# Uniqueness and short time regularity of the weak Kähler-Ricci flow

Eleonora Di Nezza  Department of Mathematics, Imperial College London London,
SW7 2AZ, United Kingdom<EMAIL_ADDRESS>and  Chinh H. Lu
Mathematical Sciences, Chalmers University of Technology, 412 96 Gothenburg,
Sweden<EMAIL_ADDRESS>
(Date:
The authors are partially supported by the French ANR project MACK. The
second-named author is supported by the European Research Councils)

###### Abstract.

Let $X$ be a compact Kähler manifold. We prove that the Kähler-Ricci flow
starting from arbitrary closed positive $(1,1)$-currents is smooth outside
some analytic subset. This regularity result is optimal meaning that the flow
has positive Lelong numbers for short time if the initial current does. We
also prove that the flow is unique when starting from currents with zero
Lelong numbers.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 A priori estimates
  4. 4 Regularity of weak solutions
  5. 5 Uniqueness and stability
  6. 6 Convergence at time zero

##  1\. Introduction

Over the last few decades, the Ricci flow introduced by Hamilton in [22]

$\frac{dg}{dt}=-2{\rm Ric}(g)$

has found spectacular applications in Riemannian geometry [27], [28], [29]. As
observed by Bando, starting from a Kähler metric, the Kähler property is
preserved by the Ricci flow and the resulting flow is called the Kähler-Ricci
flow. Its long time existence is now well known thanks to the work of Cao [6],
Tsuji [39] and Tian-Zhang [38]. Convergence of the flow at infinity is also of
great interest as shown in [6], as this effect can be used to find Kähler-
Einstein metrics if they exist. Since then, the Kähler-Ricci flow, as well as
its twisted version, has been studied intensively.

Of particular interest is the attempt to run the Kähler-Ricci flow from
singular data (see [7], [8], [34], [37], [21]), and it can be considered as an
alternative way to regularize currents. Another motivation for studying the
weak Kähler-Ricci flow comes from an analytic analogue of Mori’s Minimal Model
Program. It is conjectured (see [34], [35], [36]) that the minimal model of a
projective algebraic variety can be obtained as the eventual limit of the
Kähler-Ricci flow after passing through singularities. At each singular time,
the flow develops singularities along some analytic subset of $X$, and so to
continue through these singularities it is necessary to restart the flow from
singular objects.

In [34], Song-Tian succeeded in starting the flow from continuous initial
data. A remarkable progress has been recently made by Guedj and Zeriahi [21]
which allows to define a unique maximal flow starting from any closed positive
$(1,1)$-current. Moreover, as shown in [21], the flow slowly smooths out the
initial current and the speed of the regularizing effect depends on the value
of the Lelong number of the initial current.

The scalar parabolic Monge-Ampère equation. Before going further and stating
the main results of the paper, let us fix some notations. Let $X$ be a compact
Kähler manifold of dimension $n$ and $\alpha_{0}\in H^{1,1}(X,\mathbb{R})$ a
Kähler class. Fix $\eta$ a smooth closed $(1,1)$-form on $X$. The
$\eta$-twisted Kähler-Ricci flow, introduced in [9], is the following:

(1.1) $\frac{\partial\omega_{t}}{\partial t}=-{\rm Ric}(\omega_{t})+\eta\ ,\
\omega_{t}|_{t=0}=T_{0},$

where $T_{0}$ is a fixed closed positive $(1,1)$-current in $\alpha_{0}$. When
$T_{0}$ is a Kähler form, as mentioned above, the flow admits a unique smooth
solution on a maximal interval $[0,T_{\max})$, where

$T_{\max}:=\sup\\{t\geq 0\,|\,tK_{X}+t\\{\eta\\}+\alpha_{0}\;\;\mbox{is
nef}\\}.$

It is standard (and more convenient) to rewrite the flow as a scalar parabolic
Monge-Ampère equation. Fix $\omega$ a Kähler form in $\alpha_{0}$. Let
$\varphi_{0}$ be a global potential of $T_{0}$, i.e.
$T_{0}:=\omega+dd^{c}\varphi_{0}$. Set

$\chi:=\eta-{\rm Ric}(\omega),\ \theta_{t}:=\omega+t\chi,$

and consider the following equation

(1.2) $\frac{\partial\varphi_{t}}{\partial
t}=\log\left[\frac{(\theta_{t}+dd^{c}\varphi_{t})^{n}}{\omega^{n}}\right],\
\varphi_{t}\to\varphi_{0}\ {\rm as}\ t\to 0.$

If $\varphi_{t}$ solves (1.2) then a straightforward computation shows that
$\omega_{t}:=\theta_{t}+dd^{c}\varphi_{t}$ solves the flow (1.1). Conversely,
if $\omega_{t}$ solves the flow (1.1) then it follows from the $dd^{c}$-lemma
that we can write

$\omega_{t}=\theta_{t}+dd^{c}\varphi_{t},$

where $\varphi_{t}$ solves the parabolic Monge-Ampère equation (1.2).

The maximal Kähler-Ricci flow. Fix $T_{0}=\omega+dd^{c}\varphi_{0}$ a closed
positive $(1,1)$ current in the class $\alpha_{0}$. The integrability index of
$T_{0}$ (or $\varphi_{0}$) is defined by

$c(T_{0})=c(\varphi_{0}):=\sup\left\\{\lambda>0\ \big{|}\
e^{-2\lambda\varphi_{0}}\in L^{1}(X)\right\\}.$

Assume that $1/2c(T_{0})<T_{\max}$. Let $\varphi_{0,j}$ be a sequence of
smooth $\omega_{0}$-psh functions decreasing to $\varphi_{0}$. Such sequences
exist thanks to [11] (see also [3]). Let $\varphi_{t,j}$ be the unique
solution of the parabolic equation (1.2) with initial data $\varphi_{0,j}$. As
shown by Guedj and Zeriahi in [21], as $j\to+\infty$ the sequence
$\varphi_{t,j}$ decreases to $\varphi_{t}$ which satisfies the following:

  * •

For each $t>0$, $\varphi_{t}$ is a $\theta_{t}$-psh function. Moreover, if
$t>1/2c(T_{0})$, $\varphi_{t}$ is smooth on $X$ and solves (1.2) in the
classical sense.

  * •

$\varphi_{t}$ converges in capacity to $\varphi_{0}$ as $t\to 0$.

Remark. The assumption $1/2c(\varphi_{0})<T_{\max}$ is necessary to insure
that the maximal scalar solution $\varphi_{t}$ is well-defined. Without this
condition it can happen that the sequence $\varphi_{t,j}$ decreases to
$-\infty$ (see example 4.1).

The flow constructed by approximation as above was also shown to be maximal
meaning that it dominates any “weak" solution of (1.2). The regularity of this
flow was obtained for $t$ not too small and nothing was known when
$t<1/2c(T_{0})$. Example 6.4 in [21] suggests that there might be no
regularity at all due to the presence of positive Lelong numbers. However, as
in Demailly’s regularization theorem [12], one can expect that the
regularizing effect happens outside some analytic subset. Our first result
shows that it is indeed the case.

Theorem A. Assume that $1/2c(T_{0})<T_{\max}$. Then the maximal Kähler-Ricci
flow starting from $T_{0}$ is smooth in a Zariski open subset of $X$.

More precisely, the Zariski open subset in Theorem A is described by the
complement of Lelong superlevel sets of $T_{0}$. For each $s>0$ we define

$D_{s}:=\\{x\in X\ \big{|}\ \nu(T_{0},x)\geq s\\}.$

The precise statement of Theorem A says that for each $\varepsilon>0$ there
exists an analytic subset $Y_{\varepsilon}:=D_{\kappa(\varepsilon)}$ such that
$\omega_{t}$ is smooth in $X\setminus Y_{\varepsilon}$ for every
$t>\varepsilon$. Here the constants $\kappa(\varepsilon)$ depends only on
$\varepsilon$ and $T_{0}$ in such a way that it decreases to $0$ as
$\varepsilon$ goes to $0$. These constants $\kappa(\varepsilon)$ are
approximately $\varepsilon$, up to an error term in Demailly’s regularizing
process. An interesting particular case is when $\omega$ represents the first
Chern class of an ample holomorphic line bundle over $X$. In this case the
error term is zero and $\kappa(\varepsilon)$ can be taken to be $\varepsilon$.
Siu’s theorem [31] guarantees that $Y_{\varepsilon}$ are analytic subsets of
$X$.

We also show that our result in Theorem A is optimal in the sense that any
maximal Kähler-Ricci flow starting from currents having positive Lelong
numbers has positive Lelong numbers in short time (see Theorem 4.5).

Our previous analysis gives rise to the following definition.

Definition. A function $\varphi:(0,T_{\max})\times X\to\mathbb{R}$ is called a
weak solution of the equation (1.2) starting from $\varphi_{0}$ if the
following conditions are satisfied:

  * (i)

For each $t>0$, the function $\varphi_{t}$ is $\theta_{t}$-psh on $X$,

  * (ii)

The function $t\mapsto\varphi_{t}$ is continuous as a map from $[0,T_{\max})$
to $L^{1}(X)$.

  * (iii)

For each $\varepsilon>0$ there exists an analytic subset
$D_{\varepsilon}\subset X$ such that the function

$(t,x)\mapsto\varphi_{t}(x)$

is smooth on $[\varepsilon,T_{\max})\times(X\setminus D_{\varepsilon})$ where
the equation (1.2) is satisfied in the classical sense. Moreover
$D_{\varepsilon}=\emptyset$ if $\varepsilon>1/(2c(\varphi_{0}))$.

A family $(\omega_{t})$ of closed positive $(1,1)$ currents is called a weak
Kähler Ricci flow (1.1) starting from $T_{0}=\omega+dd^{c}\varphi_{0}$ if
$\omega_{t}=\theta_{t}+dd^{c}\varphi_{t}$ and $\varphi_{t}$ is a weak solution
of the equation (1.2).

The second goal of this paper is to study the stability and the uniqueness of
the weak Kähler-Ricci flow. As mentioned above, the flow constructed in [21]
is maximal among weak solutions of the parabolic equation (1.2). But it is not
clear whether this maximal solution is the unique weak Kähler-Ricci flow. When
the inital current has zero Lelong numbers, we prove that the uniqueness
holds:

Theorem B. Assume that $c(T_{0})=+\infty$. Then the following holds:

  * •

Uniqueness: Any weak Kähler-Ricci flow starting from $T_{0}$ is maximal. In
other words the flow is unique.

  * •

Stability: Assume that $T_{0,j}$ is a sequence of positive closed
$(1,1)$-currents converging to $T_{0}$ in the $L^{1}$ topology. Then the
corresponding Kähler Ricci flow $\omega_{t,j}$ converges to $\omega_{t}$ in
the following sense: for each $t\in(0,T_{\max})$, $\omega_{t,j}$ converges in
$\mathcal{C}^{\infty}(X)$ to $\omega_{t}$ as $j\to+\infty$.

The case when $T_{0}$ has positive Lelong number is more delicate as the
maximal flow has unbounded potential in short time. The usual method using the
classical maximum principle and the pluripotential comparison principle, which
are the main tools in studying parabolic equations, breaks down. However, we
also have a partial result:

Theorem B’. Assume that $c(T_{0})$ is finite and $1/(2c(T_{0}))<T_{\max}$.
Then any weak flow having the same singularities as the maximal flow is also
maximal.

Here, we say that two flows $\omega_{t},\Omega_{t}$ have the same singularity
if their potentials $u_{t},v_{t}$ satisfy

$u_{t}-C\leq v_{t}\leq u_{t}+C,\ \forall t$

for some constant $C$.

When the initial current $T_{0}$ has some local regularity properties it is
natural to ask for stronger convergence of the flow when the time goes to
zero. If $T_{0}$ has continuous potential it was shown in [34] that the
solution $\varphi_{t}$ converges uniformly to $\varphi_{0}$. When $T_{0}$ has
finite energy the convergence is in energy as shown in [21]. In the last
section we prove the following:

Theorem C. Let $\varphi_{0}\in{\rm PSH}(X,\omega)$ be such that
$1/2c(\varphi_{0})<T_{\max}$ and $\varphi_{t}$ be the maximal solution of
(1.2) starting from $\varphi_{0}$.

  * •

If $\varphi_{0}$ is continuous in an open subset $U\subset X$ then
$\varphi_{t}$ converges locally uniformly to $\varphi_{0}$ in $U$.

  * •

If $e^{\gamma\varphi_{0}}\in\mathcal{C}^{\infty}(X)$ for some positive
constant $\gamma$ and $\varphi_{0}$ is strictly $\omega$-plurisubharmonic on
$X$, then $\varphi_{t}$ converges in $\mathcal{C}^{\infty}_{\rm loc}(\Omega)$
to $\varphi_{0}$, where $\Omega=\\{\varphi_{0}>-\infty\\}$.

The first statement generalizes the uniform convergence of Song-Tian [34]. The
second one covers interesting particular cases of currents having analytic
singularities.

The organization of the paper is as follows. In section 2 we recall basic
pluripotential theory that will be needed later on. In Section 3 we establish
various a priori estimates. The proof of Theorem A will be given in Section 4
while the proof of Theorem B and Theorem B’ will be given in Section 5. In
Section 6 we investigate fine convergence of the flow at time $t=0$ when more
regularity on the initial current is assumed, proving Theorem C (see Theorem
6.1 and Theorem 6.2).

Acknowledgement. We are very grateful to Vincent Guedj and Ahmed Zeriahi for
useful discussions and suggestions. We also thank Bo Berndtsson for many
stimulating discussions during lunch time.

##  2\. Preliminaries

In this section we recall classical results in pluripotential theory that will
be needed in the sequel. We also review basic techniques in parabolic complex
Monge-Ampère equations.

###  2.1. Recap on pluripotential theory

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$. The
set ${\rm PSH}(X,\omega)$ contains all $\omega$-psh functions on $X$
($\omega$-psh for short). A function $u$ is called $\omega$-psh on $X$ if it
is upper semicontinuous and integrable on $X$, and $\omega+dd^{c}u\geq 0$ in
the weak sense of currents. For a bounded $\omega$-psh function $u$, it
follows from [1] that the Monge-Ampère operator $(\omega+dd^{c}u)^{n}$ is
well-defined as a non-negative Borel measure on $X$. If $u\in{\rm
PSH}(X,\omega)$ is unbounded, we can still define the non-pluripolar Monge-
Ampère measure of $u$. It follows from the local property in the plurifine
topology of the Monge-Ampère operator that the following sequence

${\bf 1}_{\\{u>-j\\}}(\omega+dd^{c}\max(u,-j))^{n}$

is non-decreasing in $j$. Its limit as $j\to+\infty$ is called the non-
pluripolar Monge-Ampère measure of $u$, denoted by

$\mathrm{MA}\,(u)=(\omega+dd^{c}u)^{n}:=\lim_{j\to+\infty}(\omega+dd^{c}\max(u,-j))^{n}.$

As shown in [20], $\mathrm{MA}\,(u)$ vanishes on pluripolar sets. Its total
mass takes values in $[0,\int_{X}\omega^{n}]$. As defined in [20], $u$ belongs
to $\mathcal{E}(X,\omega)$ if the total mass of its non-pluripolar Monge-
Ampère measure is maximal, i.e. $\int_{X}\mathrm{MA}\,(u)=\int_{X}\omega^{n}$.
As shown in [20], functions in $\mathcal{E}(X,\omega)$ have mild
singularities: if $u\in\mathcal{E}(X,\omega)$, it has zero Lelong number at
any point in $X$. This together with Skoda’s theorem [33] yield

$\int_{X}e^{-Au}dV<+\infty,\ \forall A>0.$

Observe that the integrability index $c(u)=+\infty$ if and only if
$\nu(u,x)=0$ for any $x\in X$.

###  2.2. The generalized Monge-Ampère capacity

In [14, 15] we introduce and study the so called generalized Monge-Ampère
capacity defined as

${\rm Cap}_{\psi}(E):=\sup\left\\{\int_{E}\mathrm{MA}\,(u)\,|\,u\in{\rm
PSH}(X,\omega),\;\;\psi-1\leq u\leq\psi\right\\}\qquad E\subset X$

where $\psi$ is a possibly singular $\delta\omega$-psh function,
$\delta\in(0,1)$, and $\mathrm{MA}\,(u)$ denotes the non-pluripolar Monge-
Ampère measure of $u$. Observe that if $\psi\equiv C$, then we recover the
classical Monge-Ampère capacity introduced in [1], [25], [19]. Such
generalized capacities are the key ingredient when studying singularities of
solutions of degenerate complex Monge-Ampère equations and they enjoy several
nice properties. Among those, it can be proven that when
$\psi\in\mathcal{E}(X,\omega)$ then the $\psi$-capacity characterizes
pluripolar sets, namely

${\rm Cap}_{\psi}^{*}(E)=0\Longleftrightarrow\ \text{E is a pluripolar subset
of}\ X,$

where ${\rm Cap}_{\psi}^{*}$ is the outer $\psi$-capacity defined for any
subset $E\subset X$ by

${\rm Cap}_{\psi}^{*}(E):=\inf\left\\{{\rm Cap}_{\psi}(U)\ \big{|}\ U\
\text{is an open subset of }X,\ E\subset U\right\\}.$

We refer the reader to [15] for more details.

Here below we recall a classical lemma (which goes back to [26]) that it is
used in the proof of Theorem 3.3.

######  Lemma 2.1.

Let $g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a non-increasing, right
continuous function such that $g(+\infty)=0$. Assume that there exists $C>0$
such that $g$ satisfies

(2.1) $s\,g(t+s)\leq C\,g(t)^{2}\qquad\quad\forall s\in[0,1],\,t>0.$

Let $t_{0}>0$ be such that $g(t_{0})\leq(2C)^{-1}$. Then we have

$g(t)=0,\ \ \forall t\geq 2+t_{0}.$

We refer the reader to [26] (or [16]) for a proof of the above lemma.

###  2.3. The comparison principle

One of the main tools in studying the Kähler-Ricci flow is the comparison
principle. The following version is classical:

######  Proposition 2.2.

Let $u,v:\ [0,T_{\max})\times X\rightarrow\mathbb{R}$ be smooth functions such
that $u_{t},v_{t}$ are strictly $\theta_{t}$-psh functions and

$\dot{u_{t}}\leq\log\left[\frac{(\theta_{t}+dd^{c}u_{t})^{n}}{\omega^{n}}\right]\quad\mbox{and}\quad\dot{v_{t}}\geq\log\left[\frac{(\theta_{t}+dd^{c}v_{t})^{n}}{\omega^{n}}\right].$

Then $u_{t}-v_{t}\leq\sup_{X}(u_{0}-v_{0})$. In particular if
$\varphi_{t},\psi_{t}$ are solutions of (1.2) and $\varphi_{0}\geq\psi_{0}$
then $\varphi_{t}\geq\psi_{t}$ for all $t>0$.

We refer to [21, Corollary 2.2] for a proof.

In dealing with the weak Kähler-Ricci flow it is often more convenient to use
the following generalized comparison principle, which allows the subsolution
to have singularities:

######  Proposition 2.3.

Assume that $\varphi:[0,T_{\max})\times X\rightarrow\mathbb{R}$ is a smooth
solution of the parabolic equation (1.2). Assume that $\psi$ is a weak
subsolution of (1.2) with initial data $\psi_{0}\leq\varphi_{0}$ and
$c(\psi_{0})=+\infty$. Then $\psi_{t}\leq\varphi_{t},\ \forall
t\in[0,T_{\max})$.

Here a weak subsolution with initial data $\psi_{0}$ is by definition a smooth
function $\psi:(0,T_{\max})\times X\to\mathbb{R}$ such that

$\dot{\psi_{t}}\leq\log\left[\frac{(\theta_{t}+dd^{c}\psi_{t})^{n}}{\omega^{n}}\right],\
\forall t\in(0,T_{\max}),\ \ \psi_{t}\to\psi_{0}\ \ {\rm in}\ \ L^{1}(X).$

In Proposition 2.2, the functions $\varphi,\psi$ are both smooth in
$[0,T_{\max})\times X$ while in the above statement there is no regularity
assumption on $\psi$. We also stress that the result still holds without the
assumption that $c(\varphi_{0})=+\infty$ (see Section 4 and Section 5). The
proof without this condition is however more involved and we establish it
later.

###### Proof.

Fix $\varepsilon>0$ and note that $\psi-\varphi$ is a smooth function on
$[\varepsilon,T]\times X$, for any $T<T_{\max}$. It then follows from the
maximum principle that $\psi-\varphi$ attains its maximum on
$[\varepsilon,T]\times X$ at a point $(\varepsilon,x_{\varepsilon})$, thus

$\psi_{t}-\varphi_{t}\leq\sup_{X}(\psi_{\varepsilon}-\varphi_{\varepsilon}).$

Now, the function $(\varepsilon,x)\rightarrow\varphi_{\varepsilon}(x)$ is
smooth by assumption and in particular continuous. It then follows from
Hartogs’ lemma and from the fact that $\psi_{t}$ converges to $\psi_{0}$ in
$L^{1}$ when $t\rightarrow 0$ that

$\sup_{X}(\psi_{\varepsilon}-\varphi_{\varepsilon})\rightarrow\sup_{X}(\psi_{0}-\varphi_{0})\leq
0\quad\mbox{as}\;\varepsilon\rightarrow 0.$

It then follows that $\psi_{t}\leq\varphi_{t}$ for all $t>0$. ∎

######  Remark 2.4.

One can also prove Proposition 2.3 by using the following observation: any
weak solution of the parabolic equation (1.2) is upper semi-continuous in two
variables $(t,x)$. The latter fact follows from Hartogs’ lemma as the
following lemma shows.

######  Lemma 2.5.

Let $T>0$ be a constant and let $\varphi:[0,T]\to{\rm PSH}(X,\omega)$ be a
continuous map where the latter space is equipped with the $L^{1}$ topology.
Then $\varphi(t,x)$ as a function of two variables $(t,x)$ is upper
semicontinuous on $[0,T]\times X$.

We thank Vincent Guedj for the following proof:

###### Proof.

Let $(\varphi_{t})$ be a weak solution of the equation (1.2) and assume by
contradiction that there exists a sequence $(t_{j},x_{j})$ converging to
$(t,x)$ such that

$\limsup_{j\to+\infty}\varphi_{t_{j}}(x_{j})>\varphi_{t}(x).$

Since the function $t\mapsto\varphi_{t}$ is continuous as a map from
$[0,T_{\max})$ to $L^{1}(X)$, it follows that $\varphi_{t_{j}}$ converges in
$L^{1}(X)$ to $\varphi_{t}$. We can assume that $\varphi_{t}(x)>-\infty$ (the
other case follows similarly). Take $\varepsilon>0$ small enough such that

$\limsup_{j\to+\infty}\varphi_{t_{j}}(x_{j})>\varphi_{t}(x)+\varepsilon.$

It follows from the semicontinuity of $\varphi_{t}$ that there exists a small
closed ball $B$ around $x$ such that

$\varphi_{t}(y)<\varphi_{t}(x)+\varepsilon/2,\ \forall y\in B.$

Then for $j$ big enough we will have $x_{j}\in B$. Thus

$\limsup\varphi_{t_{j}}(x_{j})\leq\limsup\sup_{B}\varphi_{t_{j}}\leq\sup_{B}\varphi_{t}<\varphi_{t}(x)-\varepsilon/2,$

as follows from Hartogs’ lemma. This yields a contradiction. ∎

###  2.4. Equisingular approximation

Let $\varphi$ be a $\omega$-psh function on $X$. We are interested in
approximating $\varphi$ by quasi-psh functions which are less singular than
$\varphi$ and such that their singularities are somehow comparable to those of
$\varphi$. In this paper we will make use of Demailly’s equisingular
approximation. For each $c>0$, define

$D_{c}:=\left\\{z\in X\ \big{|}\ \nu(\varphi,z)\geq c\right\\}.$

It follows from a result of Y.T. Siu [31] that each $D_{c}$ is an analytic
subset of $X$ (see [24] or [12] for alternative proofs). If the Lelong number
of $\varphi$ is positive at some point on $X$ it was conjectured by Demailly
and Kollár in [13] that $e^{-\alpha\varphi}$ is not integrable at the critical
value $\alpha$. This openness conjecture was recently proved by Berndtsson [2]
(see also [30], [18] for alternative proof) while the two dimensional case was
proven some years ago by Favre and Jonsson [17].

Demailly’s equisingular approximation gives us a sequence of quasi-psh
functions $(\varphi_{j})$ having analytic singularities converging to
$\varphi$. Moreover, the sequence $(\varphi_{j})$ also keeps track on the
Lelong numbers of $\varphi$.

######  Theorem 2.6 (Demailly’s equisingular approximation).

[12, 10] Let $\varphi$ be a $\omega$-plurisubharmonic function on $X$. There
exists a sequence of quasi-plurisubharmonic functions $(\varphi_{m})$ such
that

  * (i)

$\varphi\leq\varphi_{m}$ and $\varphi_{m}$ converges pointwise and in
$L^{1}(X)$ to $\varphi$ as $m\to+\infty$.

  * (ii)

$\varphi_{m}$ has logarithmic singularities, $e^{2m\varphi_{m}}$ is smooth on
$X$ and vanishes on the set of points $x$ such that $e^{-2m\varphi}$ is not
locally integrable near $x$.

  * (iii)

$\omega+dd^{c}\varphi_{m}\geq-\varepsilon_{m}\omega$, where
$\varepsilon_{m}>0$ converges to $0$ as $m\to+\infty$.

  * (iv)

$\int_{X}e^{2m(\varphi_{m}-\varphi)}dV<+\infty$.

######  Lemma 2.7.

Fix $\varphi\in{\rm PSH}(X,\omega)$. Then for each $\varepsilon>0$ there
exists $\kappa(\varepsilon)>0$ and $\psi_{\varepsilon}\in
PSH(X,\omega)\cap\mathcal{C}^{\infty}(X\setminus D_{\kappa(\varepsilon)})$
such that

$\int_{X}e^{2(\psi_{\varepsilon}-\varphi)/\varepsilon}dV<+\infty.$

Here, $\kappa(\varepsilon)>0$ decreases to $0$ as $\varepsilon\searrow 0$.

###### Proof.

Fix $\varepsilon>0$ and let $c(\varphi)$ be the integrability index of
$\varphi$, i.e.

$c(\varphi):=\sup\left\\{t>0\ \big{|}\ e^{-2t\varphi}\in L^{1}(X)\right\\}.$

We can assume that $c(\varphi)<+\infty$ (otherwise we are done). Let
$(\varphi_{m})$ be the Demailly’s approximating sequence of $\varphi$ (see
Theorem 2.6). Note that $\varphi_{m}$ is smooth outside the analytic subset
$D_{1/m}=\\{z\in X\ \big{|}\ \nu(\varphi,z)\geq\frac{1}{m}\\}$ and

$\omega+dd^{c}\varphi_{m}\geq-\varepsilon_{m}\omega,$

and the constant $\varepsilon_{m}$ decreases to $0$ as $m$ goes to $+\infty$.
We let $m(\varepsilon)>>1$ denote the smallest integer $m$ such that

$m>\frac{2}{\varepsilon(1+\varepsilon_{m})}\ \ {\rm and}\ \
\frac{2\varepsilon_{m}}{\varepsilon(1+\varepsilon_{m})}<c(\varphi).$

Hölder’s inequality shows that the function

$\psi_{\varepsilon}:=\frac{\varphi_{m}}{1+\varepsilon_{m}},$

is smooth outside $D_{\kappa(\varepsilon)}$ and satisfies our requirements
with $\kappa(\varepsilon)=m(\varepsilon)^{-1}$. ∎

######  Remark 2.8.

The constants $\varepsilon_{m}$ come from the gluing process of Demailly [10].
When $\omega$ represents the first Chern class of an ample holomorphic line
bundle over $X$, one can argue globally and avoid the gluing process. Hence
$\varepsilon_{m}$ can be taken to be zero in this case and we can choose
$k(\varepsilon)=\varepsilon$.

##  3\. A priori estimates

Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$. We want to run
the flow from a singular initial data $\varphi_{0}\in{\rm PSH}(X,\omega)$
having positive Lelong numbers. The strategy will be to pick $(\varphi_{0,j})$
a smooth sequence of strictly $\omega$-psh function decreasing to
$\varphi_{0}$ as $j\rightarrow+\infty$ and to consider $(\varphi_{t,j})$ the
smooth flow running from $\varphi_{0,j}$. The goal is to establish uniform
estimates that will allow us to pass to the limit. Thus, in the sequel we work
with $\varphi_{t,j}$ (that we will denote by $\varphi_{t}$ for simplicity) but
to get uniform estimates we should always take into account the singular
behavior of the initial data $\varphi_{0}$.

###  3.1. Notations

From now on we fix $\varepsilon_{0}>0$ and a function $\psi\in{\rm
PSH}(X,\omega)\cap\mathcal{C}^{\infty}(X\setminus D)$ for some analytic subset
$D\subset X$. Let $E_{1},E_{2}$ denote the following quantities

$E_{1}:=\int_{X}\exp{\left(\frac{2\psi-2\varphi_{0}}{\varepsilon_{0}}\right)}dV<+\infty,$
$E_{2}:=\int_{X}\exp{\left(\frac{-\psi}{T}\right)}\omega^{n}<+\infty,$

where $1/(2c(\varphi_{0}))<T<T_{\max}$. Observe that such a function $\psi$
exists thanks to Lemma 2.7 and $E_{2}$ is finite since $T>1/2c(\varphi_{0})$
and by construction $\psi$ is less singular than $\varphi_{0}$. We also
underline that $D$ is a Lelong number super-level set related to the function
$\varphi_{0}$. Since we approximate $\varphi_{0}$ from above by a smooth
sequence $(\varphi_{0,j})$, the corresponding constants $E_{1}^{j}$ are
uniform in $j$ and we can pass to the limit when $j\to+\infty$.

In the remaining part of this section $\varphi_{0}$ will be a smooth strictly
$\omega$-psh function on $X$. One should keep in mind that $\varphi_{0}$ in
this a priori estimate section plays the role of the approximating sequence
$\varphi_{0,j}$.

We set, just for simplicity,

$\theta_{t}:=\omega+t\eta-t{\rm Ric}(\omega).$

Let $\varphi_{t}$ be the solution of the parabolic Monge-Ampère equation:

$\frac{\partial\varphi_{t}}{\partial
t}=\log\left[\frac{(\theta_{t}+dd^{c}\varphi_{t})^{n}}{\omega^{n}}\right],\
\varphi_{t}|_{t=0}=\varphi_{0},$

on a maximal interval $[0,T_{\max})$. Recall that $T_{\max}$ is defined by

$T_{\max}:=\sup\\{t\geq 0\,|\,tK_{X}+t\\{\eta\\}+\alpha_{0}\;\;\mbox{is
nef}\\}.$

The existence of the solution on this maximal interval follows from [6], [39],
[38]. We fix $T$ and $S$ such that

$\frac{1}{2c(T_{0})}<T<S<T_{\max}.$

Since our purpose is to prove regularity of the maximal solution of the
equation (1.2) on $(0,T]$, it costs no generality to assume that

$\frac{2\omega}{3}\leq\theta_{t}\leq 2\omega,\forall t\in[0,S].$

Thus for each $t\in[0,S]$ we have

$\theta_{t}=\frac{t\theta_{S}}{S}+\frac{S-t}{S}\omega\geq\left(1-\frac{t}{3S}\right)\omega.$

###  3.2. The $\mathcal{C}^{0}$ estimate

We recall the following upper bound for $\varphi_{t}$ whose proof can be
found, for example, in [21].

######  Lemma 3.1.

The following estimate holds

$\varphi_{t}\leq\sup_{X}\varphi_{0}+(n\log 2)t,\ \ \forall t\in[0,T].$

As we will see in the next sections, if we start the flow from a current with
positive Lelong numbers there is no hope for the flow to be smooth. Thus a
“reasonable" $\mathcal{C}^{0}$ estimate must involve some singular quasi-
plurisubharmonic function. In our recent works [14] and [15] we have developed
the theory of generalized capacities to deal with these singular estimates. In
particular we proved that if $\mathrm{MA}\,(\varphi)\leq Ce^{-\phi}$ for some
quasi-plurisubharmoinc function $\phi$ then for any $a>0$ small enough (i.e.
$a\phi\in{\rm PSH}(X,\omega/2)$) there exists a uniform constant $A>0$ such
that

$\varphi\geq a\phi-A.$

In Theorem 3.2 and Lemma 3.3 below we use the same ideas to establish the
following $\mathcal{C}^{0}$ estimate for the complex parabolic Monge-Ampère
equation (1.2).

######  Theorem 3.2.

Fix $\varepsilon>\varepsilon_{0}$. For $t\in[\varepsilon,T]$ the following
estimate holds

$\varphi_{t}\geq\left(1-\frac{t}{2T}\right)\psi-C,$

where $C$ is a positive constant depending on $\varepsilon$, our fixed
parameters and an upper bound for $E_{1}$ and $E_{2}$.

###### Proof.

Fix $t\in[\varepsilon,T]$. It follows from [21, Proposition 3.1] that

(3.1) $\dot{\varphi}_{t}\leq\frac{\varphi_{t}-\varphi_{0}}{t}+n,$

and so

$\mathrm{MA}\,(\varphi_{t})=e^{\dot{\varphi}_{t}}\leq
e^{n+(\varphi_{t}-\varphi_{0})/t}\omega^{n}.$

If $\varphi_{0}$ has zero lelong numbers then, as shown in [21], the estimate
(3.1) is enough (and crucial) to get a $\mathcal{C}^{0}$ estimate for
$\varphi_{t}$ using Kołodziej’s theorem [26]. However, for the general case,
namely when $\varphi_{0}$ has positive Lelong numbers, this method breaks down
since the bound on $\mathrm{MA}\,(\varphi_{t})$ is no longer integrable. This
is why we need to use the generalized capacities that we introduced in [15].

Now set

$\psi_{t}=\left(1-\frac{t}{2T}\right)\psi.$

Observe that

$\theta_{t}\geq\left(1-\frac{t}{3S}\right)\omega\ \ \mbox{and}\ \
dd^{c}\psi_{t}+\left(1-\frac{t}{2T}\right)\omega\geq 0.$

It then follows that $\psi_{t}$ is $\delta\theta$-plurisubharmonic with
$\delta\in(0,1)$ depending on $\varepsilon_{0},T,S$, namely

$\delta=\frac{6TS-3S\varepsilon_{0}}{6TS-2T\varepsilon_{0}}<1.$

Now, we want to apply Lemma 3.3 below with $A=1/t$ and $u=\varphi_{0}/t$ to
get the desired estimate. In order to do that, we also need to bound the
following quantity

(3.2) $\int_{X}\exp{\left(\frac{q(\psi_{t}-\varphi_{0})}{t}\right)}\omega^{n}$

for some $q>1$. Indeed, by fixing $p>2$, $2/q$ being its conjugate such that

$\frac{pq}{\varepsilon}<\frac{2}{\varepsilon_{0}}$

and using Hölder’s inequality we obtain

$\int_{X}e^{\frac{q(\psi_{t}-\varphi_{0})}{t}}\omega^{n}=\int_{X}e^{\frac{q(\psi-\varphi_{0})}{t}}e^{\frac{-q\psi}{2T}}\omega^{n}\leq\left(\int_{X}e^{\frac{pq(\psi-\varphi_{0})}{t}}\omega^{n}\right)^{1/p}\left(\int_{X}e^{\frac{-\psi}{T}}\omega^{n}\right)^{q/2}.$

The right-hand side of the above inequality is finite as follows from (3.1)
and from the construction of $\psi$ in Lemma 2.7:

$\int_{X}e^{\frac{pq(\psi-\varphi_{0})}{t}}\omega^{n}\leq\int_{X}e^{\frac{2(\psi-\varphi_{0})}{\varepsilon_{0}}}\omega^{n}<+\infty.$

Thus we can bound the term in (3.2) in terms of $E_{1}$ and $E_{2}$.
Therefore, our $\mathcal{C}^{0}$ estimate only depends on the fixed parameters
$\varepsilon,\varepsilon_{0},T,S$ and on upper bounds for $E_{1},E_{2}$. The
proof is thus complete. ∎

######  Lemma 3.3.

Assume that $\varphi\in\mathcal{E}(X,\omega)$ and
$(\omega+dd^{c}\varphi)^{n}\leq e^{A\varphi-u}\omega^{n}$, where $A>0$, $u$ is
some measurable function such that $e^{A\psi-u}\in L^{q}(X)$ for $\psi\in{\rm
PSH}(X,\delta\omega)$ with $\delta\in(0,1)$ and for some $q>1$. Then we have
the following estimate

$\varphi\geq\psi-C,$

where $C$ is a positive constant depending only on $A,\delta,\omega,p$ (where
$p$ is the conjugate exponent of $q$) and on an upper bound for
$\int_{X}e^{q(A\psi-u)}\omega^{n}$.

###### Proof.

The proof follows the arguments in [14]. The goal is to prove that the
function

$H(t):=\left[{\rm Cap}_{\psi}(\varphi<\psi-t)\right]^{1/n}$

satisfies the inequality (2.1). Since $H(t)$ is right continuous and $H(t)$
goes to zero as $t$ goes to $+\infty$ (see [14, Lemma 2.6]) we can conclude
using Lemma 2.1 above.

Fix $s\in[0,1]$, $t>0$, then by [14, Proposition 2.8] we get

$s^{n}{\rm Cap}_{\psi}(\varphi<\psi-
t-s)\leq\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(\varphi).$

Using the assumption and Hölder’s inequality we then have

$\displaystyle\int_{\\{\varphi<\psi-t\\}}\mathrm{MA}\,(\varphi)$
$\displaystyle\leq$
$\displaystyle\int_{\\{\varphi<\psi-t\\}}e^{A\varphi-u}\,\omega^{n}$
$\displaystyle=$
$\displaystyle\int_{\\{\varphi<\psi-t\\}}e^{A(\varphi-\psi)}e^{A\psi-u}\,\omega^{n}$
$\displaystyle\leq$ $\displaystyle e^{-At}{\rm
Vol}(\\{\varphi<\psi-t\\})^{1/p}\left(\int_{\\{\varphi<\psi-t\\}}e^{q(A\psi-u)}\,\omega^{n}\right)^{1/q}.$

It follows from [19] that

${\rm Vol}(\\{\varphi<\psi-t\\})^{1/p}\leq C(p,\omega,\delta,n)\left[{\rm
Cap}_{(1-\delta)\omega}(\varphi<\psi-t)\right]^{2}.$

Thanks to [14, Lemma 2.7] we also have

${\rm Cap}_{(1-\delta)\omega}\leq{\rm Cap}_{\psi}.$

Putting all together yields

$sH(t+s)\leq C^{1/n}e^{-At/n}H(t)^{2}\leq BH(t)^{2},$

where $B=C^{1/n}$ and $C$ depends on $\omega,p,\delta,A$ and an upper bound
for $\|e^{A\psi-u}\|_{L^{q}(X)}$. Now, choose $t_{\infty}>0$ such that

$2C^{2/n}e^{-At_{\infty}/n}{\rm Vol}(X)^{2/n}<1.$

We then have $H(t_{\infty}+1)<(2B)^{-1}$. It then follows from Lemma 2.1 that

$\varphi\geq\psi-t_{\infty}-3.$

∎

###  3.3. The $\mathcal{C}^{2}$ estimate

We recall the following well-known result (see [40] for a proof).

######  Lemma 3.4.

Let $\alpha,\beta$ be positive $(1,1)$-forms. Then

$n\left(\frac{\alpha^{n}}{\beta^{n}}\right)^{\frac{1}{n}}\leq{\rm
tr}_{\beta}(\alpha)\leq
n\left(\frac{\alpha^{n}}{\beta^{n}}\right)\cdot\left({\rm
tr}_{\alpha}(\beta)\right)^{n-1}.$

Recall that $(\varphi_{t})_{0\leq t<T_{\max}}$ is a smooth solution of the
parabolic Monge-Ampère equation (1.2) starting from a smooth strictly
$\omega$-psh function $\varphi_{0}$.

We use $\Delta_{t}$ to denote the Laplacian with respect to the Kähler metric
$\omega_{t}$ which is defined by

$\omega_{t}:=\theta_{t}+dd^{c}\varphi_{t}.$

In order to establish the $\mathcal{C}^{2}$ estimates we need a lower bound
for $\dot{\varphi_{t}}$. Let us stress that during the proof of our estimates,
the constants $C_{1},C_{2}...$ stand for various positive constants which are
under control.

######  Proposition 3.5.

For all $(t,x)\in(\varepsilon,T]\times X$,

$\dot{\varphi}_{t}(x)\geq n\log(t-\varepsilon)+A(\Psi_{t}-\varphi_{t})(x)-C,$

where $A,C$ are positive constants depending on $\varepsilon$, an upper bound
for $E_{1},E_{2}$ and our fixed parameters. Here, $\Psi_{t}$ is defined by

$\Psi_{t}=\left(1-\frac{t}{2S}\right)\psi.$

###### Proof.

Consider the following function $G$ defined on $(\varepsilon,T]\times X$ by

$G(t,x):=\dot{\varphi}_{t}(x)+A(\varphi_{t}-\Psi_{t})-n\log(t-\varepsilon),\ \
\Psi_{t}:=\left(1-\frac{t}{2S}\right)\psi,$

where $A$ is a positive constant to be specified later. Observe that $G$ is
smooth on $(\varepsilon,T]\times X_{0}$, where

$X_{0}:=\\{x\in X\ \big{|}\ \psi(x)>-\infty\\}.$

In the sequel, all computations take place in $(\varepsilon,T]\times X_{0}$.
We compute

$\left(\frac{\partial}{\partial
t}-\Delta_{t}\right)G=A\dot{\varphi}_{t}-\frac{n}{t-\varepsilon}+A\frac{\psi}{2S}-nA+A{\rm
tr}_{\omega_{t}}(\theta_{t}+dd^{c}\Psi_{t})+{\rm tr}_{\omega_{t}}(\chi).$

It follows from

$\theta_{t}\geq\left(1-\frac{t}{3S}\right)\omega\ \ \mbox{and}\ \
dd^{c}\Psi_{t}+\left(1-\frac{t}{2S}\right)\omega\geq 0$

that

$\theta_{t}+dd^{c}\Psi_{t}\geq\frac{t\omega}{6S}\geq\frac{\varepsilon\omega}{6S}.$

We choose $A>0$ big enough so that

$\frac{A\varepsilon\omega}{6S}+\chi\geq\omega.$

Thus we have

(3.3) $\left(\frac{\partial}{\partial t}-\Delta_{t}\right)G\geq
A\dot{\varphi}_{t}-\frac{n}{t-\varepsilon}-nA+A\frac{\psi}{2S}+{\rm
tr}_{\omega_{t}}(\omega).$

It follows from Lemma 3.4 that

(3.4) ${\rm tr}_{\omega_{t}}(\omega)\geq
n\left(\frac{\omega^{n}}{\omega_{t}^{n}}\right)^{1/n}=n\exp{\left(\frac{-\dot{\varphi_{t}}}{n}\right)}.$

It is elementary that for each small constant $b>0$ and each big constant
$B>0$ there exists a constant $B^{\prime}$ depending only on $b$ and $B$ such
that

$be^{x}-Bx\geq-B^{\prime},\ \ \forall x\in\mathbb{R}.$

Using this observation, we can find positive constants $C_{1},C_{2}$ under
control such that

(3.5)
$A\dot{\varphi}_{t}+n\exp{\left(\frac{-\dot{\varphi_{t}}}{n}\right)}\geq\exp{\left(\frac{-\dot{\varphi_{t}}}{n}-C_{1}\right)}-C_{2}.$

From (3.3), (3.4) and (3.5) we deduce that

(3.6) $\left(\frac{\partial}{\partial
t}-\Delta_{t}\right)G\geq\exp{\left(\frac{-\dot{\varphi_{t}}}{n}-C_{1}\right)}+A\frac{\psi}{2S}-\frac{n}{t-\varepsilon}-C_{2}-An.$

Notice that the function $G$ is smooth on $(\varepsilon,T]\times X_{0}$ and it
goes to $+\infty$ on the boundary. Thus we see that $G$ attains its minimum on
$(\varepsilon,T]\times X_{0}$ at some point
$(t_{0},x_{0})\in(\varepsilon,T]\times X_{0}$. It follows from the minimum
principle and (3.6) that, at $(t_{0},x_{0})$, we have

$\dot{\varphi}_{t}\geq-n\log\left(C_{2}+\frac{n}{t_{0}-\varepsilon}-A\frac{\psi}{2S}+An\right)-nC_{1}.$

We then get

$\displaystyle G(t_{0},x_{0})$ $\displaystyle\geq$ $\displaystyle-
C_{3}-n\log\left[C_{5}(t_{0}-\varepsilon)+n-\frac{A(t_{0}-\varepsilon)\psi(x_{0})}{2S}\right]+A(\varphi_{t_{0}}-\Psi_{t_{0}})$
$\displaystyle\geq$ $\displaystyle-
C_{4}-n\log\left[C_{5}(t_{0}-\varepsilon)+n-\frac{A(t_{0}-\varepsilon)\psi(x_{0})}{2S}\right]-\delta\psi(x_{0}),$

where $\delta=\frac{At_{0}(S-T)}{2TS}>0$ and the last inequality follows from
Theorem 3.2. We then obtain a uniform lower bound for $G(t_{0},x_{0})$, and
hence the result follows. ∎

######  Theorem 3.6.

Fix $\varepsilon>\varepsilon_{0}$. For all $x\in X$ and $t\in[\varepsilon,T]$
one has

$0\leq(t-\varepsilon)\log{\rm tr}_{\omega}(\omega_{t})\leq-A\psi+C,$

where $A,C$ are positive constants depending on $\varepsilon$, an upper bound
for $E_{1},E_{2}$ and our fixed parameters.

###### Proof.

Consider

$H:=(t-\varepsilon)\log{\rm
tr}_{\omega}(\omega_{t})-A(\varphi_{t}-\Psi_{t}),t\in[\varepsilon,T],x\in X,$

where $A$ is a positive constant to be specified later and the function
$\Psi_{t}$ is defined by

$\Psi_{t}:=\left(1-\frac{t}{2S}\right)\psi.$

Notice that $H$ is smooth on $[\varepsilon,T]\times X_{0}$ and $H$ goes to
$-\infty$ on the boundary of $X_{0}$, where

$X_{0}:=\\{x\in X\ \big{|}\ \psi(x)>-\infty\\}.$

Set $u={\rm tr}_{\omega}(\omega_{t})$. We compute

$\left(\frac{\partial}{\partial t}-\Delta_{t}\right)H=\log
u+\frac{t-\varepsilon}{u}\frac{\partial u}{\partial
t}-A\dot{\varphi}_{t}-A\frac{\psi}{2S}-(t-\varepsilon)\Delta_{t}\log
u+A\Delta_{t}(\varphi_{t}-\Psi_{t}).$

As in the proof of Proposition 3.5 we have

${\rm
tr}_{\omega_{t}}(\theta_{t}+dd^{c}\Psi_{t})\geq\frac{\varepsilon}{6S}{\rm
tr}_{\omega_{t}}(\omega).$

Thus we obtain

$\Delta_{t}(\varphi_{t}-\Psi_{t})\leq n-\frac{\varepsilon}{6S}{\rm
tr}_{\omega_{t}}(\omega).$

Thanks to [5, Lemma 2.2] (which is an improvement of [32]) we also have

$\Delta_{t}\log u\geq\frac{{\rm tr}_{\omega}(dd^{c}\dot{\varphi_{t}})}{{\rm
tr}_{\omega}(\omega_{t})}-B\,{\rm tr}_{\omega_{t}}(\omega),$

where $B$ depends only on a lower bound for the holomorphic bisectional
curvature of $\omega$. Furthermore,

$\frac{t-\varepsilon}{u}\frac{\partial u}{\partial
t}=\frac{t-\varepsilon}{u}\left({\rm tr}_{\omega}(\chi)+{\rm
tr}_{\omega}(dd^{c}\dot{\varphi_{t}})\right)\leq\frac{t-\varepsilon}{u}{\rm
tr}_{\omega}(dd^{c}\dot{\varphi_{t}})+C_{1}(t-\varepsilon)\,{\rm
tr}_{\omega_{t}}(\omega)$

where in the last inequality we use the fact that ${\rm tr}_{\omega}(\chi)$ is
bounded from above together with the trivial inequality $n\leq{\rm
tr}_{\omega_{t}}(\omega){\rm tr}_{\omega}(\omega_{t})$. Thus

$\frac{t-\varepsilon}{u}\frac{\partial u}{\partial
t}-(t-\varepsilon)\Delta_{t}\log u\leq(B+C_{1})(t-\varepsilon)\,{\rm
tr}_{\omega_{t}}(\omega).$

It follows from Lemma 3.4 applied to $\alpha=\omega_{t}$ and $\beta=\omega$
that

${\rm tr}_{\omega}(\omega_{t})\leq n\,e^{\dot{\varphi_{t}}}\left({\rm
tr}_{\omega_{t}}(\omega)\right)^{n-1}.$

Using the inequality $(n-1)\log x<x+C_{n}$, we obtain

$\log u\leq\log n+\dot{\varphi_{t}}+(n-1)\log{\rm
tr}_{\omega_{t}}(\omega)\leq\dot{\varphi_{t}}+{\rm
tr}_{\omega_{t}}(\omega)+C_{2}.$

Thus we have

$\left(\frac{\partial}{\partial
t}-\Delta_{t}\right)H\leq-(A-1)\dot{\varphi}_{t}+\left[(B+C_{1})(t-\varepsilon)+1-\frac{A\varepsilon}{6S}\right]{\rm
tr}_{\omega_{t}}(\omega)-\frac{A\psi}{2S}+C_{3}.$

We will choose $A>0$ big enough such that

$(B+C_{1})T+1-\frac{A\varepsilon}{6S}\leq-1.$

Assume now that $H$ achieves its maximum at some point
$(t_{0},x_{0})\in[\varepsilon,T]\times X_{0}$. If $t_{0}=\varepsilon$ then we
are done. Assume that $t_{0}>\varepsilon$. From now on our computations take
place at this maximum point. By the maximum principle we have

$0\leq\left(\frac{\partial}{\partial t}-\Delta_{t}\right)H\leq
C_{3}-(A-1)\dot{\varphi_{t}}-\frac{A\psi}{2S}-{\rm tr}_{\omega_{t}}(\omega),$

or, equivalently

(3.7) ${\rm tr}_{\omega_{t}}(\omega)\leq
C_{3}-(A-1)\dot{\varphi_{t}}-\frac{A\psi}{2S}.$

In particular, since ${\rm tr}_{\omega_{t}}(\omega)\geq 0$, we have the
following upper bound for $\dot{\varphi_{t}}$:

(3.8) $\dot{\varphi_{t}}\leq\frac{C_{3}}{A-1}-\frac{A\psi}{2S(A-1)}.$

It follows from Lemma 3.4 that

${\rm tr}_{\omega_{t}}(\omega)\geq
n\exp\left(\frac{-\dot{\varphi_{t}}}{n}\right).$

Plugging this into (3.7) we obtain

(3.9) ${\rm tr}_{\omega_{t}}(\omega)\leq C_{4}-\frac{A\psi}{4S}.$

Thus applying again Lemma 3.4 yields

$\log u\leq\log
n+\dot{\varphi_{t}}+(n-1)\log\left[C_{4}-\frac{A\psi}{4S}\right].$

From this and from (3.8) we obtain

$H\leq
C_{5}-A\left[\varphi_{t}-\left(1-\frac{t}{2S}-\frac{t-\varepsilon}{2(A-1)S}\right)\psi\right]+(n-1)(t-\varepsilon)\log\left[C_{4}-\frac{A\psi}{4S}\right].$

Now, we increase $A$ a little bit, if necessary, so that

$\delta:=\frac{\varepsilon}{2T}-\frac{\varepsilon}{2S}-\frac{T}{2(A-1)S}>0.$

Thus since $\psi\leq 0$ we obtain

$H\leq
C_{5}-A\left[\varphi_{t}-\left(1-\frac{t}{2T}\right)\psi\right]+A\delta\psi+(n-1)(t-\varepsilon)\log\left[C_{4}-\frac{A\psi}{4S}\right].$

The second term is uniformly bounded from above thanks to Theorem 3.2. Now,
using the obvious inequality $-\varepsilon x+\log x\leq C$ we obtain a uniform
upper bound for $H$ at the point $(t_{0},x_{0})$ and the result follows. ∎

###  3.4. More estimates

In this subsection we derive some other a priori estimates for the solutions
of the parabolic equation (1.2). These estimates will be used later to prove
stronger convergence of the flow at zero when the initial data has some
regularity properties (Theorems 6.3 and 6.2). The setting and the notations
will be the same as those in the previous section.

######  Proposition 3.7.

Assume that there exist smooth $\omega$-plurisubharmonic functions
$\phi_{1},\phi_{2}$ and positive constants $0<\delta<1/2$, $C_{1}>0$ such that

$\dot{\varphi_{0}}\geq C_{1}\phi_{1}\ \ {\rm and}\ \
\varphi_{0}\geq\delta\phi_{2}.$

Then there exists a uniform constant $C_{2}$ depending on $C_{1}$ and our
fixed parameters such that

$\dot{\varphi_{t}}\geq C_{2}(\phi_{2}+1)+C_{1}\phi_{1}.$

###### Proof.

Consider the following function

$H(t,x):=\dot{\varphi_{t}}-C_{1}\phi_{1}+A(\varphi_{t}-\delta\phi_{2}),$

where $A>0$ is a big constant to be specified later. We use the same arguments
as in Proposition 3.5. Since $H$ is smooth on $[0,T]\times X$ it attains a
minimum at some point $(t_{0},x_{0})$. If $t_{0}$ is zero then we are done
since $H(0,.)\geq 0$ by our construction.

Assume now that $t_{0}>0$. We do our computation below at this minimum point
$(t_{0},x_{0})$. By the minimum principle we have

$0\geq\left(\frac{\partial}{\partial t}-\Delta_{t}\right)H\geq-
An+A\dot{\varphi_{t}}+\left(\frac{A}{6}-C_{1}-A_{1}\right){\rm
tr}_{\omega_{t}}(\omega).$

Here we use the fact that $\theta_{t}\geq 2\omega/3$ and $\chi\geq-
A_{1}\omega$, for some constant $A_{1}$ under control. Now, choose $A$ big
enough such that $\frac{A}{6}-C_{1}-A_{1}>1$ and using Lemma 3.4 combined with
the inequality $e^{x}\geq Bx-C_{B}$, we obtain a uniform lower bound for
$\dot{\varphi}_{t_{0}}(x_{0})$. The lower bound for $H(t_{0},x_{0})$ then
follows observing that, by [21, Lemma 2.9], $\varphi_{t}\geq\varphi_{0}-c(t)$,
and so $\varphi_{t}-\delta\phi_{2}\geq-c(t)$ where $c(t)\rightarrow 0$ as
$t\rightarrow 0$.

∎

######  Theorem 3.8.

Assume that there exist smooth $\omega$-plurisubharmonic function
$\phi_{1},\phi_{2}$ and a positive constants $0<\delta<1/2$ such that

$\Delta_{\omega}\varphi_{0}\leq e^{-C_{1}\phi_{1}}\ \ \ \ {\rm and}\ \
\varphi_{0}\geq\delta\phi_{2}.$

Then there exists a uniform constant $C_{2}>0$ such that

${\rm tr}_{\omega}\omega_{t}\leq C(1+e^{-C_{1}\phi_{1}-\delta\phi_{2}}),\
\forall t\in[0,T].$

###### Proof.

Consider the function

$H(t,x)=\log{\rm
tr}_{\omega}(\omega_{t})+C_{1}\phi_{1}-A(\varphi_{t}-\delta\phi_{2}),$

where $A>0$ is a big constant (under control) to be specified later. This is a
smooth function on $[0,T]\times X$ and hence it attains its maximum at some
point $(t_{0},x_{0})$. If $t_{0}=0$ we are done since by our construction
$H(0,.)\leq 0$. Assume now that $t_{0}>0$. From now on our computations take
place at this maximum point $(t_{0},x_{0})$. By the maximum principle and by
the same arguments as in Theorem 3.6 we eventually get

(3.10) $0\leq\left(\frac{\partial}{\partial t}-\Delta_{t}\right)H\leq
A_{1}-A\dot{\varphi_{t}}-\left(\frac{A}{6}-B-C_{1}-A_{2}\right){\rm
tr}_{\omega_{t}}(\omega),$

where $-B<0$ is a lower bound for the holomorphic bisectional curvature of
$\omega$, and $A_{2}$ is a constant under control. Here we use the fact that
$\delta<1/2$ and $\theta_{t}\geq\frac{2}{3}\omega$ for all $t\in[0,T]$ and

$-{\rm tr}_{\omega_{t}}(dd^{c}\phi_{1})\leq{\rm tr}_{\omega_{t}}(\omega),\ \
{\rm tr}_{\omega_{t}}(\chi)\geq-A_{2}{\rm tr}_{\omega_{t}}(\omega).$

It follows from Lemma 3.4 that

${\rm tr}_{\omega_{t}}\omega\geq
n\exp\left(\frac{-\dot{\varphi_{t}}}{n}\right).$

Plugging this into (3.10), choosing $A$ big enough and using the inequality
$-\varepsilon x+\log x\leq C_{\varepsilon}$ with $0<\varepsilon<(An)^{-1}$, we
obtain an upper bound for ${\rm tr}_{\omega_{t}}(\omega)$ at $(t_{0},x_{0})$.
We infer that there exists a uniform constant $C>0$ such that
$H(t_{0},x_{0})\leq C$. Thus $H(t,x)\leq C$ and so

$\log{\rm tr}_{\omega}(\omega_{t})\leq C-C_{1}\phi_{1}+A\delta\phi_{2}$

or equivalently

${\rm tr}_{\omega}(\omega_{t})\leq
C^{\prime}e^{-C_{1}\phi_{1}+A\delta\phi_{2}}.$

∎

##  4\. Regularity of weak solutions

In this section we study short time regularity of the Kähler-Ricci flow (1.1)
starting from a current having positive Lelong number. In particular, we give
a proof of Theorem A and show that the result is optimal.

In the whole section we assume that $\varphi_{0}$ is a $\omega$-psh function
on $X$ such that

$\frac{1}{2c(\varphi_{0})}<T_{\max}.$

This condition guarantees that the approximating smooth solutions will not
converge uniformly to $-\infty$ (see [21, Lemma 2.9]).

###  4.1. Construction of the maximal weak solution

For convenience, let us recall the construction of the maximal weak solution
due to Guedj and Zeriahi [21]. Let $(\varphi_{0,j})$ be a decreasing sequence
of smooth strictly $\omega$-plurisubharmonic functions which converges to
$\varphi_{0}$. Let $(\varphi_{t,j})$ be the sequence of smooth solutions of
the equation (1.2) starting from $\varphi_{0,j}$. When $j$ goes to $+\infty$,
for each $t>0$ fixed, the sequence $(\varphi_{t,j})$ decreases to a well-
defined $\theta_{t}$-psh function $\varphi_{t}$ (not identically $-\infty$
thanks to [21, Lemma 2.9]).

######  Example 4.1.

Let us stress that the condition $1/2c(\varphi_{0})<T_{\max}$ is necessary to
insure that the maximal flow is well-defined as the following example shows.
Consider $X=\mathbb{P}^{1}$ and $\omega=4\omega_{FS}$ whose potential is given
in homogeneous coordinate by

$\rho=2\log(|z|^{2}+|w|^{2}).$

It is well-known that ${\rm Ric}(\omega)=\omega$. Consider the Kähler-Ricci
flow

$\dot{\omega}_{t}=-{\rm Ric}\ \omega$

and observe that in this case we have $T_{\max}=1$. Consider the following
sequence of functions

$\varphi_{0,j}:=2\log(|z|^{2}+|w|^{2}/j)-\rho.$

The sequence $(\varphi_{0,j})_{j}$ decreases to a $\omega$-psh function
$\varphi_{0}$ with $c(\varphi_{0})=1/4$. Thus $1/2c(\varphi_{0})=2>T_{\max}$.
A straightforward computation shows that

$\omega+dd^{c}\varphi_{0,j}=e^{-\varphi_{0,j}-\log j}\omega.$

Let us define

$\varphi_{t,j}:=(1-t)\varphi_{0,j}-t\log j-t+(t-1)\log(1-t),\ \ t\in(0,1)$

and note that $\varphi_{t,j}$ solves the following parabolic Monge-Ampere
equation

$\omega-t{\rm Ric}(\omega)+dd^{c}u_{t}=e^{\dot{u}_{t}}\omega,$

with initial data $\varphi_{0,j}$. We can also check that $\varphi_{t,j}$
decreases to $-\infty$ if $t>0$. Hence the maximal solution constructed in
[21] is not well-defined. Note, however, that the flow $\omega_{t,j}$
converges to $\omega_{t}=(1-t)\omega_{0}$.

######  Lemma 4.2.

The limit $\varphi_{t}$ obtained as above does not depend on the choice of the
approximating sequence $(\varphi_{0,j})$.

###### Proof.

Let $\psi_{0,j}$ be another decreasing sequence of smooth strictly
$\omega$-psh functions which converges to $\varphi_{0}$. Let $(\psi_{t,j})$ be
the corresponding smooth solutions of (1.2) starting from $\psi_{0,j}$. Fix
$k\in\mathbb{N}$ and $\varepsilon>0$. By Hartogs’ lemma there exists
$j_{k}\in\mathbb{N}$ such that

$\psi_{0,j}\leq\varphi_{0,k}+\varepsilon,\ \ \forall j\geq j_{k}.$

It follows from the comparison principle (Proposition 2.2) that

$\psi_{t,j}\leq\varphi_{t,k}+\varepsilon,\ \ \forall j\geq j_{k}.$

Now, letting $k\to+\infty$ and then $\varepsilon\to 0$ we get one inequality.
The result follows since we can inverse the role of $\varphi_{t}$ and
$\psi_{t}$ in the above arguments. ∎

######  Remark 4.3.

The proof given above also shows that if $\varphi_{t}$ and $\psi_{t}$ are two
maximal solutions of (1.2) with $\varphi_{0}\leq\psi_{0}$ then
$\varphi_{t}\leq\psi_{t}$.

###  4.2. Short time singularity

It was shown in [21, Theorem 6.1] that when $t>1/2c(\varphi_{0})$,
$\omega_{t}:=\theta_{t}+dd^{c}\varphi_{t}$ is a Kähler form satisfying the
twisted Kähler-Ricci flow equation (1.1). In particular they proved that for
$t>1/2c(\varphi_{0})$ $\varphi_{t}$ is a smooth and verifies equation (1.2).

In what follows we investigate the behavior of the maximal solution in short
time. We show that the maximal solution of the equation (1.2) starting from a
current with positive Lelong numbers is unbounded for short time. This relies
on a quite simple observation that we will explain in the next lines before
entering into the details of the proof of Lemma 4.4.

Assume that $\varphi_{0}\in{\rm PSH}(X,\omega)$ has positive Lelong number at
some $x_{0}\in X$, then

$\varphi_{0}\leq\gamma\log|z|+C$

for some positive constants $\gamma,C$. Without loss of generality we can
assume that $\varphi_{0}=\phi(z)\log|z|^{2}$, where $z$ is a local coordinate
in a neighborhood $V$ around $z=0$ and $\phi$ is a cut-off function which
vanishes outside $V$ and $\phi\equiv 1$ near $z=0$. We can also assume that
$\varphi_{0}\in{\rm PSH}(X,\omega/2)$ and $T_{\max}>1$. Let $\varphi_{t}$ be
the maximal solution of the equation (1.2) with initial data $\varphi_{0}$.
Fix $\varepsilon>0$ small enough. Consider the following family

$u_{t,\varepsilon}:=\left[1-(n+1)t\right]\phi(z)\log(|z|^{2}+\varepsilon^{2})+Ct.$

We can choose $C>0$ large enough (which does not depend on $\varepsilon$) such
that $u_{t,\varepsilon}$ is a supersolution of the equation (1.2). Since
$u_{0,\varepsilon}\geq\varphi_{0}$ it follows from the comparison principle
that $u_{t,\varepsilon}\geq\varphi_{t},\ \forall t\in(0,1/(n+1))$. By letting
$\varepsilon\searrow 0$ we can conclude that $\varphi_{t}$ is unbounded and
has positive Lelong number for very short time.

In the Lemma below we prove that the solution is singular for all
$0<t<1/2nc(\varphi_{0})$. When $n=1$ this coupled with [21, Theorem 6.1] gives
a complete picture of the regularity properties of the Kähler-Ricci flow.

######  Lemma 4.4.

Let $\phi\in{\rm PSH}(X,\omega)$ be such that
$e^{\gamma\phi}\in\mathcal{C}^{\infty}(X)$ for some positive constant
$\gamma$. Assume that $\varphi_{0}\leq\phi\leq 0$. Then there exists a
positive constant $C$ depending on an upper bound for $dd^{c}e^{\gamma\phi}$
such that

$\varphi_{t}\leq(1-n\gamma t)\phi+Ct,\ \forall t\in[0,1/n\gamma].$

###### Proof.

Notice that in the statement of the lemma we impose implicitly that
$1/n\gamma<T_{\max}$. For simplicity we also assume that

$\theta_{t}\leq\omega,\ \forall t\in[0,1/n\gamma].$

In practice $\gamma$ will be very large and these conditions are irrelevant.
By replacing $\phi$ by the function
$\gamma^{-1}\log(e^{\gamma\phi}+\varepsilon)$ and letting $\varepsilon\to 0$
we can assume that $\phi$ is actually smooth on $X$. Exploiting the same
arguments in the proof of Lemma 4.2 we can also assume that $\varphi_{0}$ is
smooth.

Observe that there is a uniform constant $C>0$ depending only on an upper
bound for $dd^{c}e^{\gamma\phi}$ such that

$dd^{c}\phi\leq Ce^{-\gamma\phi}\omega.$

Set

$\phi_{t}:=(1-n\gamma t)\phi+t\log(C^{n}2^{n}).$

Since $\phi\leq 0$ and $dd^{c}\phi\leq Ce^{-\gamma\phi}\omega$ we see that

$0\leq(\omega+dd^{c}\phi_{t})\leq(1+Ce^{-\gamma\phi})\omega\leq
2Ce^{-\gamma\phi}\omega.$

It then follows that

$(\omega+dd^{c}\phi_{t})^{n}\leq
2^{n}C^{n}e^{-n\gamma\phi}\omega^{n}=e^{\dot{\phi}_{t}}\omega^{n}.$

Then $\phi_{t}$ is a supersolution of the following parabolic equation

$(\omega+dd^{c}u_{t})^{n}=e^{\dot{u}_{t}}\omega^{n},$

while $\varphi_{t}$ is a subsolution since we have assumed that
$\theta_{t}\leq\omega$. The result follows by the classical maximum principle.
∎

Note that the above result shows that for short time we have propagation of
sigularities (as it is highlighted in Theorem 4.5 below) and that our $C^{0}$
estimate in Theorem 3.2 is optimal.

######  Theorem 4.5.

Assume that $\varphi_{0}$ has positive Lelong numbers. Then we have the
following relation between the Lelong number superlevel sets of $\varphi_{t}$
and $\varphi_{0}$:

$D_{c}(\varphi_{0})\subset D_{c(t)}(\varphi_{t}),\ \ c(t)=c-2nt.$

In particular, the maximal solution $\varphi_{t}$ has positive Lelong numbers
for any $t<1/2nc(\varphi_{0})$.

###### Proof.

Let $c>0$ and $x_{0}\in D_{c}(\varphi_{0})$. It follows from the very
definition of the Lelong number that we can find a function $\phi\in{\rm
PSH}(X,A\omega)$ for some positive constant $A$ such that

$e^{2\phi/c}\in\mathcal{C}^{\infty}(X)\ \ {\rm and}\ \ \phi\geq\varphi_{0}.$

In fact, the function $\phi$ can be written locally as

$\phi(z)=c\log|z|,$

where $z$ is a local coordinate centered at $x_{0}$. Applying Lemma 4.4 we get

$\varphi_{t}\leq(1-2nt/c)\phi+Ct.$

It follows that $\nu(\varphi_{t},x_{0})\geq c-2nt$, hence $x_{0}\in
D_{c(t)}(\varphi_{t})$.

In particular, if $t<1/2nc(\varphi_{0})$ then for a constant $c$ such that
$2nt<c<1/c(\varphi_{0})$ the set $D_{c}(\varphi_{0})$ is non empty as follows
from Skoda’s integrability theorem since $e^{-\frac{2}{c}\varphi_{0}}\notin
L^{1}$. It follows from the first part of the theorem that
$D_{c(t)}(\varphi_{t})$ is nonempty for $c(t)=c-2nt>0$. ∎

Our previous analysis describes various singular situations. It also sheds
some light on the behavior of the maximal solution in short time. In fact, it
strongly suggests that the maximal solution is smooth outside analytic subsets
of $X$. We next prove Theorem A confirming this suggestion by employing our
previous a priori estimates.

###  4.3. Proof of Theorem A

Let $(\varphi_{0,j})$ be a decreasing sequence of smooth strictly $\omega$-psh
functions converging to $\varphi_{0}$. This is always possible thanks to
Demailly’s regularization (see [11], or [3]). Let $(\varphi_{t,j})$ be smooth
solutions of the parabolic equation (1.2) starting from $\varphi_{0,j}$. It
was proven by Guedj-Zeriahi in [21] that, as $j\to+\infty$, the family
$\varphi_{t,j}$ decreases to $\varphi_{t}$, which are $\theta_{t}$-psh
functions on $X$. They also proved that for each $t>(2c(\varphi_{0}))^{-1}$,
the function $\varphi_{t}$ is smooth on $X$ and satisfies (1.2) pointwise on
$X$. Moreover, $\varphi_{t}\to\varphi_{0}$ in capacity as $t\to 0$.

We investigate the regularity of $\varphi_{t}$ for small $t$. Let us fix
$\varepsilon_{0}>0$. Fix a constant $\varepsilon>\varepsilon_{0}$. Thanks to
Lemma 2.7 we fix $\psi\in{\rm PSH}(X,\omega)\cap\mathcal{C}^{\infty}_{\rm
loc}(X\setminus D_{\kappa})$ (the constant $\kappa=\kappa(\varepsilon_{0})$ is
defined by Lemma 2.7) such that

(4.1)
$\int_{X}\exp{\left(\frac{2\psi-2\varphi_{0}}{\varepsilon_{0}}\right)}dV<+\infty.$

We can always assume that $\psi\leq 0$. Recall that by the choice of $T$ and
since $\psi$ is less singular that $\varphi_{0}$, we always have

(4.2) $\int_{X}\exp{\left(\frac{-\psi}{T}\right)}\omega^{n}<+\infty.$

We will prove that $\varphi_{t}$ is smooth on $X\setminus D_{\kappa}$ for each
$t>\varepsilon$. Fix a compact subset $K\Subset X\setminus D_{\kappa}$. We
apply Lemma 3.1 and Theorem 3.2 for the flow $\varphi_{t,j}$ (and use the
function $\psi$ above) and note that, thanks to (4.1) and (4.2) the constants
in these theorems do not depend on $j$. Thus

$\sup_{t\in[\varepsilon,T]}\sup_{K}|\varphi_{t,j}|\leq C(\varepsilon,K).$

It follows from Theorem 3.6 that we also have a uniform bound on
$\Delta\varphi_{t,j}$:

$\sup_{t\in[\varepsilon,T]}\sup_{K}|\Delta_{\omega}(\varphi_{t,j})|\leq
C(\varepsilon,K).$

Now, from Proposition 3.5 and from the fact that the functions $\varphi_{t,j}$
satisfy the equation (1.2) we obtain a locally uniform bound on
$\dot{\varphi}_{t,j}$:

$\sup_{t\in[\varepsilon,T]}\sup_{K}\left|\frac{d\varphi_{t,j}}{dt}\right|\leq
C(\varepsilon,K).$

Then using the complex parabolic Evans-Krylov theory together with Schauder’s
estimate (see [4, Theorem 4.1.4]), we obtain higher order estimates on
$K\times[\varepsilon,T]$ just as in [21]. This justifies the smoothness of
$\varphi_{t}$ on $X\setminus D_{\kappa}$.

##  5\. Uniqueness and stability

###  5.1. Currents with zero Lelong numbers

In this section we study the uniqueness and stability of the Kähler-Ricci flow
starting from currents having zero Lelong number. The following estimate was
proven in [21] for the maximal flow. The ideas in [21] are the prove such
estimates (in a uniform way) for the smooth flows approximating the maximal
flow. We prove in the lemma below that, in the case of zero Lelong numbers,
the same estimate holds for any weak solution. The idea of our proof is quite
simple: we follow the same strategy of Guedj-Zeriahi, but instead of dealing
with the flow $\varphi_{t}$ itself we deal with the flow
$\varphi_{t+\varepsilon}$ (for $\varepsilon>0$) and then we let
$\varepsilon\to 0$. The key point is that the new flow is smooth up to zero
and the classical comparison principle can be applied.

######  Lemma 5.1.

Assume that $(\varphi_{t})$ is a weak solution of the parabolic equation (1.2)
with initial data $\varphi_{0}$. Assume also that $c(\varphi_{0})=+\infty$.
Let $\beta>0$ be a constant such that $2\beta>1/T_{\max}$. Then

$\varphi_{t}\geq(1-2\beta t)\varphi_{0}-C(t),\ \ 0<t<(2\beta)^{-1}.$

where $C(t)$ depends only on $t$ and $C(t)\searrow 0$ as $t\searrow 0$.

###### Proof.

Fix a positive constant $\alpha$ such that $2\beta-\alpha>1/T_{\max}$. Take a
small constant $\varepsilon>0$ such that

$(2\beta-\alpha)\omega+(1+2\beta\varepsilon)\chi>0.$

Set

$u_{t}:=\varphi_{t+\varepsilon},\ t\in[0,T_{\max}-\varepsilon).$

Since $c(\varphi_{0})=+\infty$, it follows from [21, Theorem 1.1] that $u$ is
smooth on $X\times[0,T]$. Applying [21, Lemma 2.9] we see that for any $x\in
X$ and any $t\in\left[0,(2\beta)^{-1}-\varepsilon\right)$,

(5.1) $(1-2\beta t)u_{0}(x)+\alpha tv_{\varepsilon}(x)+n(t\log t-t)\leq
u_{t}(x),$

where $v_{\varepsilon}$ is the unique continuous $\omega$-psh function
satisfying

$\alpha^{n}(\omega+dd^{c}v_{\varepsilon})^{n}=e^{\alpha
v_{\varepsilon}-2\beta\varphi_{\varepsilon}}\omega^{n}.$

Since $\varphi_{0}$ has zero Lelong number everywhere and since
$\varphi_{\varepsilon}$ converges to $\varphi_{0}$ in $L^{1}$ as
$\varepsilon\searrow 0$ it follows from [41, Theorem 3.1] that

$\int_{X}e^{-4\beta\varphi_{\varepsilon}}\omega^{n}\leq C,$

for a uniform constant $C>0$. Therefore, it follows from Kołodziej’s
$\mathcal{C}^{0}$ estimate (see [26]) that $v_{\varepsilon}$ is uniformly
bounded. Thus from (5.1) we get

$(1-2\beta t)\varphi_{\varepsilon}(x)-C(t)\leq\varphi_{t+\varepsilon}(x).$

Letting $\varepsilon\rightarrow 0$ we obtain the result. ∎

From Lemma 5.1 we immediately get the following convergence result.

######  Corollary 5.2.

Assume that $c(\varphi_{0})=+\infty$ and $\varphi_{t}$ is a weak solution of
the parabolic equation (1.2). Then $\varphi_{t}$ converges to $\varphi_{0}$ in
capacity as $t$ goes to zero.

We are now in the position to prove the following comparison principle.

######  Theorem 5.3.

Let $\varphi_{0},\psi_{0}\in{\rm PSH}(X,\omega)$ be such that
$c(\varphi_{0})=+\infty$. Let $(\varphi_{t}),(\psi_{t})$ be weak solutions of
the equation (1.2) starting from $\varphi_{0}$ and $\psi_{0}$ respectively.
Then

$\varphi_{0}\leq\psi_{0}\Longrightarrow\varphi_{t}\leq\psi_{t},\forall t>0.$

In particular, if $c(\varphi_{0})=+\infty$ then there is a unique weak
solution of the equation (1.2) with initial data $\varphi_{0}$.

###### Proof.

We devide the proof into two steps.

Step 1. We assume that $\chi\geq 0$. Fix $\varepsilon>0$ and consider the
following function

$v_{t}:=\psi_{t+\varepsilon}+C(\varepsilon),$

where $C(\varepsilon)$ is the constant defined in Lemma 5.1. Accordingly we
have that $v_{0}\geq\psi_{0}$. We compute

$\dot{v}_{t}=\dot{\psi}_{t+\varepsilon}=\log\left[\frac{(\omega+(t+\varepsilon)\chi+dd^{c}v_{t})^{n}}{\omega^{n}}\right].$

Since $\chi\geq 0$ and $\varphi_{t}$ is a weak solution of (1.2) it follows
that

$\dot{\varphi}_{t}=\log\left[\frac{(\omega+t\chi+dd^{c}\varphi_{t})^{n}}{\omega^{n}}\right]\leq\log\left[\frac{(\omega+(t+\varepsilon)\chi+dd^{c}\varphi_{t})^{n}}{\omega^{n}}\right].$

Our computations above show that $v_{t}$ is a smooth solution of the following
parabolic equation

$\dot{v}_{t}=\log\left[\frac{(\omega+(t+\varepsilon)\chi+dd^{c}v_{t})^{n}}{\omega^{n}}\right],$

while $\varphi_{t}$ is a weak subsolution. Since $\varphi_{0}\leq\psi_{0}\leq
v_{0}$ it follows from the generalized comparison principle (Proposition 2.3)
that $\varphi_{t}\leq v_{t}$. Since $C(\varepsilon)\searrow 0$ as
$\varepsilon\searrow 0$ the result follows.

Step 2. We remove the non-negative assumption on $\chi$. The idea is to make a
change of variable to get rid of the sign of $\chi$. Without loss of
generality we can assume that $T_{\max}>1$ and hence $\chi>-\omega$. We make
the following change of variable

$u_{t}:=e^{t}\varphi_{1-e^{-t}}.$

Then $u_{t}$ satisfies the following equation

$\dot{u}_{t}=u_{t}-nt+\log\left[\frac{(\omega+(e^{t}-1)\chi^{\prime}+dd^{c}u_{t})^{n}}{\omega^{n}}\right],$

where $\chi^{\prime}=\omega+\chi>0$. Thus this change of variable gives us
another equation with the twisted form $\chi^{\prime}>0$. We then can apply
our techniques in the first step. For the reader’s convenience we explain the
proof in full details.

Fix $\varepsilon>0$ small enough and consider the following families:

$u_{t}:=e^{t}\varphi_{1-e^{-t}}\ ,\
v_{t}:=e^{t+\varepsilon}\psi_{1-e^{-t-\varepsilon}}+n\varepsilon(e^{t}-1)+e^{t+\varepsilon}C(1-e^{-\varepsilon}),$

where $C(s)$ is the constant defined in Lemma 5.1. Observe that $v$ is smooth
on $[0,t_{0}]\times X$, where $t_{0}$ is a fixed positive constant such that
$1-e^{-\varepsilon-t_{0}}<T_{\max}$. We can assume that $\varphi_{t}$ and
$\psi_{t}$ are negative for small $t$. Then it follows from Lemma 5.1 that

$v_{0}\geq\varphi_{0}=u_{0}.$

Since $\chi^{\prime}:=\chi+\omega\geq 0$ it follows that

$\dot{u_{t}}\leq
u_{t}-nt+\log\left[\frac{(\omega+(e^{t+\varepsilon}-1)\chi^{\prime}+dd^{c}u_{t})^{n}}{\omega^{n}}\right],$

and

$\dot{v_{t}}=v_{t}-nt+\log\left[\frac{(\omega+(e^{t+\varepsilon}-1)\chi^{\prime}+dd^{c}v_{t})^{n}}{\omega^{n}}\right].$

It thus follows from the comparison principle (Proposition 2.3) that
$u_{t}\leq v_{t},\forall t\in[0,t_{0}]$. Letting $\varepsilon\searrow 0$ we
obtain $\varphi_{t}\leq\psi_{t}$ for any $t\in(0,t_{0})$. Then we can conclude
$\varphi_{t}\leq\psi_{t}$ for any $t\in(0,T_{\max})$. ∎

The following stability result follows from the uniqueness of the flow.

######  Theorem 5.4.

Let $\varphi_{0}$ be a $\omega$-psh function and $\varphi_{0,j}$ a sequence of
$\omega$-psh functions such that $\varphi_{0,j}\rightarrow\varphi_{0}$ in
$L^{1}$. Assume also that $c(\varphi_{0})=+\infty$. Denote by $\varphi_{t,j}$
and $\varphi_{t}$ the solutions of (1.2) with initial data $\varphi_{0,j}$ and
$\varphi_{0}$ respectively. Then for each $t>0$,

$\varphi_{t,j}\rightarrow\varphi_{t}\quad\mbox{in}\;\;\mathcal{C}^{\infty}(X)\
\mbox{as}\quad j\to+\infty.$

###### Proof.

Assume first that the sequence $(\varphi_{0,j})$ is deceasing. Then we can
find a decreasing sequence $(\psi_{0,j})$ of smooth strictly $\omega$-psh
functions which converges to $\varphi_{0}$ and satisfies

$\varphi_{0}\leq\varphi_{0,j}\leq\psi_{0,j},\ \ \forall j\in\mathbb{N}.$

Let $(\varphi_{t,j})$ and $(\psi_{t,j})$ be the corresponding weak solutions
of the equation (1.2) with initial data $\varphi_{0,j}$ and $\psi_{0,j}$
respectively. It follows from the comparison principle (Theorem 5.3) that

$\varphi_{t}\leq\varphi_{t,j}\leq\psi_{t,j},\ \ \forall j\in\mathbb{N}.$

By Lemma 4.2 we know that $\psi_{t,j}\searrow\varphi_{t}$, hence
$\varphi_{t,j}\searrow\varphi_{t}$. Moreover it follows from [21] that for
each $\varepsilon>0$, $T<T_{\max}$, $k\in\mathbb{N}$ we have

$\|\varphi_{j}\|_{\mathcal{C}^{k}([\varepsilon,T]\times X)}\leq
C(\varepsilon,T,k),$

for $j>j(\varepsilon)$ large enough. Then using Arzelà-Ascoli theorem, we know
that, for each $t>0$, $\varphi_{t,j}$ converges to $\varphi_{t}$ in
$\mathcal{C}^{\infty}(X)$ as $j$ goes to $+\infty$. Hence the result follows.

Now, let us prove the general statement. We can assume that $\varphi_{0}$ is
negative. Recall that, for each $j$ fixed, $\varphi_{t,j}$ converges in
$L^{1}(X)$ to $\varphi_{0,j}$ as $t\searrow 0$. The arguments in the previous
step also apply in this case yielding that for each $t>0$, $\varphi_{t,j}$
converges to some $\psi_{t}$ in $\mathcal{C}^{\infty}(X)$ as $j$ goes to
$+\infty$. Note that, by construction, $\psi_{t}$ is such that

$\frac{\partial\psi_{t}}{\partial
t}=\log\left[\frac{(\theta_{t}+dd^{c}\psi_{t})^{n}}{\omega^{n}}\right],\
\forall t>0.$

It follows from Lemma 5.1 that

$\varphi_{t,j}\geq(1-t)\varphi_{0,j}-C(t),\forall t\in(0,1).$

Letting $j\to+\infty$, then $t\searrow 0$ we obtain $\liminf_{t\to
0}\psi_{t}\geq\varphi_{0}$. To see the reverse inequality we set

$u_{0,j}:=\left(\sup_{k\geq j}\varphi_{0,k}\right)^{*}\searrow\varphi_{0}.$

Let $u_{t,j}$ be the maximal solution of (1.2) with initial data $u_{0,j}$. By
the comparison principle we have

$u_{t,j}\geq\varphi_{t,j}.$

It follows from the previous step that $(u_{t,j})$ decreases to $\varphi_{t}$,
hence $\psi_{t}\leq\varphi_{t}$. Since $\varphi_{t}$ converges in $L^{1}(X)$
to $\varphi_{0}$, we infer that $\psi_{t}$ converges in $L^{1}(X)$ to
$\varphi_{0}$. From the comparison principle (Theorem 5.3) we obtain
$\varphi_{t}\equiv\psi_{t}$. ∎

###  5.2. Currents with positive Lelong numbers

In this section we try and prove the uniqueness result for the general case:
initial currents with positive Lelong numbers. We fix the following notations.
Assume that $\varphi_{0}$ is a $\omega$-psh function on $X$, $\eta$ is a
smooth closed $(1,1)$-forms on $X$. Assume also that

$\frac{1}{2c(\varphi_{0})}<T_{\max}:=\sup\left\\{t>0\ \big{|}\
\\{\omega\\}+t\eta-tc_{1}(X)>0\right\\}.$

Fix two positive constants $\alpha,\beta$ such that

$\frac{1}{2c(\varphi_{0})}<\frac{1}{2\beta}<\frac{1}{2\beta-\alpha}<T_{\max}.$

Recall that the twisted Kähler-Ricci flow reads in the level of potentials as:

(5.2)
$\dot{\varphi_{t}}=\log\left[\frac{(\theta_{t}+dd^{c}\varphi_{t})^{n}}{\omega^{n}}\right],$

where $\theta_{t}:=\omega+t\eta-t{\rm Ric}(\omega)$. We recall the definition
of weak solutions of the parabolic equation (5.2).

######  Definition 5.5.

A function $\varphi:(0,T_{\max})\times X\to\mathbb{R}$ is called a weak
solution of the equation (5.2) starting from $\varphi_{0}$ if the following
conditions are satisfied:

  * (i)

For each $t>0$, the function $\varphi_{t}$ is $\theta_{t}$-psh on $X$.
Moreover the function $t\mapsto\varphi_{t}$ is continuous as a map from
$[0,T_{\max})$ to $L^{1}(X)$.

  * (ii)

For each $\varepsilon>0$, there exists an analytic subset
$D_{\varepsilon}\subset X$ such that the function

$(t,x)\mapsto\varphi_{t}(x)$

is smooth, strictly $\theta_{t}$-psh on
$[\varepsilon,T_{\max})\times(X\setminus D_{\varepsilon})$ and satisfies the
equation (5.2) in the classical sense. Moreover, $D_{\varepsilon}$ is empty if
$\varepsilon>1/(2c(\varphi_{0}))$.

We first show that the maximal solution constructed in [21] is indeed the
maximal solution meaning that it dominates every weak solutions (in the sense
of Definition 5.5).

######  Lemma 5.6.

Let $\varphi_{t}$ be the maximal solution of (5.2) starting from $\varphi_{0}$
and let $\psi_{t}$ be a weak solution with initial data $\psi_{0}$. Then

$\psi_{0}\leq\varphi_{0}\Longrightarrow\psi_{t}\leq\varphi_{t},\ \ \forall
t\in(0,T_{\max}).$

###### Proof.

Let $\varphi_{0,j}$ be the smooth sequence decreasing to $\varphi_{0}$ and
$\varphi_{t,j}$ be the smooth sequence of approximants running from
$\varphi_{0,j}$ and decreasing poitwise to $\varphi_{t}$. It thus suffices to
show that $\psi_{t}\leq\varphi_{t,j}$ for any fixed $j$. To simplify the
notation, in the remaining part of the proof we drop the index $j$. Fix
$0<T<T_{\max}$ and $\varepsilon>\delta>0$. Denote by $Y_{\epsilon}$ the
analytic set outside of which the weak solution $\psi_{t}$ is smooth for
$t\geq\varepsilon$. Consider

$H(t,x):=\psi_{t+\varepsilon}-\varphi_{t+\varepsilon+\delta}+\delta\phi\qquad\;t\in[0,T],\;x\in
X_{\varepsilon}$

where $\phi\leq 0$ is a $\omega$-psh function that is smooth on
$X_{\varepsilon}:=X\setminus Y_{\varepsilon}$ and goes to $-\infty$ on
$Y_{\varepsilon}$. Observe that by construction $H$ is upper semicontinuos on
$[0,T-\varepsilon]\times X_{\varepsilon}$ and $\phi=-\infty$ on
$Y_{\varepsilon}$. This implies that $H$ achieves its maximum at some point
$(t_{0},x_{0})\in[0,T-\varepsilon]\times X_{\varepsilon}$. We claim that
$t_{0}=0$. Indeed, arguing by contradiction we get that at $(t_{0},x_{0})$

$\displaystyle 0\leq\frac{\partial}{\partial t}H$ $\displaystyle=$
$\displaystyle\dot{\psi}_{t+\varepsilon}-\dot{\varphi}_{t+\varepsilon+\delta}$
$\displaystyle=$
$\displaystyle\log\left[\frac{(\theta_{t}+\varepsilon\chi+dd^{c}\psi_{t+\varepsilon})^{n}}{(\theta_{t}+(\varepsilon+\delta)\chi+dd^{c}\varphi_{t+\varepsilon+\delta})^{n}}\right]$
$\displaystyle\leq$
$\displaystyle\log\left[\frac{(\theta_{t}+dd^{c}\varphi_{t+\varepsilon+\delta}+\varepsilon\chi-\delta
dd^{c}\phi)^{n}}{(\theta_{t}+(\varepsilon+\delta)\chi+dd^{c}\varphi_{t+\varepsilon+\delta})^{n}}\right]<0$

where in the last inequality we assume $\chi\geq\omega$. Observe that this
last assumption costs no generality since otherwise we can perform a change of
variables and argue as in Section 5.1. Thus we have, by letting
$\delta\searrow 0$,

$\psi_{t+\varepsilon}-\varphi_{t+\varepsilon}\leq
H(0,x_{0})\leq\sup_{X}(\psi_{\varepsilon}-\varphi_{\varepsilon}).$

Furthermore, since $(t,x)\rightarrow\varphi_{t}(x)$ is smooth, and in
particular continuous, it follows from Hartogs’ lemma that

$\sup_{X}(\psi_{\varepsilon}-\varphi_{\varepsilon})\rightarrow\sup_{X}(\psi_{0}-\varphi_{0})\leq
0.$

The conclusion follows when we let $\varepsilon$ go to zero. ∎

In the remaining part of this subsection we prove a "partial" uniqueness
result in the case of positive Lelong numbers.

######  Lemma 5.7.

Assume that $\varphi_{0}$ is as above. Let $(\varphi_{t})$ be a weak solution
of the equation (5.2) starting from $\varphi_{0}$ and assume that
$\varphi_{t}$ and the maximal solution have the same singularities. Then

$(1-2\beta t)\varphi_{0}-C(t)\leq\varphi_{t},\ \ \forall 0<t<(2\beta)^{-1},$

where $C(t)$ is a uniform constant depending only on $t$, and converges to
zero as $t$ goes to zero.

###### Proof.

We apply again the techniques in [21, Lemma 2.9]. Fix two small positive
constants $\varepsilon,\delta$ such that

(5.3) $(2\beta-\alpha-\delta)\omega+(1+2\beta\varepsilon)\chi>0.$

Consider the following function

$u_{t}=(1-2\beta t)\varphi_{\varepsilon}+\alpha tv_{\varepsilon}+n(t\log
t-t),$

where $v_{\varepsilon}$ is the unique bounded function in ${\rm
PSH}(X,\omega)$ such that

(5.4) $\alpha^{n}(\omega+dd^{c}v_{\varepsilon})^{n}=e^{\alpha
v_{\varepsilon}-2\beta\varphi_{\varepsilon}}\omega^{n}.$

Observe also that $v_{\varepsilon}$ is uniformly bounded by a constant
independent of $\varepsilon$.

Let $Y_{\varepsilon}$ be the analytic subset outside of which the maximal
solution is smooth for $t\geq\varepsilon$. Since $\varphi_{t}$ has the same
singularities as the maximal solution, it ifollows that

$\sup_{t\in[0,(2\beta)^{-1}]}\sup_{X_{\varepsilon}}(u_{t}-\varphi_{t+\varepsilon})<+\infty.$

Here $X_{\varepsilon}:=X\setminus Y_{\varepsilon}$. Let $\phi$ be a negative
$\omega$-plurisubharmonic function on $X$ which is smooth on $X_{\varepsilon}$
and goes to $-\infty$ on $Y_{\varepsilon}$. By our construction the following
function

$H(t,x)=u_{t}(x)-\varphi_{t+\varepsilon}(x)+\delta t\phi,\ \
t\in[0,(2\beta)^{-1}],\ x\in X_{\varepsilon},$

is upper semi-continuous on $[0,(2\beta)^{-1}]\times X_{\varepsilon}$.
Moreover $H(t,x)$ converges to $-\infty$ uniformly in $t$ as $x$ approaches
$Y_{\varepsilon}$. It then follows that $H$ attains a maximum at some
$(t_{0},x_{0})\in[0,(2\beta)^{-1}]\times X_{\varepsilon}$. By the choice of
$\delta$ and $\varepsilon$ and by the construction of $u_{t}$ it is easy to
see that $t_{0}=0$. If it was not the case then by the maximum principle we
would have, at this point,

(5.5) $0\leq\frac{\partial}{\partial t}H=\alpha
v_{\varepsilon}-2\beta\varphi_{\varepsilon}+n\log
t-\log\left[\frac{(\omega+(t+\varepsilon)\chi+dd^{c}\varphi_{t+\varepsilon})^{n}}{\omega^{n}}\right].$

In the other hand, by (5.3) and by the construction of $u_{t}$ we have

$\alpha t(\omega+dd^{c}v_{\varepsilon})<\omega+(t+\varepsilon)\chi+\delta
tdd^{c}\phi+dd^{c}u_{t}.$

At the point $(t_{0},x_{0})$ where $H$ attains its maximum we have

(5.6) $\alpha
t(\omega+dd^{c}v_{\varepsilon})<\omega+(t+\varepsilon)\chi+dd^{c}\varphi_{t+\varepsilon}.$

Now, from (5.4), (5.5) and (5.6) we get a contradiction. Letting $\delta$ go
to $0$ we obtain $u_{t}\leq\varphi_{t+\varepsilon}$. We finally let
$\varepsilon\to 0$ to conclude. ∎

We will prove the following uniqueness result.

######  Theorem 5.8.

Assume that $\varphi_{0}$ is as above. Let $(\varphi_{t})$ be a weak solution
of the equation (5.2) starting from $\varphi_{0}$ and assume that
$\varphi_{t}$ has the same singularities as the maximal solution. Then
$\varphi_{t}$ is also maximal.

###### Proof.

It costs no generality to assume that $\varphi_{t}\leq 0$ for small $t$. We
first treat the case when $\chi\geq 0$. Let $(\psi_{t})$ be the maximal
solution of the equation (5.2) starting from $\varphi_{0}$. We prove that
$\varphi_{t}\equiv\psi_{t}$. Since $\psi_{t}$ is maximal, thanks to Lemma 5.6
it suffices to prove that $\psi_{t}\leq\varphi_{t}$.

Fix $T\in(0,T_{\max})$, $\varepsilon>0$ and consider the following function

$u_{t}:=\varphi_{t+\varepsilon}+C(\varepsilon),\ t\in[0,T-\varepsilon],\ x\in
X_{\varepsilon},$

where $C(\varepsilon)$ and $X_{\varepsilon}$ are defined as in Lemma 5.7. We
remark that by this construction and by Lemma 5.7 it holds that

$u_{0}=\varphi_{\varepsilon}+C(\varepsilon)\geq\varphi_{0}.$

Fix a small positive constant $\delta<\varepsilon$ such that
$\delta\omega<\varepsilon\chi$. Consider now the following function

$G(t,x):=\psi_{t}(x)-u_{t}(x)+\delta\phi,\ \ t\in[0,T-\varepsilon],\ x\in
X_{\varepsilon},$

where $\phi$ is also defined in Lemma 5.7. Since $(\varphi_{t})$ and
$(\psi_{t})$ have the same singularity, meaning that the difference is
uniformly bounded in $t$, one can apply the arguments in Lemma 5.7 to see that
$G$ attains a maximum at some point $(t_{0},x_{0})$. Again, $t_{0}$ is forced
to be $0$ by the maximum principle. Indeed, if $t_{0}$ were not $0$, by the
maximum principle we would have, at $(t_{0},x_{0})$,

$0\leq\frac{\partial}{\partial
t}G=\log\left[\frac{(\theta_{t}+dd^{c}\psi_{t})^{n}}{(\theta_{t}+\varepsilon\chi+dd^{c}u_{t})^{n}}\right]<\log\left[\frac{(\theta_{t}+\varepsilon\chi+\delta
dd^{c}\phi+dd^{c}\psi_{t})^{n}}{(\theta_{t}+\varepsilon\chi+dd^{c}u_{t})^{n}}\right]\leq
0,$

which is impossible.

Change of variables: In the previous step we have assumed that $\chi\geq 0$.
To remove this assumption we perform a change of variables exactly as in the
case of zero Lelong numbers. It is harmless to assume that
$\chi^{\prime}:=\omega+\chi>0$. Now, let $(\varphi_{t})$ be a weak solution
and consider

$u_{t}:=e^{t}\varphi_{1-e^{-t}},t\in[0,+\infty).$

We compute

(5.7)
$\dot{u_{t}}=u_{t}-nt+\log\left[\frac{\omega+(e^{t}-1)\chi^{\prime}+dd^{c}u_{t})^{n}}{\omega^{n}}\right].$

Thus, any weak (maximal) solution of the equation (5.2) is in one-to-one
correspondence with a weak (maximal) solution of the equation (5.7) by a
change of variables. Now, we combine the previous step with the proof of
Theorem 5.3 to conclude. ∎

###  5.3. Proof of Theorem B and Theorem B’

Theorem B follows from Theorem 5.3 and Theorem 5.4. Theorem B’ follows from
Theorem 5.8.

##  6\. Convergence at time zero

In this section we investigate the convergence at zero of weak solutions of
the Kähler-Ricci flow. Let $\varphi_{0}$ be a $\omega$-plurisubharmonic
function on $X$ and assume that

$\frac{1}{2c(\varphi_{0})}<T_{\max}.$

This condition insures that the maximal flow constructed in [21] is well-
defined.

Let $(\varphi_{t})$ be the maximal solution of the equation (5.2) starting
from $\varphi_{0}$. If $\varphi_{0}$ is continuous Song-Tian [34] proved that
$\varphi_{t}$ converges uniformly to $\varphi_{0}$. When $\varphi_{0}$ has
finite energy, it was shown in [21] that $\varphi_{t}$ converges in energy
while the convergence in capacity always holds. If $\varphi_{0}$ is continuous
in some open subset $U\subset X$ we prove in the following that the
convergence is locally uniform in $U$. This generalizes the uniform
convergence of [34] when $U=X$.

######  Theorem 6.1.

Assume that $\varphi_{0}$ is continuous in an open subset $U$ of $X$. Then
$\varphi_{t}$ converges in $L^{\infty}_{\rm loc}(U)$ to $\varphi_{0}$.

###### Proof.

It costs no generality to assume that $\varphi_{t}\leq 0$ for small $t>0$. It
follows from [21, Lemma 2.9] that there exist a uniform constant $C$ and a
small time $t_{0}$ such that

$\varphi_{t}\geq\varphi_{s}+n(t-s)\left[\log(t-s)-C\right],\ \forall 0\leq
s\leq t\leq t_{0}.$

Set $\psi_{t}:=\varphi_{t}+nCt-nt\log t$. Using the convexity of the function
$x\log x$ we obtain

$\psi_{t}\geq\psi_{s}-nt\log 2.$

Now, set $u_{t}=\psi_{t}+nt\log 4$ and observe that

$u_{t}\geq u_{t/2}.$

Thus for each $t>0$ the sequence $u_{t_{0}2^{-j}}$ decreasingly converges to
$u_{0}=\varphi_{0}$ as $j$ goes to $+\infty$. By Dini’s theorem the
convergence is locally uniform in $U$. ∎

When the initial current has analytic singularities we obtain the following
expected convergence in its regular locus:

######  Theorem 6.2.

Assume that $e^{\gamma\varphi_{0}}$ is smooth on $X$ for some positive
constant $\gamma$ and $\varphi_{0}$ is strictly $\omega$-psh in $X$. Then the
maximal solution $\varphi_{t}$ of the equation (5.2) converges in
$\mathcal{C}^{\infty}_{\rm loc}(\Omega)$ to $\varphi_{0}$ as $t$ goes to $0$.
Here, $\Omega:=\\{\varphi_{0}>-\infty\\}$.

###### Proof.

First observe that, since $e^{\gamma\varphi_{0}}$ is smooth on $X$,
$\Omega=\\{\varphi_{0}>-\infty\\}$ is an open set. Thanks to Lemma 4.2 the
maximal solution does not depend on the choice of the approximating sequence.
Thus we can take

$\varphi_{0,j}:=\frac{1}{\gamma}\log\left(e^{\gamma\varphi_{0}}+\frac{1}{j}\right)$

as the smooth sequence decreasing to $\varphi_{0}$. To simplify the notation
we can assume that $\gamma=1$. For each $j$, let $\varphi_{t,j}$ be the smooth
solution of (5.2) starting from $\varphi_{0,j}$. Denote by
$\omega_{t}:=\theta_{t}+dd^{c}\varphi_{t,j}$. We want to prove that there
exists a positive constant $C$ such that

$\frac{1}{C}\omega\leq\omega_{t}\leq C\omega$

on each compact subset $K\subset\Omega$. The result will then follow from [4,
Theorem 3.2.17]. We start proving the right-hand side inequality. Since, in
$K$, $\omega_{t}$ and $\omega$ are both strictly positive forms, it suffices
to prove that ${\rm tr}_{\omega}{(\omega_{t})}\leq C$ in $K\subset\Omega$.
Note that on each compact $K\subset\Omega$, $\varphi_{0}$ is a
$\delta\omega$-psh function for some $\delta\in(0,1)$. Furthermore,
$\varphi_{0}$ is smooth on $\Omega$ and by construction we have
$\varphi_{0,j}\geq\varphi_{0}$. Moreover

$\Delta\varphi_{0,j}\leq Be^{-\varphi_{0}},$

for some uniform constant $B>0$. Indeed by a simple computation we get

$dd^{c}e^{\varphi_{0,j}}=e^{\varphi_{0,j}}dd^{c}\varphi_{0,j}+e^{\varphi_{0,j}}d\varphi_{0,j}\wedge
d^{c}\varphi_{0,j}\leq B\omega$

where the last inequality follows from the fact that $e^{\varphi_{0}}$ is
smooth on $X$. We now apply Theorem 3.8 to get the uniform estimate on ${\rm
tr}_{\omega}(\omega_{t})$. It remains to prove that
$C^{-1}\omega\leq\omega_{t}$ on each $K\subset\Omega$. By construction and
from the fact that $\varphi_{0}$ is strictly $\omega$-psh we get

$c\omega^{n}<(\omega+dd^{c}\varphi_{0,j})^{n}=e^{\dot{\varphi}_{0,j}}\omega^{n}$

where $c$ is a uniform positive constant. This easily implies
$\dot{\varphi}_{0,j}>\log c$. Applying Proposition 3.7, we infer that there
exists a uniform cosntant $C_{1}>0$ such that

$\dot{\varphi}_{t,j}\geq C_{1}(\varphi_{0}+1).$

By construction $\omega_{t}^{n}=e^{\dot{\varphi}_{t,j}}\omega^{n}$ and so,
thanks to the above estimate, we infer that there exists a uniform constant
$C_{2}=C(K)$ such that $\omega_{t}^{n}\geq{C_{2}^{-1}}\omega^{n}$ on each
$K\subset\Omega$. Let now $\lambda_{1},\cdots,\lambda_{n}$ be the eigenvalues
of the matrix associated to $\omega_{t}$ with respect to $\omega$. Then the
latter inequality means that
$\lambda_{1}\cdots\lambda_{n}\geq\frac{1}{C_{2}}$. This combined with the
previous estimate (i.e. $\max_{i}\lambda_{i}\leq C$) gives that
$\min_{i}\lambda_{i}\geq\frac{1}{C_{3}}$ for some uniform constant $C_{3}$.
Hence the conclusion. ∎

The same arguments as above can also be applied in other contexts. Assume that
$0<f\in L^{1}(X)$ is smooth in the complement of some closed subset $D\subset
X$. In [14] we say that $f$ satisfies the condition $\mathcal{H}_{f}$ if

$f=e^{\psi^{+}-\psi^{-}},\ \psi^{\pm}\in{\rm PSH}(X,C\omega),\ \psi^{-}\in
L^{\infty}_{\rm loc}(X\setminus D),$

for some positive constant $C$. Assume also that $f$ satisfies the
compatibility condition $\int_{X}f\omega^{n}=\int_{X}\omega^{n}$, the latter
will be normalized to be $1$. Let $\varphi_{0}\in\mathcal{E}(X,\omega)$ be the
unique normalized solution of the complex Monge-Ampère equation

$(\omega+dd^{c}\varphi_{0})^{n}=f\omega^{n}.$

We are going to prove the following convergence result:

######  Theorem 6.3.

Assume the above setting and let $(\varphi_{t})$ be the unique weak solution
of the equation (5.2) with initial data $\varphi_{0}$. Then $\varphi_{t}$
converges to $\varphi_{0}$ in $\mathcal{C}^{\infty}_{\rm loc}(X\setminus D)$.

###### Proof.

We approximate $\psi^{\pm}$ by using Demailly’s technique (see [11]). Let
$\psi^{\pm}_{j}$ denote these approximants. We then use Yau’s work [40] to
find $\varphi_{0,j}\in{\rm PSH}(X,\omega)\cap\mathcal{C}^{\infty}(X)$ such
that

$(\omega+dd^{c}\varphi_{0,j})^{n}=c_{j}e^{\psi^{+}_{j}-\psi^{-}_{j}}\omega^{n},$

where $c_{j}$ are normalization constants. It was proved in [14] that $c_{j}$
converges to $1$ as $j$ goes to $+\infty$ and the following estimates hold:

  * •

For each $\varepsilon>0$ there exists $C_{1}=C(\varepsilon)>0$ such that

$\varphi_{0,j}\geq\varepsilon\psi^{-}_{j}-C_{1}.$

  * •

There exists a uniform constant $C_{2}>0$ such that

$\Delta_{\omega}(\varphi_{0,j})\leq e^{-C_{2}\psi^{-}_{j}}.$

Now, for each $j$ let $(\varphi_{t,j})$ be the unique smooth solution of the
parabolic equation (5.2) with initial data $\varphi_{0,j}$. It follows from
Theorem 5.4 that $\varphi_{t,j}$ converges to $\varphi_{t}$ the unique weak
solution of (5.2). Using Theorem 3.8 we get a uniform estimate for ${\rm
tr}_{\omega}(\omega_{t,j})$ on each compact subset $K\Subset X\setminus D$.
This together with Proposition 3.7 and [4, Theorem 3.2.17] yield locally
uniform bounds on all derivatives of $\varphi_{t}$. This explains the
convergence in $\mathcal{C}^{\infty}_{\rm loc}(X\setminus D)$ of $\varphi_{t}$
to $\varphi_{0}$ and the result follows. ∎

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