\varphi_{\gamma,\varepsilon}(t)\|_{C^{0}(M)}$ and then $\|\dot{\phi}_{\gamma,\varepsilon}(t)-\mu_{\gamma}\phi_{\gamma,\varepsilon}(t)\|_{C^{0}(M)}$ are uniform bounded. So

[TABLE]

Hence for any $\varepsilon\in(0,\delta(\lambda))$, $\gamma\in(\beta,\beta+\delta(\lambda))$ and $t\in[\frac{1}{\delta(\lambda)},+\infty)$, we have

[TABLE]

Let $(M,d_{\beta})$ be the metric completion of $(M,\omega_{\beta})$, which is of finite diameter. Therefore,

[TABLE]

are uniformly bounded from above (see also Proposition $2.4$ in [6]). So we prove Theorem 1.5 for $\gamma\in(\beta,\beta+\delta(\lambda))$. Fix $\gamma_{0}\in(\beta,\beta+\delta(\lambda))$, we can get Theorem 1.5 for any $\varepsilon\in(0,1)$, $\gamma\in(\gamma_{0},1)$ and $t\in[1,+\infty)$ by following the arguments in [36]. Combining these two parts, we complete the proof of Theorem 1.5.

4 Proof of the case $\mu_{\beta}>0$

In this section, we always assume that $\mu_{\beta}$ is positive and there is a conical Kähler-Einstein metric $\omega_{\varphi_{\beta}}$ $(0<\beta<1)$ with cone angle $2\pi\beta$ along $D$. We first deduce some uniform regularities along the twisted Kähler-Ricci flows $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$, and then prove Lemma 1.8 and Lemma 1.10. For the sake of brevity, we only consider the case $\lambda=1$, that is, $\mu_{\gamma}=\gamma$. Our arguments are also valid for any $\lambda>0$, only if the coefficient $\gamma$ before $\omega_{\gamma}(t)$ in the case of $\lambda=1$ is replaced by $\mu_{\gamma}=1-(1-\gamma)\lambda$.

Denote $\omega_{\varphi_{\beta}}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\varphi_{\beta}$ with $\varphi_{\beta}\in L^{\infty}(M)\bigcap PSH(M,\omega_{0})$. Assume that $\sup\limits_{M}\varphi_{\beta}\leqslant-1$ and satisfies

[TABLE]

where $\xi_{\beta}$ is the constant such that $\frac{1}{V}\int_{M}e^{-\beta\varphi_{\beta}-F_{0}+\xi_{\beta}}\frac{\omega_{0}^{n}}{|s|_{h}^{2(1-\beta)}}=1$. From Kołodziej’s $L^{p}$-estimates [28, 29], $\varphi_{\beta}$ is Hölder continuous. By Demailly’s regularization result [15], we approximate $\varphi_{\beta}$ with a decreasing sequence of smooth $\omega_{0}$-psh functions $\varphi_{\varepsilon}$. Then Dini’s theorem implies that $\varphi_{\varepsilon}$ converge to $\varphi_{\beta}$ in $L^{\infty}$-sense on $M$. Without loss of generality, we assume that

[TABLE]

for $\varepsilon\in(0,1]$. In the case $\mu_{\beta}>0$, we normalize $\varphi_{0}$ by

[TABLE]

where $\varphi_{1}$ comes from the approaching sequence $\varphi_{\varepsilon}$ with $\varepsilon=1$. Since the twisted Kähler-Ricci flow preserves the Kähler class, we write $(TKRF^{\beta}_{\gamma,\varepsilon})$ as the parabolic Monge-Ampère equation on potentials,

[TABLE]

We rewrite equation (120) as

[TABLE]

where $F^{\beta}_{\gamma,\varepsilon}=F_{0}+\log(\frac{(\omega^{\beta}_{\varepsilon})^{n}}{\omega_{0}^{n}}\cdot(\varepsilon^{2}+|s|_{h}^{2})^{1-\beta})+(\beta-\gamma)\varphi_{\varepsilon}$ and $\phi^{\beta}_{\gamma,\varepsilon}(t)=\varphi^{\beta}_{\gamma,\varepsilon}(t)-k\chi_{\beta}$. By the same arguments as in Lemma 3.1 and 3.2, we have

Proposition 4.1**.**

$R(\omega^{\beta}_{\gamma,\varepsilon}(t))-(\beta-\gamma)tr_{\omega^{\beta}_{\gamma,\varepsilon}(t)}\omega_{\varphi_{\varepsilon}}-(1-\beta)tr_{\omega^{\beta}_{\gamma,\varepsilon}(t)}\theta_{\varepsilon}$* are uniformly bounded from below by $-4n$ along the twisted Kähler-Ricci flows $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$, that is, for any $\gamma\in[0,1]$, $\beta\in[0,1]$, $\varepsilon>0$ and $t\geqslant 1$,*

[TABLE]

Lemma 4.2**.**

For any $T>0$, there exists uniform constant $C$, such that for any $t\in[0,T]$, $\varepsilon>0$ and $\gamma\in[0,1]$,

[TABLE]

where constant $C$ depends only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $n$, $\omega_{0}$ and $T$.

By using the arguments in section $2$ of [37], we get the following uniform Laplacian $C^{2}$-estimates on $M$ along the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\gamma,\varepsilon})$.

Lemma 4.3**.**

For any $T>0$, there exists constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$, $\omega_{0}$ and $T$, such that for any $t\in(0,T]$, $\varepsilon>0$ and $\gamma\in[0,1]$,

[TABLE]

Lemma 4.4**.**

For any $T>0$, there exists constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$, $\omega_{0}$ and $T$, such that for any $t\in(0,T]$, $\varepsilon>0$ and $\gamma\in[0,1]$,

[TABLE]

Beweis.

Since there exists term $(\beta-\gamma)\varphi_{\varepsilon}$ in equation $(\ref{TKRF4})$, from the proofs of Proposition $3.1$ in [36] and Lemma $2.3$ in [37], we need only deal with the term

[TABLE]

From [4], there exists a uniform constant $A$ depending on $\beta$ and $\omega_{0}$ such that

[TABLE]

When $\gamma>\beta$, by $(\ref{25})$ and $n\leqslant tr_{\omega^{\beta}_{\gamma,\varepsilon}(t)}\omega_{\varepsilon}^{\beta}\cdot tr_{\omega_{\varepsilon}^{\beta}}\omega^{\beta}_{\gamma,\varepsilon}(t)$, we have

[TABLE]

When $\gamma\leqslant\beta$, we have

[TABLE]

Hence we have

[TABLE]

Let

[TABLE]

be the uniformly bounded function introduced by Guenancia-P$\breve{a}$un in [21]. Denote functions $H_{1}=t\log tr_{\omega^{\beta}_{\varepsilon}}\omega^{\beta}_{\gamma,\varepsilon}(t)+t\Psi^{\rho}_{\varepsilon}$ when $\gamma>\beta$ and $H_{2}=t\log tr_{\omega^{\beta}_{\varepsilon}}\omega^{\beta}_{\gamma,\varepsilon}(t)+t\Psi^{\rho}_{\varepsilon}+t(\beta-\gamma)\varphi_{\varepsilon}$ when $\gamma\leqslant\beta$. By choosing suitable $B$ and $\rho$, and following the arguments in [37], we have

[TABLE]

where constant $C$ depends only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$, $\omega_{0}$ and $T$. Then by the arguments as that in the proof of Lemma $2.3$ in [37], we get the uniform Laplacian $C^{2}$-estimates for $\omega^{\beta}_{\gamma,\varepsilon}(t)$. ∎

Hence away from time [math], on any compact subset in $M\setminus D$, $\omega^{\beta}_{\gamma,\varepsilon}$ are uniformly equivalent to $\omega_{0}$. Then Evans-Krylov-Safonov’s estimates (see [31]) implies the following proposition.

Proposition 4.5**.**

For any $0<\delta<T<\infty$, $k\in\mathbb{N}^{+}$ and $B_{r}(p)\subset\subset M\setminus D$, there exists constant $C_{\beta,\delta,T,k,p,r}$, such that for any $\varepsilon>0$ and $\gamma\in[0,1]$,

[TABLE]

where constant $C_{\beta,\delta,T,k,p,r}$ depends on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$, $\delta$, $k$, $T$, $\omega_{0}$ and $dist_{\omega_{0}}(B_{r}(p),D)$.

Straightforward calculation shows that the twisted Ricci potential $u^{\beta}_{\gamma,\varepsilon}(t)$ with respect to $\omega^{\beta}_{\gamma,\varepsilon}(t)$ at $t=\frac{1}{2}$ can be written as

[TABLE]

where $C^{\beta}_{\gamma,\varepsilon}(\frac{1}{2})$ is a normalization constant such that $\frac{1}{V}\int_{M}e^{-u^{\beta}_{\gamma,\varepsilon}(\frac{1}{2})}dV^{\beta}_{\gamma,\varepsilon}(\frac{1}{2})=1$. By $(\ref{04})$, Lemma 4.2 and Lemma 4.3, $C^{\beta}_{\gamma,\varepsilon}(\frac{1}{2})$ and $u^{\beta}_{\gamma,\varepsilon}(\frac{1}{2})$ are uniformly bounded. Hence $A^{\beta}_{\gamma,\varepsilon}(\frac{1}{2})$ are uniformly bounded from below for any $\gamma\in(0,1)$ and $\varepsilon\in(0,1)$. Since $A^{\beta}_{\gamma,\varepsilon}(t)$ is increasing along $(TKRF^{\beta}_{\gamma,\varepsilon})$ when $\gamma\in(0,\beta]$, combining this with Jensen’s inequality, we have

Lemma 4.6**.**

There exists uniform constant $C$, such that

[TABLE]

for any $t\geqslant\frac{1}{2}$, $\gamma\in(0,\beta]$ and $\varepsilon\in(0,1]$.

We consider the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\gamma,\varepsilon})$ starting at $t=\frac{1}{2}$. Following the arguments in section $4$ of [36], we have the following uniform Perelman’s estimates.

Theorem 4.7**.**

Let $\omega^{\beta}_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler Ricci flow $(TKRF^{\beta}_{\gamma,\varepsilon})$. Then for any $0<\beta_{1}<\beta_{2}\leqslant\beta$, there exists a uniform constant $C$, such that

[TABLE]

hold for any $\gamma\in[\beta_{1},\beta_{2}]$, $t\geq 1$ and $\varepsilon>0$.

By using these uniform Perelman’s estimates, we can get the following estimates and convergence for $A^{\beta}_{\gamma}(t)$.

Theorem 4.8**.**

For any $\gamma\in(0,\beta]$, there exists uniform constant $C$, such that

[TABLE]

Furthermore, if $\mathcal{M}^{\beta}_{\gamma}$ is bounded form below, then $A^{\beta}_{\gamma}(t)$ converge to [math] as $t\rightarrow+\infty$.

We denote constant

[TABLE]

where F^{\beta}_{\gamma,\varepsilon}(1)=F_{0}+(\beta-\gamma)\varphi_{\varepsilon}+\log\Big{(}\frac{(\omega^{\beta}_{\gamma,\varepsilon}(1))^{n}(\varepsilon^{2}+|s|_{h}^{2})^{1-\beta}}{\omega_{0}^{n}}\Big{)}. From previous discussions, $\tilde{C}^{\beta}_{\gamma,\varepsilon}$ is well-defined and uniformly bounded. Next, we consider the solution $\psi^{\beta}_{\gamma,\varepsilon}(t)=\varphi^{\beta}_{\gamma,\varepsilon}(t)+\tilde{C}^{\beta}_{\gamma,\varepsilon}e^{\gamma t}$ to the equation

[TABLE]

By using the same arguments as in the proof of Proposition $4.5$ in [37], we have the following two propositions.

Proposition 4.9**.**

For any $\beta_{0}\in(0,\beta)$, there exists uniform constant $\hat{C}_{\beta_{0}}$ such that

[TABLE]

for any $\varepsilon>0$, $\gamma\in[\beta_{0},\beta]$ and $t\geqslant 1$.

Proposition 4.10**.**

For any $\beta_{0}\in(0,1)$, there exists uniform constant $C$ such that

[TABLE]

for any $\varepsilon>0$, $\gamma\in[\beta_{0},1)$ and $t\geqslant 1$, where $\psi_{\gamma,\varepsilon}(t)$ is the solution of equation $(\ref{TKRF6})$.

Next, we prove the continuity of $\varphi^{\beta}_{\gamma,\varepsilon}(t)$ with respect to variables $\gamma$ and $\varepsilon$.

Proposition 4.11**.**

There exists a constant $\hat{\delta}$, such that for any $t\in(0,\hat{\delta})$, $\varepsilon\in[0,1)$ and $z\in M$, $\varphi^{\beta}_{\gamma,\varepsilon}(t)$ is increasing with respect to $\gamma\in[0,1]$.

Beweis.

We first consider the case $\varepsilon\in(0,1)$. By Kołodziej’s results [28, 29], there exists a Hölder continuous solution $u^{\beta}_{\gamma,\varepsilon}$ to the equation

[TABLE]

and $u^{\beta}_{\gamma,\varepsilon}$ satisfies

[TABLE]

where the normalization constant $\hat{C}$ is uniformly bounded independent of $\varepsilon$ and $\gamma$, constant $C$ depends only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$ and $F_{0}$. We define function

[TABLE]

where

[TABLE]

From similar arguments in the proof of Proposition $3.5$ in [37], we know that $\psi^{\beta}_{\gamma,\varepsilon}(t)$ is a subsolution of equation $(\ref{TKRF2})$ and hence

[TABLE]

for any $t\in(0,1)$, where $h_{1}(t)$ is a continuous function independent of $\gamma$ and $\varepsilon$ such that $h(0)=0$. Hence there exists $\hat{\delta}$, such that $h_{1}(t)>-\frac{1}{2}$ for any $t\in[0,\hat{\delta}]$. By the normalization $(\ref{nom})$, we have

[TABLE]

for any $\gamma\in[0,1]$, $\varepsilon\in(0,1)$, $t\in(0,\hat{\delta})$ and $z\in M$. For $\gamma_{1}\leqslant\gamma_{2}$, on $[0,\hat{\delta}]\times M$,

[TABLE]

Then by the maximum principle, we have $\varphi^{\beta}_{\gamma_{1},\varepsilon}(t)\leqslant\varphi^{\beta}_{\gamma_{2},\varepsilon}(t)$ for any $\varepsilon\in(0,1)$, $t\in(0,\hat{\delta})$ and $z\in M$. Then letting $\varepsilon\rightarrow 0$, we get the $\varepsilon=0$ case. ∎

Since $\varphi_{\varepsilon}$ decrease to $\varphi_{\beta}$ as $\varepsilon\searrow 0$, then for fix $\gamma\in[0,\beta]$, $t\in[0,\infty)$ and $z\in M$, $\varphi^{\beta}_{\gamma,\varepsilon}(t)$ is decreasing as $\varepsilon\searrow 0$ (see Proposition $3.3$ in [37]). Then by using the same arguments as in section $3$, we have the following results.

Proposition 4.12**.**

Let $\{\varepsilon_{i}\}\in[0,1)$ and $\{\gamma_{i}\}\in[0,\beta]$. We assume that $\varepsilon_{i}$ and $\gamma_{i}$ converge to $\varepsilon_{\infty}$ and $\gamma_{\infty}$ respectively. Then for any $[\delta,T]$ with $0<\delta<T<\infty$, there exists $\alpha\in(0,1)$ such that $\varphi^{\beta}_{\gamma_{i},\varepsilon_{i}}(t)$ converge to $\varphi^{\beta}_{\gamma_{\infty},\varepsilon_{\infty}}(t)$ on $[\delta,T]\times M$ in $C^{\alpha^{\prime}}$-sense for any $\alpha^{\prime}\in(0,\alpha)$, and on $(0,\infty)\times(M\setminus D)$ the convergence is in $C^{\infty}_{loc}$-sense.

Remark 4.13**.**

From Proposition 4.12, we know that for any $[\delta,T]$ with $0<\delta<T<\infty$, the $C^{\alpha}$-norm of $\varphi^{\beta}_{\gamma,\varepsilon}(t)$ on $[\delta,T]\times M$ for some $\alpha\in(0,1)$ is continuous with respect to $\gamma\in[0,\beta]$ and $\varepsilon\in[0,1)$. If we fix $\varepsilon\in(0,1)$, then the $C^{\infty}$-norm of $\varphi^{\beta}_{\gamma,\varepsilon}(t)$ on $[\delta,T]\times M$ is continuous with respect to $\gamma\in[0,\beta]$.

By following the arguments as that in Theorem 3.6 and Theorem 3.7, we have the following two theorems.

Theorem 4.14**.**

There exists uniform constant $D^{\beta}_{0}$ such that

[TABLE]

for any $\varepsilon\in(0,\frac{1}{2})$ and $t\geqslant 1$.

Beweis.

From the proof of Theorem 3.6, we need only prove that $\frac{1}{V}\int_{M}\phi^{\beta}_{0,\varepsilon}(t)e^{-F_{0}+C_{\beta,\varepsilon}}\frac{dV_{0}}{(\varepsilon^{2}+|s|_{h}^{2})^{1-\beta}}$ is decreasing with respect to $t$. By Jensen’s inequality and $(\ref{04})$, we have

[TABLE]

Then we can get this theorem by following the arguments in Theorem 3.6. ∎

Theorem 4.15**.**

The conical Kähler-Ricci flow $(CKRF^{\beta}_{0})$ converges to the conical Kähler-Einstein metric $\omega_{\varphi_{\beta}}$ in $C_{loc}^{\infty}$-topology outside divisor $D$ and globally in $C^{\alpha,\beta}$-sense for any $\alpha\in(0,\min\{1,\frac{1}{\beta}-1\})$.

From the arguments in [35, 36, 60, 61], we have the following Sobolev inequalities along $(TKRF^{\beta}_{\gamma,\varepsilon})$ by using Theorem 2.8, Theorem 2.9 and Proposition 4.1.

Theorem 4.16**.**

Let $M$ be a Fano manifold with complex dimension $n\geq 2$ and $\omega^{\beta}_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\gamma,\varepsilon})$. There exist uniform constants A and B depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$ and $\omega_{0}$ such that

[TABLE]

for any $v\in W^{1,2}(M,\omega^{\beta}_{\gamma,\varepsilon}(t))$, $\gamma\in(0,1]$, $\varepsilon>0$ and $t\geq 1$.

Theorem 4.17**.**

Let $M$ be a Fano manifold with complex dimension $1$ and $\omega^{\beta}_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\mu_{\gamma},\varepsilon})$. For any $n_{0}>1$, there exist uniform constants A and B depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n_{0}$, $\beta$ and $\omega_{0}$ such that

[TABLE]

for any $v\in W^{1,2}(M,\omega^{\beta}_{\gamma,\varepsilon}(t))$, $\gamma\in(0,1]$, $\varepsilon>0$ and $t\geq 1$.

Combining these uniform Sobolev inequalities with Proposition 4.1, we have the following Theorem by following Jiang’s work (Theorem $1.12$ in [27]).

Theorem 4.18**.**

Let $M$ be a Fano manifold of complex dimension $n$ and $\omega^{\beta}_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler-Ricci flow $(TKRF^{\beta}_{\gamma,\varepsilon})$. Let $f$ be a non-negative Lipschitz continuous function on $[0,\infty)\times M$ satisfying

[TABLE]

on $[0,\infty)\times M$ in the weak sense, where $a\geqslant 0$. For any $p>0$, there exists a constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $\beta$, $a$, $p$, $\omega_{0}$ and $n_{0}$ ($n_{0}=n$ if $n\geqslant 2$, $n_{0}>1$ if $n=1$) such that

[TABLE]

holds for any $T<t<T+1$ with $T\geqslant 1$, $\gamma\in(0,1]$ and $\varepsilon>0$.

Denote $L^{\beta}_{0}=\max(\sup\limits_{M}tr_{\omega_{\beta}}\omega_{\varphi_{\beta}}+osc_{M}\varphi_{\beta},D^{\beta}_{0})$. By using the arguments as that in the proof of Lemma 1.4, we have

Lemma 4.19**.**

There exists a constant $\tilde{\delta}>0$ such that

[TABLE]

for any $\varepsilon\in(0,\tilde{\delta})$, $\gamma\in(0,\tilde{\delta})$ and $t\in[\frac{1}{\tilde{\delta}},+\infty)$.

Following the arguments in the proof of Theorem 1.5, we improve the uniform Perelman’s estimates in Theorem 4.7 independent of $\gamma\in(0,\beta]$ along $(TKRF^{\beta}_{\gamma,\varepsilon})$.

Theorem 4.20**.**

Let $\omega^{\beta}_{\gamma,\varepsilon}(t)$ be a solution of the twisted Kähler Ricci flow $(TKRF^{\beta}_{\gamma,\varepsilon})$. There exists a uniform constant $C$, such that

[TABLE]

hold for any $\gamma\in(0,\beta]$, $t\geqslant 1$ and $\varepsilon\in(0,\tilde{\delta})$, where $\tilde{\delta}$ is the constant in Lemma 4.19.

From the definitions $(\ref{9090990})$ and $(\ref{9090991})$, for $\gamma\in(0,\beta]$, we have the following inequalities.

[TABLE]

where constant $C$ independent of $\gamma$.

Theorem 4.21**.**

Fix a $\beta_{0}\in(0,\tilde{\delta})$ obtained in Lemma 4.19, there exists a constant $C_{\beta_{0}}$ such that

[TABLE]

for any $\varepsilon\in(0,\tilde{\delta})$ and $t\in[0,\infty)$.

Proof of Lemma 1.8.  We denote $A=\max(\|\varphi_{\beta}\|_{C^{0}(M)}+\frac{\hat{C}_{\beta_{0}}+|\xi_{\beta}|}{\beta_{0}},C_{\beta_{0}})$. If the lemma is not true. For $\delta_{1}=\min(1,\tilde{\delta})$, there are $\varepsilon_{1}\in(0,\delta_{1})$, $\gamma^{\prime}_{1}\in(\beta_{0},\beta)$ and $t^{\prime}_{1}\in[\frac{1}{\delta_{1}},\infty)$, such that

[TABLE]

Assume that $t^{\prime}_{1}\in[1,T_{1}]$, we have

[TABLE]

Combining Remark 4.13 with $(\ref{52})$, there exists $\gamma_{1}\in(\beta_{0},\gamma^{\prime}_{1})$, such that

[TABLE]

Then there exists $t_{1}\in[1,T_{1}]$ such that

[TABLE]

For $\delta_{2}=\min(\frac{1}{2},\frac{1}{T_{1}+1},\varepsilon_{1})$, there exist $\varepsilon_{2}\in(0,\delta_{2})$, $\gamma^{\prime}_{2}\in(\beta_{0},\beta)$ and $t^{\prime}_{2}\in[\frac{1}{\delta_{2}},\infty)$, such that

[TABLE]

Assume $t^{\prime}_{2}\in[T_{1}+1,T_{2}]$, then we have

[TABLE]

Combining Remark 4.13 with $(\ref{52})$ again, there exists $\gamma_{2}\in(\beta_{0},\gamma^{\prime}_{2})$, such that

[TABLE]

and then there exists $t_{2}\in[T_{1}+1,T_{2}]$ such that

[TABLE]

After repeating above process, we get a subsequence $\psi^{\beta}_{\gamma_{i},\varepsilon_{i}}(t_{i})$ with $\varepsilon_{i}\searrow 0$, $\gamma_{i}\in(\beta_{0},\beta)$, $t_{i}\in[T_{i-1}+1,T_{i}]$ and $t_{i}\nearrow\infty$ satisfying

[TABLE]

For any $t\in[T_{i-1},T_{i-1}+1]$, we assume the maximum (or minimun) point of $\psi^{\beta}_{\gamma_{i},\varepsilon_{i}}(t)$ on $M$ is $z_{1}$ (or $z_{2}$), then by Proposition 4.9 and $(\ref{58})$,

[TABLE]

where constant $C$ independent of $\gamma_{i}$, $\varepsilon_{i}$ and $t\in[T_{i-1},T_{i-1}+1]$. On the other hand, we can write equation $(\ref{TKRF5})$ as

[TABLE]

We integrate above equality on both sides, there exists a uniform constant $C$ such that

[TABLE]

for any $\gamma\in[\beta_{0},\beta]$, $\varepsilon\in(0,1)$ and $t\geqslant 1$. Then by $(\ref{59})$, we have

[TABLE]

for any $t\in[T_{i-1},T_{i-1}+1]$, $\gamma_{i}$ and $\varepsilon_{i}$. Hence there exists uniform constant $C$ such that

[TABLE]

for any $\gamma_{i}$ and $\varepsilon_{i}$. By Lemma 4.4 and Proposition 4.9, there exists constant $C$ such that

[TABLE]

Then for any $\gamma_{i}$ and $\varepsilon_{i}$, at $t=T_{i-1}+\frac{1}{4}$,

[TABLE]

Then by Proposition $3.1$ in [36], $(\ref{62})$ and Proposition 4.9, on $[T_{i-1}+\frac{1}{4},T_{i}]\times M$, we have

[TABLE]

By Evans-Krylov-Safonov’s estimates, for any $B_{r}(p)\subset\subset M\setminus D$, there is uniform constant $C_{k,p,r}$ such that

[TABLE]

for any $t\in[T_{i-1}+\frac{1}{2},T_{i}]$, $\gamma_{i}$ and $\varepsilon_{i}$. Hence, for any $K\subset\subset M\setminus D$, we have

[TABLE]

for any $i$ and $k\in\mathbb{N}^{+}$. Then $\psi^{\beta}_{\gamma_{i},\varepsilon_{i}}(t_{i})$ (by taking a subsequence if necessary) converge to a function $\psi^{\beta}_{\gamma_{\infty}}\in C^{0}(M)\bigcap C^{\infty}(M\setminus D)$.

Let $u^{\beta}_{\gamma_{\infty}}$ be the twisted Ricci potential of $\omega^{\beta}_{\gamma_{\infty}}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\psi^{\beta}_{\gamma_{\infty}}$. The Lebesgue Dominated Convergence theorem implies that $\frac{1}{V}\int_{M}e^{-u^{\beta}_{\gamma_{\infty}}}dV^{\beta}_{\gamma_{\infty}}=1$. Since $A^{\beta}_{\gamma,\varepsilon}(t)$ is increasing along the flow $(TKRF^{\beta}_{\gamma,\varepsilon})$, for any $j<i$,

[TABLE]

the second inequality due to Jensen’s inequality. Let $i\rightarrow\infty$, we have

[TABLE]

Since $\mathcal{M}^{\beta}_{\gamma_{\infty}}$ is bounded from below (see $(\ref{20190213133})$), by Theorem 4.8, we obtain

[TABLE]

after letting $j\rightarrow\infty$. Since function $\log$ is strictly concave, we conclude $u^{\beta}_{\gamma_{\infty}}=0$ by using Jensen’s inequality and the normalization of $u^{\beta}_{\gamma_{\infty}}$. This means that $\omega^{\beta}_{\gamma_{\infty}}$ is a twisted conical Kähler-Einstein metric with cone angle $2\pi\beta$ along $D$ and satisfies

[TABLE]

At the same time, we also have

[TABLE]

Both $\psi^{\beta}_{\gamma_{\infty}}$ and $\varphi_{\beta}$ are bounded (in fact, they are Hölder continuous). Berndtsson’s uniqueness theorem [2] implies that

[TABLE]

where $F\in Aut(M)$ is generated by a holomorphic vector $X$ on $M$. Since $(\beta-\gamma_{\infty})\omega_{\varphi_{\beta}}$ has zero Lelong number, then by Siu’s decomposition for a positive closed current, we have

[TABLE]

which implies that $X$ is tangential to $D$. Then $f=id$ because we assume that there is no nontrivial holomorphic vector fields tangent to $D$. Hence $\omega^{\beta}_{\gamma_{\infty}}=\omega_{\varphi_{\beta}}$, and then $\psi^{\beta}_{\gamma_{\infty}}=\varphi_{\beta}+C_{1}$. On the other hand, we have

[TABLE]

Let $\dot{\psi}^{\beta}_{\gamma_{i},\varepsilon_{i}}(t_{i})=u^{\beta}_{\gamma_{i},\varepsilon_{i}}(t_{i})+c^{\beta}_{\gamma_{i},\varepsilon_{i}}(t_{i})$, where constants $c^{\beta}_{\gamma_{i},\varepsilon_{i}}(t_{i})$ are uniformly bounded and hence converge to a constant $C_{2}$ (by taking a subsequence if necessary). Since $u^{\beta}_{\gamma_{i},\varepsilon_{i}}(t_{i})$ converge to [math] in $C^{\infty}_{loc}$-topology in $M\setminus D$, $\dot{\psi}^{\beta}_{\gamma_{i},\varepsilon_{i}}(t_{i})$ converge to $C_{2}$ in $C^{\infty}_{loc}$-topology in $M\setminus D$. Letting $i\rightarrow\infty$ in above equation, we have

[TABLE]

Equation $(\ref{2019022301})$ implies that $C_{1}=\frac{C_{2}-\xi_{\beta}}{\gamma_{\infty}}$. From Proposition 4.9, we conclude that $|C_{1}|\leqslant\frac{\hat{C}_{\beta_{0}}+|\xi_{\beta}|}{\beta_{0}}$. Letting $i\rightarrow\infty$ in $(\ref{58})$, we have

[TABLE]

This leads to a contradiction. Thus Lemma 1.8 is proved.

By using Lemma 1.8 and Remark 4.13, we get the uniform $C^{0}$-estimates for $\psi^{\beta}_{\beta,\varepsilon}(t)=\psi_{\beta,\varepsilon}(t)$.

Proposition 4.22**.**

There exists a uniform constant $B_{\beta}$ such that

[TABLE]

for any $\varepsilon\in(0,\delta_{\beta_{0}})$ and $t\in[0,+\infty)$.

At last, we prove Lemma 1.10.

Proof of Lemma 1.10.   We only prove that there exists $\delta>0$ such that

[TABLE]

for any $\varepsilon\in(0,\delta)$, $\gamma\in[\beta,\beta+\delta)$ and $t\in[\frac{1}{\delta},+\infty)$, where $\xi_{\beta}$ is the constant in $(\ref{2019022301})$, $B_{\beta}$ is the constant in $(\ref{731})$ and $C_{\beta}$ comes from Proposition 4.10. The other direction is similar.

Denote $L=\max(\|\varphi_{\beta}\|_{C^{0}(M)}+\frac{C_{\beta}+|\xi_{\beta}|}{\beta},B_{\beta})$. If this lemma is not true. For $\delta_{1}=\min(1,\delta_{\beta_{0}})$, there exist $\varepsilon_{1}\in(0,\delta_{1})$, $\gamma^{\prime}_{1}\in(\beta,\beta+\delta_{1})$ and $t^{\prime}_{1}\in[\frac{1}{\delta_{1}},\infty)$, such that

[TABLE]

Assume $t^{\prime}_{1}\in[1,T_{1}]$, then we have

[TABLE]

Combining Remark 3.10 with $(\ref{731})$, there exists $\gamma_{1}\in(\beta,\gamma^{\prime}_{1})$, such that

[TABLE]

Then there exists $t_{1}\in[1,T_{1}]$ such that

[TABLE]

For $\delta_{2}=\min(\frac{1}{2},\frac{1}{T_{1}+1},\varepsilon_{1},\gamma_{1}-\beta)$, there exist $\varepsilon_{2}\in(0,\delta_{2})$, $\gamma^{\prime}_{2}\in(\beta,\beta+\delta_{2})$ and $t^{\prime}_{2}\in[\frac{1}{\delta_{2}},\infty)$, such that

[TABLE]

Assume $t^{\prime}_{2}\in[T_{1}+1,T_{2}]$, then we have

[TABLE]

Combining Remark 3.10 with $(\ref{731})$ again, there exists $\gamma_{2}\in(\beta,\gamma^{\prime}_{2})$, such that

[TABLE]

and then there exists $t_{2}\in[T_{1}+1,T_{2}]$ such that

[TABLE]

After repeating above process, we get a subsequence $\psi_{\gamma_{i},\varepsilon_{i}}(t_{i})$ with $\varepsilon_{i}\searrow 0$, $\gamma_{i}\searrow\beta$, $t_{i}\in[T_{i-1}+1,T_{i}]$ and $t_{i}\nearrow\infty$ satisfying

[TABLE]

By the similar arguments as in the proof of Lemma 1.8, we conclude that there exists uniform constants $C$ independent of $\varepsilon_{i}$ and $\gamma_{i}$, such that

[TABLE]

The Evans-Krylov-Safonov’s estimates imply that for any $K\subset\subset M\setminus D$ and $k\in\mathbb{N}^{+}$, there exists uniform constant $C_{k,K}$ such that for any $\gamma_{i}$, $\varepsilon_{i}$ and $t\in[T_{i-1}+\frac{3}{4},T_{i}]$,

[TABLE]

Then we can conclude that $\psi_{\gamma_{i},\varepsilon_{i}}(t_{i})$ (by taking a subsequence if necessary) converge to a function $\psi_{\beta}\in C^{0}(M)\bigcap C^{\infty}(M\setminus D)$.

Let $u_{\beta}$ be the twisted Ricci potential of $\omega_{\psi_{\beta}}=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\psi_{\beta}$, then $u_{\beta}$ satisfies $\frac{1}{V}\int_{M}e^{-u_{\beta}}dV_{\psi_{\beta}}=1$. Since $A_{\gamma,\varepsilon}(t)$ is increasing along the flow $(TKRF_{\gamma,\varepsilon})$, for any $j<i$,

[TABLE]

the second inequality due to Jensen’s inequality. Let $i\rightarrow\infty$, we have

[TABLE]

Since the Log Mabuchi energy $\mathcal{M}_{\beta}$ is bounded from below, by Theorem 2.14, we obtain

[TABLE]

after letting $j\rightarrow\infty$. Since $\log$ is strictly concave, we conclude $u_{\beta}=0$ by using Jensen’s inequality and its normalization. This means that $\omega_{\psi_{\beta}}$ is a conical Kähler-Einstein metric with cone angle $2\pi\beta$ along $D$. Since both $\psi_{\beta}$ and $\varphi_{\beta}$ are bounded (in fact, they are Hölder continuous), Berndtsson’s uniqueness theorem [2] implies that $\omega_{\psi_{\beta}}=\omega_{\varphi_{\beta}}$, and then $\psi_{\beta}=\varphi_{\beta}+C_{1}$. On the other hand, we have

[TABLE]

Let $\dot{\psi}_{\gamma_{i},\varepsilon_{i}}(t_{i})=u_{\gamma_{i},\varepsilon_{i}}(t_{i})+c_{\gamma_{i},\varepsilon_{i}}(t_{i})$, where constants $c_{\gamma_{i},\varepsilon_{i}}(t_{i})$ are uniformly bounded and hence converge to a constant $C_{2}$ (by taking a subsequence if necessary). Since $u_{\gamma_{i},\varepsilon_{i}}(t_{i})$ converge to [math] in $C^{\infty}_{loc}$-topology in $M\setminus D$, $\dot{\psi}_{\gamma_{i},\varepsilon_{i}}(t_{i})$ converge to $C_{2}$ in $C^{\infty}_{loc}$-topology in $M\setminus D$. Letting $i\rightarrow\infty$ in $(\ref{1922001111})$, we have

[TABLE]

Equation $(\ref{2019022301})$ implies that $C_{1}=\frac{C_{2}-\xi_{\beta}}{\beta}$. From Proposition 4.10, we conclude that $|C_{1}|\leqslant\frac{C_{\beta}+|\xi_{\beta}|}{\beta}$. Hence the following inequalities are deduced after we let $i\rightarrow\infty$ in $(\ref{82})$.

[TABLE]

This leads to a contradiction. Thus Lemma 1.10 is proved.

Fix a $\gamma\in(\beta-\delta,\beta+\delta)$ obtained in Lemma 1.10, combining Proposition 4.10 with Proposition $3.1$ in [36], we know that there exists uniform constant $C$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$ and $\omega_{0}$ such that

[TABLE]

for any $\varepsilon\in(0,\delta)$. Then for $k\in\mathbb{N}^{+}$ and $K\subset\subset M\setminus D$, there exists constant $C_{k,K}$ depending only on $\|\varphi_{0}\|_{L^{\infty}(M)}$, $n$, $\beta$, $k$, $\omega_{0}$ and $dist_{\omega_{0}}(K,D)$, such that for $\varepsilon\in(0,\delta)$ and $t\geqslant 1$, we have

[TABLE]

At the same time, the twisted Mabuchi energy $\mathcal{M}_{\gamma,\varepsilon}$ are uniformly bounded from below along the twisted Kähler-Ricci flow $(TKRF_{\gamma,\varepsilon})$, that is, there exists uniform constant $C$ such that

[TABLE]

for any $\varepsilon\in(0,\delta)$ and $t\geqslant 1$. Then by using the arguments in [36, 37], we can prove that the conical Kähler-Ricci flow $(CKRF_{\gamma})$ converges to a conical Kähler-Einstein metric with cone angle $2\pi\gamma$ along $D$ in $C_{loc}^{\infty}$-topology outside $D$ and globally in $C^{\alpha,\gamma}$-sense for any $\alpha\in(0,\min\{1,\frac{1}{\gamma}-1\})$.

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