\section{Conic K\"ahler metrics}

Let $S$ be a defining function of $D$ and $H_0$ a Hermitian metric on $D$ induced by $\omega_0$. Then  it is easy to see that  $|S|^{2\beta} =|S|_{H_0}^{2\beta}\in C^{2,\alpha;\beta} (M)$  for any $\alpha\in (0,1)$
  in sense of [Do11].

Moreover, one can check that
$\omega^*=\omega_0+\delta\sqrt{-1}\partial\bar{\partial} |S|^{2\beta}$ is a conic K\"ahler metric with cone angle $2\pi\beta$ along $D$,  as long as  the number $\delta$ is  sufficiently small  (cf. [Br11]).
There is an important property of $\omega^*$  shown in [JMR11]
 that the  bisectional holomorphic curvature of  $\omega^*$ is uniformly bounded from above on $M\setminus D$.


Let  $h^*$ be a log Ricci potential of $\omega^*$ defined by
\begin{align}\label{ricci-potential-omega-star}
\sqrt{-1}\partial\bar{\partial}h^*={\rm Ric}(\omega^*)-\mu\omega^*-2\pi(1-\beta)[D].
\end{align}
Then  we have
\begin{align}\label{ricci-potential-relation}
h^*= h_0-\mu\delta |S|^{2\beta}-\log\frac{|S|^{2(1-\beta)}(\omega^*)^n}{\omega_0^n}+const,
\end{align}
where $h_0$ is a Ricci potential of $\omega_0$.  A direct computation shows that  $h^*\in C^{,\gamma;\beta}(M)$, where $\gamma=\min(\frac{2}{\beta}-2,1).$


In general,   a K\"ahler potential of conic K\"ahler metric is not necessary  in $C^{2,\alpha;\beta} (M)$.  But, for a  conic K\"ahler-Einstein metric
 $\omega_{CKE}=\omega_\phi=\omega_0+\sqrt{-1}\partial\bar{\partial}\phi$ with angle $2\pi\beta$ along $D$, $\phi$ should be in   $C^{2,\alpha_0;\beta} (M)$
 for some positive number  $\alpha_0 \le\gamma.$  This is because
$\omega_{CKE}$ satisfies  a  conic K\"ahler-Einstein  metric  equation,
\begin{align}\label{conic-KE-equation}
{\rm Ric}(\omega)-\mu\omega-2\pi(1-\beta)[D]=0.
\end{align}
Then   $\phi$ satisfies  a  non-degenerate  complex Monge-Amp\`ere equation,
 \begin{align}\label{complex-MA-equation-conic}
(\omega^*+\sqrt{-1}\partial\bar\partial\phi)^n=e^{h^*-\mu\phi}(\omega^*)^n.
\end{align}
The  $C^{2,\alpha;\beta}$-regularity theorem established in [JMR11] (also in [GP13]) implies that $ \phi\in C^{2,\alpha_0;\beta} (M)$ for some $\alpha_0\le \gamma$.


For any  positive number $\alpha\le \alpha_0$,  we introduce a  space of  $C^{2,\alpha; \beta}$ K\"ahler potentials by
  \begin{align}&\mathscr{H}^{2,\alpha; \beta}(M,\omega_0)\notag\\
&=\{\psi \in C^{2,\alpha; \beta}(M)|~  \omega_\psi=\omega_0+\sqrt{-1}\partial\bar\partial\psi ~{\rm ~is ~  a ~conic~ Kaehler~  metric~ on} ~M\}.
\notag\end{align}
One can show that both functionals $F_{\omega_0,\mu}(\cdot)$ and $I_{\omega_0}(\cdot)$ are well-defined on $\mathscr{H}^{2,\alpha;\beta}$~~
\newline $(M,\omega_0)$.

\begin{lem}\label{functional-smoothing} For any $\psi\in\mathcal H(M,\omega_0) $, there a sequence of $\psi_\delta\in \mathscr{H}^{2,\alpha;\beta}(M,\omega_0)$ such that
$$ F_{\omega_0,\mu}(\psi)=\lim_{\delta\to 0}  F_{\omega_0,\mu}(\psi_\delta)$$
and
$$I_{\omega_0}(\psi)=\lim_{\delta\to 0} I_{\omega_0}(\psi_\delta).$$
\end{lem}

\begin{proof} In fact, one can choose $\psi_\delta=\psi+\delta |S|^{2\beta}$ to verify the lemma.
\end{proof}.


\section{Lower bound of  $F_{\omega_0,\mu}(\cdot)$}

In this section, we  use  the continuity method of Ding-Tian  in  [DT92] ( also see  [Ti97]) to  study   the lower bound of  log $F$-functional  $F_{\omega_0,\mu}(\cdot)$.   This method will be extended to prove
Main Theorem \ref{tian-zhu-conic-ke-metric}  in this paper.
It is worthy to mention  that   the lower bound of  $F_{\omega_0,\mu}(\cdot)$  can be   obtained by using  a general theorem of  Berndtsson for the uniqueness problem of
special K\"ahler potetials  in [Be11].  Berndtsson's method is based on applications of  geodesic theory  about  K\"ahler potentials space studied in  [Se92], [Do98], [Ch00], etc..


\begin{theo}\label{Bern_thm}
Let  $\omega=\omega_{CKE}=\omega_\phi$  be a conic K\"ahler-Einstein metric  on $M$ with cone angle $2\pi\beta$ along $D$. Then $\phi$ obtains the minimum
 of $F_{\omega_0,\mu}(\cdot)$ on $\mathscr{H}^{2,\alpha, \beta}(M,\omega)$.  In particular,
\begin{equation}
F_{\omega_0,\mu}(\psi)\ge-c(\omega_0,\mu),~\forall~ \psi\in\mathscr{H}^{2,\alpha;\beta}(M,\omega_0).
\end{equation}
\end{theo}

\begin{proof}
For any $\psi\in \mathscr{H}^{2,\alpha; \beta}(M,\omega_0)$,  log Ricci potential of  $\hat\omega=\omega_{\psi}$ is given by
 $$h_{\hat\omega}=-\log\frac{\omega_\psi^n}{(\omega^*)^n} -\mu\psi+h^*+const.$$
 Then $h_{\hat\omega}\in  C^{,\alpha;\beta}(M)$.  We consider the following  complex Monge-Amp\`ere  equations with a parameter $t\in [0,\mu]$:
\begin{equation}\label{backward_MAE}
({\hat\omega}+\sqrt{-1}\partial\bar\partial\varphi)^n=e^{h_{\hat\omega}-t\varphi}{\hat\omega}^n.
\end{equation}
  By the assumption,  there exists  a solution $\varphi_\mu=\phi-\psi+const$ at $t=\mu=1-(1-\beta)\lambda$.   Note that  the kernel of operator $(\Delta_{\omega}+\mu)$ is zero (cf. [Do11]).
Then  by the Donaldson's linear theory for Laplace operators associated to conic  metrics, we can apply  Implicity   Function Theorem to
show that there exists  a $\delta>0$ such that  (\ref{backward_MAE}) is solvable in the space  $\mathscr{H}^{2,\alpha;\beta}(M,\omega_0)$ on  any $t\in (\mu-\delta, \mu]$.

Set
$$E=\{s\in [0,\mu]|~  (\ref{backward_MAE}) \mbox{ is solvable on }~t=s ~{\rm in } ~ \mathscr{H}^{2,\alpha'; \beta}(M,\omega_0) ~{\rm for ~some}~\alpha'\le \alpha \}.$$
We  want  to prove $E=[0,\mu].$
Clearly,  $E$ is non-empty since $(\mu-\delta, \mu]\subset E$.   On the other hand,  it is easy to see that   (\ref{backward_MAE}) are equivalent to Ricci curvature equations,
\begin{align}\label{log-twisted-equation-t}{\rm Ric}(\hat \omega_{\varphi})=t\hat\omega_{\varphi}+ (\mu-t)\hat\omega+ 2\pi (1-\beta)[D], ~t~\in [0,\mu].
\end{align}
Then
 \begin{align}\label{ricci-equation-t}{\rm Ric}(\hat\omega_{\varphi_t})>t\hat\omega_{\varphi_t}, ~{\rm in}~ M\setminus D.
\end{align}
Thus  the first non-eigenvalue of $\Delta_t$ is strictly  bigger than $t$ [JMR11],
where  ${\Delta}_t$ is the Laplace operator associated to $\omega_t$  and ${\omega}_t=\hat\omega_{\varphi_t}=\hat{\omega}+\sqrt{-1}\partial\bar\partial\varphi_t$.
 It follows that  the kernel of operator $(\Delta_{t}+t)$ is zero on  any $t\in E$.   By  Implicity   Function Theorem,  $E$ is  an  open set.
 It  remains  to prove that   $E$ is  also a closed set.  This is related to apriori estimates for solution $\varphi_t$ of   (\ref{backward_MAE}) on  $t\in E$ below.

First we deal with the  $C^0$-estimate.  We may assume that $t\ge \delta$   by the implicity theorem   since  (\ref{backward_MAE}) is solvable  at  $t=0$ [JMR11].
By a direct computation,  we have
\begin{align}\nonumber
 \frac{d}{dt}(I_{\hat\omega}(\varphi_t)-J_{\hat{\omega}}(\varphi_t))&=-\frac{1}{V}\int_M\varphi_t{\Delta}_t\dot{\varphi_t}{\omega}_t^n\\ \nonumber
 &=\frac{1}{V}\int_M({\Delta}_t\dot{\varphi_t}+t\dot{\varphi_t}){\Delta}_t\dot{\varphi_t}{\omega}_t^n.\nonumber
\end{align}
  Note that
$${\Delta}_t\dot{\varphi_t}=-t\dot{\varphi_t}-\varphi_t$$
 by differentiating $\int_M e^{h_{\hat{\omega}}-t\varphi}\hat{\omega}^n=V$.  By the fact that  the first non-eigenvalue of $\Delta_t$ is strictly  bigger than $t$, we get
$$\frac{d}{dt}(I_{\hat{\omega}}(\varphi_t)-J_{\hat{\omega}}(\varphi_t))\ge 0.$$
 This means that   $I_{\hat{\omega}}(\varphi_t)-J_{\hat{\omega}}(\varphi_t)$ is  increasing in $t$. Thus
 $$I_{\hat{\omega}}(\varphi_t)\le (n+1)I_{\hat{\omega}}(\varphi_\mu)\le C.$$
By using the Green formula [JMR11],  we derive
$${\rm osc}(\varphi_t)\le C.$$

To get the $C^2$-estimate, we rewrite (\ref{backward_MAE}) as
\begin{equation}\label{backward_MAE-2}
(\omega^*+\sqrt{-1}\partial\bar\partial\varphi^*)^n=e^{h^*-t\varphi^*-(\mu-t)(\psi-\delta |S|^{2\beta})  } (\omega^*)^n,
\end{equation}
where $\varphi^*=\varphi-\delta |S|^{2\beta}+\psi$ and $h^*$ is the log Ricci potential of $\omega^*$ as in (\ref{ricci-potential-omega-star}).
Since ${\rm Ric}(\omega_{\varphi})>0$,  by  the Chern-Lu inequality [Cher68], [Lu68],   we have
\begin{align}\label{Chern-Lu inequality-1}
\Delta_t \log{\rm  tr}_{  \omega_{t}}(\omega^*)\ge -a(\omega^*)\, {\rm  tr}_{  \omega_{t}}(\omega^*), ~{\rm in}~M\setminus D,
\end{align}
where $a=a(\omega^*)$ is a uniform constant  which depends only on the upper bound of bisectional holomorphic curvature of  $\omega^*$,  and so  it
depends only on  $\omega_0$ and the divisor $D$.  Set
$$u=\log{\rm  tr}_{  \omega_{t}}(\omega^*) -(a+1)\varphi^*.$$
Then  there exists a uniform constant  $C=C(\sup_M  \phi, \sup_M  \psi)$ such that
$$\Delta_t  u\ge e^{u-C(a+1)}-n(a+1).$$
By the maximum principle as in [JMR11],  it follows
\begin{align}\label{upper-bound-phi-1}\omega_t\ge C^{-1}\omega^*.
\end{align}
By (\ref{backward_MAE-2}), we also get
\begin{align}\label{lower-bound-phi-1}\omega_t\le C \omega^*.
\end{align}

Once (\ref{upper-bound-phi-1}) and (\ref{lower-bound-phi-1}) hold,   we can apply  the $C^{2,\alpha;\beta}$-regularity theorem  in  [JMR11] to  show
$$\|\varphi\|_{C^{2,\alpha';\beta}(M)}\le C$$
 for some $\alpha'\le \alpha$. Thus   $\varphi_t\in  \mathscr{H}^{2,\alpha'; \beta}(M,\omega_0)$.  This implies that $E$ is a closed set and so $E=[0,\mu]$.

 By a direct computation as in [DT92], we have
 \begin{align}\label{integral-dt} t\left(J_{\hat{\omega}}(\varphi_\mu)-\frac{1}{V}\int_M\varphi_\mu\hat{\omega}^n\right)=-\int_0^t  (I-J)_{\hat{\omega}}(\varphi_s)ds\le 0, ~\forall~t\le \mu.
\end{align}
Note that  $\int_M e^{h_{\hat{\omega}}-\mu\varphi_\mu}\hat{\omega}^n=V$.  Thus
$$F_{\hat{\omega},\mu}(\varphi_\mu)=J_{\hat{\omega}}(\varphi_\mu)-\frac{1}{V}\int_M\varphi_\mu\hat{\omega}^n\le 0.$$
By  the cocycle condition  of  log  $F$-functional,  it follows
\begin{align}
F_{\omega,\mu}({\psi-\phi})=-F_{\hat{\omega},\mu}(\varphi_\mu)\ge 0.\notag
\end{align}
Again by  the cocycle condition,   we get
\begin{align}\label{cocycle-condition}
F_{\omega_0,\mu}(\psi)=F_{\omega,\mu}(\psi-\phi)+F_{\omega_0,\mu}(\phi)\ge F_{\omega_0,\mu}(\phi).
\end{align}
 Hence we prove that  $F_{\omega_0,\mu}(\cdot)$ takes the minimum at  $\phi$.
\end{proof}

\begin{cor}\label{Bern_thm-coro}
Suppose that there exists   a conic K\"ahler-Einstein metric  on $M$ with  cone angle $2\pi\beta$ along $D$.  Then
\begin{align}
F_{\omega_0,\mu}(\psi)\ge-c(\omega_0,\mu),~\forall~ \psi\in\cup_{0<\alpha'\le \alpha_0}\mathscr{H}^{2,\alpha';\beta}(M,\omega_0).\notag
\end{align}
\end{cor}


\section{Approximation of conic K\"ahler  metrics}

In this section, we construct approximated smooth K\"ahler potentials  of solution $\varphi_t$ of (\ref{backward_MAE}) on  each $t\in (0,\mu)$ by solving certian complex Monge-Amp\`ere equation.
 First we shall smooth the conic metric
 $\hat \omega=\omega_{\psi}$.  Note
$$(\hat\omega)^n=f_0\omega_0^n, $$
where $f_0=g\frac{1}{|S|^{2-2\beta}}$  for some  $L^\infty$-function $g$.   In particular,   $f_0$ is a $L^p$-function.

Take a family of smooth functions $f_\delta$ with $\int_M f_\delta\omega_0^n=\int_M \omega_0^n$ such that $f_\delta$ converge to $f_0$ in  $L^p$ as $\delta\to 0$.
Then by  the Yau's solution to Calabi's  problem,
there are a family of  K\"ahler potentials $\Psi_\delta$, which  solve equations $(\delta>0)$,
$$(\omega_0+\sqrt{-1}\partial\bar\partial\Psi_{\delta})^n=f_\delta\omega_0^n.$$
By  the Kolodziej's  H\"older estimate  [Kol08], $\Psi_{\delta}$  converge to $\psi$ in  the $C^\alpha$-norm modulo constants as $\delta\to 0.$
For simplicity, we set  $\omega_\delta=\omega_0+\sqrt{-1}\partial\bar\partial\Psi_{\delta}.$


We  modify  (\ref{log-twisted-equation-t}) by a family of Ricci curvature equations with parameter $\delta\in (0,\delta_0]$ for each  $t\in [0,\mu]$,
\begin{align}\label{twisted-equation-t}{\rm Ric}(\omega^\delta_{\varphi_{\delta}})=t\omega^\delta_{\varphi_{\delta}}+ (\mu-t)\omega_\delta+ (1-\beta)\eta_\delta,
\end{align}
where $\omega^\delta_{\varphi_{\delta}}=\omega_\delta+\sqrt{-1}\partial\bar\partial\varphi_{\delta}$ and $\eta_\delta=\lambda\omega_0+\sqrt{-1}\partial\bar\partial \log(\delta+|S|^2).$
(\ref{twisted-equation-t}) are in fact a family of twisted K\"ahler-Einstein metric eqautions  associated to  positive $(1,1)$-forms $\Omega= (\mu-t)\omega_\delta+ (1-\beta)\eta_\delta$ [ST12].
One can check that (\ref{twisted-equation-t}) are equivalent to the following  complex Monge-Amp\`ere equations,
\begin{equation}\label{smooth-continuity-modified}
(\omega_\delta+\sqrt{-1}\partial\bar\partial\varphi_\delta)^n=e^{h_\delta-t\varphi_\delta}\omega_\delta^n,
\end{equation}
where $h_\delta$ are  twisted Ricci potentials of $\omega_\delta$ defined by
\begin{align}\label{twisted-h-delta}\sqrt{-1}\partial\bar\partial h_\delta= {\rm Ric}(\omega_\delta)-\mu\omega_\delta- (1-\beta)\eta_\delta.
\end{align}
We shall study the solutions of  (\ref{smooth-continuity-modified})  and their convergence as $\delta\to 0.$


Rewrite   (\ref{twisted-equation-t})  as
\begin{align}\label{twisted-equation-t-2}{\rm Ric}(\omega^\delta_{\varphi_{\delta}})=t\omega^\delta_{\varphi_{\delta}}+(\mu-t)\omega_0+ (1-\beta)\eta_\delta  +(\mu-t)\sqrt{-1}\partial\bar\partial  \Psi_{\delta}.
\end{align}
 Then  equations (\ref{smooth-continuity-modified})  are equivalent to
\begin{equation}\label{twisted-MA-equation}
(\omega_0+\sqrt{-1}\partial\bar\partial\hat\varphi_\delta)^n=\frac{1}{(\delta+|S|^2)^{1-\beta}}e^{h_0-t\hat\varphi_\delta -(\mu-t)\Psi_\delta}\omega_0^n,~~t\in[0,\mu],
\end{equation}
where  $\hat \varphi_{\delta}= \hat \varphi_{t, \delta}=\varphi_{t, \delta} + \Psi_{\delta}$.

As in [Ti12],  for a fixed $\delta>0$,  we define a family of twisted F-functionals with parameter  $t\in (0,\mu ]$ as follows,
 \begin{align}\label{twisted-f-functional-t}
F_{t, \delta}(\varphi)=J_{\omega_0}(\varphi)-\frac{1}{V}\int_M\varphi\omega^n_0-\frac{1}{t}\log
\left(\frac{1}{V}\int_Me^{\hat h_\delta-t\varphi}\omega_0^n\right),
\end{align}
where
\begin{align}
\hat h_\delta=h_0-(1-\beta)\log (\delta+|S|_0^2)+(t-\mu)\Psi_{\delta}+C_\delta,~~\int_M(e^{\hat h_\delta}-1)\omega^n_0=0.\notag
\end{align}
Then all $F_{t, \delta}(\cdot)$ are proper for  any  $t\in (0,\mu ), \delta\in (0,\delta_0]$ since  log $F$-functionals $F_{\omega_0, t}(\cdot)$ defined in (\ref{log-functional})  are  proper   for  any  $t\in (0,\mu )$. The latter follows from a result in  [LS12]  by  using the fact that
$F_{\omega_0, \mu }(\cdot)$  is bounded from below according to Theorem  \ref{Bern_thm}.  By the Green formula,  we get
  $${\rm osc}_{M}\hat \varphi_{t,\delta} \le C(I_{\omega_0}(\hat \varphi_{t,\delta})+1) \le C',~\forall ~t\in (0,\mu ), $$
where the constants $C,C'$ depend only on $t$.   Note that  all higher order estimates
for solutions  $\hat \varphi_{t,\delta}$ depend only on $\delta$ and  their $C^0$-norm.  Thus  by using the continuity method as in the proof of Theorem  2.5 in [Ti12],
(\ref{twisted-MA-equation}), and so (\ref{smooth-continuity-modified})  are solvable on  any  $t\in (0,\mu), \delta\in (0,\delta_0]$.



Next we improve   higher order estimates for solutions  $\hat \varphi_{t,\delta}$  to show  that they are independent of $\delta>0$.   Let's  introduce a family of smooth  K\"ahler potentials $\Phi^\beta_\delta$ ($\delta >0)$ constructed by
Guenancia-Paun   in [GP13].    Such  $\Phi^\beta_\delta$ have property:

1) $\Phi^\beta_\delta$ converge to   $\Phi^\beta_0=k|S|^{2\beta}$  as $\delta\to 0$ in  sense of H\"older-norm.

2) Let
$$h_{\kappa_\delta}=-\log\frac{\kappa_\delta^n}{\omega_0^n} -\Phi^\beta_\delta+h_0$$
   be  Ricci potentials of $\kappa_\delta=\omega_0+ \sqrt{-1}\partial\bar\partial \Phi^\beta_\delta$.  Then
$\frac{1}{\delta+|S|^{2\beta}} e^{h_\kappa}$ is uniformly bounded on $\delta$.

3) The  bisectional holomorphic curvatures  $R_{\delta i\bar ij\bar j}$ of  $\kappa_\delta$ satisfy: for any K\"ahler metric
$\omega_{\phi+ \Phi^\beta_\delta}=\kappa_\delta+\sqrt{-1}
\partial\bar\partial\phi$, it holds
\begin{align}\label{gp-inequality} &\sum_{i<j} (\frac{1+\phi_{ i\bar i}} {1+\phi_{ j\bar j}} +\frac{1+\phi_{ j\bar j}} {1+\phi_{ i\bar i}}  -2)R_{\delta i\bar ij\bar j} -C_0{\rm tr}_{\kappa_\delta}(\omega_{\phi+ \Phi^\beta_\delta})
 \Delta_{ \omega_{\phi+ \Phi^\beta_\delta}} \Phi^\beta_\delta\notag\\
&+ \Delta_{ \kappa_{\delta}}  \log(\frac{\kappa_\delta^n}{\omega_0^n}\times (\delta+|S|^2)^{1-\beta})\notag\\
&\le C \sum_{i<j} (\frac{1+\phi_{ i\bar i}} {1+\phi_{ j\bar j}} +\frac{1+\phi_{j\bar j}}{1+\bar\phi_{ i\bar i}}) +C {\rm tr}_{\kappa_\delta}(\omega_{\phi+ \Phi^\beta_\delta })\times {\rm tr}_{\omega_{\phi+ \Phi^\beta_\delta}}(\kappa_\delta)+C,
\end{align}
where $C_0$ and $C$ are two uniform constants.

The following is about uniform aprior $C^2$-estimate for $\varphi= \varphi_{t,\delta}$.

\begin{lem}\label{c2-estimate-twisted-metrics-lemma} For any  $t\in (0,\mu),\delta\in(0,\delta_0]$,  it holds
\begin{align}\label{c2-estimate-twisted-metrics}
C^{-1}  \kappa_\delta\le \omega^\delta_{\varphi_{t,\delta}} \le C \kappa_\delta.
\end{align}
Here $C$ is a uniform constant  which depends only on the metric  $\hat\omega$ and $t$.

\end{lem}

\begin{proof} Let $\bar\varphi=\bar\varphi_\delta=\hat\varphi_\delta-\Phi^{\beta}_\delta$. Then (\ref{twisted-MA-equation}) are  equivalent to
\begin{equation}\label{twisted-MA-equation-3}
(\kappa_\delta+\sqrt{-1}\partial\bar\partial\bar\varphi)^n=\frac{1}{(\delta+|S|_0^2)^{1-\beta}} e^{h_{\kappa_\delta}-t\bar\varphi -(\mu-t)\Psi_{\delta} -(1-t)\Phi^\beta_\delta }\kappa_\delta^n,~~t\in (0,\mu_0).
\end{equation}
Following Yau's  $C^2$-estimate in [Yau78], we have
\begin{align}\label{c2-yau} &  -\Delta_{\omega_{\varphi_\delta}^\delta} \log {\rm tr}_{\kappa_\delta} (  \omega^\delta_{\varphi_{\delta}} )\notag\\
&\le\frac{1}{{\rm tr}_{\kappa_\delta}  (  \omega^\delta_{\varphi_{\delta}} ) }\sum_{i<j} (\frac{1+\bar\varphi_{ i\bar i}} {1+\bar\varphi_{ j\bar j}} +\frac{1+\bar\varphi_{ j\bar j}}{1+\bar\varphi_{ i\bar i}} -2)R_{\delta i\bar ij\bar j} \notag\\
&+\frac{1}{{\rm tr}_{\kappa_\delta}  (  \omega^\delta_{\varphi_{\delta}} )} \Delta_{ \kappa_{\delta}}( t\bar\varphi +(\mu-t)\Psi_{\delta} +(1-t)\Phi^\beta_\delta - \bar h_{\kappa_\delta} ).
\end{align}
  On the other hand,  by
$$B\kappa_\delta + \sqrt{-1}\partial\bar\partial \Psi_{\delta}\ge 0,$$
it is easy to see
\begin{align}\Delta_{   \omega^\delta_{\varphi_{\delta}} } \Psi_{\delta}\ge \frac{  \Delta_{ \kappa_{\delta}} \Psi_{\delta}  }  {{\rm tr}_{\kappa_\delta}    (  \omega^\delta_{\varphi_{\delta}} )}
-nB {\rm tr}_{ \omega^\delta_{\varphi_{\delta}}}(\kappa_\delta).\notag
\end{align}
 Using the fact
\begin{align}\label{laplace-h}\Delta_{ \kappa_{\delta}} h_0 \ge  -A,
\end{align}
we get
\begin{align} &\frac{1}{{\rm tr}_{\kappa_\delta} (   \omega^\delta_{\varphi_{\delta}} )} \Delta_{ \kappa_{\delta}}( t\bar\varphi +(\mu-t)\Psi_{\delta} +(1-t)\Phi^\beta_\delta  - \bar h_{\kappa_\delta} ) -\Delta_{ \omega^\delta_{\varphi_{\delta}}  } \Psi_{\delta} \notag\\
&\le \frac{1}{{\rm tr}_{\kappa_\delta} (    \omega^\delta_{\varphi_{\delta}})} \Delta_{ \kappa_{\delta}}  \log(\frac{\kappa_\delta^n}{\omega_0^n}\times (\delta+|S|^2_0)^{1-\beta})
+\frac{n(1-t) +A}{{\rm tr}_{\kappa_\delta} (    \omega^\delta_{\varphi_{\delta}})}
+t +nB{\rm tr}_{   \omega^\delta_{\varphi_{\delta}}}(\kappa_\delta).
\notag
\end{align}
Thus   by  the Guenancia-Paun inequality (\ref{gp-inequality}) for metrics   $\omega^\delta_{\varphi_{\delta}}$,  we deduce from (\ref{c2-yau}),
\begin{align}& - \Delta_{    \omega^\delta_{\varphi_{\delta}}} (\log {\rm tr}_{\kappa_\delta} (  \omega^\delta_{\varphi_{\delta}}) +C_0 \Phi^\beta_\delta - \Psi_{\delta} ) \notag\\
&\le C'  {\rm tr}_{   \omega^\delta_{\varphi_{\delta}}}(\kappa_\delta)+C'.\notag
\end{align}

Let
$$u=\log{\rm  tr}_{\kappa_\delta}(    \omega^\delta_{\varphi_{\delta}})+C_0 \Phi^\beta_\delta - \Psi_{\delta} -(C'+1)\bar\varphi_\delta.$$
Then
$$\Delta_{    \omega^\delta_{\varphi_{\delta}}} u\ge {\rm  tr}_{   \omega^\delta_{\varphi_{\delta}}} (\kappa_\delta)-C''.$$
By the maximum principle,  it follows
\begin{align}\label{upper-bound-phi-delta}    \omega^\delta_{\varphi_{\delta}}=\kappa_{\delta} +\sqrt{-1}\partial\bar\partial\bar\varphi_\delta\ge C^{-1}\kappa_\delta.
\end{align}
By (\ref{twisted-MA-equation-3}), we can also get
 \begin{align}\label{lower-bound-phi-delta}      \omega^\delta_{\varphi_{\delta}}\le C\kappa_\delta.
\end{align}

\end{proof}



\begin{theo}\label{phi-delta-convergence} For any  $t\in (0,\mu)$,  it holds
\begin{align}
\lim_\delta  \varphi_{t, \delta}=\varphi_t\notag
\end{align}
in sense of H\"older-norm.
\end{theo}

\begin{proof} First we claim that $\varphi_{t, \delta}$ converges  to a  $C^{2,\alpha;\beta}$-solution  of (\ref{backward_MAE}) as $\delta\to 0$.
In fact, by  the Kolodziej's H\"older estimate,  we see that  $\hat \varphi_{t,\delta}$ converge to a H\"older continuous  solution  $\phi'$ of following complex Monge-Amp\`ere equation in  the current sense,
\begin{equation}\label{smooth-continuity-2}
(\omega_0+\sqrt{-1}\partial\bar\partial\phi')^n= \frac{1}{|S|^{2-2\beta}}e^{h_0- t \phi' +( t-\mu)\psi}\omega_0^n.
\end{equation}
Clearly,  (\ref{smooth-continuity-2}) is nothing, just  (\ref{backward_MAE}).  Since $\omega^*=\omega_0+\sqrt{-1}\partial\bar\partial\Phi_0^\beta$ is
equivalent to $\hat\omega$,  by Lemma \ref{c2-estimate-twisted-metrics-lemma},  we get
\begin{align}\label{c2-limit-phi-delta-t}
C^{-1}  \hat\omega\le \omega_{\phi'} \le C \hat\omega, ~{\rm in}~ M\setminus D,
\end{align}
where $C$ is a uniform constant.   Note that  (\ref{smooth-continuity-2}) implies that $\phi'- \psi$  is a solution of  (\ref{backward_MAE}).  Thus by the $C^{2,\alpha;\beta}$  reguarity  theorem,
 $\phi'- \psi$ is a $C^{2,\alpha;\beta}$-solution of   (\ref{backward_MAE}).   This proves the claim.

On the other hand,  according to  the proof in Theorem \ref{Bern_thm}, it is easy to see that $C^{2,\alpha;\beta}$-solution of  (\ref{backward_MAE})
 as a  twisted K\"ahler-Einstein metric  is unique. Thus  $\phi'- \psi$ must be  $\varphi_{ t}$. The theorem is proved.


\end{proof}

\section{Smoothing of twisted Ricci potentials}

Define a Log Ricci potential $h_t$  of  $\hat\omega_{\varphi_t}$ of solution of (\ref{backward_MAE}) on  $t$ by
$$\sqrt{-1}\partial\bar\partial h_t={\rm Ric}(\hat\omega_{\varphi_{t}}) - \mu\hat\omega_{\varphi_t}- 2\pi (1-\beta)[D], ~\int_M  e^{ h_t} \hat\omega_{\varphi_{t}}^n=V, $$
and  define a twisted  Ricci potential of  $   \omega^\delta_{\varphi_{t, \delta}}= \omega_\delta+\sqrt{-1}\partial\bar\partial  \varphi_{t,\delta} $ of solution of (\ref{smooth-continuity-modified}) on  $t$ by
$$\sqrt{-1}\partial\bar\partial h_{t,\delta}=  {\rm Ric}(\omega_{\varphi_{t, \delta}}^\delta) -\mu\omega_{\varphi_{t, \delta}}^\delta-(1-\beta)\eta_\delta, ~\int_M  e^{ h_{t,\delta}}  ( \omega^\delta_{\varphi_{t,\delta}})^n=V.$$
 Then it is easy to see
$$  h_t=-(\mu-t)\varphi_t+ const, $$
and
 $$ h_{t, \delta}=-(\mu-t)\varphi_{t, \delta} + const.$$
Thus by Theorem \ref{phi-delta-convergence}, we have

\begin{lem}For any  $t\in (0,\mu)$,  it holds
\begin{align}
\lim_{\delta\to 0}  h_{t,\delta}=h_t\notag
\end{align}
in sense of H\"older-norm.
\end{lem}


To smooth $ h_{t,\delta}$ for each  fixed $\delta\in (0,\delta_0)$,  we introduce the following  twisted K\"ahler-Ricci flow,
 \begin{align}\label{twisted-K-R-flow}
&\frac{\partial}{\partial s} \omega_{\psi}^\delta= -{\rm Ric} (\omega_{\psi}^\delta ) +\mu\omega_{\psi}^\delta+(1-\beta)\eta_\delta,\notag\\
& \omega_{\psi}^\delta(\cdot, 0)=\omega^\delta_{\psi_{0, \delta}}=\omega^\delta_{\psi_{s, \delta}}|_{s= 0}=\omega_{\varphi_{t,\delta}}^\delta.
\end{align}
Clearly,   the  twisted  Ricci potential  $h_{t,s, \delta}$ of  $\omega_{\psi_{s,\delta}}^\delta$ is given by
$$ h_{t, s, \delta}=- \frac{\partial}{\partial s} \psi_{s,\delta}+ const.$$
In particular,
$$ h_{t, \delta}=- \frac{\partial}{\partial s} \psi_{s,\delta}|_{s=0}+ const.$$

(\ref{twisted-K-R-flow}) reduces to a complex Monge-Amp\`ere flow,
\begin{align}\label{smooth-flow-MA-0}
& \frac{\partial}{\partial s} \tilde \psi =\log \frac{( \omega_{\varphi_{t, \delta}+\tilde\psi}^\delta)^n}{(\omega_{\varphi_{t, \delta}}^\delta)^n}+\mu\tilde  \psi  -  h_{t,\delta},\notag\\
& \tilde \psi_{0,\delta}=\tilde\psi_{\delta}(0,\cdot)= 0.
\end{align}
Here $h_{t,\delta}$ can be normalized so that $ h_{t, \delta}=-(\mu-t)\varphi_{t, \delta}$. Then $ h_{t, \delta}=- \frac{\partial}{\partial s} \tilde\psi_{s,\delta}|_{s=0}$
 Similarly  to  K\"ahler-Ricci flow  in [Ti97],  applying   the maximum principle to (\ref{smooth-flow-MA-0}), we have   following estimates (also see [LZ14]).
\begin{lem}\label{gradient-laplace-flow}

\begin{align} &1)~| \frac{\partial}{\partial s} \tilde\psi_{s,\delta}|^2+s|\nabla '\frac{\partial}{\partial s} \tilde\psi_{s,\delta}|^2\le e^{2\mu s} (\mu-t)^2 \|\varphi_{t,\delta}\|_{C^0(M)}^2.\notag\\
&2)  ~\Delta' (- \frac{\partial}{\partial s} \tilde\psi_{s,\delta})\ge e^{\mu s} \Delta h_{t,\delta}.\notag
\end{align}
Here $\Delta'$, $ \Delta$ are Laplace operators associated  to metrics $\omega_{\psi_{s, \delta}}^\delta, \omega_{\varphi_{t,\delta}}^\delta$, respectively.
  \end{lem}

\begin{lem}\label{holder-estimate-perturbation-flow}Let $v=v_{t, s,\delta}$  be a normalization of $h_{t,s,\delta}$ by  adding  a suitable constant  such  that
$$\int_M v(\omega_{\psi_{s,\delta }}^\delta)^n=0.$$
Let $\gamma=\frac{1}{8+8n}$.  Then there exists a
small number $\epsilon>0$ such that for any $t$ and $\varphi_{t,\delta}$
satisfying
\begin{align}\label{t-condition-phi}(\mu-t)^{1+
\frac{\gamma}{2}}\|\varphi_{t,\delta}\|_{C^0(M)}\le\epsilon,\end{align}
we have
 \begin{align}\label{holder-estimate-h-delta}\|v\|_{C^{\frac{1}{2}}(M)} \le
C\epsilon (\mu-t)^{\frac{\gamma}{2}}
\end{align}
  and
 \begin{align}\label{c0-perturbation}{\rm osc}_M \tilde\psi_{s,\delta}\le C\epsilon(\mu -t)^{\frac{\gamma}{2}}, ~\forall~ s\in [(\mu -t)^{2\gamma}, 1],
  \end{align}
    provided that  for any $s\in [(\mu -t)^{2\gamma}, 1]$ the first non-zero eigenvalue
    \begin{align}\label{large-eigenvalue}\lambda_1\ge \lambda_0>0
\end{align}
    of Laplace operator $\Delta'$  associated to the
metric $\omega_{\psi_{s,\delta}}^\delta$  and the following condition holds: there exists a
constant  $a>0$ such that for any $x_0\in M$ and $0<r<1$,
\begin{align}\label{volume-condition}{\rm vol} (B_r(x_0)) \ge ar^{2n}
\end{align}
 with respect to  $\omega_{\psi_{s,\delta}}^\delta$. Here $C=C(
a,\lambda_0)$ denotes a uniform constant depending only on the
constants $ a$ and $\lambda_0$.

\end{lem}

\begin{proof} Lemma \ref{holder-estimate-perturbation-flow} can be proved following the argument  of  smoothing lemma  in [Ti97] (also see Proposition 4.1 in [CTZ05]). In fact, under the  conditions (\ref{c0-perturbation})
and (\ref{large-eigenvalue}),  using the estimates  1) and 2)  in  Lemma \ref{twisted-K-R-flow},  we get
$$|v|
\le C( a,\lambda_0)
(1+\|h_{t,\delta}\|_{C^0(M)})(\mu -t)^{\frac
{3}{8(n+1)}},~\forall~
s\in[(\mu -t)^{2\gamma}, 1]. $$
 On the other hand,  again by the  estimate 1)  in  Lemma \ref{twisted-K-R-flow},  we have
 \begin{align}  \|\nabla' v\|^2_{C^0(M)}  \leq \frac{1}{s}
 e^2\|h_{t,\delta}\|^2_{C^0(M)},\notag
~\forall ~s>0.
\end{align}
 Combining  these two relations,   we derive
\begin{align}
\|v_s\|_{C^{\frac{1}{2}}(M)}
&\le C( a,
\lambda_0)(1+\|h_{t,\delta}\|_{C^0(M)})(\mu -t)^{\frac
{1}{8(n+1)}}\notag\\
&\le C [\epsilon+\|h_{t,\delta}\|_{C^0(M)} (\mu -t)^{\frac
{\gamma}{2}}] (\mu -t)^{\frac
{\gamma}{2}} ,~\forall~
s\in[(\mu -t)^{2\gamma}, 1]\notag.
\end{align}
 Note
  $$-(\mu-t)\varphi_{t,\delta}=h_{t,\delta}.$$
Thus under the assumption (\ref{t-condition-phi}),  we get (\ref{holder-estimate-h-delta}) immediately.

By the estimate 1) in Lemma \ref{gradient-laplace-flow},   we have
 \begin{align}| \frac{\partial
}{\partial s}
\tilde\psi|\le
e\|h_{t,\delta}\|_{C^0(M)},~\forall~s\le
1.\notag
\end{align}
 Note
 $$\tilde\psi=(\int_0^{(\mu-t)^{2\gamma}}+\int_{(\mu-t)^{2\gamma}}^s)(\frac{\partial }{\partial s}
 \tilde\psi).$$
Then by the assumption  (\ref{t-condition-phi}),  we obtain
\begin{align}\label{c0-perturbation-2} {\rm osc}_M \tilde\psi &\le (\mu-t)^{2\gamma} \sup_{s\in [0,1]} \|\frac{\partial }{\partial s}
 \tilde\psi \|_{C^0(M)}\notag\\
 & + \sup_{s\in [(\mu -t)^{2\gamma} ,1]} \|  \frac{\partial }{\partial s}
 \tilde\psi \|_{C^0(M)}\notag\\
 &\le C\epsilon (\mu -t)^{\frac{\gamma}{2}}.
\end{align}

\end{proof}





\section{Convergence of twisted K\"ahler-Ricci flows}

In this section, we  deal with  the local convergence of flows (\ref{twisted-K-R-flow}). First, similarly  to Lemma \ref{c2-estimate-twisted-metrics-lemma}, we have

\begin{lem}\label{c2-parabolic-twisted-KR} For any  $t\in (0,\mu),\delta\in(0,\delta_0]$,  it holds
\begin{align}\label{c2-parabolic-estimate-twisted-metrics}
C^{-1}  \kappa_\delta\le \omega^\delta_{\psi_{s,\delta}} \le C \kappa_\delta.
\end{align}
Here $C$ is a uniform  constant depends only on  metrics  $\hat\omega, \omega^*$ and norms of  $ \|\psi_{s, \delta}\|_{C^0(M)}$,  $\|\frac{\partial}{\partial s}\psi_{s,\delta}\|_{C^0(M)}.$
\end{lem}


\begin{proof}
Let $\bar\psi_\delta=\bar\psi_{s, \delta}=\psi_{s, \delta}+ \Psi_{\delta}-\Phi_{\delta}^\beta$. Then  by (\ref{smooth-flow-MA-0}), $\bar\psi=\bar\psi_\delta$ satisfies the following complex Monge-Amp\`ere flow,
\begin{align}\label{smooth-flow-MA}
& \frac{\partial}{\partial s} \bar \psi =\log \frac{(\kappa_\delta+\sqrt{-1}\partial\bar\partial\bar\psi)^n}{\kappa_\delta^n}+\mu\bar  \psi  - \bar h_{\kappa_\delta},\notag\\
& \bar \psi_{0,\delta}=\bar\psi_{\delta}(0,\cdot)= \varphi_{t,\delta}+\Psi_{\delta} -\Phi_{\delta}^\beta.
\end{align}

Following the estimate of (\ref{c2-yau}),  for the parabolic  equation  (\ref{smooth-flow-MA}),  we  get  Yau's $C^2$-estimate,
\begin{align} &(\frac{\partial}{\partial s}  -\Delta_{ \omega_{\psi_\delta}}^\delta) \log {\rm tr}_{\kappa_\delta} (\omega_{\psi_\delta}^\delta)\notag\\
&\le\frac{1}{{\rm tr}_{\kappa_\delta} (\omega_{\psi_\delta}^\delta)}\sum_{i<j} (\frac{1+\bar\psi_{ i\bar i}} {1+\bar\psi_{ j\bar j}} +\frac{1+\bar\psi_{ j\bar j}}{1+\bar\psi_{ i\bar i}} -2)R_{\delta i\bar ij\bar j} +\frac{1}{{\rm tr}_{\kappa_\delta} (\omega_{\psi_\delta}^\delta)} \Delta_{ \kappa_{\delta}}(\mu\bar  \psi_\delta - \bar h_{\kappa_\delta} )\notag
\end{align}
On the other hand, by  (\ref{laplace-h}), we have
\begin{align} &\frac{1}{{\rm tr}_{\kappa_\delta} (\omega_{\psi_\delta}^\delta)} \Delta_{ \kappa_{\delta}}(\mu\bar  \psi_\delta - \bar h_{\kappa_\delta} )\notag\\
&\le \frac{1}{{\rm tr}_{\kappa_\delta} (\omega_{\psi_\delta}^\delta)} \Delta_{ \kappa_{\delta}}  \log(\frac{\kappa_\delta^n}{\omega_0^n}\times (\delta+|S|^2_0)^{1-\beta}) +\frac{A}{{\rm tr}_{\kappa_\delta} (\omega_{\psi_\delta}^\delta)}
+\mu.
\notag
\end{align}
By  the Guenancia-Paun  inequality (\ref{gp-inequality}), it follows
\begin{align}&(\frac{\partial}{\partial s}  - \Delta_{ \omega_{\psi_\delta}^\delta}) (\log {\rm tr}_{\kappa_\delta} (\omega_{\psi_\delta}^\delta) +C_0 \Phi^\beta_\delta) \notag\\
&\le C_0' {\rm tr}_{\omega_{\psi_\delta}^\delta}(\kappa_\delta)+C_0', \notag
\end{align}
where $C_0$ and $C_0'$ are two uniform constants depending only on  metrics $\hat\omega$ and $\omega^*$. Hence by choosing a large number $B$, we deduce
 \begin{align}&(\frac{\partial}{\partial s}  - \Delta_{ \omega_{\psi_\delta}^\delta}) (\log {\rm tr}_{\kappa_\delta} (\omega_{\psi_\delta}^\delta) +C_0 \Phi^\beta_\delta -B\bar \psi_\delta ) \notag\\
&\le - {\rm tr}_{\omega_{\psi_\delta}^\delta}(\kappa_\delta)+C_0''. \notag
\end{align}
Now  we can apply the maximum principle to see that there exists a  uniform constant $C$, which depends only on $\hat\omega, \omega^*,   \|\psi_{s,\delta}\|_{C^0(M)}$ and  $\|\frac{\partial}{\partial s}\psi_{s,\delta}\|_{C^0(M)}$, such that
\begin{align}\omega_{\psi_{s,\delta}}^\delta=\kappa_\delta+ \sqrt{-1}\partial\bar\partial \bar\psi_{s,\delta} \ge C^{-1}\kappa_\delta.\notag
\end{align}
By (\ref{smooth-flow-MA}), we  also obtain
 \begin{align}\kappa_\delta  \ge  C'^{-1} \omega_{\psi_{s, \delta}}^\delta,\notag
\end{align}
where $C'=C'( \hat\omega, \omega^*, \|\psi_{s,\delta}\|_{C^0(M)}, \|\frac{\partial}{\partial s}\psi_{s,\delta}\|_{C^0(M)}).$
\end{proof}


\begin{theo}\label{perturbation-limit-from-flow}  For any  $s\in (0,1]$,  $\omega_{\psi_{s,\delta}}^\delta$ converge to a conic K\"ahler metric $\hat\omega_{\tilde\phi_{t,s}}=\hat\omega+\sqrt{-1}\partial\bar\partial \tilde \phi_{t,s}$
in sense of $C^{2,\alpha;\beta}$ K\"ahler potentials.

\end{theo}

\begin{proof}  By  the estimate 1) in  Lemma \ref{gradient-laplace-flow},  we have
\begin{align}\label{c0-psi-s} \|\psi_{s, \delta}\|_{C^0(M)}, \|\frac{\partial}{\partial s}\psi_{s,\delta}\|_{C^0(M)} \le e(\mu-t) \|\varphi_{t,\delta}\|_{C^0(M)}.
\end{align}
Then by   Lemma \ref {c2-parabolic-twisted-KR},  we see that there exists a  uniform constant  $C$, which depends only on $\hat\omega,\varphi_t$,  such that
$$C^{-1}  \kappa_\delta\le \omega_{\psi_{s, \delta}}^\delta \le C \kappa_\delta,~\forall ~\delta\in (0,\delta_0].$$
Thus the Sobolev constant  associated to  $\omega_{\psi_{s, \delta}^\delta}$  is uniformly bounded above  as same as  the metric $\kappa_\delta$ (cf. [LZ14]).
 Derivativing (\ref{smooth-flow-MA}) on $s$, we have
$$(\frac{\partial}{\partial s}  -\Delta_{ \omega_{\psi_{\delta}}^\delta }) \dot\psi_\delta=\mu\dot\psi_\delta,$$
where $\dot\psi_\delta=\dot\psi_{s,\delta}=\frac{\partial }{\partial s} \psi_{s,\delta}$.   Hence   the standard Moser iteration  method for the parabolic equation   implies
that there exist  a  positive number $\alpha$ and  a uniform constant $C$ such that
$$\sup_{x,y\in M} \frac{ |\dot \psi_{\delta}(x)-\dot\psi_{\delta}(y)|} {|x-y|_{\kappa_{\delta}}^\alpha}\le C.$$
As a consequece, $\dot \psi_{\delta}$ converges  to a  H\"older continueous function $f$ associated to the metric $\omega$ as $\delta\to 0$.  Namely,
$$\sup_{x,y\in M} \frac{ |f(x)-f(y)|} {|x-y|_{\omega}^\alpha}\le C.$$
  On the other hand, by the Kolodziej's H\"older estimate,  $ \psi_{\delta}$ are  uniformly  H\"older continuous functions, so they converge to
 a H\"older continuous function $\tilde\phi_t=\tilde\phi_{t,s}$ as $\delta\to 0$.  Moreover, $\tilde\phi_t$ is a current solution of following complex Monge-Amp\`ere equation,
\begin{equation}\label{limit-phi-delta-t-flow}
(\hat\omega+\sqrt{-1}\partial\bar\partial\phi)^n=e^{f+h_{\hat\omega}-\mu\phi}(\hat\omega)^n.
\end{equation}
By the reguarity theorem  in [JMR11],  it follows  that  $\tilde\phi_t$ is a $C^{2,\alpha';\beta}$-solution.  Hence $\hat\omega_{\tilde\phi_t}$ is a  conic K\"ahler metric.


\end{proof}

\begin{lem}\label{small-c0-puterbation-limit}For any $t $ and $\varphi_t$
satisfying
\begin{align}\label{t-condition}(\mu-t)^{1+
\frac{\gamma}{2}}\|\varphi_{t}\|_{C^0(M)}\le\epsilon,
\end{align}
 it holds
\begin{align}\label{c0-perturbation-2-limit} {\rm osc }_M (\tilde\phi_{t,s}-\varphi_t)\le C\epsilon(\mu -t)^{\frac{\gamma}{2}}, ~\forall ~s\in (0,1].
\end{align}
\end{lem}

\begin{proof} In  (\ref{c0-perturbation-2}),    we in fact prove
$$ {\rm osc }_M \tilde\psi_{s,\delta}\le C\epsilon(\mu -t)^{\frac{\gamma}{2}}, ~\forall ~s\in (0,1].$$
Then  (\ref{c0-perturbation-2-limit}) follows immediately from Theorem \ref{phi-delta-convergence} and Theorem \ref{perturbation-limit-from-flow} by taking $\delta\to 0$.
 \end{proof}



\begin{lem}\label{holder-estimate-perturbation}Let  $\tilde v=\tilde v_{t,s}$  be a normalization of $h_{\tilde\phi_{t,s}}$ by adding a  suitable constant such  that
$$\int_M  \tilde v (\hat\omega_{\tilde \phi_{t,s }})^n=0.$$
Then for any $t $ and $\varphi_t$
satisfying (\ref{t-condition}),
it holds
\begin{align}\label{l2-v}
 \|\tilde v_{t,s}\|_{C^{\frac{1}{2}}(M)} \le
C\epsilon (1-t)^{\frac{\gamma}{2}}, ~\forall ~s\in  [(\mu -t)^{2\gamma}, 1].
\end{align}

\end{lem}

\begin{proof} We claim: for the metric $\hat\omega_{ \tilde\phi_{t,s}}$, it holds
\begin{align}\label{metric-equivalence-t}
\frac{1}{2} \hat\omega\le \hat\omega_{\tilde\phi_{t,s}}\le 2\hat\omega.
\end{align}

By the above  claim together Theorem \ref{perturbation-limit-from-flow}, we see  that  there exists a small $\delta_0$ such that the conditions (\ref{large-eigenvalue}) and (\ref{volume-condition})  in Lemma \ref{holder-estimate-perturbation-flow}
are satisfied for metrics $\omega_{\psi_{s,\delta}}^\delta$ with $\delta\in(0,\delta_0]$. Note that  (\ref{t-condition-phi})  also holds by Theorem \ref{phi-delta-convergence}.
Then by    Lemma \ref{c2-parabolic-twisted-KR},   it follows that   (\ref{holder-estimate-h-delta})  holds for $v=v_{t,s,\delta}$ with   $\delta\in (0, \delta_0]$. Taking the limit of $v_{t,s,\delta}$ as $\delta\to 0$, we get  (\ref{l2-v}).

We prove   (\ref{metric-equivalence-t}) by contradiction. If  (\ref{metric-equivalence-t}) is not true, then there exists a $\psi\in\mathscr{H}^{2,\alpha;\beta}(M,\omega_0)$ such that
the solution $\varphi_t$ of  (\ref{backward_MAE})  on $t$ satisfies (\ref{t-condition}) and the  $C^{2,\alpha;\beta}$-norm of  K\"ahler potential $\tilde\phi_{t,s}$ in Theorem \ref{perturbation-limit-from-flow} satisfies
\begin{align}\label{big-estimate}\|\tilde\phi_{t,s}\|_{C^{2,\alpha;\beta}(M)}\ge A_0,
\end{align}
where $A_0$ is a positive  number.
On the other hand, from the proof of Theorem \ref{perturbation-limit-from-flow}, the  $C^{2,\alpha;\beta}$-norm of  $\tilde\phi_{t,s}$ depends on $\varphi_t$ continuously. Thus we may aslso assume that
$$\|\tilde\phi_{t,s}\|_{C^{2,\alpha;\beta}(M)}\le 2A_0$$
and
\begin{align}\label{metric-equivalence-t-2}
\frac{1}{4}  \hat\omega\le \hat\omega_{\tilde\phi_{t,s}}\le  4\hat\omega.
\end{align}

Once  (\ref{metric-equivalence-t-2}) holds,  we can use the above argument again to
 conclude that  there exists a small $\delta_0$ such that   (\ref{holder-estimate-h-delta})  holds for $v=v_{t,s,\delta}$ with   $\delta\in (0, \delta_0]$. Taking the limit of $v_{t,s,\delta}$ as $\delta\to0$, we get  (\ref{l2-v})
 for   $\hat\omega_{\tilde\phi_{t,s}}$.
Applying  Implicity Function  Theorem to  (\ref{limit-phi-delta-t-flow}), we  obtain
\begin{align}\|\tilde\phi_{t,s}\|_{C^{2,\alpha;\beta}(M)}\le C(\epsilon)\to 0,~{\rm as}~\epsilon\to 0.\notag
\end{align}
But this is impossible by (\ref{big-estimate}). The claim is proved.

\end{proof}

\section{Properness of  $F_{\omega_0,\mu}(\cdot)$}

By using the estimates at  last section,  we can improve Theorem \ref{Bern_thm} to

\begin{theo}\label{tian-zhu-conic-ke-metric-weak}    Suppose that  there exists  a conic   K\"ahler-Einstein metric $\omega=\omega_{CKE}$ on $M$  along
$D$ with cone angle $2\pi\beta\in (0,2\pi).$ Then there exists two uniform constants $\delta$  and $C$ such that
\begin{align}\label{nonlinear-ms}F_{\omega_0,\mu}(\psi)\ge \delta I(\psi)^{\frac{1}{8n+9}} -C,~\forall~\psi\in \mathscr{H}^{2,\alpha; \beta}(M,\omega_0).
\end{align}
\end{theo}


\begin{proof}

First,  by  the first relation in  (\ref{integral-dt}),  we get an identity
 $$F_{\omega,\mu}(\psi-\phi)=F_{\omega,\mu}(-\varphi_\mu)=\frac{1}{\mu}\int_0^\mu(I-J)_{\hat{\omega}}(\varphi_s)ds.$$
Then as in [TZ00], [CTZ05],  we obtain
\begin{align}\label{f-inequiality}&F_{\omega,\mu}(\psi-\phi)\notag\\
&\ge \frac{1}{\mu}(\mu-t)(I-J)_{\hat\omega}(\varphi_t)\notag\\
&\ge \frac{1}{n\mu}(\mu-t)J_{\hat\omega}(\varphi_t)\notag\\
&\ge  \frac{1}{n\mu}(\mu-t)J_{\hat\omega}(\varphi_\mu)   -  \frac{1}{n\mu}(\mu-t) {\rm osc}_M(\varphi_t-\varphi_\mu)\notag\\
&\ge  \frac{1}{n\mu(n+1)}(\mu-t)I_{\hat\omega}(\varphi_\mu)   -  \frac{1}{n\mu}(\mu-t) {\rm osc}_M(\varphi_t-\varphi_\mu).
\end{align}

Next,   for a small $\epsilon$,  we choose  a $t$
such that
\begin{align}\label{special-condition} (\mu-t)^{1+\frac{\gamma}{2}}\|\varphi_t\|_{C^0(M)}=\epsilon.
\end{align}
 Without loss of generality, we may assume that the above inequality can be obtained,
  otherwise $\|\varphi_t\|_{C^0(M)}$ is  unifromly bonuded and  the situation will be simple.
   Then by   Theorem \ref{perturbation-limit-from-flow} and  Lemma \ref{small-c0-puterbation-limit},   there exists  a  $C^{2,\alpha'; \beta}$ K\"ahler potential  $\tilde\varphi_t$ such that
\begin{align}{\rm osc}_M(\varphi_t-\varphi_\mu)&  \le {\rm osc}_M (\tilde\varphi_t -\varphi_\mu)+ {\rm osc}_M( \tilde \varphi_t -\varphi_t)\notag\\
&\le {\rm osc}_M (\tilde\varphi_t -\varphi_\mu)+  C\epsilon(\mu -t)^{\frac{\gamma}{2}}.
\notag
\end{align}
On the other hand,   by   Lemma \ref{holder-estimate-perturbation},  we can apply  Implicity Function  Theorem to  (\ref{limit-phi-delta-t-flow}) to get
 $${\rm osc}_M (\tilde\varphi_t -\varphi_\mu)\le C(\epsilon)\to 0,~{\rm as}~ \epsilon\to 0.$$
 Thus
\begin{align}\label{osc-phi-t-phi-mu}{\rm osc}_M(\varphi_t-\varphi_\mu)\le C(\epsilon).
\end{align}

Combining (\ref{f-inequiality}) and (\ref{osc-phi-t-phi-mu}), we have
 \begin{align}\label{f-proper-1}F_{\omega,\mu}(\psi-\phi)\ge  \frac{1}{n\mu(n+1)}
(\mu-t)I(\varphi_\mu)-C.
\end{align}
 Note
 $$\|\varphi_t\|_{C^0(M)}\le {\rm osc}_M(\varphi_t).$$
Then by (\ref{osc-phi-t-phi-mu}), we get
  \begin{align}\label{osc-phi-t}\|\varphi_t\|_{C^0(M)}\le {\rm osc}_M(\varphi_\mu)+1.
\end{align}

In a special case, we assume that the K\"ahler potential $\psi$ satisfies
\begin{align}\label{osc-condition}
{\rm osc}_M\psi\le I_{\omega_0}(\psi)+C_0,
\end{align}
where $C_0$ is a uniform constant.
Then by the relation (\ref{special-condition}) and (\ref{osc-phi-t}), a simple computation shows
\begin{align}\label{special-case-1} F_{\omega,\mu}(\psi-\phi)&\ge \delta
I_{\omega_0}(\varphi_\mu)^{\frac{1}{8n+9}}-C'\notag\\
&\ge \delta
I_{\omega_0}(\psi)^{\frac{1}{8n+9}}-C',
\end{align}
 where $\delta,  C'>0$ are two uniform constants which depending only on the choice of $\epsilon$ in (\ref{special-condition}).   Using  the cocycle condition in  (\ref{cocycle-condition}), we derive immediately,
\begin{align}\label{special-case}F_{\omega_0,\mu}(\psi)\ge \delta
I_{\omega_0}(\psi)^{\frac{1}{8n+9}}-C''.
\end{align}
In  general case, we can use a trick in [TZ00] to derive (\ref{special-case}) for $\psi$. In fact, we can first  apply  (\ref{special-case})  for solutions $\varphi_t$ with $t\ge\epsilon_0>0$ to get an estimate for
${\rm osc}_M(\varphi_t-\varphi_\mu)$,  then by  the relation in  (\ref{f-inequiality})
 we   obtain  (\ref{special-case}) for $\psi$.

\end{proof}


\begin{proof}[End of proof of  Theorem  \ref{tian-zhu-conic-ke-metric}]Theorem  \ref{tian-zhu-conic-ke-metric} is an improvement of Theorem \ref{tian-zhu-conic-ke-metric-weak}.  By Lemma \ref{functional-smoothing},
 we  suffice  to obtain  the esitimate (\ref{proper-inequality}) in Theorem  \ref{tian-zhu-conic-ke-metric} for
K\"ahler potentials in $\mathscr{H}^{2,\alpha; \beta}(M,\omega_0)$.
 It  was observed by Phong-Song-Strum-Weinkove that (\ref{proper-inequality})  can  come from  (\ref{nonlinear-ms})
in case of K\"ahler-Einstein metric [PSSW08].   In fact, as in [TZ00], by   (\ref{nonlinear-ms}) for  solutions $\varphi_t$ with $t\ge\epsilon_0>0$, they further show that there exists a $t_0$ with $\mu-t_0\ge \delta_0>0$ (where $\mu=1$)
  for some  uniform constant $\delta_0$  such that
$${\rm osc}_M(\varphi_{t_0}-\varphi_\mu)\le A,$$
where $A$ is  a uniform constant which depends only on  the K\"ahler-Einstein metric.  We show  that such choice of $t_0$ can be done similarly  in our case of  conic K\"ahler-Einstein metric $\omega_\phi$ as follows.

By  the first  relation in (\ref{integral-dt}) together with  the equation (\ref{backward_MAE}), we have
\begin{align} &F_{\omega,\mu}(\varphi_t-\varphi_\mu)=F_{\hat\omega,\mu}(\varphi_t)-F_{\hat\omega,\mu}(\varphi_\mu)\notag\\
&= -\frac{1}{t}\int_0^t (I-J)_{\hat{\omega}}(\varphi_s)ds+\frac{1}{\mu}\int_0^{\mu} (I-J)_{\hat{\omega}}(\varphi_s)ds -\frac{1}{\mu} \log(\frac{1}{V}\int_M e^{ (t-\mu)\varphi_t}\hat\omega_{\varphi_t}^n)\notag\\
&\le-\frac{1}{t}\int_0^t (I-J)_{\hat{\omega}}(\varphi_s)ds+\frac{1}{\mu}\int_0^{\mu} (I-J)_{\hat{\omega}}(\varphi_s)ds+ \frac{\mu-t}{\mu V}\int_M  \varphi_t\hat\omega_{\varphi_t}^n.\notag
\end{align}
Note that  the first  relation in (\ref{integral-dt}) is equivalent to
$$- \frac{1}{V} \int_M \varphi_t\hat\omega_{\varphi_t}^n= (I-J)_{\hat{\omega}}(\varphi_t)-\frac{1}{t} \int_0^t  (I-J)_{\hat{\omega}}(\varphi_s)ds.$$
It follows
\begin{align}\label{f-upper-bound} F_{\omega,\mu}(\varphi_t-\varphi_\mu)&\le  \frac{1}{\mu} \int_t^{\mu}  (I-J)_{\hat{\omega}}(\varphi_s)ds -\frac{\mu-t}{\mu} (I-J)_{\hat{\omega}}(\varphi_t) \notag\\
 &\le \frac{\mu-t}{\mu} [(I-J)_{\hat{\omega}}(\varphi_\mu) - (I-J)_{\hat{\omega}}(\varphi_t)]\notag\\
&\le \frac{n(\mu-t)}{\mu}{\rm osc}_M(\varphi_t-\varphi_\mu).
\end{align}

On the other hand,  by the Green formula in [JMR11],  (\ref{osc-condition}) holds for $\varphi_t-\varphi_\mu$ whenever $t\ge \epsilon_0>0$
 since  Ricci curvature of $\hat\omega_{\varphi_t}=\omega+\sqrt{-1}\partial\bar\partial (\varphi_t-\varphi_\mu)$  is strictly positive.
Then applying  (\ref{special-case-1}) for $\varphi_t-\varphi_\mu$, we see that there exist two constants $A_0, C>0$  such that
$$F_{\omega,\mu}(\varphi_t-\varphi_\mu)\ge  A_0
I_\omega(  \varphi_t-\varphi_\mu)^{\frac{1}{8n+9}}-C, ~\forall ~t\ge\epsilon_0.$$
Combining this with (\ref{f-upper-bound}), we derive
\begin{align}\label{i-functional-relation}
I_{\omega} ( \varphi_t-\varphi_\mu)^{\frac{1}{8n+9}} [A_0-\frac{n(\mu-t)}{\mu} I_{\omega} (  \varphi_t-\varphi_\mu)^{1-\frac{1}{8n+9}}]\le C'.
\end{align}

Case 1:  For any $t\in [\frac{\mu}{2},\mu]$,  it holds
 $$(\mu-t) I_{\omega} (  \varphi_{t}-\varphi_\mu)^{1-\frac{1}{8n+9}}<\frac{A_0\mu}{2n}.$$
Then   we can choose $t=t_0=\frac{\mu}{2}$ so that
$ I_{\omega} (  \varphi_{t_0}-\varphi_\mu)$,  and also ${\rm osc}_M( \varphi_{t_0}-\varphi_\mu)$ is uniformly bounded.
 Thus by  the relation in (\ref{f-inequiality}), we get
 (\ref{proper-inequality}).  The proof is finished.

Case 2:  There exists a $t_0\in [\frac{\mu}{2},\mu]$ such that
  \begin{align}\label{t-condition-2}  (\mu-t_0) I_{\omega} (  \varphi_{t_0}-\varphi_\mu)^{1-\frac{1}{8n+9}}=\frac{A_0\mu}{2n}.
\end{align}
By the  above choice of $t_0$, from (\ref{i-functional-relation})  it is easy to see that   ${\rm osc}_M( \varphi_{t_0}-\varphi_\mu)$ is uniformly bounded.
Again by  (\ref{t-condition-2}), we get $\mu-t_0\ge \delta_0>0$ for some  uniform constant $\delta_0$. The theorem is proved.


There is another way to get (\ref{proper-inequality}) by using the Donaldson's openness   theorem,  Theorem \ref{Do_openness} in next section. This is observed in [LS12].


\end{proof}




\section{A new proof of Donaldson's openness   theorem}

In this section, we apply Theorem \ref{tian-zhu-conic-ke-metric-weak}  to  prove  the following Donaldson's openness   theorem.

\begin{theo}\label{Do_openness}Let $D$ be a smooth divisor of a Fano manifold $M$ with $[D]\in \lambda c_1(M)$ for some $\lambda>0$ such  that there is no non-zero holomorphic field
which is tangent to $D$ along $D$. 
Suppose that there exists  a conic  K\"ahler-Einstein metric on $M$ with  cone angle $2\pi\beta_0\in (0,2\pi)$ along $D$. 
 Then for any $\beta$ close to $\beta_0$ there exists a conic K\"ahler-Einstein metric  with  cone angle $2\pi\beta$.
\end{theo}

\begin{proof}  Let $\mu_0= 1-\lambda(1-\beta_0)$.   Then  $F_{\omega_0,\mu_0}(\cdot)$ is proper by Theorem \ref{tian-zhu-conic-ke-metric-weak}. Thus
   twisted $F$-functionals $F_{\mu_0,\delta}(\cdot)$ defined by (\ref{twisted-f-functional-t})  are all proper for
any $\delta\in (0,\delta_0]$.  By the argument in Section 3,   it follows  that  there exists a solution of  (\ref{smooth-continuity-modified})  on $t=\mu_0$ for any  $\delta\in (0,\delta_0]$.  Hence,  for a fixed $\delta=\frac{\delta_0}{2}$, we
apply  Implicit  Function  Theorem to see that there exists an $\epsilon_0$ such that (\ref{smooth-continuity-modified})   is  solvable for any  $\mu\in [\mu_0,\mu_0+\epsilon_0)$. Note that  the  twisted Ricci potential
  $h_\delta$ of $\omega_\delta$
in  (\ref{smooth-continuity-modified}) satisfies
(\ref{twisted-h-delta}) with $\beta=1- \frac{1-\mu}{\lambda}$.
This means that there exists a twisted K\"ahler-Einstein metric
associated to positive $(1,1)$-form $\Omega=  (1-\beta)\eta_{\frac{\delta_0}{2}}$ for any $\mu\in [\mu_0,\mu_0+\epsilon_0)$.   By a result of  X. Zhang and X. W. Zhang [ZhZ13],  the twisted $F$-functionals $F_{\mu,\frac{\delta_0}{2}}(\cdot)$ is  proper for any $\mu\in [\mu_0,\mu_0+\epsilon_0)$. In fact,  they prove a version of Theorem \ref{tian-zhu-conic-ke-metric-weak} on a  Fano manifold  which admits a twisted K\"ahler-Einstein metric.

 By  a direct computation,  it is easy to see that for any $\delta\in (0,\delta_0)$ it holds
$$|F_{\mu,\frac{\delta_0}{2}}(\psi)-F_{\mu,\delta}(\psi)|\le C(\|\Psi_{\delta}\|_{C^0(M)}, \|\Psi_{\frac{\delta_0}{2}}\|_{C^0(M)})\le C, ~\forall~\psi\in  \mathcal H(M,\omega_0).$$
This implies that $ F_{\mu,\delta}(\cdot)$ are all proper for any  $\delta\in (0,\delta_0)$ and $\mu\in [\mu_0,\mu_0+\epsilon_0)$.
Thus by the argument in Section 3,  we see that  there exists a solution $\varphi_{\mu, \delta}$
 of  (\ref{smooth-continuity-modified})  on $t=\mu\in  [\mu_0,\mu_0+\epsilon_0)$ for any  $\delta\in (0,\delta_0)$.
Moreover,
$${\rm osc}_M\varphi_{\mu, \delta}\le C,$$
  where  $C$   is a uniform constant   independent of $\mu$ and $\delta$.
On the other hand,  the estimate (\ref{c2-estimate-twisted-metrics})  in Lemma \ref{c2-estimate-twisted-metrics-lemma}  also holds for metrics $\omega_{\varphi_{\mu, \delta}}^\delta$.
 By taking a sequence $\delta_i\to 0$,   $\varphi_{\mu,\delta_i}$ converge   to a H\"older continueous function $\varphi_\infty -\psi$  which satisfies the   weak conic K\"ahler-Einstein metric equation,
$${\rm Ric}(\omega_{\varphi_\infty})=\mu\omega_{\varphi_\infty}+2\pi  (1-\beta)[D], ~\mu\in [\mu_0, \mu_0+\epsilon_0)$$
with property:
$$C^{-1}  \hat\omega\le \omega_{\varphi_\infty} \le C \hat\omega, ~{\rm in}~ M\setminus D$$
for some uniform positive number  $C$.
By  the regularity theorem in [JMR11],   $\omega_{\varphi_\infty}$ is a conic K\"ahler-Einstein metric in sense of $C^{2,\alpha;\beta}$ K\"ahler potentials.
The proof of  Theorem \ref{Do_openness} is completed.

\end{proof}


\vskip20mm

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