# Supremum of Perelman’s entropy and Kähler-Ricci flow on a Fano manifold

Gang Tian  ,  Shijin Zhang  ,  Zhenlei $\text{Zhang}^{*}$ and  Xiaohua
$\text{Zhu}^{**}$ Gang Tian
School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871,
China
and Department of Mathematics, Princeton University, New Jersey, NJ 02139, USA
<EMAIL_ADDRESS>Shijin Zhang, School of Mathematical Sciences and BICMR,
Peking University, Beijing, 100871, P.R.China<EMAIL_ADDRESS>Zhenlei
Zhang
Department of Mathematics, Beijing Capital Normal University, Beijing, China
<EMAIL_ADDRESS>Xiaohua Zhu
School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871,
China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we extend the method in [TZhu5] to study the energy level
$L(\cdot)$ of Perelman’s entropy $\lambda(\cdot)$ for Kähler-Ricci flow on a
Fano manifold. Consequently, we first compute the supremum of $\lambda(\cdot)$
in Kähler class $2\pi c_{1}(M)$ under an assumption that the modified
Mabuchi’s K-energy $\mu(\cdot)$ defined in [TZhu2] is bounded from below.
Secondly, we give an alternative proof to the main theorem about the
convergence of Kähler-Ricci flow in [TZhu3].

###### Key words and phrases:

Kähler-Ricci flow, Kähler-Ricci solitons, Perelman’s entropy

###### 1991 Mathematics Subject Classification:

Primary: 53C25; Secondary: 53C55, 58E11

* Partially supported by a grant of BMCE 11224010007 in China

** Partially supported by the NSFC Grant 10990013

## Introduction

In this paper, we extend the method in [TZhu5] to study the energy level
$L(\cdot)$ of Perelman’s entropy $\lambda(\cdot)$ for Kähler-Ricci flow on an
$n$-dimensional compact Kähler manifold $(M,J)$ with positive first Chern
class $c_{1}(M)>0$ (namely called a Fano manifold). We will show that
$L(\cdot)$ is independent of choice of initial Kähler metrics in $2\pi
c_{1}(M)$ under an assumption that the modified Mabuchi’s K-energy
$\mu(\cdot)$ is bounded from below (cf. Proposition 3.1 in Section 3). The
modified Mabuchi’s K-energy $\mu(\cdot)$ is a generalization of Mabuchi’s
K-energy. It was showed in [TZhu2] that $\mu(\cdot)$ is bounded from below if
$M$ admits a Kähler-Ricci soliton.

As an application of Proposition 3.1, we first compute the supremum of
Perelman’s entropy $\lambda(\cdot)$ in Kähler class $2\pi c_{1}(M)$ [Pe]. More
precisely, we prove that

######  Theorem 0.1.

Suppose that the modified Mabuchi’s K-energy is bounded from below. Then

(0.1)
$\sup\\{\lambda({g^{\prime}})|~{}g^{\prime}\in\mathcal{K}_{X}\\}=(2\pi)^{-n}[nV-
N_{X}(c_{1}(M))].$

Here the quantity $N_{X}(c_{1}(M))$ is a nonnegative invariance in
$\mathcal{K}_{X}$ and it is zero iff the Futaki-invariant vanishes [Fu]. We
denote $\mathcal{K}_{X}$ to be a class of $K_{X}$-invariant Kähler metrics in
$2\pi c_{1}(M)$, where $K_{X}$ is an one-parameter compact subgroup of
holomorphisms transformation group generated by an extremal holomorphic vector
field $X$ for Kähler-Ricci solitons on $M$ [TZhu2]. We note that we do not
need to assume an existence of Kähler-Ricci solitons in Theorem 0.1. In fact,
if we assume the existence of Kähler-Ricci solitons then we can use a more
direct way to prove Theorem 0.1 and that the supremum of $\lambda(\cdot)$ can
be achieved in $\mathcal{K}_{X}$ (cf. Section 1). It seems that the supremum
of $\lambda(\cdot)$ can be achieved in the total space of Kähler potentials in
$2\pi c_{1}(M)$ if $M$ admits a Kähler-Ricci soliton. In a special case of
small neighborhood of a Kähler-Ricci soliton the positivity has been verified
by computing the second variation of $\lambda(\cdot)$ in [TZhu4].

As another application of Proposition 3.1, we prove the following convergence
result for Kähler-Ricci flow.

######  Theorem 0.2.

Let $(M,J)$ be a compact Kähler manifold which admits a Kähler-Ricci soliton
$(g_{KS},X)$. Then Kähler-Ricci flow with any initial Kähler metric in
$\mathcal{K}_{X}$ will converge to a Kähler-Ricci soliton in $C^{\infty}$ in
the sense of Kähler potentials. Moreover, the convergence can be made
exponentially.

We note that that without loss of generality we may assume that a Kähler-Ricci
soliton $g_{KS}$ on $M$ is corresponding to the above $X$ (cf. [TZhu1],
[TZhu2]). Theorem 0.2 was first proved by Tian and Zhu in [TZhu3] by using an
inequality of Moser-Trudinger type established in [CTZ]111We need to add more
details about how to use the Moser-Trudinger typed inequality in general
case.. Here we will modify arguments in [TZhu5] in our general case that
$(M,J)$ admits a Kähler-Ricci soliton to give an alternative proof of this
theorem. This new proof does not use such an inequality of Moser-Trudinger
type. Moreover, in particular, in case that $(M,J)$ admits a Kähler-Einstein
metric this new proof allows us to avoid to use a deep result recentlly proved
by Chen and Sun in [CS] for the uniqueness of Kähler-Einsteins in the sense of
orbit space to give a self-contained proof to the main theorem in [TZhu5].

The organization of paper is as follows. In Section 1, we discuss an upper
bound of $\lambda(\cdot)$ in general case-without any condition for
$\mu(\cdot)$ and show that the quantity $(2\pi)^{-n}[nV-N_{X}(c_{1}(M))]$ is
an upper bound of $\lambda(\cdot)$ in $\mathcal{K}_{X}$ (cf. Proposition 1.4).
In Section 2, we will summarize to give some estimates for modified Ricci
potentials of evolved Kähler metrics along Kähler-Ricci flow (cf. Proposition
2.3). In Section 3, we prove Proposition 3.1 and so do Theorem 0.1. Theorem
0.2 will be proved in Section 6. In Section 4, we improve our key Lemma 3.2 in
Section 3 independent of time $t$ (cf. Proposition 4.2). Section 5 is a
discussion about an upper bound of $\lambda(\cdot)$ in $\mathcal{K}_{Y}$ for a
general holomorphic vector field $Y\in\eta_{r}(M)$. Section 7 is an appendix
where we discuss the gradient estimate and Laplace estimate for the minimizers
of Perelman’s $W$-functional along the Kähler-Ricci flow.

##  1\.  An upper bound of $\lambda(\cdot)$

In this section, we first review Perelman’s $W$-functional for triples
$(g,f,\tau)$ on a closed $m$-demensional Riemannian manifold $M$ (cf. [Pe],
[TZhu5]). Here $g$ is a Riemannian metric, $f$ is a smooth function and $\tau$
is a constant. In our situation, we will normalize volume of $g$ by

(1.1) $\int_{M}dV_{g}\equiv\,V$

and so we can fix $\tau$ by $\frac{1}{2}$. Then the $W$-functional depends
only on a pair $(g,f)$ and it can be reexpressed as follows:

(1.2) $\displaystyle W(g,f)=(2\pi)^{-m/2}\int_{M}[\frac{1}{2}(R(g)+|\nabla
f|^{2})+f]e^{-f}dV_{g},$

where $R(g)$ is a scalar curvature of $g$ and $(g,f)$ satisfies a
normalization condition

(1.3) $\int_{M}e^{-f}dV_{g}\,=V.$

Then Perelman’s entropy $\lambda(g)$ is defined by

$\lambda(g)=\inf_{f}\\{W(g,f)|~{}(g,f)~{}\text{satisfies}~{}(\ref{norm-1})\\}.$

It is well known that $\lambda(g)$ can be attained by some smooth function $f$
(cf. [Ro]). In fact, such a $f$ satisfies the Euler-Lagrange equation of
$W(g,\cdot)$,

(1.4) $\triangle f+f+\frac{1}{2}(R-|\nabla
f|^{2})=(2\pi)^{m/2}V^{-1}\lambda(g).$

Following Perelman’s computation in [Pe], we can deduce the first variation of
$\lambda(g)$,

(1.5) $\delta\lambda(g)=-(2\pi)^{-m/2}\int_{M}<\delta
g,\mathrm{Ric}(g)-g+\nabla^{2}f>e^{-f}dV_{g},$

where $\mathrm{Ric}(g)$ denotes the Ricci tensor of $g$ and $\nabla^{2}f$ is
the Hessian of $f$. Hence, $g$ is a critical point of $\lambda(\cdot)$ if and
only if $g$ is a gradient shrinking Ricci-soliton which satisfies

(1.6) $\mathrm{Ric}(g)+\nabla^{2}f=g,$

where $f$ is a minimizer of $W(g,\cdot)$. The following lemma was proved in
[TZhu5] for the uniqueness of solutions (1.4) when $g$ is a gradient shrinking
Ricci soliton.

######  Lemma 1.1.

If $g$ satisfies (1.6) for some $f$, then any solution of (1.4) is equal to
$f$ modulo a constant. Consequently, a minimizer of $W(g,\cdot)$ is unique if
the metric $g$ is a gradient shrinking Ricci-soliton. Conversely, if $f$ is a
function in (1.6) for $g$, then $f$ satisfies (1.4).

In case that $(M,J)$ is an $n$-dimensional Fano manifold, for any Kähler
metric $g$ in $2\pi c_{1}(M)$, (1.1) is equal to

(1.7)
$\int_{M}dV_{g}=\int_{M}\omega_{g}^{n}\,=\,(2\pi)^{n}\int_{M}c_{1}(M)^{n}\,\equiv\,V.$

Moreover, (1.6) becomes an equation for Kähler-Ricci solitons,

$Ric(\omega_{g})-\omega_{g}=\mathrm{L}_{X}\omega_{g},$

where $Ric(\omega_{g})$ is a Ricci form of $g$ and $L_{X}$ denotes the Lie
derivative along a holomorphic vector field $X$ on $M$. By the uniqueness of
Kähler-Ricci solitons [TZhu1], [TZhu2], we may assume that $X$ lies in a
reductive Lie subalgebra $\eta_{r}(M)$ of $\eta(M)$ after a holomorphism
transformation, where $\eta(M)$ consists of all holomorphic vector fields on
$M$. Such a $X$ ( we call it an extremal holomorphic vector field for Kähler-
Ricci solitons ) can be determined as follows.

Let $\text{Aut}_{r}(M)$ be a connected Lie subgroup of automorphisms group of
$M$ generated by $\eta_{r}(M)$. Let $K$ be a maximal compact subgroup of
$\text{Aut}_{r}(M)$. Without loss of generality, we may choose a $K$-invariant
background metric $g$ with its Kähler form $\omega_{g}$ in $2\pi c_{1}(M)$. In
[TZhu2], as an obstruction to Kähler-Ricci solitons, Tian and Zhu introduced a
modified Futaki-invariant $F_{X}(v)$ for any $X,v~{}\in~{}\eta(M)$ by

(1.8)
$F_{X}(Z)=\int_{M}Z(h_{g}-\hat{\theta}_{X,\omega_{g}})e^{\hat{\theta}_{X,\omega_{g}}}\omega_{g}^{n},\quad\forall~{}Z\in\eta(M),$

where $h_{g}$ is a Ricci potential of $g$ and $\hat{\theta}_{X,\omega_{g}}$ is
a real-valued potential of $X$ associated to $g$ defined by
$L_{X}\omega_{g}=\sqrt{-1}\partial\bar{\partial}\hat{\theta}_{X,\omega_{g}}$
with a normalization condition

(1.9) $\int_{M}\hat{\theta}_{X,\omega_{g}}e^{h_{g}}\omega_{g}^{n}=0.$

It was showed that there exists a unique $X~{}\in~{}\eta_{r}(M)$ such that

$F_{X}(v)\equiv 0,~{}\forall~{}v\in~{}\eta_{r}(M).$

Moreover, $F_{X}(v)\equiv 0$, for any $v\in~{}\eta(M)$ if $(M,J)$ admits a
Kähler-Ricci soliton.

Let $K_{X}$ be an one-parameter compact subgroup of holomorphisms
transformation group generated by $X$. We denote $\mathcal{K}_{X}$ to be a
class of $K_{X}$-invariant Kähler metrics in $2\pi c_{1}(M)$. Let
$\theta_{X,\omega_{g}}$ be a real-valued potential of $X$ associated to $g$
with a normalization condition

(1.10)
$\int_{M}e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}=\int_{M}\omega_{g}^{n}=V.$

Clearly, $\theta_{X,\omega_{g}}=\hat{\theta}_{X,\omega_{g}}-c_{X}$ for some
constant $c_{X}$ which is independent of $g\in\mathcal{K}_{X}$ .

######  Definition 1.2.

For $g\in\mathcal{K}_{X}$, define $N_{X}(\omega_{g})$ by

$N_{X}(\omega_{g})=\int_{M}\theta_{X,\omega_{g}}e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}.$

By Jensen’s inequality, it is easy to see

$\displaystyle\begin{aligned}
&\frac{1}{V}\int_{M}(-\theta_{X,\omega_{g}})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&\leq\text{log}\\{\frac{1}{V}\int_{M}e^{-\theta_{X,\omega_{g}}}e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\}=0.\end{aligned}$

The equality holds if and only if $\theta_{X,\omega_{g}}=0$. This shows that
$N_{X}(\omega_{g})$ is nonnegative and it is zero if and only if the Futaki-
invariant vanishes [Fu]. Moreover, we have

######  Lemma 1.3.

$N_{X}(\omega_{g})$ is independent of choice of $g$ in $\mathcal{K}_{X}$.

###### Proof.

Choose a $K$-invariant Kähler form $\omega$ in $2\pi c_{1}(M)$. Then for any
Kähler metric $g$ in $\mathcal{K}_{X}$ there exists a Kähler potential
$\varphi$ such that the imaginary part of $X(\varphi)$ vanishes and Kähler
form of $g$ satisfies

$\omega_{g}=\omega_{\varphi}=\omega+\sqrt{-1}\partial\bar{\partial}\varphi.$

Thus we suffice to prove

$N_{X}(\omega_{\varphi})=N_{X}(\omega_{t\varphi}),~{}~{}\forall~{}t\in~{}[0,1],$

where
$\omega_{t\varphi}=\omega+t\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\varphi.$
This follows from

$\displaystyle\begin{aligned}
\frac{dN_{X}(\omega_{t\varphi})}{dt}&=\int_{M}X(\varphi)e^{\theta_{X,\omega_{t\varphi}}}\omega_{t\varphi}^{n}+\int_{M}\theta_{X,\omega_{t\varphi}}(X+\triangle)(\varphi)e^{\theta_{X,\omega_{t\varphi}}}\omega_{t\varphi}^{n}\\\
&=\int_{M}X(\varphi)e^{\theta_{X,\omega_{t\varphi}}}\omega_{t\varphi}^{n}-\int_{M}\nabla^{i}\varphi\nabla_{\bar{i}}\theta_{X,\omega_{t\varphi}}\omega_{t\varphi}^{n}\\\
&=0.\end{aligned}$

Here we have used the fact

$\theta_{X,\omega_{t\varphi}}=\theta_{X,\omega_{0}}+tX(\varphi).$

∎

By the above lemma, $N_{X}(\cdot)$ is an invariance on $\mathcal{K}_{X}$,
which is independent of choice of $g$. For simplicity, we denote this
invariance by $N_{X}(c_{1}(M))$. The following proposition gives an upper
bound of $\lambda(\cdot)$ in $\mathcal{K}_{X}$ related to $N_{X}(c_{1}(M))$.

######  Proposition 1.4.

$\sup_{g\in\mathcal{K}_{X}}\lambda(g)\leq(2\pi)^{-n}[nV-N_{X}(c_{1}(M))].$

###### Proof.

Since $\lambda(g)\leq W(g,-\theta_{X,\omega_{g}})$, we suffice to prove

(1.11) $W(g,-\theta_{X,\omega_{g}})=(2\pi)^{-n}[nV-N_{X}(c_{1}(M))].$

In fact, by using the facts $R(g)=2n+\Delta h_{g}$ and

$\int_{M}(\Delta\theta_{X,\omega_{g}}+|\nabla\theta_{X,\omega_{g}}|^{2})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}=0,$

we have

$\displaystyle\begin{aligned}
&\int_{M}(R(g)+|\nabla\theta_{X,\omega_{g}}|^{2})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&=2nV+\int_{M}(\Delta
h_{g}-\Delta\theta_{X,\omega_{g}})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&=2nV-\int_{M}<\nabla(h_{g}-\theta_{X,\omega_{g}}),\nabla\theta_{X,\omega_{g}}>e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&=2nV-2\int_{M}X(h_{g}-\theta_{X,\omega_{g}})e^{\theta_{X,\omega_{g}}}\omega_{g}^{n}\\\
&=2nV-2e^{-c_{X}}F_{X}(X).\end{aligned}$

In the last equality above, we used the relation (1.8). Since $X$ is extremal
, we have

$F_{X}(X)=0.$

Thus by (1.2) for $f=-\theta_{X,\omega}$ together with Lemma 1.3, one will get
(1.11). ∎

In case that $M$ admits a Kähler-Ricci soliton $g_{KS}$, by Lemma 1.1, a
minimizer $f$ of $W(g_{KS},\cdot)$ in $\mathcal{K}_{X}$ must be $-\theta_{X}$.
Thus for any $g\in\mathcal{K}_{X}$, by Proposition 1.4, we have

$\displaystyle\begin{aligned} \lambda(g_{KS})&=W(g_{KS},-\theta_{X})\\\
&=(2\pi)^{-n}[nV-N_{X}(c_{1}(M))]\geq\lambda(g).\end{aligned}$

Therefore we get the following corollary.

######  Corollary 1.5.

Suppose that $(M,J)$ admits a Kähler-Ricci soliton $g_{KS}$. Then $g_{KS}$ is
a global maximizer of $\lambda(\cdot)$ in $\mathcal{K}_{X}$ and

(1.12) $\lambda(g_{KS})=(2\pi)^{-n}[nV-N_{X}(c_{1}(M))].$

######  Remark 1.6.

Corollary 1.5 implies that a Kähler-Einstein metric is a global maximizer of
$\lambda(\cdot)$ in $2\pi c_{1}(M)$ even with varying complex structures and
supremum of $\lambda(\cdot)$ is $(2\pi)^{-n}nV$ since $N_{X}(c_{1}(M))=0$.
Note that $N_{X}(c_{1}(M))>0$ if the Futaki-invariant does not vanish. Thus
Corollary 1.5 also implies that the supremum of $\lambda(\cdot)$ in case that
$(M,J)$ admits a Kähler-Ricci soliton is strictly less than one in case that
$(M,J)$ admits a Kähler-Einstein metric.

##  2\. Estimates for modified Ricci potentials

In this section, we summarize some apriori estimates for modified Ricci
potentials of evolved Kähler metrics along Kähler-Ricci flow. Some similar
estimates have been also discussed in [TZhu3] and [PSSW], we refer the readers
to those two papers. We consider the following (normalized) Kähler-Ricci flow:

(2.1) $\frac{\partial g(t,\cdot)}{\partial t}\,=\,-{\rm
Ric}(g(t,\cdot))+g(t,\cdot),~{}~{}~{}g(0)=g,$

where Kähler form of $g$ is in $2\pi c_{1}(M)$. It was proved in [Ca] that
(2.1) has a global solution $g_{t}=g(t,\cdot)$ for all time $t>0$. For
simplicity, we denote by $(g_{t};g)$ a solution of (2.1) with initial metric
$g$. Since the flow preserves the Kähler class, we may write Kähler form of
$g_{t}$ as

$\omega_{\phi}\,=\,\omega_{g}+\sqrt{-1}\partial\overline{\partial}\phi$

for some Käher potential $\phi=\phi_{t}$.

Let $X\in\eta_{r}(M)$ be the extremal holomorphic vector field on $M$ as in
Section 1 and $\sigma_{t}=\exp\\{tX\\}$ an one-parameter subgroup generated by
$X$. Let $\phi^{\prime}=\phi_{\sigma_{t}}$ be corresponding Kähler potentials
of $\sigma_{t}^{\star}\omega_{\phi_{t}}$. Then $\omega_{\phi^{\prime}}$ will
satisfy a modified Kähler-Ricci flow,

(2.2) $\frac{\partial}{\partial
t}\omega_{\phi^{\prime}}=-\text{Ric}(\omega_{\phi^{\prime}})+\omega_{\phi^{\prime}}+L_{X}\omega_{\phi^{\prime}}.$

Equation (2.2) is equivalent to the following Monge-Ampére flow for
$\phi^{\prime}$ (modulo a constant),

(2.3) $\frac{\partial\phi^{\prime}}{\partial
t}=\log\frac{\omega_{\phi^{\prime}}^{n}}{\omega_{g}^{n}}+\phi^{\prime}+\theta_{X,\omega_{\phi^{\prime}}}-h_{g},~{}~{}~{}~{}\phi^{\prime}(0,\cdot)=c,$

where $c$ is a constant and all Kähler potentials
$\phi^{\prime}=\phi_{t}^{\prime}=\phi^{\prime}(t,\cdot)$ are in a space given
by

$\mathcal{P}_{X}(M,\omega)=\\{\varphi\in
C^{\infty}(M)|~{}\omega_{\varphi}=\omega+\sqrt{-1}\partial\overline{\partial}\varphi>0,\quad\mathrm{Im}(X(\varphi))=0\\}.$

By using the maximum principle to (2.2) or (2.3), we get

(2.4)
$h_{\phi^{\prime}}-\theta_{X,\omega_{\phi^{\prime}}}=-\frac{\partial}{\partial
t}\phi^{\prime}+c_{t},$

for some constants $c_{t}$. Here $h_{\phi^{\prime}}$ are Ricci potentials of
$\omega_{\phi^{\prime}}$ which are normalized by

(2.5) $\int_{M}e^{h_{\phi^{\prime}}}\omega_{\phi^{\prime}}^{n}=V.$

The following estimates are due to G. Perelman. We refer the readers to [ST]
for their proof.

######  Lemma 2.1.

There are constants $c$ and $C$ depending only on the initial metric $g$ such
that (a) $\text{diam}(M,\omega_{\phi^{\prime}})\leq C$; (b)
$\text{vol}(B_{r}(p),\omega_{\phi^{\prime}})\geq cr^{2n}$; (c)
$\|h_{\phi^{\prime}}\|_{C^{0}(M)}\leq C$; (d) $\|\nabla
h_{\phi^{\prime}}\|_{\omega_{\phi^{\prime}}}\leq C$; (e) $\|\Delta
h_{\phi^{\prime}}\|_{C^{0}(M)}\leq C$.

Recall that the modified Mabuchi’s K-energy $\mu(\cdot)$ is defined in
$\mathcal{P}_{X}(M,\omega)$ by

$\displaystyle\begin{aligned}
\mu(\varphi)=~{}&-\frac{n}{V}\int_{0}^{1}\int_{M}\dot{\psi}[\text{Ric}(\omega_{\psi})-\omega_{\psi}-\sqrt{-1}\partial\overline{\partial}\theta_{X,\omega_{\psi}}\\\
&+\sqrt{-}\overline{\partial}(h_{\omega_{\psi}}-\theta_{X,\omega_{\psi}})\wedge\partial\theta_{X,\omega_{\psi}})]\wedge
e^{\theta_{X,\omega_{\psi}}}\omega_{\psi}^{n-1}\wedge dt,\end{aligned}$

where $\psi=\psi_{t}$ $(0\leq t\leq 1)$ is a path connecting $0$ to $\varphi$
in $\mathcal{P}_{X}(M,\omega)$. If $X=0$, then $\mu_{\omega_{g}}(\phi)$ is
nothing but Mabuchi’s $K$-energy [Ma]. Then by (2.2), we have

(2.6)
$\frac{d\mu(\phi^{\prime})}{dt}=-\frac{1}{V}\int_{M}\|\overline{\partial}\frac{\partial\phi^{\prime}}{\partial
t}\|_{\omega_{\phi^{\prime}}}^{2}e^{\theta_{X,\omega_{\phi^{\prime}}}}(\omega_{\phi^{\prime}})^{n}\leq
0.$

This implies that $\mu(\phi^{\prime})$ is uniformly bounded if $\mu(\cdot)$ is
bounded from below in $\mathcal{P}_{X}(M,\omega)$.

Let
$u_{X,\phi^{\prime}}=u_{X,\omega_{g_{t}^{\prime}}}=h_{\phi^{\prime}}-\theta_{X,\omega_{\phi^{\prime}}}$.
Then

######  Lemma 2.2.

There exists a uniform $C$ such that

$\|\nabla u_{X,\phi^{\prime}}\|_{\omega_{\phi^{\prime}}}\leq C.$

###### Proof.

First we note that $\theta_{X,\omega_{\phi^{\prime}}}$ is uniformly bounded in
$\mathcal{P}_{X}(M,\omega)$ (cf. [Zhu1], [ZZ]). Then by (c) of Lemma 2.1, we
have

$\|u_{X,\phi^{\prime}}\|_{C^{0}}=\|u_{X,\omega_{g_{s}^{\prime}}}\|_{C^{0}}\leq
C,~{}\forall~{}s>0$

for some uniform constant $C$. Now we consider the flow (2.3) with zero as an
initial Kähler potential and the background Kähler form $\omega_{g}$ replaced
by $\omega_{g_{s}^{\prime}}$. By an estimate in Lemma 4.3 in [CTZ], we see

$t\|\nabla
u_{X,\omega_{g_{s+t}^{\prime}}}\|_{\omega_{g_{s+t}^{\prime}}}^{2}\leq
e^{2t}\|u_{X,\omega_{g_{s}^{\prime}}}\|_{C^{0}},~{}~{}\forall~{}t>0.$

In particular, we get

$\|\nabla
u_{X,\omega_{g_{s+t}^{\prime}}}\|_{\omega_{g_{s+t}^{\prime}}}^{2}\leq
C^{\prime},~{}~{}\forall~{}t\in[1,2].$

Since the above estimate is independent of $s$, we conclude that the lemma is
true.

∎

Now we begin to prove the main result in this section.

######  Proposition 2.3.

Suppose that $\mu(\cdot)$ is bounded from below in
$\mathcal{P}_{X}(M,\omega)$. Then we have:
(a) $\lim_{t\rightarrow\infty}||u_{X,\phi^{\prime}}||_{C^{0}}=0$;
(b) $\lim_{t\rightarrow\infty}||\nabla
u_{X,\phi^{\prime}}||_{\omega_{\phi^{\prime}}}=0$;
(c) $\lim_{t\rightarrow\infty}||\triangle u_{X,\phi^{\prime}}||_{C^{0}}=0$.

###### Proof.

Let $H(t)=\int_{M}|\nabla
u_{X,\omega_{g_{t}}^{\prime}}|^{2}e^{\theta_{X,\omega_{g_{t}}^{\prime}}}\omega_{g_{t}^{\prime}}^{n}$.
Then by (2.6), one sees that there exists a sequence of $t_{i}\in[i,i+1]$ such
that

$\lim_{i\rightarrow\infty}H(t_{i})=0.$

Thus by using a differential inequality

$\frac{dH(t)}{dt}\leq CH(t),$

where $C$ is a uniform constant (cf. [PSSW]), we get

(2.7) $\lim_{t\rightarrow\infty}\int_{M}|\nabla
u_{X,\omega_{g_{t}^{\prime}}}|_{g_{t}^{\prime}}^{2}e^{\theta_{X,\omega_{g_{t}^{\prime}}}}\omega_{g_{t}^{\prime}}^{n}=0.$

Let

$\tilde{u}_{t}=u_{X,\omega_{g_{t}^{\prime}}}-\frac{1}{V}\int_{M}u_{X,\omega_{g_{t}^{\prime}}}e^{h_{t}^{\prime}}\omega_{g_{t}^{\prime}}^{n},$

where $h_{t}^{\prime}=h_{\phi^{\prime}(t,\cdot)}$. Then by using the weighted
Poincaré inequality in [TZhu3] together with (c) of Lemma 2.1, we obtain from
(2.7),

$\int_{M}\tilde{u}_{t}^{2}e^{h_{t}^{\prime}}\omega_{g_{t}^{\prime}}^{n}\leq\int_{M}|\nabla
u_{X,\omega_{g_{t}^{\prime}}}|_{g_{t}^{\prime}}^{2}e^{h_{t}^{\prime}}\omega_{g_{t}^{\prime}}^{n}\to
0,~{}~{}\text{as }~{}t\to\infty.$

Consequently, we derive

(2.8)
$\lim_{t\rightarrow\infty}\int_{M}\tilde{u}_{t}^{2}\omega_{g_{t}^{\prime}}^{n}=0.$

We claim

$\lim_{t\rightarrow\infty}\|\tilde{u}_{t}\|_{C^{0}}=0.$

The claim immediately implies (a) of Proposition 2.3 by the normalization
conditions

$\int_{M}e^{\theta_{X,\omega_{g_{t}^{\prime}}}}\omega_{g_{t}^{\prime}}^{n}=\int_{M}e^{h_{t}^{\prime}}\omega_{g_{t}^{\prime}}^{n}=V.$

To prove the claim, we need to use an inequality

(2.9) $\|\tilde{u}_{t}\|_{C^{0}}^{n+1}\leq C\|\nabla
u_{X,\omega_{g_{t}^{\prime}}}\|^{n}_{g_{t}^{\prime}}[\int_{M}\tilde{u}_{t}^{2}\omega_{g_{t}^{\prime}}^{n}]^{\frac{1}{2}}.$

(2.9) can be proved by using the non-collapsing estimate (b) in Lemma 2.1 (cf.
[PSSW], [Zhu2]). Thus by Lemma 2.2 and (2.8), the claim is proved.

By (a) we can show that after a suitable choice of constant $c$ in the flow
(2.3) it holds

$\lim_{t\to\infty}\|\frac{\partial}{\partial t}\phi^{\prime}\|_{C^{0}}=0.$

In fact under the assumption of lower bound of modified $K$-energy, one can
choose such a $c$ (cf. [TZhu3]) such that

$\lim_{t\to\infty}\int_{M}\frac{\partial}{\partial
t}\phi^{\prime}e^{\theta_{X,\omega_{\phi^{\prime}}}}\omega_{\phi^{\prime}}^{n}=0.$

Then by (2.4), we will get the conclusion. On the other hand, by Lemma 2.2 and
(d) of Lemma 2.1, we have

$\sup_{t\in[0,\infty)}\|X\|_{g_{t}^{\prime}}<C$

for some uniform constant $C$. Therefore, by using the following lemma we
prove (b) and (c).

∎

######  Lemma 2.4.

([PSSW]) There exist $\delta,K>0$ depending only on $n$ and the constant
$C_{X}=\sup_{t\in[0,\infty)}\|X\|_{g_{t}^{\prime}}$ with the following
property. For $\epsilon$ with $0<\epsilon\leq\delta$ and any $t_{0}>0$, if

$\|\frac{\partial\phi^{\prime}}{\partial t}\|_{C^{0}}(t_{0})\leq\epsilon,$

then

$\|\nabla
u_{X,\omega_{g_{t_{0}+2}^{\prime}}}\|^{2}_{g_{t_{0}+2}^{\prime}}+\|\Delta
u_{X,\omega_{g_{t_{0}+2}^{\prime}}}\|_{C^{0}}\leq K\epsilon.$

##  3\. Proof of Theorem 0.1

According to [TZhu5], an energy level $L(g)$ of entropy $\lambda(\cdot)$ along
Kähler-Ricci flow $(g_{t};g)$ is defined by

$L(g)=\lim_{t\rightarrow\infty}\lambda(g_{t}).$

By the monotonicity of $\lambda(g_{t})$, we see that $L(g)$ exists and it is
finite. In this section, our goal is to prove

######  Proposition 3.1.

Suppose that the modified Mabuchi’s K-energy is bounded from below in
$\mathcal{K}_{X}$. Then for any $g\in\mathcal{K}_{X}$.

(3.1) $L(g)=(2\pi)^{-n}(nV-N_{X}(c_{1}(M)).$

The above proposition shows that the energy level $L(g)$ of entropy
$\lambda(\cdot)$ does not depend on the initial Kähler metric
$g\in\mathcal{K}_{X}$. Thus by using the Kähler-Ricci flow $(g_{t};g)$ for any
Kähler metric $g\in\mathcal{K}_{X}$ and the monotonicity of $\lambda(g_{t})$,
we will get Theorem 0.1.

To prove Proposition 3.1, we need the following key lemma.

######  Lemma 3.2.

Let $f_{t}$ be a minimizer of $W(g_{t},\cdot)$-functional associated evolved
Kähler metric $g_{t}$ of (2.1) at time $t$ and $h_{t}$ a Ricci potential of
$g_{t}$ which satisfying the normalization (2.5) . Then there exists a
sequence of $t_{i}\in[i,i+1]$ such that

(a)
$\lim_{t_{i}\to\infty}\|\Delta(f_{t_{i}}+h_{t_{i}})\|_{L^{2}(M,\omega_{g_{t_{i}}})}\,=\,0$;

(b)
$\lim_{t_{i}\to\infty}\|\nabla(f_{t_{i}}+h_{t_{i}})\|_{L^{2}(M,\omega_{g_{t_{i}}})}\,=\,0$;

(c) $\lim_{t_{i}\to\infty}\|f_{t_{i}}+h_{t_{i}}\|_{C^{0}}\,=\,0$.

###### Proof.

Lemma 3.2 is a generalization of Proposition 4.4 in [TZhu5]. We will follow
the argument there. First by (1.5), it is easy to see that

$\displaystyle\begin{aligned}
\frac{d}{dt}\lambda(g_{t})=(2\pi)^{-n}\int_{M}|\mathrm{Ric}(g_{t})-g_{t}+\nabla^{2}f_{t}|_{g_{t}}^{2}e^{-f_{t}}\omega_{g_{t}}^{n}.\end{aligned}$

It follows that

$\frac{d}{dt}\lambda(g_{t}))\geq(2\pi)^{-n}\frac{1}{2n}\int_{M}|\triangle(h_{t}+f_{t})|^{2}e^{-f(t)}\omega_{g_{t}}^{n}.$

Since $\lambda(g_{t})\leq W(g_{t},0)=(2\pi)^{-n}nV$ are uniformly bounded, we
see that there exists a sequence of $t_{i}\in[i,i+1]$ such that

$\lim_{i\rightarrow\infty}\int_{M}|\triangle(h_{t_{i}}+f_{t_{i}})|^{2}e^{-f_{t_{i}}}\omega_{g_{t_{i}}}^{n}=0.$

Note that $f_{t}$ is uniformly bounded [TZhu5]. Hence we see that that (a) of
the lemma is true. By (a), we also get

(3.2) $\displaystyle\begin{aligned}
&\lim_{t_{i}\to\infty}\|\nabla(f_{t_{i}}+h_{t_{i}})\|_{L^{2}(M,\omega_{g_{t_{i}}})}\\\
&\leq\lim_{t_{i}\to\infty}\int_{M}|f_{t_{i}}+h_{t_{i}}||\triangle(f_{t_{i}}+h_{t_{i}})|\omega_{g_{t_{i}}}^{n}\\\
&\leq
C\lim_{t_{i}\to\infty}\|\Delta(f_{t_{i}}+h_{t_{i}})\|_{L^{2}(M,\omega_{g_{t_{i}}})}=0.\end{aligned}$

This proves (b) of the lemma. It remains to prove (c).

Let $q_{t}=f_{t}+h_{t}$. Then

(3.3) $\displaystyle\begin{aligned} -\Delta{q_{t}}&=-\Delta f_{t}-\Delta
h_{t}\\\ &=f_{t}+\frac{1}{2}(R-|\nabla
f_{t}|^{2})-(2\pi)^{2n}V^{-1}\lambda(g_{t})-\Delta h_{t}\leq C.\end{aligned}$

Define

$\tilde{q_{t}}=q_{t}-\frac{1}{V}\int_{M}q_{t}e^{h_{t}}\omega_{g_{t}}^{n}.$

By using the weighted Poincaré inequality (cf. [TZhu3]), we have

$\int_{M}\tilde{q_{t}}^{2}e^{h_{t}}\omega_{g_{t}}^{n}\leq\int_{M}|\nabla
q_{t}|^{2}e^{h_{t}}\omega_{g_{t}}^{n}.$

It follows by (b),

(3.4)
$\lim_{i\rightarrow\infty}\int_{M}\tilde{q}_{t_{i}}^{2}\omega_{g_{t_{i}}}^{n}=0.$

Hence, following an argument in the proof of Proposition 4.4 in [TZhu5], we
will get estimates

(3.5) $\|\tilde{q_{t_{i}}}^{+}\|_{C^{0}}\leq
C\|\tilde{q}_{t_{i}}\|_{L^{2}(M,\omega_{g_{t_{i}}})}\to
0,~{}~{}\text{as}~{}i\to\infty,$

and

(3.6)
$\lim_{i\rightarrow\infty}\int_{M}\tilde{q_{t_{i}}}^{-}\omega_{g_{t_{i}}}^{n}=0,$

where $q_{t}^{+}=\max\\{q_{t},0\\}$ and $q_{t}^{-}=\min\\{q_{t},0\\}$.
Consequently, we derive

$\int_{M}\tilde{q_{t_{i}}}e^{-f_{t_{i}}}\omega_{g_{t_{i}}}^{n}=0.$

This implies

(3.7)
$\lim_{i\rightarrow\infty}\int_{M}q_{t_{i}}e^{-f_{t_{i}}}\omega^{n}_{g_{t_{i}}}=0$

according to the normalization
$\int_{M}e^{-f_{t}}\omega^{n}_{g_{t}}=\int_{M}e^{h_{t}}\omega^{n}_{g_{t}}=V$.

Next we improve that

(3.8) $\lim_{i\rightarrow\infty}|q_{t_{i}}|=0.$

Let $u_{t}=e^{-\frac{f_{t}}{2}}-e^{\frac{h_{t}}{2}}$. We claim

(3.9)
$\lim_{i\rightarrow\infty}\|u_{t_{i}}\|_{L^{2}(M,\omega_{g_{t_{i}}})}=0.$

In fact, by Jensen’s inequality and (3.7), one sees

$\displaystyle\begin{aligned}
\frac{1}{V}\int_{M}e^{-\frac{f_{t_{i}}}{2}}e^{\frac{h_{t_{i}}}{2}}\omega_{g_{t_{i}}}^{n}&=\frac{1}{V}\int_{M}e^{\frac{f_{t_{i}}+h_{t_{i}}}{2}}e^{-f_{t_{i}}}\omega_{g_{t_{i}}}^{n}\\\
&\geq
e^{\frac{1}{2V}\int_{M}(f_{t_{i}}+h_{t_{i})}e^{-f_{t_{i}}}\omega_{g_{t_{i}}}^{n}}\to
1,~{}\text{as}~{}i\to\infty.\end{aligned}$

On the other hand,

$\int_{M}e^{-\frac{f_{t}}{2}}e^{\frac{h_{t}}{2}}\omega_{g_{t}}^{n}\leq(\int_{M}e^{-f_{t}}\omega_{g_{t}}^{n})^{\frac{1}{2}}(\int_{M}e^{h_{t}}\omega_{g_{t}}^{n})^{\frac{1}{2}}=V.$

Hence

$\lim_{i\rightarrow\infty}\int_{M}e^{-\frac{f_{t_{i}}}{2}}e^{\frac{h_{t_{i}}}{2}}\omega_{g_{t_{i}}}^{n}=V.$

It follows

$\lim_{i\rightarrow\infty}\int_{M}u_{t_{i}}^{2}\omega_{g_{t_{i}}}^{n}=2V-2\lim_{i\rightarrow\infty}\int_{M}e^{-\frac{f_{t_{i}}}{2}}e^{\frac{h_{t_{i}}}{2}}\omega_{g_{t_{i}}}^{n}=0.$

This completes the proof of claim.

Since equation (1.4) is equivalent to

(3.10) $\Delta
v_{t}-\frac{1}{2}f_{t}v_{t}-\frac{1}{4}R(g_{t})v_{t}=\frac{1}{2V}(2\pi)^{n}\lambda(g_{t})v_{t},$

where $v_{t}=e^{\frac{-f_{t}}{2}}$, by Lemma 2.1, it is easy to see

$|\Delta u_{t}|\leq C.$

Then by the standard Moser’s iteration, we get from (3.9),

$\|u_{t_{i}}\|_{C^{0}}\leq C\|u_{t_{i}}\|_{L^{2}(M,\omega_{g_{t_{i}}})}\to
0,~{}\text{as}~{}i\to\infty.$

This implies (3.8), so we obtain (c) of the lemma.

∎

###### Proof of Proposition 3.1.

Note that $\frac{R(g_{t})}{2}=n+\frac{1}{2}\Delta h_{t}$, where $\Delta$ is
the Beltrima-Laplacian operator associated to the Riemannian metric $g_{t}$.
Then

$\int_{M}\frac{1}{2}(R(g_{t})+|\nabla
f_{t}|^{2})e^{-f_{t}}dV_{g_{t}}=nV+\frac{1}{2}\int_{M}\Delta(f_{t}+h_{t})e^{-f_{t}}dV_{g_{t}}.$

Thus by (a) of Lemma 3.2, one sees that there exists a sequence of time
$t_{i}$ such that

(3.11) $\lim_{i\rightarrow\infty}\int_{M}\frac{1}{2}(R(g_{t_{i}})+|\nabla
f_{t_{i}}|^{2})e^{-f_{t_{i}}}dV_{g_{t_{i}}}=nV.$

On the other hand, since the modified Mabuchi’s K-energy is bounded from
below, we see that (a) of Proposition 2.3 is true. Then by (c) of Lemma 3.2,
it follows

(3.12)
$\lim_{i\to\infty}\|f_{t_{i}}+\theta_{X,\omega_{g_{t_{i}}}}\|_{C^{0}}=0.$

Here we used a fact
$\sigma_{t}^{\star}\theta_{X,\omega_{g_{t}}}=\theta_{X,\omega_{g_{t}^{\prime}}}$
since $X$ lies in the center of $\eta_{r}(M)$ [TZhu1]. Hence

(3.13) $\displaystyle\begin{aligned}
&\lim_{i\to\infty}\int_{M}f_{t_{i}}e^{-f_{t_{i}}}dV_{g_{t_{i}}}\\\
&=-\lim_{i\to\infty}\int_{M}\theta_{X,\omega_{g_{t_{i}}}}e^{\theta_{X,\omega_{g_{t_{i}}}}}\omega^{n}_{g_{t_{i}}}=-N_{X}(c_{1}(M)).\end{aligned}$

By combining (3.11) and (3.13), we get

$\displaystyle\begin{aligned}
&\lim_{i\to\infty}\lambda(g_{t_{i}})=\lim_{i\to\infty}\int_{M}[\frac{1}{2}(R(g_{t_{i}})+|\nabla
f_{t_{i}}|^{2})+f_{t_{i}}]e^{-f_{t_{i}}}dV_{g_{t_{i}}}\\\ &=nV-
N_{X}(c_{1}(M))\end{aligned}$

Therefore, by using the monotonicity of $\lambda(g_{t})$ along the flow
$(g_{t};g)$, we obtain (3.1).

∎

It was showed in [TZhu4] that a Kähler-Ricci soliton is a local maximizer of
$\lambda(\cdot)$ in the Kähler class $2\pi c_{1}(M)$. Together with Corollary
1.5, one may guess that a Kähler-Ricci soliton is a global maximizer of
$\lambda(\cdot)$. More general, according to Theorem 0.1 , we propose the
following conjecture.

######  Conjecture 3.3.

Suppose that the modified Mabuchi’s K-energy is bounded from below. Then

$\sup_{\omega_{g^{\prime}}\in 2\pi
c_{1}(M)}\lambda({g^{\prime}})=(2\pi)^{-n}[nV-N_{X}(c_{1}(M))].$

##  4\. Improvement of Lemma 3.2

In this section, we use Perelman’s backward heat flow to improve estimate (c)
in Lemma 3.2 independent of $t$. Moreover, we show the gradient estimate of
$f_{t}+h_{t}$ also holds. Although Lemma 3.2 is sufficient to be applied to
prove Theorem 0.1 and Theorem 0.2, results of this section are independent of
interests. We hope that these results will have applications in the future.

Fix any $t_{0}\geq 1$. We consider a backward heat equation in
$t\in[t_{0}-1,t_{0}]$,

(4.1) $\frac{\partial}{\partial t}f_{t_{0}}(t)=-\triangle f_{t_{0}}(t)+|\nabla
f_{t_{0}}(t)|^{2}-\triangle h_{t}$

with an initial $f_{t_{0}}(t_{0})=f_{t_{0}}$. Clearly, the equation preserves
the normalizing condition
$\frac{1}{V}\int_{M}e^{-f_{t_{0}}(t)}\omega_{g_{t}}^{n}=1$. Moreover, by the
maximum principle, we have

(4.2) $\|f_{t_{0}}(t)\|_{C^{0}}\leq C(g),\hskip
8.5359pt\forall~{}t\in[t_{0}-1,t_{0}],$

since $\triangle h_{t}$ are uniformly bounded. Here the constant $C(g)$
depends only on the initial metric $g$ of (2.1).

Similarly to (1.5), we can compute

(4.3) $\displaystyle\begin{aligned} &\frac{d}{dt}W(g_{t},f_{t_{0}})\\\
&=(2\pi)^{-n}\int_{M}(\|\partial\overline{\partial}(h_{t}+f_{t_{0}}(t))\|^{2}+\|\partial\partial
f_{t_{0}}(t)\|^{2})e^{-f_{t_{0}}(t)}\omega_{{}_{g_{t}}}^{n}.\end{aligned}$

By using (4.3), we want to prove

######  Lemma 4.1.

(4.4) $\|f_{t}+h_{t}-c_{t}\|_{L^{2}(M,g_{t})}\rightarrow
0,~{}\text{as}~{}t\rightarrow\infty,$

where $c_{t}=\frac{1}{V}\int_{M}(f_{t}+h_{t})e^{h_{t}}\omega_{g_{t}}^{n}$.

###### Proof.

First by (4.3), one sees

$\displaystyle\lambda(g_{t_{0}})-\lambda(g_{t_{0}-1})$ $\displaystyle\geq$
$\displaystyle
W(g_{t_{0}},f_{t_{0}}(t_{0}))-W(g_{t_{0}-1},f_{t_{0}}(t_{0}-1))$
$\displaystyle\geq$
$\displaystyle(2\pi)^{-n}\frac{1}{2n}\int_{t_{0}-1}^{t_{0}}\int_{M}|\triangle(f_{t_{0}}(t)+h_{t})|^{2}e^{-f_{t_{0}}(t)}\omega_{g_{t}}^{n}dt.$

It follows

$\int_{t_{0}-1}^{t_{0}}\int_{M}|\triangle(f_{t_{0}}(t)+h_{t})|^{2}\omega^{n}_{g_{t}}dt\rightarrow
0,~{}\text{as }~{}t_{0}\rightarrow\infty.$

Thus by using the weighted Poincaré inequality as in (3.4) in last section ,
we will get

(4.5) $\displaystyle\begin{aligned}
&\int_{t_{0}-1}^{t_{0}}dt\int_{M}(f_{t_{0}}(t)+h_{t}-c_{t_{0}}(t))^{2}\omega^{n}_{g_{t}}\\\
&\leq
C(g_{0})[\int_{t_{0}-1}^{t_{0}}dt\int_{M}|\triangle(f_{t_{0}}(t)+h_{t})|^{2}\omega^{n}_{g_{t}}]^{1/2}\to
0,~{}\text{as}~{}t_{0}\to\infty,\end{aligned}$

where
$c_{t_{0}}(t)=\frac{1}{V}\int_{M}(f_{t_{0}}(t)+h_{t})e^{h_{t}}\omega_{g_{t}}^{n}.$

Next, since $\frac{dh_{t}}{dt}=\Delta h_{t}+h_{t}-a_{t}$, where
$a_{t}=\frac{1}{V}\int_{M}h_{t}e^{h_{t}}\omega^{n}_{g_{t}}$, by a
straightforward calculation, we see

$\displaystyle\frac{d}{dt}\int_{M}(h_{t}+f_{t_{0}}-c_{t_{0}}(t))^{2}\omega^{n}_{g_{t}}$
$\displaystyle=\int_{M}[2(h_{t}+f_{t_{0}}(t)-c_{t_{0}}(t))(\triangle
f_{t_{0}}-|\nabla f_{t_{0}}(t)|^{2}+h_{t}-a_{t}-\frac{dc_{t_{0}}}{dt})$
$\displaystyle-(h_{t}+f_{t_{0}}(t)-c_{t_{0}}(t))^{2}\triangle
h_{t}]\omega^{n}_{g_{t}}$

Then by Lemma 3.2, we get

$\displaystyle|\frac{d}{dt}\int_{M}(h_{t}+f_{t_{0}}(t)-c_{t_{0}})^{2}\omega^{n}_{g_{t}}|$
$\displaystyle\leq C+C\int_{M}\big{(}|\triangle f_{t_{0}}(t)|+|\nabla
f_{t_{0}}(t)|^{2}+|\frac{dc_{t_{0}}(t)}{dt}|\big{)}\omega^{n}_{g_{t}}$
$\displaystyle\leq
C+C\int_{M}\big{(}|\nabla\bar{\nabla}(f_{t_{0}}(t)+h_{t})|^{2}+|\frac{dc_{t_{0}}(t)}{dt}|\big{)}\omega^{n}_{g_{t}}.$

Notice that

$\displaystyle\frac{dc_{t_{0}}}{dt}=\frac{1}{V}\int_{M}[\triangle
f_{t_{0}}(t)-|\nabla
f_{t_{0}}(t)|^{2}-(h_{t}+f_{t_{0}}(t))(h_{t}+a_{t})]e^{h_{t}}\omega^{n}_{g_{t}}.$

We can also estimate

$|\frac{dc_{t_{0}}}{dt}|\leq
C+C\int_{M}|\nabla\bar{\nabla}(f_{t_{0}}(t)+h_{t})|^{2}\omega^{n}_{g_{t}}.$

Hence we derive

(4.6)
$\big{|}\frac{d}{dt}\int_{M}(f_{t_{0}}(t)+h_{t}-c_{t_{0}}(t))^{2}dv\big{|}\leq
C+C\int_{M}|\nabla\bar{\nabla}(f_{t_{0}}(t)+h_{t})|^{2}\omega^{n}_{g_{t}}.$

Therefore, according to

$\displaystyle\int_{t_{0}-1}^{t_{0}}dt\int_{M}|\nabla\bar{\nabla}(f_{t_{0}}(t)+h_{t})|^{2}e^{-f_{t_{0}}(t)}\omega^{n}_{g_{t}}$
$\displaystyle\leq(2\pi)^{n}(\lambda(g_{t_{0}})-\lambda(g_{t_{0}-1}))\to
0,~{}\text{as}~{}t_{0}\rightarrow\infty,$

(4.5) and (4.6) will implies

$\|f_{t_{0}}(t)+h_{t}-c_{t_{0}}(t)\|_{L^{2}(g_{t},M)}\rightarrow
0,~{}\text{as}~{}t_{0}\rightarrow\infty,~{}\forall~{}t\in[t_{0}-1,t_{0}].$

Consequently, we get (4.4). ∎

######  Proposition 4.2.

(4.7) $\|f_{t}+h_{t}\|_{C^{0}}+\|\nabla(f_{t}+h_{t})\|_{g_{t}}\rightarrow
0,~{}\text{ as}~{}t\rightarrow\infty.$

###### Proof.

With the help of Lemma 4.1, by using same argument in the proof of (c) in
Lemma 3.2, we can prove that

(4.8) $\|f_{t}+h_{t}\|_{C^{0}}\rightarrow 0,~{}\text{
as}~{}t\rightarrow\infty.$

So we suffice to prove

(4.9) $\|\nabla(f_{t}+h_{t})\|_{g_{t}}\rightarrow 0,~{}\text{
as}~{}t\rightarrow\infty.$

We will use the Moser’s iteration to obtain (4.9 ) as in lemma 7.2 in
Appendix. We note by (4.8) and Theorem 7.1 that

(4.10) $\int_{M}|\nabla
q_{t}|^{2}\omega_{g_{t}}^{n}=|\int_{M}-\Delta(f_{t}+h_{t})(f_{t}+h_{t})\omega_{g_{t}}^{n}|\leq
C\|f_{t}+h_{t}\|_{C^{0}}\to 0,$

where $q_{t}=f_{t}+h_{t}$ satisfies an equation

$\triangle q_{t}=\frac{1}{2}(|\nabla f_{t}|^{2}-2f_{t}+\triangle
h_{t})+(2\pi)^{n}V^{-1}\lambda(g_{t})-n.$

Let $w_{t}=|\nabla q_{t}|^{2}$. Then by the Bochner formula, we have

$\displaystyle\triangle w_{t}$ $\displaystyle=$ $\displaystyle|\nabla\nabla
q_{t}|^{2}+|\nabla\bar{\nabla}q_{t}|^{2}+\nabla_{i}\triangle
q_{t}\nabla_{\bar{i}}q_{t}+\nabla_{\bar{i}}\triangle
q_{t}\nabla_{i}q_{t}+R_{i\bar{j}}\nabla_{\bar{i}}q_{t}\nabla_{j}{q_{t}}.$

Hence for any $p\geq 2$, it follows

(4.11) $\displaystyle\begin{aligned} &\frac{4(p-1)}{p^{2}}\int_{M}|\nabla
w_{t}^{p/2}|^{2}\omega_{g_{t}}^{n}=-\int_{M}w_{t}^{p-1}\triangle
q_{t}\omega_{g_{t}}^{n}\\\ &=-\int_{M}w_{t}^{p-1}(|\nabla\nabla
q_{t}|^{2}+|\nabla\bar{\nabla}q_{t}|^{2})\omega_{g_{t}}^{n}\\\
&-2\text{Re}\int_{M}q_{t}^{p-1}\nabla_{i}\triangle
q_{t}\nabla_{\bar{i}}q_{t}\omega_{g_{t}}^{n}-\int_{M}q_{t}^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}q_{t}\nabla_{j}q_{t}\omega_{g_{t}}^{n}.\end{aligned}$

On the other hand, by Lemma 2.1 and Theorem 7.1, we estimate

$\displaystyle\begin{aligned}
&\quad-2\text{Re}\int_{M}w_{t}^{p-1}\nabla_{i}\triangle
q_{t}\nabla_{\bar{i}}w_{t}\omega_{g_{t}}^{n}\\\
&=-\text{Re}\int_{M}w_{t}^{q-1}\nabla_{i}(|\nabla f_{t}|^{2}-2f+\triangle
h_{t})\nabla_{\bar{i}}q_{t}\omega_{g_{t}}^{n}\\\ &=-\text{Re}\int_{M}(|\nabla
f_{t}|^{2}-2f_{t}+\triangle
h_{t})(\frac{2(p-1)}{p}w_{t}^{\frac{p}{2}-1}\nabla_{i}w_{t}^{p/2}\nabla_{\bar{i}}q_{t}+w_{t}^{q-1}\triangle
q_{t})\omega_{g_{t}}^{n}\\\ &\leq
C(g)[\int_{M}\frac{2(p-1)}{p}w_{t}^{\frac{p-1}{2}}|\nabla
w_{t}^{p/2}|\omega_{g_{t}}^{n}+\int_{M}w_{t}^{p-1}|\triangle
q_{t}|]\omega_{g_{t}}^{n}\\\ &\leq\frac{p-1}{p^{2}}\int_{M}|\nabla
w^{p/2}|^{2}\omega_{g_{t}}^{n}+C(g)^{\prime}p\int_{M}w^{p-1}\omega_{g_{t}}^{n}\end{aligned}$

and

$\displaystyle\begin{aligned}
&\quad-\int_{M}w_{t}^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}q_{t}\nabla_{j}q_{t}\omega_{g_{t}}^{n}\\\
&=-\int_{M}w_{t}^{p}\omega_{g_{t}}^{n}-\int_{M}w_{t}^{p-1}\nabla_{i}\nabla_{\bar{j}}h_{t}\nabla_{\bar{i}}q_{t}\nabla_{j}q_{t}\omega_{g_{t}}^{n}\\\
&=-\int_{M}w_{t}^{p}\omega_{g_{t}}^{n}+\int_{M}w_{t}^{p-1}\nabla_{\bar{j}}h_{t}(\nabla_{\bar{i}}q_{t}\nabla_{i}\nabla_{j}q_{t}\omega_{g_{t}}^{n}+\triangle
q_{t}\nabla_{j}q_{t})\omega_{g_{t}}^{n}\\\
&\quad+\frac{2(p-1)}{p}\int_{M}w_{t}^{\frac{p}{2}-1}\nabla_{i}w_{t}^{p/2}\nabla_{\bar{j}}h_{t}\nabla_{\bar{i}}q_{t}\nabla_{j}q\omega_{g_{t}}^{n}\\\
&\leq\int_{M}w_{t}^{p-1}(|\nabla\nabla
q_{t}|^{2}+\frac{1}{2}|\nabla\bar{\nabla}q_{t}|^{2})\omega_{g_{t}}^{n}+\frac{p-1}{p^{2}}\int_{M}|\nabla
w_{t}^{p/2}|^{2}\omega_{g_{t}}^{n}\\\
&+C(g)(p-1)\int_{M}w_{t}^{p}\omega_{g_{t}}^{n}.\end{aligned}$

Then substituting the above two inequalities into (4.11), we get

$\displaystyle\int_{M}|\nabla w_{t}^{p/2}|^{2}\omega_{g_{t}}^{n}\leq
C(g)(p-1)^{2}\int_{M}w_{t}^{p-1}\omega_{g_{t}}^{n},\hskip
5.69046pt\forall~{}p\geq 2.$

By using Zhang’s Sobolev inequality [Zha], we deduce

(4.12)
$\displaystyle\big{(}\int_{M}w_{t}^{p\nu}\big{)}^{1/\nu}\omega_{g_{t}}^{n}\leq
C(g)C_{s}(q-1)^{2}\int_{M}w_{t}^{p-1}\omega_{g_{t}}^{n},\quad\forall~{}p\geq
2,$

where $\nu=\frac{n}{n-1}$. To run the iteration we put $p_{0}=1$ and
$p_{k+1}=p_{k}\nu+\nu$, $k\geq 0$. Hence

$\displaystyle\|w_{t}\|_{L^{p_{k+1}}}$ $\displaystyle\leq$
$\displaystyle(CC_{s})^{\frac{1}{p_{k}+1}}p_{k}^{\frac{2}{p_{k}+1}}\|w\|_{L^{p_{k}}}^{\frac{p_{k}}{p_{k}+1}}$
$\displaystyle\leq$
$\displaystyle(CC_{s})^{\sum_{i=0}^{i=k}\frac{\nu^{k-i}}{p_{k}+1}}\prod_{i=0}^{i=k}p_{i}^{\frac{2\nu^{k-i}}{p_{k}+1}}\|w_{t}\|_{L^{1}}^{\prod\frac{p_{i}}{p_{i}+1}}$
$\displaystyle\leq$ $\displaystyle
C(n,g)C_{s}^{\frac{n}{2}}\|w_{t}\|_{L^{1}}^{\gamma(n)}$

for a constant $\gamma(n)$ depending only on $n$, where we have used the fact
$p_{k}\leq 2\nu^{k}$ for $k\geq 1$. Therefore by (4.10), we prove

$\|w_{t}\|_{C^{0}}\leq
C(n,g)C_{s}^{\frac{n}{2}}\|w_{t}\|_{L^{1}}^{\gamma(n)}\to
0,~{}\text{as}~{}t\to\infty.$

∎

##  5\. Another version of the invariance $N_{X}(\omega_{g})$

Let $Y\in\eta_{r}(M)$ so that $\text{Im}(Y)$ generates an one-parameter
compact subgroup of $K$. Denote $\mathcal{K}_{Y}$ to be a class of
$K_{Y}$-invariant Kähler metrics in $2\pi c_{1}(M)$. Then according to the
proof of Proposition 1.4, we actually prove

(5.1) $\displaystyle\begin{aligned}
\sup_{g\in\mathcal{K}_{Y}}\lambda(g)\leq(2\pi)^{-n}[nV-\tilde{F}_{Y}(Y)-N_{Y}(c_{1}(M))].\end{aligned}$

Note that

$\tilde{F}_{Y}(Y)=\int_{M}Y(h_{g}-\theta_{Y,\omega})e^{\theta_{Y,\omega_{g}}}\omega_{g}^{n}$

and

$N_{Y}(c_{1}(M))=\int_{M}\theta_{Y,\omega_{g}}e^{\theta_{Y,\omega_{g}}}\omega_{g}^{n}$

are both holomorphic invariances of $M$. In this section, we want to show

######  Proposition 5.1.

Let $H(Y)=\tilde{F}_{Y}(Y)+N_{Y}(c_{1}(M))$. Then

$\sup_{Y\in\eta_{r}(M)}H(Y)=N_{X}(c_{1}(M)),$

where $X$ is the extremal vector field as in Section 1.

###### Proof.

Choose a constant $c_{Y}$ so that
$\hat{\theta}_{Y,\omega_{g}}=\theta_{Y,\omega_{g}}+c_{Y}$ satisfies a
normalization condition

(5.2) $\displaystyle\begin{aligned}
\int_{M}\hat{\theta}_{Y,\omega_{g}}e^{h_{g}}\omega_{g}^{n}=0.\end{aligned}$

Then $\hat{\theta}_{Y,\omega_{g}}$ satisfies an equation

$\Delta\hat{\theta}_{X,\omega_{g}}+X(h_{g})+\hat{\theta}_{X,\omega_{g}}=0$

Thus using the integration by part, we have

$\tilde{F}_{Y}(Y)+\int_{M}\hat{\theta}_{Y,\omega_{g}}e^{\theta_{Y,\omega_{g}}}\omega_{g}^{n}=0$

It follows

(5.3) $H(Y)=-c_{Y}V=\int_{M}\theta_{Y,\omega_{g}}e^{h_{g}}\omega_{g}^{n}.$

We compute the first variation of $H(Y)$ in $\eta_{r}(M)$. By the definition
of $\theta_{Y+tY^{\prime}}$, we see that there exist constants $b(t)$ such
that $\theta_{Y+tY^{\prime}}=\theta_{Y}+t\theta_{Y^{\prime}}+b(t)$. Since
$\int_{M}e^{\theta_{Y+tY^{\prime}}}\omega_{g}^{n}=V$, we have

$e^{-b(t)}=\frac{1}{V}\int_{M}e^{\theta_{Y}+t\theta_{Y^{\prime}}}\omega_{g}^{n}.$

Thus we get

(5.4)
$\frac{dH(Y+tY^{\prime})}{dt}|_{t=0}=\int_{M}\theta_{Y^{\prime}}e^{h_{g}}\omega_{g}^{n}-\int_{M}\theta_{Y^{\prime}}e^{\theta_{Y}}\omega_{g}^{n}=\tilde{F}_{Y}(Y^{\prime}).$

Therefore, by [TZhu2], we see that there exists a unique critical
$X\in\eta_{r}(M)$ of $H(\cdot)$ such that

(5.5) $\tilde{F}_{X}(Y^{\prime})=F_{X}(Y^{\prime})\equiv
0,~{}~{}\forall~{}Y^{\prime}\in\eta_{r}(M).$

Similarly, we have

$\theta_{tY+(1-t)Y^{\prime}}=t\theta_{Y}+(1-t)\theta_{Y^{\prime}}+b(t)^{\prime},~{}\forall~{}t\in[0,1]$

for some constants $b(t)^{\prime}$. Then

$\displaystyle\begin{aligned}
V=\int_{M}e^{\theta_{tY+(1-t)Y^{\prime}}}\omega_{g}^{n}&=e^{b(t)^{\prime}}\int_{M}e^{t\theta_{Y}+(1-t)\theta_{Y^{\prime}}}\omega_{g}^{n}\\\
&\leq
e^{b(t)^{\prime}}[t\int_{M}e^{\theta_{Y}}\omega_{g}^{n}+(1-t)\int_{M}e^{\theta_{Y^{\prime}}}\omega_{g}^{n}]\\\
&=e^{b(t)^{\prime}}V.\end{aligned}$

Thus $b(t)^{\prime}\geq 0$. Consequently

$H(tX+(1-t)Y)\geq tH(X)+(1-t)H(Y).$

This means that $H(\cdot)$ is a concave functional on $\eta_{r}(M)$. It
follows that $X$ is a global maximizer of $H(\cdot)$. Therefore we prove the
proposition by using the fact $H(X)=N_{X}(c_{1}(M))$.

∎

######  Corollary 5.2.

Let $\mathcal{K}_{K}$ be a class of $K$-invariant Kähler metrics in $2\pi
c_{1}(M)$. Suppose that

(5.6) $\displaystyle\begin{aligned}
\sup_{g\in\mathcal{K}_{K}}\lambda(g)<\inf_{Y\in\eta_{r}(M)}(2\pi)^{-n}[nV-
F_{Y}(Y)-N_{Y}(c_{1}(M))].\end{aligned}$

Then $(M,J)$ could not admit any Kähler-Ricci soliton. Furthermore, the
modified Mabuchi’s K-energy could not be bounded from below.

###### Proof.

The first part of corollary follows from Proposition 5.1 and Corollary 1.5.
The second part follows from Proposition 5.1 and Theorem 0.1.

∎

The above corollary gives a new obstruction to the existence of Kähler-Ricci
solitons.

##  6\.  Proof of Theorem 0.2

In this section, we will modify the proof of Main Theorem in [TZhu5] to prove
Theorem 0.2. The proof in [TZhu5] depends on a generalized uniqueness theorem
for Kähler-Einsteins recently proved by Chen and Sun in [CS]. Here we avoid to
use their theorem so that we can generalize the proof to the case of Kähler-
Ricci solitons by applying Proposition 3.1. As in [TZhu5], we write an initial
Kähler form $\omega_{g}$ of Kähler-Ricci flow (2.1) by

$\omega_{g}=\omega_{\varphi}=\omega_{g_{KS}}+\sqrt{-1}\partial\overline{\partial}\varphi\in
2\pi c_{1}(M)$

for a Kähler potential $\varphi$ on $M$. We define a path of Kähler forms

$\omega_{g^{s}}=\omega_{g_{KS}}+s\sqrt{-1}\partial\overline{\partial}\varphi$

and set

$\displaystyle\begin{aligned}
I=\\{s\in~{}[0,1]|~{}&(g_{t}^{s};g^{s})~{}\text{converges to a K\"{a}hler-
Ricci soliton in }~{}C^{\infty}~{}~{}\\\ &\text{in sense of K\"{a}hler
potentials}\\}.\end{aligned}$

Clearly, $I$ is not empty by the assumption of existence of Kähler-Ricci
solitons on $M$. We want to show that $I$ is in fact both open and closed.
Then it follows that $I=[0,1]$. This will finish the proof Theorem 0.2.

The openness of $I$ is related to the following stability theorem of Kähler-
Ricc flow, which was proved in [Zhu2].

######  Lemma 6.1.

Let $(M,J)$ be a compact Kähler manifold which admits a Kähler-Ricci soliton
$(g_{KS},X)$. Let $\psi$ be a Kähler potential of a $K_{X}$-invariant initial
metric $g$ of (2.1). Then there exists a small $\epsilon$ such that if

$\|\psi\|_{C^{3}}\leq\epsilon,$

the solution $g(t,\cdot)$ of (2.1) will converge to a Kähler-Ricci soliton
with respect to $X$ in $C^{\infty}$ in the sense of Kähler potentials.
Moreover, the convergence can be made exponentially.

######  Remark 6.2.

Lemma 6.1 is still true if the $K_{X}$-invariant condition is removed for the
initial metric $g$ (cf. [Zhu2]). But we do not know whether the convergence is
exponentially fast or not.

###### Proof of openness of $I$.

Suppose that $s_{0}\in I$. Then by the uniqueness of Kähler-Ricci solitons
[TZhu1], the flow $(g_{t}^{s_{0}};\omega_{s_{0}})$ converges to $g_{KS}$ after
a holomorphism transformation in $\text{Aut}_{r}(M)$. Namely, there exists a
$\sigma\in\text{Aut}_{r}(M)$ such that
$\sigma^{\star}\omega_{g_{t}^{s_{0}}}=\omega_{KS}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}^{s_{0}})_{\sigma}$
with property

$\|(\varphi_{t}^{s_{0}})_{\sigma}\|_{C^{k}}\leq C_{k}e^{-\alpha_{k}t},$

where $C_{k},\alpha_{k}>0$ are two uniform constants. Then we can choose $T$
sufficiently large such that

$\|(\varphi_{t}^{s_{0}})_{\sigma}\|_{C^{3}(M)}<\frac{\delta}{2},$

where $\delta$ is a small number determined in Lemma 6.1. Since the Kähler-
Ricci flow is stable for any fixed finite time, there is a small $\epsilon>0$
such that

$\|\varphi_{T}^{s}-(\varphi_{T}^{s_{0}})_{\sigma}\|_{C^{3}(M)}\,<\,\frac{\delta}{2},~{}\forall~{}s\in[s_{0},s_{0}+\epsilon],$

where $\varphi_{T}^{s}$ is a Kähler potential of evolved Kähler metric
$g_{T}^{s}$ of Kähler-Ricci flow $(g_{t}^{s};\sigma^{\star}\omega_{s})$ at
time $T$. Hence, we have

(6.1)
$\displaystyle\|\varphi_{T}^{s}\|_{C^{3}(M)}\,<\,\delta,~{}\forall~{}s\in[s_{0},s_{0}+\epsilon].$

Then the flow $(g_{t};g_{T}^{s})$ with initial $g_{T}^{s}$ will converge to a
Kähler-Ricci soliton in $C^{\infty}$ according to Lemma 6.1. This shows $s\in
I$ for any $s\in[s_{0},s_{0}+\epsilon]$

∎

Let $\varphi_{t}^{s}$ be a family of Kähler potentials of evolved Kähler
metric $g_{t}^{s}$ of Kähler-Ricci flow $(g_{t}^{s};\omega_{s})$. To make
potentials $\varphi^{s}_{t}$ more smaller to control, we need the following
lemma, which was proved in [TZhu1].

######  Lemma 6.3.

Let $M$ be a compact Kähler manifold which admits a Kähler-Ricci soliton
$(g_{KS},X)$. Let $\varphi$ be a $K_{X}$-invariant Kähler potential. Then
there exists a unique holomorphism transformation $\sigma\in\text{Aut}_{r}(M)$
such that $\varphi_{\sigma}\in\Lambda^{\perp}(\omega_{KS})$ with property

$J(\varphi_{\sigma})=\inf_{\tau\in\text{Aut}_{r}(M)}J(\varphi_{\tau}),$

where $\Lambda^{\perp}(\omega_{KS})$ is an orthogonal space to kernel space of
linear operator $(\Delta_{g_{KS}}+X+\text{Id})(\psi)$ and

$J(\varphi)=-\int_{M}\varphi
e^{\theta_{X,\omega_{\varphi}}}\omega_{\varphi}^{n}+\int_{0}^{1}\int_{M}\varphi
e^{\theta_{X,\omega_{\lambda\varphi}}}\omega_{\lambda\varphi}^{n}\wedge
d\lambda\geq 0.$

Moreover,

$\|\sigma-\text{Id}\|\leq C(\|\varphi\|_{C^{5}}),$

where $\|\sigma-\text{Id}\|$ denotes the distance norm in Lie group
$\text{Aut}_{r}(M)$ .

###### Proof of closedness of $I$.

By the openness of $I$, we see that there exists a $\tau_{0}\leq 1$ with
$[0,\tau_{0})\subset I$. We need to show that $\tau_{0}\in I$. In fact we want
to prove that for any $\delta>0$ there exists a large $T$ such that

(6.2)
$\displaystyle\|(\phi^{s}_{t})_{\sigma_{s,t}}\|_{C^{5}}\leq\delta,~{}\forall~{}t\geq
T~{}~{}\text{and}~{}s<\tau_{0},$

where $\sigma_{s,t}$ are some holomorphisms in $\text{Aut}_{r}(M)$. We will
use an argument by contradiction as in [TZhu5]. On contrary, we can find a
sequence of evolved Kähler metrics $g_{t_{i}}^{s_{i}}$ of Kähler-Ricci flows
$(g_{t}^{s_{i}};g^{s_{i}})$, where $s_{i}\to\tau_{0}$ and $t_{i}\to\infty$,
and a sequence of unique holomorphisms
$\sigma_{s_{i},t_{i}}\in\text{Aut}_{r}(M)$ for pairs $(s_{i},t_{i})$ such that
$(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}\in\Lambda^{\perp}(\omega_{KS})$
and

(6.3)
$\displaystyle\|(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}\|_{C^{5}}\geq\delta_{0}>0,$

for some constant $\delta_{0}$. Since the Kähler-Ricci flow
$(g_{t}^{s_{i}};g^{s_{i}})$ converges to some Kähler-Ricci soliton, by the
uniqueness of Kähler-Ricci solitons [TZhu1], the flow after a holomorphism
transformation in $\text{Aut}_{r}(M)$ converges to $g_{KS}$. So we may further
assume that $\phi^{s_{i}}_{t_{i}}$ satisfy

(6.4)
$\displaystyle\|(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}\|_{C^{5}}\leq
2\delta_{0}.$

Then there exists a subsequence
$(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}$ (still used by
$(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}$ ) of
$(\phi^{s_{i}}_{t_{i}})_{\sigma_{s_{i},t_{i}}}$ converging to a potential
$\phi_{\infty}\in\Lambda^{\perp}(\omega_{KS})$ with property

(6.5) $\displaystyle 2\delta_{0}\geq\|\phi_{\infty}\|_{C^{5}}\geq\delta_{0}.$

We want to show that

(6.6)
$\displaystyle\lambda(\omega_{\phi_{\infty})}=\lambda(g_{KS})=(2\pi)^{-n}(nV-
N_{X}(c_{1}(M)).$

First we note that the modified $K$-energy is bounded from below since $M$
admits a Kähler-Ricci soliton [TZhu2]. Then by Proposition 3.1 and the
monotonicity of $\lambda(g_{t}^{\tau_{0}})$, we see that for any $\epsilon>0$,
there exists a large $T>0$ such that

$\lambda(g^{\tau_{0}}_{t})\geq(2\pi)^{-n}(nV-
N_{X}(c_{1}(M))-\frac{\epsilon}{2},~{}~{}\forall~{}t\geq T.$

Since Kähler-Ricci flow is stable in finite time and $\lambda(g_{t}^{s})$ is
monotonic in $t$, there is a small $\delta>0$ such that for any
$s\geq\tau_{0}-\delta$, we have

(6.7) $\displaystyle\lambda(g_{t}^{s})\geq(2\pi)^{-n}(nV-
N_{X}(c_{1}(M))-\epsilon,~{}~{}\forall~{}t\geq T.$

Since $s_{i}\to\tau_{0}$ and $t_{i}\to\infty$, we conclude that

$\lim_{s_{i}\to\tau_{0},t_{i}\to\infty}\lambda(\sigma_{s_{i},t_{i}}^{\star}g^{s_{i}}_{t_{i}})=\lim_{s_{i}\to\tau_{0},t_{i}\to\infty}\lambda(g^{s_{i}}_{t_{i}})=(2\pi)^{-n}(nV-
N_{X}(c_{1}(M)).$

By the continuity of $\lambda(\cdot)$, we will get $(\ref{identity-3})$.

Now by Corollary 1.4 together with $(\ref{identity-3})$, we see that
$\omega_{\phi_{\infty}}$ is a global maximizer of $\lambda(\cdot)$ in
$\mathcal{K}_{X}$, so it is a critical point of $\lambda(\cdot)$. Then it is
easy to show that $\omega_{\phi_{\infty}}$ a Kähler-Ricci soliton with respect
to $X$ by computing the first variation of $\lambda(\cdot)$ as done in (1.5).
Thus by the uniqueness result for Kähler-Ricci solitons in [TZhu1], we get

$\omega_{\phi_{\infty}}=\sigma^{\star}\omega_{KS},$

where $\sigma\in\text{Aut}_{r}(M)$. Since
$\phi_{\infty}\in\Lambda^{\perp}(\omega_{KS})$, by Lemma 6.3, $\phi_{\infty}$
must be zero. This is a contradiction to $(\ref{contradiction-2})$. The
contradiction implies that (6.2) is true.

By $(\ref{claim-5})$, we see that for any $\delta>0$ there exists a large
$T_{0}$ and $\sigma_{0}\in\text{Aut}_{r}(M)$ such that

(6.8)
$\displaystyle\|(\phi_{T_{0}}^{\tau_{0}})_{\sigma_{0}}\|_{C^{5}}\leq\delta.$

Then by Lemma 6.1, the Kähler-Ricci flow
$(g_{t};\omega_{(\phi_{T_{0}}^{\tau_{0}})_{\sigma_{0}}})$ converge to a
Kähler-Ricci soliton. Thus, we prove that $\tau_{0}\in I$.

∎

######  Remark 6.4.

According to the proof of Theorem 0.2 and Remark 6.2, Theorem 0.2 will be
still true if the $K_{X}$-invariant condition is removed for the initial
metric $g$ of (2.1) assuming that Conjecture (3.3) is true.

##  7\. Appendix

In [TZhu5], it was proved the minimizer $f_{t}$ of $W(g_{t},\cdot)$-functional
associated to evolved Kähler metric $g_{t}$ of Kähler-Ricci flow (2.1) is
uniformly bounded (see also [TZha]). In this appendix, we show that the
gradients of $f_{t}$ are also uniformly bounded, and so are $\triangle f_{t}$
by (1.4). Namely, we prove

######  Theorem 7.1.

There is a uniform constant $C$ such that

$\|f_{t}\|+\|\nabla f_{t}\|+\|\triangle f_{t}\|\leq C,~{}~{}\forall~{}t>0.$

We will derive $\|\nabla f_{t}\|$ in Theorem 7.1 by studying a general
nonlinear elliptic equation as follows:

(7.1) $\triangle w(x)=w(x)F(x,w(x))$

where the Laplace operator $\triangle$ is associated to a Kähler metric $g$ in
$2\pi c_{1}(M)$ and $F$ is a smooth function on $M\times\mathbb{R}^{+}$, which
satisfies a structure condition:

(7.2) $-A-Bt^{\alpha}\leq F(\cdot,t)\leq H(t).$

Here $0\leq A,B\leq\infty,0\leq\alpha<\frac{2}{n}$ are constants, and $H$ is a
proper function on $\mathbb{R}^{+}$ which satisfies a growth control at $0$:

(7.3) $\limsup_{t\rightarrow 0}\big{(}tH(t)\big{)}<\infty.$

######  Lemma 7.2.

Let $w$ is a positive solution of (7.1). Then

(7.4) $\|\nabla w\|_{C^{0}}\leq C(n)C_{s}^{\frac{n}{2}}\big{(}\|\nabla
h\|_{C^{0}}+\|wF\|_{C^{0}}\big{)}^{n}\big{(}\int_{M}(1+|\nabla
w|^{2})dV_{g}\big{)}^{1/2},$

where $C_{s}$ is a Sobelev constant of $g$ and $h$ is a Ricci potential of
$g$.

###### Proof.

We will use the Moser’s iteration to $L^{p}$-estimate of $|\nabla w|$. By the
Bochner formula, we have

$\displaystyle\begin{aligned} \triangle|\nabla w|^{2}&=|\nabla\nabla
w|^{2}+|\nabla\bar{\nabla}w|^{2}+\nabla_{i}\triangle
w\nabla_{\bar{i}}w+\nabla_{i}w\nabla_{\bar{i}}\triangle
w+R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}w\\\ &=|\nabla\nabla
w|^{2}+|\nabla\bar{\nabla}w|^{2}+\nabla_{i}(wF)\nabla_{\bar{i}}w+\nabla_{i}w\nabla_{\bar{i}}(wF)+R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}w.\end{aligned}$

Put $\eta=|\nabla w|^{2}+1$ . Then for $p\geq 2$, it follows

(7.5) $\displaystyle\begin{aligned}
&\frac{4(p-1)}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}\\\
&=-\int_{M}\eta^{p-1}\triangle\eta dV_{g}\\\
&=-\int_{M}\eta^{p-1}\big{(}|\nabla\nabla
w|^{2}+|\nabla\bar{\nabla}w|^{2}\big{)}dV_{g}-\int_{M}\eta^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}wdV_{g}\\\
&-\int_{M}\eta^{p-1}\big{(}\nabla_{i}(wF)\nabla_{\bar{i}}w+\nabla_{\bar{i}}(wF)\nabla_{i}w\big{)}dV_{g}.\end{aligned}$

The last term on the right hand side can be estimate as follows.

$\displaystyle-\int_{M}\eta^{p-1}\big{(}\nabla_{i}(wF)\nabla_{\bar{i}}w+\nabla_{\bar{i}}(wF)\nabla_{i}w\big{)}dV_{g}$
$\displaystyle=\int_{M}wF\big{(}\nabla_{i}\eta^{p-1}\nabla_{\bar{i}}w+\nabla_{\bar{i}}\eta^{p-1}\nabla_{i}w+2\eta^{p-1}\triangle
w\big{)}dV_{g}$
$\displaystyle=\frac{2(p-1)}{p}\int_{M}wF\eta^{\frac{p}{2}-1}\big{(}\nabla_{i}\eta^{p/2}\nabla_{\bar{i}}w+\nabla_{\bar{i}}\eta^{p/2}\nabla_{i}w\big{)}dV_{g}$
$\displaystyle+2\int_{M}wF\eta^{p-1}\triangle wdV_{g}.$

Then

(7.6)
$\displaystyle-\int_{M}\eta^{p-1}\big{(}\nabla_{i}(wF)\nabla_{\bar{i}}wdV_{g}+\nabla_{\bar{i}}(wF)\nabla_{i}w\big{)}dV_{g}$
$\displaystyle\leq\frac{2(p-1)}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}+2(p-1)p\int_{M}(wF)^{2}\eta^{p-2}(\eta-1)dV_{g}$
$\displaystyle+\int_{M}\eta^{p-1}\big{(}\frac{(\triangle
w)^{2}}{2n}+2n(wF)^{2}\big{)}dV_{g}$
$\displaystyle\leq\frac{2(p-1)}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}+\frac{1}{2n}\int_{M}\eta^{p-1}(\triangle
w)^{2}dV_{g}$
$\displaystyle+2[p(p-1)+n]\|wF\|_{C^{0}}^{2}\int_{M}\eta^{p}dV_{g}.$

For the second term on the right hand side, we note

$R_{i\overline{j}}=g_{i\overline{j}}+h_{i\overline{j}}.$

Then

$\displaystyle\begin{aligned}
&-\int_{M}\eta^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}wdV_{g}\\\
&=\int_{M}\eta^{p-1}\nabla_{i}\nabla_{\bar{j}}h\nabla_{\bar{i}}w\nabla_{j}wdV_{g}-\int_{M}\eta^{p}dV_{g}\\\
&=\frac{2(p-1)}{p}\int_{M}\eta^{p/2-1}\nabla_{\bar{j}}h\nabla_{i}\eta^{p/2}\nabla_{\bar{i}}w\nabla_{j}wdV_{g}\\\
&+\int_{M}\eta^{p-1}\nabla_{\bar{j}}h\big{(}\triangle
w\nabla_{j}w+\nabla_{\bar{i}}w\nabla_{i}\nabla_{j}w\big{)}dV_{g}-\int_{M}\eta^{p}dV_{g}.\end{aligned}$

Thus

(7.7)
$\displaystyle-\int_{M}\eta^{p-1}R_{i\bar{j}}\nabla_{\bar{i}}w\nabla_{j}wdV_{g}$
$\displaystyle\leq\frac{p-1}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}+p(p-1)\|\nabla
h\|_{C^{0}}^{2}\int_{M}\eta^{p-2}(\eta-1)^{2}dV_{g}$
$\displaystyle+\frac{1}{2n}\int_{M}\eta^{p-1}(\triangle
w)^{2}dV_{g}+\frac{n}{2}\|\nabla
h\|_{C^{0}}^{2}\int_{M}\eta^{p-1}(\eta-1)dV_{g}$
$\displaystyle+\frac{1}{2n}\int_{M}\eta^{p-1}|\nabla\nabla
w|^{2}dV_{g}+\frac{n}{2}\|\nabla
h\|_{C^{0}}^{2}\int_{M}\eta^{p-1}(\eta-1)dV_{g}$
$\displaystyle\leq\frac{p-1}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}+\frac{1}{2n}\int_{M}\eta^{p-1}(\triangle
w)^{2}dV_{g}$ $\displaystyle+\frac{1}{2n}\int\eta^{p-1}|\nabla\nabla
w|^{2}dV_{g}+[p(p-1)+n]\|\nabla h\|_{C^{0}}^{2}\int_{M}\eta^{p}dV_{g}.$

Substituting (7.6) and (7.7) into (7.5), we get

$\displaystyle\begin{aligned}
&\frac{p-1}{p^{2}}\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}\\\
&\leq-\int_{M}\eta^{p-1}\big{(}|\nabla\nabla
w|^{2}+|\nabla\bar{\nabla}w|^{2}\big{)}dV_{g}\\\
&+\frac{1}{n}\int_{M}\eta^{p-1}(\triangle
w)^{2}dV_{g}+\frac{1}{2n}\int_{M}\eta^{p-1}|\nabla\nabla w|^{2}dV_{g}\\\
&+[2(p-1)p+n]\big{(}\|\nabla
h\|_{C^{0}}^{2}+\|wF\|_{C^{0}}^{2}\big{)}\int_{M}\eta^{p}dV_{g}\\\ &\leq
C(n)p^{2}\big{(}\|\nabla
h\|_{C^{0}}^{2}+\|wF\|_{C^{0}}^{2}\big{)}\int_{M}\eta^{p}dV_{g}.\end{aligned}$

It follows

$\int_{M}|\nabla\eta^{p/2}|^{2}dV_{g}\leq C(n)p^{3}\big{(}\|\nabla
h\|_{C^{0}}^{2}+\|wF\|_{C^{0}}^{2}\big{)}\int_{M}\eta^{p}dV_{g},\hskip
8.5359pt\forall~{}p\geq 2.$

Therefore, by iteration, we derive

$\sup\eta\leq C(n)D^{n/2}\big{(}\int_{M}\eta^{2}dV_{g}\big{)}^{1/2},$

where $D=C_{s}\big{(}\|\nabla h\|_{C_{0}}^{2}+\|wF\|_{C^{0}}^{2}\big{)}$. This
implies (7.4) . ∎

######  Proposition 7.3.

(7.8) $\|\nabla w\|_{C^{0}}\leq C(\|w\|_{L^{2}}),$

where the constant $C$ depends only on $n,C_{s},A,B,\alpha,H$,
$\text{Vol}(g)$, $\|\nabla h\|_{C^{0}}$ and $\|w\|_{L^{2}}$.

###### Proof.

First we note that by using the standard Moser’s iteration to equation

$\triangle w(x)\geq-A-Bw^{\alpha},$

it is easy to see

$\sup w\leq C(1+\|w\|_{L^{2}}^{\gamma})$

for some constants $C$ and $\gamma$ which depend only on
$n,C_{s},A,B,\alpha,H$ and $\rm{Vol}(g)$. On the other hand, by (7.1), we have

$\int_{M}|\nabla w|^{2}dV_{g}=-\int_{M}w\triangle wdV_{g}=-\int_{M}wFdV_{g}.$

Then we see that $\|\nabla w\|_{L^{2}}$ is bounded by $\|w\|_{L^{2}}$. Thus
the proposition follows from Lemma 7.2.

∎

Since $v_{t}=e^{-\frac{f_{t}}{2}}$ satisfies (3.10) which is a type of
equation (7.1), then by Perelman’s estimates (d) in Lemma 2.1 and Zhang’s
estimate for Sobelev constants associated to $g_{t}$ in [Zha] together with
$C^{0}$-estimate for $f_{t}$ in [TZhu5], we obtain a uniform gradient estimate
for $v_{t}$ from Proposition 7.3, and so for $f_{t}$ . By equation (1.4), we
also derive a a uniform Laplacian estimate for $f_{t}$. Thus Theorem 7.1 is
true. Theorem 7.1 will be used in Section 4.

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