#  Complete non-compact gradient Ricci solitons with nonnegative Ricci
curvature

Yuxing Deng  and  Xiaohua $\text{Zhu}^{*}$ Yuxing Deng
School of Mathematical Sciences, Peking University, Beijing, 100871, China
Xiaohua Zhu
School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871,
China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we give a delay estimate of scalar curvature for a complete
non-compact expanding (or steady) gradient Ricci soliton with nonnegative
Ricci curvature. As an application, we prove that any complete non-compact
expanding (or steady) gradient Kähler-Ricci solitons with positively pinched
Ricci curvature should be Ricci flat. The result answers a question in case of
Kähler-Ricci solitons proposed by Chow, Lu and Ni in a book.

###### Key words and phrases:

Ricci soliton, Ricci flow, pinched Ricci curvature

###### 2000 Mathematics Subject Classification:

Primary: 53C25; Secondary: 53C55, 58J05

* Partially supported by the NSFC Grants 11271022 and 11331001

##

##  1\. Introduction

Ricci soliton plays an important role in the study of Hamilton’s Ricci flow,
in particular in the singularities analysis of Ricci flow [15], [3], [21]. In
case of shrinking gradient Ricci solitons with positive curvature, Hamilton
proved that the solitons should be isometric to a standard sphere in
$\mathbb{R}^{3}$ in two dimensional case [15]. Perelman generalized Hamilton’s
result to three dimensional case [21]. Later on, Nabor proved that any four-
dimensional shrinking gradient Ricci soliton with positive bounded curvature
operator should be a standard sphere in $\mathbb{R}^{5}$ [17]. On the other
hand, Perelman and Brendle proved that any steady gradient Ricci soliton with
nonnegative sectional curvature should be a Bryant’s soliton in case of
2-dimension and 3-dimension, respectively [21], [6], [2], [1]. However, to
author’s acknowledge, there is rarely understanding in case of expanding
gradient Ricci solitons even for lower dimensional manifolds. For example, how
to classify complete non-compact gradient expanding (or steady) Ricci solitons
under a suitable curvature condition. The purpose of this paper is to give a
rigidity theorem for a class of expanding (or steady) gradient Kähler-Ricci
solitons with nonnegative Ricci curvature.

######  Definition 1.1.

A complete Riemannian metric $g$ on $M$ is called a gradient Ricci soliton if
there exists a smooth function $f$ ( which is called a defining function) on
$M$ such that

(1.1) $R_{ij}+\rho g_{ij}=\nabla_{i}\nabla_{j}f,$

where $\rho\in\mathbb{R}$ is a constant. The gradient Ricci soliton is called
expanding, steady and shrinking according to the sign $\rho>,=,<0$,
respectively.

For simplicity, we normalize $\rho=1,0,-1$. In addition, $g$ is a Kähler
metric on a complex manifold $M$, we call $g$ is a Kähler-Riccoi soliton.
Since $\overline{\partial}f$ induces a holomorphic vector field on $M$, (1.1)
was usually written in a complex version,

(1.2) $R_{i\bar{j}}+\rho g_{i\bar{j}}=\nabla_{i}\nabla_{\bar{j}}f,$

A gradient soliton $(M,g,f)$ is called complete if $g$ and $\nabla f$ are both
complete. It is known that the completeness of $(M,g)$ implies the
completeness of $\nabla f$ [26]. Throughout this paper, we always assume the
soliton is complete. If there is a point $o\in M$ such that $\nabla f(o)=0$,
we call $o$ an equilibrium point of $(M,g)$. By studying the existence of
equilibrium points, we prove the boundedness of scalar curvature of $g$.

######  Theorem 1.2.

Let $(M,g)$ be a complete non-compact expanding gradient Ricci soliton with
nonnegative Ricci curvature or a complete non-compact steady gradient Kähler-
Ricci soliton with nonnegative bisectional curvature and positive Ricci
curvature. Then the scalar curvature of $g$ is bounded and it attains the
maximum at the unique equilibrium point.

The proof of Theorem 1.2 will be given in case of expanding Ricci solitons in
next section. For the steady Ricci solitons, the proof for the existence of
equilibrium points is a bit different, although the boundedness of scalar
curvature is directly from an identity (4.2). We will use a result of local
convergence for Kähler-Ricci flow by Chau and Tam to prove the existence in
Section 5 [11].

Theorem 1.2 will be applied to prove the following rigidity theorem for
Kähler-Ricci solitons with nonnegative Ricci curvature.

######  Theorem 1.3.

Let $(M^{n},g)$ be a complete non-compact gradient Kähler-Ricci soliton with
non-negative Ricci curvature. Suppose that there exists a point $p\in M$ such
that the scalar curvature $R$ of $g$ satisfies

(1.3) $\displaystyle\frac{1}{{\rm vol}(B_{r}(p))}\int_{B_{r}(p)}R{\rm
dr}\leq\frac{\varepsilon(r)}{1+r^{2}},~{}\text{if}~{}g~{}\text{ is
expending};$

or

(1.4) $\displaystyle\frac{1}{{\rm vol}(B_{r}(p))}\int_{B_{r}(p)}R{\rm
dr}\leq\frac{\varepsilon(r)}{1+r},~{}\text{if}~{}g~{}\text{ is steady},$

where $\varepsilon(r)\rightarrow 0$ as $r\to\infty$. Then $g$ is Ricci-flat.
Moreover, $(M,g)$ is isometric to $\mathbb{C}^{n}$ if $g$ is expending.

We note that under the condition of nonnegative sectional curvature (or
nonnegative holomorphic bisectional curvature for Kähler manifolds) several
rigidity theorems were obtained in [15], [19], [22], etc. For Ricci solitons,
we are able to use the Ricci flow to weaken the condition of curvature to
nonnegative Ricci curvature.

As a corollary, we obtain a version of Theorem 1.3 under the pointed-wise
Ricci decay condition.

######  Theorem 1.4.

Let $(M^{n},g)$ be a complete non-compact gradient Kähler-Ricci soliton with
non-negative Ricci curvature. Suppose that $g$ satisfies

(1.5) $\displaystyle
R(x)\leq\frac{\varepsilon(r(x))}{1+r(x)^{2}}~{}(\varepsilon(r)\rightarrow
0,\text{ as}~{}r\to\infty),~{}\text{if}~{}g~{}\text{ is expending};$

or

(1.6) $\displaystyle R(x)\leq\frac{C}{1+r(x)^{2n+\epsilon}}~{}\text{for
some}~{}C,\epsilon>0,~{}\text{if}~{}g~{}\text{ is steady}.$

Then $g$ is Ricci-flat. Moreover, $(M,g)$ is isometric to $\mathbb{C}^{n}$ if
$g$ is expending.

In case of steady solitons in Theorem 1.4, if we assume that $(M,g)$ has
nonnegative bisectional curvature instead of nonnegative Ricci curvature, then
the condition (1.6) can be weakened as

(1.7) $\displaystyle R(x)\leq\frac{C}{1+r(x)^{1+\epsilon}}.$

In fact, we can prove that $(M,g)$ is isometric to $\mathbb{C}^{n}$ by Theorem
1.2, see Proposition 5.1. Proposition 5.1 is an analogy of Hamilton’s result
for Kähler manifolds [13].

A Riemannian metric is called with property of positively pinched Ricci
curvature if there is a uniform constant $\delta>0$ such that
$\text{Ric}(g)\geq\delta Rg$ [13], [15]. It was proved that the scalar
curvature of complete non-compact expanding (or steady) gradient Ricci
solitons with positively pinched Ricci curvature has exponential decay (cf.
Theorem 9.56, [6]). Thus as a direct consequence of Theorem 1.3, we obtain

######  Corollary 1.5.

Non-trivial complete non-compact expanding or steady gradient Kähler-Ricci
soliton with positively pinched Ricci curvature doesn’t exist for $n\geq 2$.

Corollary 1.5 answers a question in case of Kähler-Ricci solitons proposed by
Chow, Lu and Ni in their book [6] (cf. page 390). They asked whether there
exists an expanding gradient Ricci soliton with positively pinched Ricci
curvature when $n\geq 3$.

Theorem 1.3 and 1.4 will be proved in Section 4 and Section 5 according to
expending or steady solitons, respectively.

##  2\.  Boundedness of scalar curvature–I

In this section, we prove the boundedness of scalar curvature in case of
expending Ricci solitons. Let $(M^{n},g)$ be a Riemannian manifold. In local
coordinates $(x^{1},x^{2},\cdots,x^{n})$, curvature tensor Rm of $g$ is
defined by

${\rm Rm}(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}})\frac{\partial}{\partial x^{k}}\triangleq\sum
R^{l}_{ijk}\frac{\partial}{\partial x^{l}}$

and $R_{ijkl}\triangleq\sum g_{lm}R_{ijk}^{m}$. Then the Ricci curvature is
given by

$R_{jk}=\sum R_{ijk}^{i}.$

Thus by the commutation formula,

(2.1)
$\displaystyle(\nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i})\alpha_{k_{1}\cdots
k_{r}}=-\sum_{l=1}^{r}R^{m}_{ijk_{l}}\alpha_{k_{1}\cdots k_{l-1}mk_{l+1}\cdots
k_{r}},$

we get from the Bianchi identity,

(2.2) $2\sum\nabla_{i}R_{ij}=\nabla_{j}R.$

Let $(M^{n},g,f)$ be an expanding gradient Ricci soliton and $\phi_{t}$ be a
family of diffeomorphisms generated by $-\nabla f$. Then the induced metrics
$g(t)=\phi_{t}^{*}g$ satisfy

(2.3) $\frac{\partial}{\partial t}g=-2\text{Ric}(g)-2g.$

(2.3) is equivalent to

(2.4) $R_{ij}(t)+g_{ij}(t)=\nabla_{i}\nabla_{j}f(t),$

where $f(t)=\phi_{t}^{*}f$ and $\nabla$ is taken w.r.t $g(t)$.

######  Lemma 2.1.

$\frac{\partial}{\partial t}R=2{\rm Ric}(\nabla f(t),\nabla f(t)).$

###### Proof.

Differentiating $(\ref{expanding-soliton})$ on both sides, we have

$\displaystyle\nabla_{k}R_{ij}=\nabla_{k}\nabla_{i}\nabla_{j}f.$

It follows from (2.1),

$\displaystyle\nabla_{i}R_{jk}-\nabla_{j}R_{ik}=-\sum R_{ijkl}\nabla_{l}f.$

Thus by (2.2), we get

(2.5) $\displaystyle\nabla_{j}R=-2R_{jl}\nabla_{l}f.$

Hence

$\displaystyle\frac{\partial}{\partial t}R(x,t)=\frac{\partial}{\partial
t}R(\phi_{t}(x),0)=-\langle\nabla R,\nabla f\rangle=2\text{Ric}(\nabla
f,\nabla f).$

∎

Let $B_{r}(o,t)$ be a $r$-geodesic ball centered at $o\in M$ w.r.t $g(t)$.
Then

######  Lemma 2.2.

Let $g(x,t)$ be a solution of (2.3) with nonnegative Ricci curvature for any
$t\in(0,\infty)$. Then for any $r>0$ and $\delta>0$, there exists a
$T_{0}=T_{0}(r,\delta)>0$ such that $B_{r}(o,0)\subset B_{\delta}(o,t)$ for
any $t\geq T_{0}$.

###### Proof.

By (2.3), it is easy to see that

$\displaystyle\frac{{\rm d}|v|^{2}_{t}}{{\rm
d}t}\leq-2|v|^{2}_{t},~{}\forall~{}t\geq 0,$

where $v\in T_{p}^{(1,0)}M$ for any $p\in M$. Then

$|v|_{t}^{2}\leq e^{-2t}|v|_{0}^{2}.$

Connecting $o$ and $p$ by a minimal geodesic curve $\gamma(s)$ with an arc-
parameter $s$ w.r.t the metric $g(x,0)$ in $B_{r}(o,0)$, we get

(2.6) $d_{t}(o,p)\leq\int_{0}^{l}|\gamma^{\prime}(s)|_{t}{\rm
ds}\leq\int_{0}^{l}|\gamma^{\prime}(s)|_{0}e^{-2t}{\rm ds}\leq re^{-2t},$

where $l$ is the length of $\gamma(s)$. Therefore, by taking $t$ large enough.
we see that $B_{r}(o,1)\subset B_{\delta}(o,t)$. ∎

Taking an integration along a geodesic curve on both sides of (1.1), on can
show that $f(x)\geq\frac{r(x)^{2}}{4}$ under the assumption of nonnegative
Ricci curvature. This implies that $f(x)$ attains the minimum at some point
$o\in M$. Thus $\nabla f(o)=0$. Moreover the equilibrium point $o$ is unique.
This is because, if there is another equilibrium point $p$, then
$\phi_{t}(o)=o$ and $\phi_{t}(p)=p$. In particular $d_{t}(o,p)=d_{0}(o,p)$. On
the other hand, by (2.6), $d_{t}(o,p)\leq e^{-2t}d_{0}(o,p),~{}\forall~{}t>0$.
Hence, $d_{0}(o,p)=0$, and consequently $o=p$.

Now we begin to prove Theorem 1.2.

###### Proof of Theorem 1.2 (the expanding case).

Let $o$ be the unique equilibrium point. Then by Lemma 2.2, for any
$r>\delta>0$, there exists $T_{0}$ such that

$B_{r}(o,0)\subset B_{\delta}(o,t),\mbox{\quad}\forall\mbox{\ }t\geq T_{0}.$

On the other hand, by Lemma 2.1, we see that $R(x,t)$ is nondecreasing in $t$.
Thus

(2.7) $\sup_{x\in B_{r}(o,0)}R(x,0)\leq\sup_{x\in
B_{\delta}(o,t)}R(x,t),\mbox{\quad}\forall\mbox{\ }r>0,\mbox{\ }\delta>0.$

Note that $\phi_{t}:(M^{n},g(t))\rightarrow(M^{n},g(0))$ are a family of
isometric deformations. It follows

(2.8) $\sup_{x\in B_{\delta}(o,0)}R(x,0)=\sup_{x\in
B_{\delta}(o,t)}R(x,t),\mbox{\quad}\forall\mbox{\ }\delta>0.$

Hence, combining $(\ref{T1-1})$ and $(\ref{T1-2})$, we get

$\sup_{x\in B_{r}(o,0)}R(x,0)=\sup_{x\in
B_{\delta}(o,0)}R(x,0),\mbox{\quad}\forall\mbox{\ }r>\delta>0.$

Let $\delta\rightarrow 0$, we derive

$\sup_{x\in B_{r}(o,0)}R(x,0)=R(o,0),\mbox{\quad}\forall\mbox{\ }r>0.$

This proves the theorem. ∎

##  3\. Expanding Kähler-Ricci solitons

In this section, we prove both Theorem 1.3 and Theorem 1.4 in case of
expanding Kähler-Ricci solitons. Theorem 1.4 is a consequence of Theorem 1.3
by the following lemma.

######  Lemma 3.1.

Let $(M,g)$ be an expanding gradient Kähler-Ricci soliton which satisfies
(1.5) in Theorem 1.4. Then there exists a function $\varepsilon^{\prime}(r)$ (
$\varepsilon^{\prime}(r)\to 0$ as $r\to\infty$) such that

$\frac{1}{{\rm vol}(B_{r}(p))}\int_{B_{r}(p)}R{\rm
dv}\leq\frac{\varepsilon^{\prime}(r)}{1+r^{2}}.$

###### Proof.

Note that an expanding Ricci soliton with nonnegative Ricci curvature has
maximal volume growth (cf. [7] or [12]). Namely, there exists a uniform
constant $\delta>0$ such that

$\text{vol}(B_{r}(p))\geq\delta r^{2n}.$

On the other hand, by the volume comparison theorem, we have

$\text{vol}(\partial B_{r}(p))\leq n\frac{\text{vol}(B_{r}(p))}{r}\leq
Cr^{2n-1},$

where $C$ is a uniform constant. Thus

$\displaystyle\frac{1}{\text{vol}(B_{r}(p))}\int_{B_{r}(p)}R{\rm dv}=$
$\displaystyle\frac{1}{\text{vol}(B_{r}(p))}\int_{0}^{r}{\rm ds}\int_{\partial
B_{s}(p)}R{\rm d\sigma}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\delta
r^{2n}}\int_{0}^{r}\frac{\varepsilon(s)}{1+s^{2}}\text{vol}(\partial
B_{s}(p)){\rm ds}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\delta
r^{2n}}\int_{0}^{r}C(s+1)^{2n-3}\varepsilon(s){\rm ds}$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon^{\prime}(r)}{1+r^{2}},$

where $\varepsilon^{\prime}(r)\rightarrow 0$ as $r\rightarrow 0$. ∎

###### Proof of Theorem 1.3 (the expanding case).

Ricci-Flat: Let $\phi_{t}$ be a family of diffeomorphisms generated by
$-\nabla f$. Let $g(t)=\phi_{t}^{*}g$ and $\widehat{g}(\cdot,t)=tg(\cdot,\ln
t)$. Then $\widehat{g}(\cdot,t)$ satisfies

(3.1) $\left\\{\begin{aligned} \frac{\partial}{\partial
t}\widehat{g}_{i\bar{j}}(x,t)&=-\widehat{R}_{i\bar{j}}(x,t)\\\
\widehat{g}_{i\bar{j}}(x,1)&=g_{i\bar{j}}(x).\end{aligned}\right.$

Let
$F(x,t)=\ln\det(\widehat{g}_{i\bar{j}}(x,t))-\ln\det(\widehat{g}_{i\bar{j}}(x,1))$.
By $(\ref{kr-flow})$, it is easy to see

$\displaystyle F(x,t)=-\int_{1}^{t}\widehat{R}(x,s)\rm{ds}\leq 0.$

Since

$t\widehat{R}(o,t)=R(o,\ln t)=R(o,0),~{}\text{and}~{}t\widehat{R}(x,t)=R(x,\ln
t)\leq R(o,0),$

where $o$ is the equilibrium point of $M$, by Theorem 1.2, $F$ is uniformly
bounded on $x$. Moreover, we have

(3.2) $\displaystyle M(t)\doteq-\inf_{x\in M}F(x,t)=R(o,0)\ln t.$

In the following, we shall estimate the upper bound of $M(t)$ by using the
Green integration as in [24] (also see [18]).

By a direct computation, we have

(3.3)
$\Delta_{1}F(x,t)=\widehat{R}(x,1)-g^{i\bar{j}}(x,1)\widehat{R}_{i\bar{j}}(x,t)\leq\widehat{R}(x,1)+\frac{\partial}{\partial
t}e^{F(x,t)},$

where the Lapalace $\Delta_{1}$ is w.r.t $\widehat{g}(x,1)$. Let $G_{r}(x,y)$
be a positive Green’s function with zero boundary value w.r.t
$\widehat{g}(x,1)$ on $\hat{B}_{r}(x,1)$. Note that

$\int_{\hat{B}_{r}(x_{0},1)}\frac{\partial G_{r}(x_{0},y)}{\partial\nu}{\rm
ds}=-1\text{ and}~{}\frac{\partial G_{r}(x_{0},y)}{\partial\nu}\leq 0.$

By integrating (3.3) on both sides, we have

$\displaystyle\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)(1-e^{F(x,t)}){\rm dv}$
$\displaystyle\leq$ $\displaystyle\int_{1}^{t}{\rm
ds}\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)(-\Delta_{1}F(x,s)){\rm dv}$
$\displaystyle+t\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}$ $\displaystyle=$
$\displaystyle\int_{1}^{t}\Big{(}F(x_{0},s)+\int_{\hat{B}_{r}(x_{0},1)}\frac{\partial
G_{r}(x_{0},y)}{\partial\nu}F(x,s){\rm dv}\Big{)}{\rm ds}$
$\displaystyle+t\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}$ (3.4) $\displaystyle\leq$ $\displaystyle
t\Big{(}M(t)+\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}\Big{)}.$

On the other hand, by the Green function estimate (cf. Lemma 1.1 in [25]),

$G_{r}(x,y)\geq
C_{1}^{-1}\int_{d(x,y)}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x,1))}{\rm
ds},\mbox{\quad}\forall\mbox{\ }y\in\hat{B}_{\frac{r}{5}}(x,1),$

where $C_{1}$ is a uniform constant, we get

$\displaystyle\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)(1-e^{F(x,t)}){\rm dv}$
$\displaystyle\geq$ $\displaystyle
C_{1}^{-1}\int_{0}^{\frac{r}{5}}\Big{(}\int_{\tau}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x_{0},1))}{\rm
ds}\Big{)}\Big{(}\int_{\partial\hat{B}_{\tau}(x_{0},1)}(1-e^{F(x,t)}){\rm
d\sigma}\Big{)}{\rm d\tau}$ $\displaystyle\geq$ $\displaystyle
C_{1}^{-1}\int_{0}^{\frac{r}{5}}\Big{(}\int_{\frac{r}{5}}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x_{0},1))}{\rm
ds}\Big{)}\Big{(}\int_{\partial\hat{B}_{\tau}(x_{0},1)}(1-e^{F(x,t)}){\rm
d\sigma}\Big{)}{\rm d\tau}$ $\displaystyle\geq$
$\displaystyle\frac{C_{2}^{-1}r^{2}}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}(1-e^{F(x,t)}){\rm
dv}$ $\displaystyle\geq$
$\displaystyle\frac{eC_{2}^{-1}r^{2}}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}\frac{-F(x,t)}{1-F(x,t)}{\rm
dv}.$

It follows

$\displaystyle\frac{r^{2}}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}(-F(x,t)){\rm
dv}$ $\displaystyle\leq
C_{3}(1+M(t))\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)(1-e^{F(x,t)}){\rm dv}$

Hence by (3), we derive

$\displaystyle\frac{r^{2}}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}(-F(x,t)){\rm
dv}$ (3.5) $\displaystyle\leq
C_{3}t(1+M(t))\Big{(}M(t)+\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}\Big{)}.$

By $(\ref{T2-1})$, we have $\Delta_{1}(-F(x,t))\geq-\hat{R}(x,1)$. Then by the
mean value inequality (cf. Lemma 2.1 of [18]), we see

$\displaystyle-F(x_{0},t)\leq\frac{C(n)}{\text{vol}(\hat{B}_{\frac{r}{5}}(x_{0},1))}\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}(-F(x,t)){\rm
dv}$ (3.6)
$\displaystyle+\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}G_{\frac{r}{5}}(x_{0},y)\widehat{R}(y,1){\rm
dv}.$

Hence, to get an upper bound of $-F(x_{0},t)$, by $(\ref{inequality-2})$ and
$(\ref{inequality-3})$, we shall estimate the integral
$\int_{\hat{B}_{\frac{r}{5}}(x_{0},1)}G_{\frac{r}{5}}(x_{0},y)\widehat{R}(y,1)\rm{dv}$.

Recall the Li-Yau’s estimate for the Green function: There exits a positive
Green’s function $G(x,y)$ such that (cf. Theorem 5.2 in [16])

(3.7) $\displaystyle C(n)^{-1}\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm
dt}}{\text{vol}(\hat{B}_{\sqrt{t}}(x))}\leq G(x,y)\leq
C(n)\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm
dt}}{\text{vol}(\hat{B}_{\sqrt{t}}(x))}.$

Then

$\displaystyle\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}\leq\int_{0}^{r}{\rm
ds}\int_{\partial\hat{B}_{s}(x_{0},1)}G(x_{0},y)\widehat{R}(y,1){\rm d\sigma}$
$\displaystyle\leq$ $\displaystyle C(n)\int_{0}^{r}{\rm
ds}\Big{(}\int_{\partial\hat{B}_{s}(x_{0},1)}\widehat{R}(y,1){\rm
d\sigma}\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm
dt}}{\text{vol}(\hat{B}_{\sqrt{t}}(x))}\Big{)}$ $\displaystyle=$
$\displaystyle C(n)\Big{(}\int_{r^{2}}^{+\infty}\frac{{\rm
dt}}{\text{vol}(\hat{B}_{\sqrt{t}}(x))}\int_{\hat{B}_{r}(x_{0},1)}\widehat{R}{\rm
dv}$ (3.8) $\displaystyle+$
$\displaystyle\int_{0}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x_{0},1))}\int_{\hat{B}_{s}(x_{0},1)}\widehat{R}{\rm
dvds}\Big{)},\mbox{\quad}\forall\mbox{\ }r>0.$

Since $(M,g)$ has the maximal volume growth, there exists a uniform constant
$\delta>0$ such that

$\frac{\text{vol}(B_{s}(x))}{\text{vol}(B_{t}(x))}\geq\delta\Big{(}\frac{s}{t}\Big{)}^{2n},~{}\forall~{}s\geq
t\geq c_{0}.$

It follows

$\displaystyle\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm
dt}}{\text{vol}(B_{\sqrt{t}}(x))}\leq
C_{4}\frac{d^{2}(x,y)}{\text{vol}(B_{d(x,y)}(x))}\mbox{,\quad}\forall\mbox{\
}d(x,y)\geq c_{0}.$

Hence, we get from (3),

$\displaystyle\int_{\hat{B}_{r}(x_{0},1)}G_{r}(x_{0},y)\widehat{R}(y,1){\rm
dv}$ $\displaystyle\leq
C(n)\Big{(}C_{4}\frac{r^{2}}{\text{vol}(\hat{B}_{r}(x_{0},1)}\int_{\hat{B}_{r}(x_{0},1)}\widehat{R}{\rm
dv}$ (3.9)
$\displaystyle+\int_{0}^{r}\frac{s}{\text{vol}(\hat{B}_{s}(x_{0},1)}\int_{\hat{B}_{s}(x_{0},1)}\widehat{R}{\rm
dvds}\Big{)},\mbox{\quad}\forall\mbox{\ }r>0.$

By the volume comparison theorem and the condition (1.5), we have

$\displaystyle\frac{1}{\text{vol}(B_{r}(o))}\int_{B_{r}(o)}R(x){\rm dv}\leq$
$\displaystyle\frac{\text{vol}(B_{r+d}(p))}{\text{vol}(B_{r}(o))}\cdot\frac{1}{\text{vol}(B_{r+d}(p))}\int_{B_{r+d}(p)}R(x)\rm{dv}$
$\displaystyle\leq$
$\displaystyle\frac{\text{vol}(B_{r+2d}(o))}{\text{vol}(B_{r}(o))}\cdot\frac{\varepsilon(r+d)}{1+(r+d)^{2}}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{r+2d}{r}\Big{)}^{2n}\frac{\varepsilon(r+d)}{1+(r+d)^{2}},$

where $d=d(o,p)$. Then there exists another function $\varepsilon^{\prime}(r)$
($\varepsilon^{\prime}(r)\to 0$ as $r\to\infty$) such that

$\frac{1}{\text{vol}(B_{r}(o))}\int_{B_{r}(o)}R(x){\rm
dv}\leq\frac{\varepsilon^{\prime}(r)}{1+r^{2}}.$

Thus inserting the above inequality into (3), we derive at $x_{0}=o$,

(3.10) $\int_{\hat{B}_{r}(o,1)}G_{r}(o,y)\widehat{R}(y,1){\rm
dv}\leq\varepsilon^{\prime\prime}(r)+\varepsilon^{\prime\prime}(r)\ln(1+r^{2}),$

where $\varepsilon^{\prime\prime}(r)\to 0$ as $r\to\infty$.

Combining $(\ref{inequality-1})$, $(\ref{inequality-2})$ and
$(\ref{inequality-5})$, it is easy to see

$\displaystyle-F(o,t)\leq
r^{-2}C(n)t(M(t)+1)\Big{(}M(t)+C_{5}\varepsilon^{\prime\prime}(r)+C_{6}\varepsilon^{\prime\prime}(r)\ln(1+r^{2})\Big{)}$
$\displaystyle+\Big{(}C_{7}\varepsilon^{\prime\prime}(r)+C_{8}\varepsilon^{\prime\prime}(r)\ln(1+r^{2})\Big{)},\mbox{\quad}\forall\mbox{\
}r>0.$

Note that $-F(o,t)=M(t)=R(o,0)\ln t$. Then by taking $r=t$, we obtain

$\displaystyle R(o,0)\ln t\leq$ $\displaystyle
C_{9}\varepsilon^{\prime\prime}(t)+C_{10}\varepsilon^{\prime\prime}(t)\ln(1+t)+C_{11}\frac{\ln
t}{t},\mbox{\quad}\forall\mbox{\ }t\geq 1.$

Dividing by $\ln t$ on both sides of the above inequality and letting
$t\rightarrow\infty$, we deduce $R(o,0)=0$. Hence we prove that $g$ is Ricci
flat

Flatness: We shall further prove that $g$ is a flat metric on
$\mathbb{C}^{n}$. Note that

(3.11) $\displaystyle g=\text{hess}f$

since $g$ is Ricci flat. Then $f$ is strictly convex and $f$ attains the
minimum at $o$. By a direct computation, we have

$\displaystyle\langle\nabla f,X\rangle=(\nabla{\rm d}f)(\nabla
f,X)=\langle\nabla_{\nabla f}\nabla f,X\rangle,\mbox{\ }\forall\mbox{\
}X\in\Gamma^{\infty}(TM).$

It follows

$\nabla_{\nabla f}\nabla f=\nabla f.$

Thus

$\nabla_{\frac{\nabla f}{|\nabla f|}}\Big{(}\frac{\nabla f}{|\nabla
f|}\Big{)}=\frac{1}{|\nabla f|}(\frac{\nabla_{\nabla f}\nabla f}{|\nabla
f|}-\frac{\langle\nabla_{\nabla f}\nabla f,\nabla f\rangle}{|\nabla
f|^{3}}\nabla f)=0,~{}x\in M\setminus\\{o\\}.$

This implies that any integral curve generated by $\frac{\nabla f}{|\nabla
f|}$ ($x\in M\setminus\\{o\\}$) is geodesic.

Let $\phi_{t}$ and $\varphi_{t}$ be one-parameter diffeomorphisms groups
generated by $-\nabla f$ and $-\frac{\nabla f}{|\nabla f|}$, respectively.
Then as in the proof of (2.6), we have $d(\phi_{t}(x),o)=e^{-t}d(x,o)$. Thus
$\langle\nabla f,\nabla r\rangle=-\frac{{\rm d}}{{\rm dt}}d(\phi_{t}(x),o)>0$.
This shows that $\varphi_{s}(x)$ is a geodesic curve from $x$ to $o$. Let
$\gamma(s)=\varphi_{d(x,o)-s}(x)$. Then $\gamma(s)$ is a minimal geodesic
curve from $o$ to $x$ as long as $\text{dist}(o,x)\leq r_{0}<<1.$ Moreover, we
have

$\displaystyle\left\\{\begin{aligned} &\frac{{\rm d^{2}}}{{\rm
ds^{2}}}f(\gamma(s))=1,\\\ &\frac{{\rm d}}{{\rm ds}}f(\gamma(s))=|\nabla
f|\rightarrow 0,\mbox{\ as\ }s\rightarrow 0,\\\
&f(\gamma(0))=f(o)=0.\end{aligned}\right.$

Therefore, we deduce

$\displaystyle f(x)=\frac{1}{2}r^{2}(x),~{}\text{if}~{}r(x)\leq r_{0}.$

In particular,

(3.12) $\displaystyle|\nabla f(x)|=r.$

We claim that $g$ is flat on $B_{r_{0}}(o)$. Since $\partial B_{r_{0}}(o)$ is
diffeomorphic to $\mathbb{S}^{2n-1}$, we can choose an orthonormal basis
$\\{e_{1},\cdots,e_{2n-1}\\}$ on $\partial B_{r_{0}}(o)$. Let
$X_{i}(\varphi_{t}(x))=(\varphi_{t})_{*}e_{i}$ for $x\in\partial
B_{r_{0}}(o)$, $1\leq i\leq 2n-1$. Then $\\{\nabla r=\frac{\nabla f}{|\nabla
f|},X_{1},\cdots,X_{2n-1}\\}$ is a global frame on
$B_{r_{0}}(o)\setminus\\{o\\}$. Clearly, $\mathscr{L}_{\nabla r}X_{i}=0$,
$1\leq i\leq 2n-1$. Thus by (3.12) and (3.11), it follows

$\displaystyle\frac{\partial}{\partial r}\langle\nabla r,X_{i}\rangle=$
$\displaystyle\mathscr{L}_{\nabla r}\langle\nabla r,X_{i}\rangle$
$\displaystyle=$ $\displaystyle(\mathscr{L}_{\nabla r}g)(\nabla
r,X_{i})+\langle\mathscr{L}_{\nabla r}\nabla r,X_{i}\rangle+\langle\nabla
r,\mathscr{L}_{\nabla r}X_{i}\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{r}{\rm Hess}f(\nabla r,X_{i})$ $\displaystyle=$
$\displaystyle\frac{2}{r}\langle\nabla r,X_{i}\rangle.$

Since $\langle\nabla r,X_{i}\rangle|_{r=r_{0}}=0$, we get $\langle\nabla
r,X_{i}\rangle=0$ for any $x\in B_{r_{0}}(o)\setminus\\{o\\}$. Similarly, we
have

(3.13) $\left\\{\begin{aligned} \frac{\partial}{\partial r}\langle
X_{i},X_{j}\rangle=&\frac{2}{r}\langle X_{i},X_{j}\rangle,\\\ \langle\nabla
r,X_{i}\rangle|_{r=r_{0}}&=\delta_{ij}.\end{aligned}\right.$

Consequently, $\langle X_{i},X_{j}\rangle=\frac{r^{2}}{r_{0}^{2}}\delta_{ij}$
for any $x\in B_{r_{0}}(o)\setminus\\{o\\}$. Hence,

$g={\rm dr}\otimes{\rm dr}+\frac{r^{2}}{r_{0}^{2}}\sum_{i,j=1}^{2n-1}{\rm
d\theta^{i}}\otimes{\rm d\theta^{j}},$

where $\\{dr,\theta^{1},\cdots,\theta^{2n-1}\\}$ are the corresponding coframe
of $\\{\nabla r=\frac{\nabla f}{|\nabla f|},X_{1},$
$\cdots,X_{2n-1}\\}$. This proves that $g$ is isometric to an Euclidean metric
on $B_{r_{0}}(o)$. Therefore, $g$ is flat on $B_{r_{0}}(o)$. The claim is
true.

At last, we show that $g$ is globally flat. Since $\phi_{t}$ is an isometric
diffeomorphism from $(B_{r_{0}}(o,t),g(x,t))$ to $(B_{r_{0}}(o,0),g(x,0))$. We
see that $(B_{r_{0}}(o,t),g(x,t))$ is flat by the above claim. On the other
hand, by the flow $(\ref{normalized-ricci-flow})$ and the fact that $g$ is
Ricci-flat, we have $g(x,t)=e^{-2t}g(x,0)$. Hence, $(B_{r_{0}}(o,t),g(x,0))$
is also flat. Since $M$ is exhausted by $B_{r_{0}}(o,t)$ as
$t\rightarrow\infty$ according to Lemma 2.2, we see that $g$ is globally flat
and $M$ is simply connected. As a consequence, $(M,g)$ is isometric to
$\mathbb{C}^{n}$. ∎

##  4\. Steady Kähler-Ricci solitons

In this section, we deal with steady gradient Ricci solitons $(M^{n},g,f)$. As
in Section 3, we let $\phi_{t}$ be a family of diffeomorphisms generated by
$-\nabla f$ and $g(\cdot,t)=\phi_{t}^{*}g$. Then $g(\cdot,t)$ satisfies

(4.1) $\frac{\partial}{\partial t}g_{ij}=-2R_{ij}.$

It turns

$R_{ij}(t)=\nabla_{i}\nabla_{j}f(t),$

where $f(t)=\phi_{t}^{*}f$ and $\nabla$ is taken w.r.t $g(t)$. Hence by the
Bianchi identity (2.2), one can obtain

(4.2) $R+|\nabla f|^{2}=\text{const}.$

This shows that the scalar curvature of $g$ is uniformly bounded.

Analogous to Lemma 2.1, we have

######  Lemma 4.1.

$\frac{\partial}{\partial t}R=2{\rm Ric}(\nabla f(t),\nabla f(t))$.

In general, we do not know whether a steady gradient Ricci soliton admits an
equilibrium point. However, we can still prove Rigidity Theorem 1.3 in case of
steady gradient Kähler-Ricci solitons by using the fact of boundedness of
scalar curvature.

###### Proof of Theorem 1.3 (the steady case).

Since $(M,g)$ is Kählerian, we may rewrite (4.1) as,

(4.3) $\left\\{\begin{aligned} \frac{\partial}{\partial
t}g_{i\bar{j}}(x,t)&=-R_{i\bar{j}}(x,t)\\\
g_{i\bar{j}}(x,0)&=g_{i\bar{j}}(x).\end{aligned}\right.$

In order to get the estimate for the Green function as in Section 3, we use a
trick in [24] to consider a product space $\widehat{M}=M\times\mathbb{C}^{2}$
with a product metric $\widehat{g}=g+dw^{1}\wedge d\bar{w}^{1}+dw^{2}\wedge
d\bar{w}^{2}$. Then $\widehat{g}(x,t)=g(x,t)+dw^{1}\wedge
d\bar{w}^{1}+dw^{2}\wedge d\bar{w}^{2}$ is a solution of (4.3) on
$\widehat{M}$ with the initial metric $\widehat{g}$.

It was proved by Shi that for any $s>t$ and $B_{t}(x)\subset
B_{s}(x)\subset(\widehat{M},\widehat{g})$ (cf. Section 6 in [24]), it holds

$\frac{\text{vol}(B_{s}(x))}{\text{vol}(B_{t}(x))}\geq
C_{0}^{-1}\Big{(}\frac{s}{t}\Big{)}^{4}.$

Then

$\int_{d^{2}(x,y)}^{+\infty}\frac{{\rm dt}}{\text{vol}(B_{\sqrt{t}}(x))}\leq
C_{0}\frac{d^{2}(x,y)}{\text{vol}(B_{d(x,y)}(x))}\mbox{,\quad}\forall\mbox{\
}d(x,y)\geq c_{0}.$

Thus by the Li-Yau estimate [16], there exists a global Green’s function $G$
on $(\widehat{M},\widehat{g})$ which satisfies (3.7).

Let
$F(x,t)=\ln\det(\widehat{g}_{i\bar{j}}(x,t))-\ln\det(\widehat{g}_{i\bar{j}}(x,0))$.
By (4.3), it is easy to see

(4.4) $F(x,t)=-\int_{0}^{t}\widehat{R}(x,s){\rm ds}.$

Then by Lemma 4.1, we have

$t\widehat{R}(x,0)\leq-F(x,t)\leq C_{0}t,$

where $C_{0}=\sup_{\widehat{M}}\widehat{R}(x,t).$ Thus as in Section 3, to
prove that $R\equiv 0$, we shall give a growth estimate of $-F(x,t)$ on $t$.

Fix an arbitrary point $x_{0}\in M$. For convenience, we denote a $r$-geodesic
ball $B_{r}(x_{0},t)$ of $(\widehat{M},\widehat{g}(t))$ centered at
$(x_{0},0,0)$. As in Section 3, by using the Green formula, we can estimate

$\displaystyle-F(x_{0},t)\leq\frac{C_{1}t(1+M(t))}{r^{2}}$
$\displaystyle\Big{(}M(t)+\int_{B_{r}(x_{0},0)}G(x_{0},y)\widehat{R}(y,0){\rm
dv}\Big{)}$ (4.5)
$\displaystyle+\int_{B_{\frac{r}{5}}(x_{0},0)}G(x_{0},y)\widehat{R}(y,0){\rm
dv}.$

Moreover,

$\displaystyle\int_{B_{r}(x_{0},0)}G_{r}(x_{0},y)\widehat{R}(y,0){\rm dv}$
$\displaystyle\leq
C(n)\Big{(}\frac{r^{2}}{\text{vol}(B_{r}(x_{0},0))}\int_{B_{r}(x_{0},1)}\widehat{R}{\rm
dv}$ (4.6)
$\displaystyle+\int_{0}^{r}\frac{s}{\text{vol}(B_{s}(x_{0},0))}\int_{B_{s}(x_{0},0)}\widehat{R}\rm{dvds}\Big{)},~{}\forall~{}r>0.$

On the other hand, by the volume comparison together with $(\ref{condition-
steady})$, we have

(4.7) $\frac{1}{\text{vol}(B_{r}(x_{0},0)}\int_{B_{r}(x_{0},0)}\widehat{R}{\rm
dv}\leq\frac{C}{\text{vol}(B_{r}(x_{0}))}\int_{B_{r}(x_{0})}R{\rm
dv}\leq\frac{\varepsilon_{1}(r)}{1+r},$

where the function $\varepsilon_{1}(r)\rightarrow 0$ as $r\rightarrow\infty$.
Thus combining (4) and (4.7), we get from (4),

$\displaystyle-F(x_{0},t)\leq
r^{-2}C(n)t(C^{\prime}t+1)(C^{\prime}t+r\varepsilon_{2}(r))+r\varepsilon_{2}(r),\mbox{\quad}\forall\mbox{\
}r>0.$

where $\varepsilon_{2}(r)\rightarrow 0$ as $r\rightarrow\infty$. Consequently,

(4.8) $\displaystyle tR(x_{0},0)\leq
r^{-2}C(n)t(C^{\prime}t+1)(C^{\prime}t+r\varepsilon_{2}(r))+r\varepsilon_{2}(r),\mbox{\quad}\forall\mbox{\
}r>0.$

Now we choose a monotonic $\varepsilon_{3}(r)$ such that
$\varepsilon_{3}(r)\rightarrow 0$ and
$\frac{\varepsilon_{2}(r)}{\varepsilon_{3}(r)}\rightarrow 0$ as
$r\rightarrow\infty$. Let $r=t\varepsilon_{3}^{-1}(t)$. Then by (4.8 ), we get

$tR(x_{0},0)\leq
C_{1}\varepsilon_{3}^{2}(t)(C^{\prime}t+t\frac{\varepsilon_{2}(t\varepsilon_{3}^{-1}(t))}{\varepsilon_{3}(t\varepsilon_{3}^{-1}(t))})+t\frac{\varepsilon_{2}(t\varepsilon_{3}^{-1}(t))}{\varepsilon_{3}(t\varepsilon_{3}^{-1}(t))},\mbox{\quad}\forall\mbox{\
}t\gg 1.$

By dividing by $t$ on both sides of the above inequality and then letting
$t\rightarrow\infty$, it is easy to see that $R(x_{0},0)=0$. Since $x_{0}$ is
an arbitrary point, we prove that $R(x)\equiv 0$. ∎

By Theorem 1.3 , we can finish the proof of Theorem 1.4.

###### Proof of Theorem 1.4 (the steady case).

Since $(M,g)$ is a complete non-
compact manifold with nonnegative Ricci curvature, the volume growth of $g$ is
at least linear. Then by (1.6), it is easy to see that the average curvature
condition (1.4) is satisfied in Theorem 1.3 as in the proof of Lemma 3.1.
Hence by Theorem 1.3, we get Theorem 1.4 immediately. ∎

##  5\. Boundedness of scalar curvature–II

In this section, we prove the existence and uniqueness of equilibrium point
for the steady Kähler-Ricci soliton $(M^{n},g,f)$ in Theorem 1.2. As a
consequence, the maximum of scalar curvature of $g$ can be attained.

###### Proof of Theorem 1.2 (the steady case).

Existence: Let $g(\cdot,t)=\phi_{t}^{*}g$ be a family of steady solitons
generated by $-\nabla f$. Then $g(\cdot,t)$ is an eternal solution of (4.3).
Since $g(\cdot,t)$ has uniformly positive holomorphic bisectional curvature in
space time $M\times(-\infty,\infty)$, we apply Theorem 2.1 in [11] to see that
there exists a sequence of solutions
$g_{\alpha}(\cdot,t)=g(\cdot,t_{\alpha}+t)$ on $\Phi_{\alpha}(D(r))(\subset
M)$ such that $\Phi_{\alpha}^{*}(g_{\alpha}(\cdot,t))$ converge to a smooth
solution $h(x,t)$ of (4.3) uniformly and smoothly on a compact subset $D(r)$
for any $t\in(-1,\infty)$, where $D(r)$ is an Euclidean ball centered at the
origin with radius $r$ and $\Phi_{\alpha}$ are local biholomorphisms from
$D(r)$ to $M$. Moreover, by using the Cao’s argument in [3], it was proved
that $h(x,t)$ is generated by a steady Kähler-Ricci soliton $(D(r),h,f^{h})$
with $\nabla f^{h}(o)=0$. Namely, $h(x,t)$ satisfies

$R_{i\bar{j}}(h(t))=\nabla_{i}\nabla_{\bar{j}}f^{h}(t),\mbox{\quad}\nabla_{i}\nabla_{j}f^{h}(t)=0,$

where $f^{h}(t)$ are induced functions of $f^{h}$ and $\nabla f^{h}(t)$ vanish
at the origin for any $t\in(-1,\infty)$.

On the other hand, similar to (2.5), we have for solitons $(M,g(t),f(t))$,

$R_{,i}(t)+R_{i\bar{j}}(t)\nabla_{\bar{j}}f(t)=0.$

Then $\nabla f(t)$ is determined by the curvature tensor. Define a sequence of
a family of holomorphic vector fields $V(t_{\alpha})$ on $D(r)$ by

$\Phi_{\alpha}^{*}R_{,i}(t_{\alpha})+\Phi_{\alpha}^{*}R_{i\bar{j}}(t_{\alpha})V(t_{\alpha})_{j}=0.$

Clearly, $(\Phi_{\alpha})_{*}V(t_{\alpha})=\nabla f(t_{\alpha})$. By the
convergence of $g_{\alpha}(\cdot,t)$, holomorphic vector fields
$V(t_{\alpha})$ converge to $\nabla f^{h}$ in $C^{\infty}$-topology on $D(r)$
for any $t\in(-1,\infty)$. Since the eigenvalues of ${\rm Ric}(h(t))$ are
positive at $0$ by Proposition 2.2 in [11], the integral curves of
$-\nabla^{h}f^{h}$ will converge to $0$ in $D(r)$ when $r$ is sufficiently
small by the soliton equation. By the convergence of $V(t_{\alpha})$, the
integral curve of $-V(t_{\alpha})$ will also converge to a point $q$ in
$D(r_{1})$ for some $r_{1}<r$ when $\alpha$ is large enough (cf. Page 9 of
[12]). As a consequence, $q$ is a zero point of $V(t_{\alpha})$ in $D(r_{1})$.
This proves that there exists a zero point of $\nabla f(t_{\alpha})$ in $M$
for each $\alpha$ since $\Phi_{\alpha}^{*}$ is a local biholomorphism.

Uniqueness: Suppose that $p$ and $q$ are two equilibrium points. Then
$d_{0}(p,q)=d_{t}(p,q)$. Choose $l>0$ such that $q\in B_{l}(p,0)$. Note that
$\phi_{t}:(M^{n},g(t))\rightarrow(M^{n},g(0))$ are a family of isometric
deformations. Thus

$C=\inf_{x\in B_{l}(p,t)}\mu_{1}(x,t)=\inf_{x\in
B_{l}(p,0)}\mu_{1}(x,0)>0,\mbox{\ }\forall x\in B_{l}(p,0),$

where $\mu_{1}(x,t)$ is the smallest eigenvalue of $\text{Ric}(x,t)$ w.r.t
$g(x,t)$. Since the metric is decreasing along the flow, we see that
$B_{l}(p,0)\subset B_{l}(p,t)$. Hence by (4.1), we get

$\displaystyle\frac{{\rm
d}|v_{x}|^{2}_{t}}{\rm{dt}}\leq-\mu_{1}(x,t)|v_{x}|^{2}_{t}\leq-C|v_{x}|^{2}_{t},~{}\forall~{}t\geq
0,$

where $x\in B_{l}(p,0)$ and $v_{x}\in T_{x}^{(1,0)}M$. Therefore, if we let
$\gamma(s)$ be a minimal geodesic curve connecting $p$ and $q$ with an arc-
parameter $s$ w.r.t the metric $g(x,0)$ in $B_{l}(p,0)$, we deduce

$\displaystyle d_{t}(p,q)\leq\int_{0}^{d}|\gamma^{\prime}(s)|_{t}{\rm
ds}\leq\int_{0}^{d}|\gamma^{\prime}(s)|_{0}e^{-Ct}{\rm ds}=d_{0}(p,q)e^{-Ct}.$

Letting $t\to\infty$, we see that $d_{t}(p,q)=d_{0}(p,q)=0$. This proves that
$p=q.$ ∎

######  Proposition 5.1.

Let $(M^{n},g,f)$ be a simply connected complete non-comp-act steady gradient
Kähler-Ricci soliton with nonnegative bisectional curvature. Suppose that $g$
satisfies

(5.1) $\displaystyle R(x)\leq\frac{C}{1+r(x)^{1+\epsilon}},$

for some $\epsilon>0,C$. Then $(M,g)$ is isometric to $\mathbb{C}^{n}$.

###### Proof.

We suffice to show that $g$ is Ricci flat. On the contrary, we may assume that
the Ricci curvature of $g(\cdot,t)$ is positive everywhere by a dimension
reduction theorem of Cao for Kähler-Ricci flow on a simply connected complete
Kähler manifold with nonnegative bisectional curvature [4], where
$g(\cdot,t)=\phi_{t}^{*}g$ is the generated solution of (4.3) as in Section 4.
Let $o$ be the unique equilibrium point of $g$ according to Theorem 1.2. In
the following we use an argument of Hamilton in [13] to show that there exists
a pointedwise backward limit $g_{\infty}(x)$ of $g(x,t)$ on
$M\setminus\\{o\\}$ and $(M\setminus\\{o\\},g_{\infty})$ is a complete flat
Riemannian manifold.

Since

$R(x)+|\nabla f|^{2}=R(o),$

by (5.1), we see

$\lim_{d(x,o)\rightarrow\infty}|\nabla f|^{2}(x)=R(o)>0.$

Note the equilibrium point is unique. It follows

$C_{\delta}=\inf_{M\setminus B_{\delta}(o)}|\nabla
f|^{2}>0,\mbox{\quad}\forall\mbox{\ }\delta>0.$

This implies

$d(\phi_{t}(x),o)\geq C_{\delta}|t|\mbox{\quad}\forall\mbox{\ }x\in M\setminus
B_{\delta}(o)\mbox{,\ }t\leq 0.$

Hence by (5.1), we get from equation (4.3),

$\displaystyle 0$ $\displaystyle\leq-\frac{\partial}{\partial t}g(x,t)\leq
R(g(x,t))g(x.t)$
$\displaystyle\leq\frac{C}{1+d^{1+\epsilon}(\phi_{t}(x),o)}g(x,t)\leq\frac{C_{\delta}^{\prime}}{1+|t|^{1+\epsilon}}g(x,t).$

Therefore, we derive

(5.2) $g(x,0)\leq g(x,t_{1})\leq g(x,t_{2})\leq C_{\delta}^{\prime}g(x,0),$

for any $x\in M\setminus B_{\delta}(o,0)$ and $-\infty<t_{2}\leq t_{1}\leq 0$.

By (5.2) and Shi’s higher order estimate for curvatures, we see that $g(x,t)$
converge locally to a limit Kähler metric $g_{\infty}(x)$ on
$M\setminus\\{o\\}$ as $t\to-\infty$. Clearly, $g_{\infty}(x)$ is Ricci-flat
since

$0=\lim_{t\to-\infty}-\frac{\partial}{\partial
t}g(x,t)=\lim_{t\to-\infty}\text{Ric}(g(\cdot,t))=\text{Ric}(g_{\infty}).$

Consequently, $g_{\infty}$ is flat. Moreover, $g_{\infty}$ is a complete,
because

$\displaystyle\lim_{x^{\prime}\to
o}d_{g_{\infty}}(x,x^{\prime})=\lim_{t\rightarrow-\infty}d_{g(\cdot,t)}(x,o)=\lim_{t\rightarrow-\infty}d_{g}(\phi_{t}(x),o)=\infty,$

where $x\in M\setminus\\{o\\}$. On the other hand, it was proved by Chau and
Tam that $M$ is biholomorphic to $\mathbb{C}^{n}$ since $M$ is a a simply
connected complete non-compact steady gradient Kähler-Ricci soliton with
positive Ricci curvature [9]. Thus, $M\setminus\\{o\\}$ is simply connected.
Hence, $M\setminus\\{o\\}$ is also biholomorhic to $\mathbb{C}^{n}$. This is a
contradiction! Therefore, $g$ is Ricci flat and consequently, $(M,g)$ is
isometric to $\mathbb{C}^{n}$.

∎

## References

## References

  * 1 Brendle, S., Rotational symmetry of self-similar solutions to the Ricci flow, Invent. Math. 194 No.3 (2013), 731-764.
  * 2 Bryant, R., Gradient Kähler Ricci solitons, arXiv:math.DG/0407453.
  * 3 Cao, H-D., Limits of solutions to the Kähler-Ricci flow, J. Diff. Geom. 45 (1997),257-272.
  * 4 Cao, H-D., On dimension reduction in the Kähler-Ricci flow, Comm. Anal. Geom. 12, No. 1, (2004), 305-320.
  * 5 Cao, H-D., Chen, B-L and Zhu, X-P., Recent developments on Hamilton’s Ricci flow, Surveys in J. Diff. Geom. 12 (2008), 47-112.
  * 6 Chow, B., Lu, P. and Ni, L., Hamilton’s Ricci flow in: Lectures in Contemporary Mathematics 3 ,Science Press, Beijing $\&$ American Mathematical Society, Providence, Rhode Island (2006).
  * 7 Carrillo, J. and Ni, L., Sharp logarithmic Sobolev Inequalities on solition and applications arXiv.0806.2417.v3.
  * 8 Chau, A. and Tam, L-F., Grandient Kähler-Ricci Solitons and a uniformization conjecture, arXiv:math/0310198v1.
  * 9 Chau, A. and Tam, L-F., A note on the uniformization of gradient K ahler-Ricci solitons, Math. Res. Lett. 12 (2005), no. 1, 19-21.
  * 10 Chau, A. and Tam, L-F., On the complex structure of Kähler manifolds with nonnegative curvature, J. Diff. Geom. 73 (2006), 491-530.
  * 11 Chau, A. and Tam, L-F., Non-negatively curved K ahler manifolds with average quadratic curvature decay, Comm. Anal. Geom. 15 (2007), no. 1, 121-146.
  * 12 Chau, A. and Tam, L-F., On the simply connectedness of nonnegatively curved Kähler manifolds and applications, Trans. Amer. Math. Soc. 363  (2011), 6291-6308.
  * 13 Hamilton, R.S., Three manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982),255-306.
  * 14 Hamilton, R.S., The Harnack estimate for the Ricci flow, J. Diff. Geom. 37 (1993), 225-243.
  * 15 Hamilton, R.S., Formation of singularities in the Ricci flow, Surveys in Diff. Geom. 2 (1995), 7-136.
  * 16 Li, P. and Yau, S-T., On the parabolic kernel of the Schr odinger operator, Acta Math. 156 (1986), 139-168.
  * 17 Naber, A., Noncompact shrinking four solitons with nonnegative curvature, Journal f r die reine und angewandte Mathematik, 645 (2010), 125-153.
  * 18 Ni, L., Kähler-Ricci flow and Poincaré-Lelong equation, Comm. Anal. Geom. 12 No.1 (2004), 111-141.
  * 19 Ni, L., An optimal gap theorem, Invent. Math. 189 No.3 (2012), 737-761.
  * 20 Ni, L., Shi, Y-G and Tam, L-F., Poisson equation, Poincaré-Lelong equation and curvature decay on complete Kähler manifolds, J. Diff. Geom. 57 (2001), 339-388.
  * 21 Perelman, G., Ricci flow with surgery on three-manifolds,
arXiv:math/0303109v1.

  * 22 Petrunin, A. and Tuschmann, W., Asymptotical flatness and cone structure at infinity, Math. Ann. 321 (2001), 775-788.
  * 23 Shi, W-X., Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Diff. Geom. 30 (1989), 223-301.
  * 24 Shi, W-X., Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Diff. Geom. 45 (1997), 94-220.
  * 25 Tam, L-F., Liouville properties of harmonic maps, Math. Res. Lett. 2 (1995), 719-735.
  * 26 Zhang, Z-H., On the Completeness of Gradient Ricci Solitons. Proc. Amer. Math. Soc. 137 (2009), 2755-2759.