# Hamilton-Ivey estimates for gradient Ricci solitons

Pak-Yeung Chan, Zilu Ma, and Yongjia Zhang

###### Abstract

We first show that any $4$-dimensional non-Ricci-flat steady gradient Ricci
soliton singularity model must satisfy $|{\operatorname{Rm}}|\leq cR$ for some
positive constant $c$. Then, we apply the Hamilton-Ivey estimate to prove a
quantitative lower bound of the curvature operator for $4$-dimensional steady
gradient solitons with linear scalar curvatrue decay and proper potential
function. The technique is also used to establish a sufficient condition for a
$3$-dimensional expanding gradient Ricci soliton to have positive curvature.
This sufficient condition is satisfied by a large class of conical expanders.
As an application, we remove the positive curvature condition in a
classification result by [Cho14] in dimension three and show that any
$3$-dimensional gradient Ricci expander $C^{2}$ asymptotic to
$\left(C(\mathbb{S}^{2}),dt^{2}+\alpha t^{2}g_{\mathbb{S}^{2}}\right)$ is
rotationally symmetric, where $\alpha\in(0,1]$ is a constant and
$g_{\mathbb{S}^{2}}$ is the standard metric on $\mathbb{S}^{2}$ with constant
curvature $1$.

##  1 Introduction

A triple $(M^{n},g,f)$ consisting of a connected smooth Riemannian manifold
$(M^{n},g)$ and a function $f\in C^{\infty}(M)$ is called a gradient Ricci
soliton, if the following equation is satisfied for some constant $\kappa$:

$\operatorname{Ric}+\nabla^{2}f=\tfrac{\kappa}{2}g.$ (1.1)

The soliton is called _shrinking_ if $\kappa>0$, _steady_ if $\kappa=0$, and
_expanding_ if $\kappa<0$. By scaling the metric with a constant, we may
always assume that $\kappa\in\\{-1,0,1\\}$. A complete gradient Ricci soliton
induces a self-similar solution to the Ricci flow, called the _canonical
form_. More precisely, if $\Phi_{t}$ is a family of self-diffeomorphisms on
$M$ and $g_{t}$ is a family of metrics given by

$\displaystyle\frac{\partial}{\partial t}\Phi_{t}$
$\displaystyle=\frac{1}{1-\kappa t}\nabla f\circ\Phi_{t},$ (1.2)
$\displaystyle\Phi_{0}$ $\displaystyle=\text{ id},$ $\displaystyle g_{t}$
$\displaystyle=(1-\kappa t)\Phi_{t}^{*}g,$

then $g_{t}$ evolves by the Ricci flow with $g_{0}=g$. When $\kappa>0$,
$\kappa=0$, or $\kappa<0$, the canonical form is _ancient_ , _eternal_ , or
_immortal_ , respectively. The Ricci soliton is an important field of study,
since they arise naturally as rescaled limits of Ricci flows near
singularities. The blow-up limit at a Type I finite-time singularity, or the
backward scaled limit of a Type I ancient solution, is the canonical form of a
Ricci shrinker (see [Per02, Nab10, EMT11, MZ21] and the references therein),
whereas Type II and Type III scaled limits of Ricci flows are closely related
to steady and expanding solitons, respectively (see [Ham93, Ham95, Cao97,
CZ00, Lot07, GZ08]). Moreover, a non-Ricci-flat steady soliton may appear as a
rescaled limit at the spatial infinity of a $4$-dimensional shrinking soliton
singularity model with unbounded curvature [CFSZ20]. Hence, the investigation
of geometric properties of Ricci solitons will shed much light on the
singularity analysis of the Ricci flow.

The classification of 3-dimensional shrinking gradient Ricci solitons has been
completed. Any complete gradient shrinker in dimension three is isometric to
either $\mathbb{R}^{3}$, or a quotient of $\mathbb{S}^{3}$, or a quotient of
$\mathbb{R}\times\mathbb{S}^{2}$ (see [Ive93, Ham95, Per03a, CCZ08, NW08,
Che09, Nab10]). However, the general picture of $3$-dimensional steady
solitons is far from clear. One significant result is due to Brendle [Br13].
He proved a conjecture due to Perelman [Per02], namely, that any nonflat and
noncollapsed 3-dimensional steady gradient Ricci soliton is the Bryant soliton
up to scaling (see also [Br14, Br20]). Deng-Zhu [DZ19, DZ20b] generalized
Brendle’s result and classified steady gradient solitons with at least linear
curvature decay in dimension three (see also [MSW19] and references therein).
One may wonder whether the Bryant soliton and $\mathbb{R}\times\Sigma_{0}$ are
the only non-trivial $3$-dimensional steady gradient solitons, where
$\Sigma_{0}$ denotes Hamilton’s cigar soliton. Indeed, $3$-dimensional steady
solitons are more complicated than their shrinking counterparts. Very
recently, Lai [Lai20] has resolved a conjecture due to Hamilton and
constructed a family of $3$-dimensional steady solitons that are flying wings.
These examples are collapsed, positively curved, and on them the curvature
does not decay uniformly to $0$ at infinity (new noncollapsed examples of
steady solitons in higher dimensions are also constructed in [Lai20]).

One important feature which facilitates the study of shrinking and steady
solitons in dimension three is that these solitons are nonnegatively curved,
which is a consequence due to the celebrated Hamilton-Ivey pinching estimate
[Ive93, Ham95, Che09] (see also [CXZ13]). As a consequence of this estimate,
any $3$-dimensional complete ancient Ricci flow must have nonnegative
sectional curvature. Since the canonical form of a shrinking or steady
gradient Ricci soliton is an ancient Ricci flow, these two types of solitons
are nonnegatively curved.

######  Theorem 1.1.

[Ham95, Ive93, Che09] Any nonflat $3$-dimensional complete shrinking or steady
gradient Ricci soliton must have nonnegative sectional curvature.
Consequently,

$|{\operatorname{Rm}}|\leq CR,$ (1.3)

for some numerical constant $C>0$, where $R$ is the scalar curvature.

In view of Theorem 1.1, two natural questions arise immediately. The first one
is whether or not higher dimensional shrinking or steady solitons have
nonnegative sectional curvature. Another one is under what condition does a
$3$-dimensional expanding soliton have nonnegative curvature. The answer to
the first question is negative. Some counterexamples are the shrinkers
constructed by Feldman-Ilmanen-Knopf [FIK03] on $O(-k)$, where $1\leq k<n$,
the steady solitons constructed by Cao [Cao94], and the steady solitons
constructed by Appleton [Ap17], where the latter two are constructed on line
bundles over the complex projective space. All of these examples do not have
nonnegative sectional curvature on the entire manifold. Nevertheless, one may
still propose related questions, such as, whether a $4$-dimensional Ricci
soliton has bounded curvature, or whether an estimate like (1.3) is true in
dimension four. One of the best results for shrinking solitons in these
directions is obtained by Munteanu-Wang [MW15]:

######  Theorem 1.2.

[MW15] Let $(M^{4},g,f)$ be a complete $4$-dimensional shrinking gradient
Ricci soliton with bounded scalar curvature. Then the curvature operator
$\operatorname{Rm}$ has bounded norm, and

$|{\operatorname{Rm}}|\leq cR\quad\text{ on }\quad M,$ (1.4)

for some positive constant $c$.

The curvature estimate (1.4) also has additional interest to Munteanu-Wang’s
program of classifying 4-dimensional noncompact shrinkers according to the
behavior of scalar curvature at infinity [KW15, MW15, MW17, MW19, CDS19].
Moreover, the estimate (1.4) can be interpreted as a weak notion of curvature
nonnegativity, namely, that the sectional curvatures of different $2$-planes
are pinched in a way that the negative ones are outweighed by the positive
ones, and that a scalar multiple of the average (i.e., the scalar curvature)
bounds the entire tensor norm $|{\operatorname{Rm}}|$. The estimate of
$|{\operatorname{Rm}}|$ in the potentially unbounded curvature setting is
lately studied by Chow, Freedman, Shin, and the third-named author [CFSZ20],
and Cao, Ribeiro, and Zhou [CRZ20].

In the $4$-dimensional steady case, Cao-Cui [CC20] showed that
$|{\operatorname{Rm}}|\leq cR^{\alpha}$, where $\alpha$ is any number in
$(0,1/2)$, provided that the scalar curvature $R$ coverges to $0$ at infinity
and that $R\geq Cr^{-k}$ near infinity for some $k>0$. Later, the first-named
author [Cha19] removed the condition $R\geq Cr^{-k}$ and proved (1.4) for non-
trivial steady gradient soliton with $\lim_{x\to\infty}R=0$ (see also [Cha20,
Che20, CZh21]). However, there are steady solitons on which $R$ does not go to
$0$ at infinity, for instance, $\mathbb{R}\times B_{3}$,
$\mathbb{R}^{2}\times\Sigma_{0}$ or $\Sigma_{0}\times\Sigma_{0}$, where
$\Sigma_{0}$ and $B_{3}$ denote Hamilton’s cigar soliton and the
$3$-dimensional Bryant soliton. It is interesting to see if (1.4) is still
true without curvature decay assumption. Very recently, (1.4) has been
established by Cao-Liu [CL21] on $4$-dimensional expanding solitons under the
assumptions that $\operatorname{Ric}\geq 0$ and that $R$ has at most
polynomial decay. See also [Der17, Cha20, Che20, Zhl21] for other estimates on
the expanding solitons. Ancient Ricci flows and steady solitons satisfying the
estimate (1.4) have been recently studied in [MZ21, CMZ21a].

We first investigate the curvature of $4$-dimensional steady Ricci soliton
with bounded Riemann curvature tensor.

######  Theorem 1.3.

Let $(M^{4},g,f)$ be a $4$-dimensional complete and non-Ricci-flat steady
gradient soliton with bounded Riemann curvature. Then there exists a positive
constant $c$ such that

$|{\operatorname{Rm}}|\leq cR\quad\text{ on }\quad M,$

where $\operatorname{Rm}$ and $R$ denote the Riemann curvature tensor and the
scalar curvature, respectively.

The main difference between Theorem 1.3 and the previous results in [CC20,
Cha19] is that, under the assumption $\sup_{M}|{\operatorname{Rm}}|<\infty$,
the scalar curvature decaying or the Ricci curvature positivity condition is
no longer required. Bounded curvature is also a natural assumption. Indeed,
among different Ricci solitons, the ones arising as a scaled limit of a
compact Ricci flow with finite-time singularity are of particular interest,
since they reveal the asymptotic behavior of the flow near the singular time.
In [CFSZ20], it is proved that a $4$-dimensional shrinking soliton singularity
model has at most quadratic curvature growth, and that a $4$-dimensional
steady soliton singularity model has bounded Riemann curvature (see [CFSZ20]
for the precise definition of Ricci soliton singularity models). Applying the
result [CFSZ20] and Theorem 1.3, we see that

######  Corollary 1.4.

Let $(M^{4},g,f)$ be a $4$-dimensional complete non-Ricci-flat gradient steady
Ricci soliton singularity model. Then there exists a positive constant $c$
such that

$|{\operatorname{Rm}}|\leq cR\quad\text{ on }\quad M.$

On the other hand, Ricci soliton singularity models have also been extensively
studied in the literature; see [Per02, Zhz07, CDM20, Bam20a, Bam20b, Bam20c,
Bam21, BCDMZ21, BCMZ21], to list but a few.

Munteanu-Wang [MW15] also generalized the Hamilton-Ivey estimate to dimension
four and gave a quantitative lower bound for $4$-dimensional shrinkers with
bounded scalar curvature. Previous works on Hamilton-Ivey type estimates under
vanishing Weyl tensor condition in higher dimensions included [ELM08, Zh09].

######  Theorem 1.5.

[MW15] Suppose that $(M^{4},g,f)$ is a $4$-dimensional complete and noncompact
shrinking gradient Ricci soliton with bounded scalar curvature. Then there
exists a positive constant $C$ such that

$\operatorname{Rm}\geq-\left(\tfrac{C}{\ln(r+1)}\right)^{1/4}\quad\text{ on
}\quad M,$

where $r$ is the distance function from a fixed base point.

Next, we shall prove an analogous result for steady solitons with proper
potential function and linear scalar curvature decay.

######  Theorem 1.6.

Let $(M^{4},g,f)$ be a $4$-dimensional complete and non-Ricci-flat steady
gradient Ricci soliton with proper potential function and linear scalar
curvature decay, i.e., $\lim_{x\to\infty}f(x)=-\infty$ and $R\leq C/(r+1)$ for
some constant $C$. Then

$-Cr^{-3/2}\leq\operatorname{Rm}\leq Cr^{-1},$ (1.5)

where $r$ is the distance function from a fixed base point.

The $\nu$-entropy of a Riemannian manifold was introduced by Perelman in
[Per02]:

$\displaystyle\nu=\nu(M,g)$
$\displaystyle:=\inf\left\\{\overline{\mathcal{W}}(g,u,\tau):\tau>0,u\geq
0,\sqrt{u}\in C_{0}^{\infty}(M),\int_{M}u\,dg=1\right\\},$

where $\overline{\mathcal{W}}$ is Perelman’s $\mathcal{W}$-functional
([Per02]) defined as

$\overline{\mathcal{W}}(g,u,\tau):=\int_{M}\left(\tau\left(|\nabla\log
u|^{2}+R\right)-\log u\right)u\,dg-\frac{n}{2}\log(4\pi\tau)-n.$

Under the non-collapsing condition $\nu>-\infty$, the diameter estimate of the
level set of $f$ in [DZ19, DZ20b] and the argument in [BCDMZ21] can be used to
obtain the following result. The details of the proof are left to the readers.

######  Theorem 1.7.

Let $(M^{4},g,f)$ be a $4$-dimensional complete and non-Ricci-flat steady
gradient Ricci soliton with proper potential function and linear scalar
curvature decay, i.e., $R\leq C/(r+1)$ for some constant $C$. Suppose, in
addition, that $\nu$ is bounded from below, i.e.,

$\nu(M,g)>-\infty.$ (1.6)

Then the canonical form of $(M,g,f)$ has
$(\mathbb{S}^{3}/\Gamma)\times\mathbb{R}$ as the tangent flow at infinity for
some finite group $\Gamma$. Thus, by [BCDMZ21], $(M,g)$ has positive curvature
operator outside a compact set and $R\sim r^{-1}$ near infinity.

######  Remark 1.8.

By [CMZ21b, Theorem 1.1] or [CMZ21c, Theorem 1.14], it can be seen that the
same conclusion in Theorem 1.7 holds if the condition (1.6) is replaced by
_bounded Nash entropy_ on the canonical form of the steady soliton.

Under the Riemann curvature linear decay assumption, the condition
$\operatorname{Ric}\geq 0$ near infinity, and the $\kappa$-noncollapsing
condition, Deng-Zhu proved a dimension reduction principle for steady solitons
[DZ20a, DZ19, DZ20b], namely, that the rescaled limits of the canonical form
of the soliton at spatial infinity split like $\left(\mathbb{R}\times
P^{n-1},dr^{2}+g_{P}(t)\right)$, where $g_{P}(t)$ is a Type I noncollapsed
ancient Ricci flow on the compact manifold $P$. Moreover, when $n=4$ and
${\operatorname{Ric}}>0$, they [DZ20a] showed that $(P,g_{P}(t))$ is the
shrinking sphere and the steady soliton have positive sectional curvature
outside compact subset (see also [CDM20, BCDMZ21]). Under the same assumptions
as Theorem 1.6, the first-named author and Zhu [CZh21] applied Deng-Zhu’s
method to prove a dichotomy for the asymptotic geometry at the spatial
infinity of a steady soliton without volume noncollapsing condition. As an
application of Theorem 1.6, we have a slightly more precise description on the
asymptotic limits in dimension four. We refer the reader to [CZh21] for the
definition of being smoothly asymptotic to a cylinder at the exponential rate.

######  Corollary 1.9.

Under the conditions in Theorem 1.6, either one of the following holds:

  1. 1.

For any sequence $p_{i}\to\infty$ in $M^{4}$, after passing to a subsequence,
$(M^{4},d_{R(p_{i})g},p_{i})$ converges in the pointed Gromov-Hausdorff sense
to a cylinder $\left(\mathbb{R}\times
Y,\sqrt{d_{e}^{2}+d_{Y}^{2}},p_{\infty}\right)$, where $d_{e}$ is the flat
metric on $\mathbb{R}$, $(Y,d_{Y})$ denotes a compact Alexandrov space with
nonnegative curvature and Hausdorff dimension $\leq 3$, and
$\sqrt{d_{e}^{2}+d_{Y}^{2}}$ is the product metric.

  2. 2.

For any sequence $p_{i}\to\infty$ in $M^{4}$, $(M^{4},d_{R(p_{i})g},p_{i})$
converges in the pointed Gromov-Hausdorff sense (without passing to
subsequence) to the ray $([0,\infty),d_{e},0)$, where $d_{e}$ is the flat
metric restricted on $[0,\infty)$. In this case, $(M,g)$ is smoothly
asymptotic to the flat cylinder
$\mathbb{R}\times\left(\mathbb{T}^{3}\big{/}\sim\right)$ at exponential rate.

######  Remark 1.10.

It is unclear at this point whether or not the Alexandrov space $(Y,d_{Y})$ in
Corollary 1.9 has nonempty boundary. The uniqueness of $(Y,d_{Y})$ is also
unknown. If a $4$-dimensional non-Ricci-flat steady gradient Ricci soliton is
$\kappa$-noncollapsed, satisfies $\operatorname{Ric}\geq 0$ outside a compact
set, and satisfies $|{\operatorname{Rm}}|\leq C/(r+1)$, then Deng-Zhu [DZ20a]
proved that the level sets near infinity are diffeomorphic to a quotient of
$\mathbb{S}^{3}$. Under the assumptions in Corollary 1.9, it follows from
Theorem 1.6, [CZh21, Corollary 1.6] and (3.41) that the level sets $\Sigma$ of
$f$ have almost nonnegative sectional curvature near infinity. Thus, by [FY92,
Per02, Per03a, Per03b], the level set $\Sigma$ of $f$ is either homeomorphic
to a quotient of $\mathbb{T}^{3}$, $\mathbb{S}^{3}$,
$\mathbb{S}^{1}\times\mathbb{S}^{2}$, or a nilmanifold (see [DZ20a] for result
in higher dimensions under the noncollapsing condition).

As mentioned above, motivated by Theorem 1.1, one may wonder if a
$3$-dimensional expanding gradient Ricci soliton must be nonnegatively curved.
Bryant constructed a family of $3$-dimensional negatively curved complete
rotationally symmetric gradient expanders on $\mathbb{R}^{3}$ (see
[CCGGIIKLLN07]). However, we will show that the sectional curvature of a
$3$-dimensional expanding gradient soliton is positive if the scalar curvature
satisfies certain decay conditions outside a compact set. Let $h$ be any
function on a noncompact manifold $M$ and $\alpha\in\mathbb{R}$ be a positive
constant. We say that $h=o(r^{-\alpha})$ if $\lim_{x\to\infty}r^{\alpha}h=0$,
$h=O(r^{-\alpha})$ if $\limsup_{x\to\infty}r^{\alpha}|h|<\infty$, where $r$ is
the distance function based at a fixed point.

######  Theorem 1.11.

Let $(M^{3},g,f)$ be a $3$-dimensional complete and noncompact expanding
gradient Ricci soliton. Assume that there exist nonnegative functions
$h_{1}=o(r^{-2})$ and $h_{2}=o(r^{-1})$ near infinity such that

$-h_{1}\leq R\leq h_{2}$ (1.7)

outside a compact set of $M$. Then $M$ has positive sectional curvature
everywhere unless it is flat.

######  Remark 1.12.

In view of the Bryant expanding solitons with negative curvature
[CCGGIIKLLN07], we see that the lower bound $h_{1}=o(r^{-2})$ in (1.7) is
sharp and cannot be relaxed. Theorem 1.11 also fails in higher dimensions,
since the expanding Kähler Ricci soliton constructed by Feldman-Ilmanen-Knopf
[FIK03] with complex dimension $m\geq 2$ has mixed Ricci curvature signs and
positive scalar curvature which decays exponentially in $r$.

It can also be seen from the proof of Theorem 1.11 that any $3$-dimensional
complete and noncompact expanding gradient Ricci soliton with nonnegative
sectional curvature outside a compact set has nonnegative sectional curvature
everywhere (see Remark 2.3). The same holds if one replaces the sectional
curvature by the Ricci curvature (see Remark 2.5).

A special phenomenon arising from $3$-dimensional Ricci expanders is that
there exist asymptotically conical examples; the same is not true for
$3$-dimensional steadies and shrinkers, though higher dimensional
asymptotically conical examples can be found in [FIK03, AK19]. With the help
of this extra structure at infinity, it is slightly more tractable to classify
$3$-dimensional asymptotically conical expanders according to their asymptotic
cones. We first recall the definition of asymptotically conical expanding
soliton. Let $X$ be a smooth $(n-1)$-dimensional closed manifold with
Riemannian metric $g_{X}$. We shall denote by $C(X)$ the cone over $X$, i.e.
$\\{(t,\omega):$ $t>0,\omega\in X\\}$. Let $g_{C}$ and $\nabla_{C}$ be the
metric $dt^{2}+t^{2}g_{X}$ on $C(X)$ and its Levi-Civita connection,
respectively. For any positive constant $S_{0}$,
$\overline{B(o,S_{0})}\subseteq C(X)$ is the subset given by $\\{(t,\omega):$
$S_{0}\geq t>0,\,\omega\in X\\}$. $X$ is also called _the link_ of the cone
$C(X)$.

######  Definition 1.13.

[Cho14, Definition 1.1] Let $k\in\mathbb{N}\cup\\{\infty\\}$. A complete
noncompact expanding gradient Ricci soliton $(M,g,f)$ is $C^{k}$-asymptotic to
the cone $(C(X),g_{C})$, if there exist constants $\varepsilon>0,S_{0}>0$, and
$c_{0}$, a compact set $K\subset M$, and a diffeomorphism $\phi:$
$C(X)\setminus\overline{B(o,S_{0})}\longrightarrow M\setminus K$, such that
for any $l=0,1,2,\cdots,k$, there is a constant $C_{l}>0$ such that for all
$t>S_{0}$

$\displaystyle\sup_{\omega\,\in
X}\left|\nabla_{C}^{l}\left(\phi^{*}g-g_{C}\right)\right|_{g_{C}}(t,\omega)\leq
C_{l}\,t^{-3\varepsilon-l};$ (1.8) $\displaystyle
f\circ\phi(t,\omega)=-\frac{t^{2}}{4}+c_{0}\quad\text{ and
}\quad\frac{2}{t}\frac{\partial\phi}{\partial t}=-\frac{\nabla f}{|\nabla
f|^{2}}.$ (1.9)

######  Remark 1.14.

Note that the notion of conical expander in Definition 1.13 is slightly more
restrictive than the one in [Cho14, Definition 1.1], as here
$\left|\nabla_{C}^{l}\left(\phi^{*}g-g_{C}\right)\right|_{g_{C}}$ is also
assumed to be bounded for $t$ close to $S_{0}$ in (1.8). Hence the
classification result in [Cho14] also holds for conical expanding solitons
satisfying Definition 1.13. Moreover, Definition 1.13 and the corresponding
definition in [Cho14] are equivalent if $f$ is proper, i.e.,
$\lim_{x\to\infty}f(x)=-\infty$, which holds in particular when
$\operatorname{Ric}\geq(\varepsilon_{0}-1/2)g$ outside a compact set for some
positive constant $\varepsilon_{0}$ (see (2.6)). For general gradient
expanding solitons, $f$ may not be proper and $M$ can have two ends (see [R13,
BM15]).

Cheeger-Colding (see [CC96] and the references therein) showed that the
tangent cones at infinity of a complete noncompact manifold with
$\operatorname{Ric}\geq 0$ and maximal volume growth are metric cones. In the
case of expanding solitons, some sufficient conditions for the existence of an
asymptotic metric cone were given in [CD15, Der17, CL21]: if on an expander it
holds that $r^{2}|{\operatorname{Rm}}|$ is bounded, then one can always find
an asymptotic metric cone at infinity in the Gromov-Hausdorff sense (c.f.
[CD15, CL21]). Moreover, the asymptotic convergence can be made to be
$C^{\infty}$ (as in Definition 1.13), if
$r^{2+k}|\nabla^{k}\operatorname{Ric}|$ is also bounded for all integer $k\geq
0$ [Der17].

Given a cone $C(X)$, it has long been an interesting problem to find expanding
solitons asymptotic to (smoothing out) $C(X)$ and to establish the uniqueness
of these expanders under different conditions ([SS13, Sie13, Der16, CDS19,
CoD20]). By constructing Killing vector fields as in [Br13], Chodosh [Cho14]
proved the following uniqueness result for asymptotically conical expanding
soliton (see also [CCCMM14] for classification result under the Bach flat
condition).

######  Theorem 1.15.

[Cho14] Let $n\geq 3$ be an integer. If $(M^{n},g,f)$ is an $n$-dimensional
complete expanding gradient Ricci soliton which has positive sectional
curvature and is $C^{2}$-asymptotic to the cone
$\left(C(\mathbb{S}^{n-1}),dt^{2}+\alpha t^{2}g_{\mathbb{S}^{n-1}}\right)$,
where $\alpha\in(0,1]$ is a constant, and $g_{\mathbb{S}^{n-1}}$ is the
standard metric on $\mathbb{S}^{n-1}$ with constant curvature $1$. Then M is
rotationally symmetric.

The uniqueness of $C^{2}$-asymptotically conical expanding Kähler soliton with
positive bisectional curvature was also shown by Chodosh-Fong [CF16]. The
proofs in [Br14], as well as the ones in [Cho14, CF16], require the
information on the sectional (or bisectional) curvature for a maximum
principle argument showing a Liouville-type theorem for a Lichnerowicz PDE.
The existence and uniqueness of expander with prescribed asymptotic cone has
been studied by Deruelle [Der16] in a more general setting. Namely if
$(X^{n-1},g_{X})$ is a simply connected closed manifold with

$\operatorname{Rm}(g_{X})>\text{id}_{\Lambda^{2}TX},$ (1.10)

then there exists a positively curved gradient Ricci expander which is
$C^{\infty}$-asymptotic to $C(X)$ [Der16]. Moreover, two gradient expanders
that satisfy $\operatorname{Rm}>0$ and are $C^{\infty}$-asymptotic to $C(X)$
must be isometric to each other [Der16]. It is natural to ask whether the
uniqueness result holds within larger category of expanding solitons (see
Remark 1.17 below). It follows from (1.8) that the decay condition (1.7) is
fulfilled by those conical gradient expanders with link $X$ satisfying (1.10)
(or more generally $\operatorname{Rm}(g_{X})\geq\text{id}_{\Lambda^{2}TX}$).
Consequently, we have the following corollary.

######  Corollary 1.16.

Suppose that $(M^{3},g,f)$ is a $3$-dimensional complete non-compact expanding
gradient Ricci soliton which is $C^{2}$-asymptotic to a cone
$\left(C(X^{2}),dt^{2}+t^{2}g_{X}\right)$, where $(X^{2},g_{X})$ is a
connected and closed $2$-dimensional Riemannian manifold. Then the following
are equivalent:

  1. 1.

$M$ has nonnegative sectional curvature;

  2. 2.

$M$ has nonnegative scalar curvature;

  3. 3.

$C(X)$ has nonnegative scalar curvature;

  4. 4.

$\operatorname{Rm}(g_{X})\geq{\rm id}_{\Lambda^{2}TX}$.

######  Remark 1.17.

If instead we assume that $M^{3}$ in Corollary 1.16 is $C^{\infty}$-asymptotic
to $C(X)$ with $\operatorname{Rm}(g_{X})>\text{id}_{\Lambda^{2}TX}$ and that
$X$ is simply connected, then by Theorem 1.11 and the uniqueness result in
[Der16], $M^{3}$ is isometric to the positively curved conical expander (with
asymptotic cone $C(X)$) constructed by Deruelle in [Der16, Theorem 1.3].

######  Remark 1.18.

Corollary 1.16, as well as Theorem 1.11, is not true in higher dimensions. The
expanding gradient Kähler Ricci soliton constructed by Feldman-Ilmanen-Knopf
[FIK03] with complex dimension $m\geq 2$ has mixed Ricci curvature signs,
positive scalar curvature, and flat asymptotic cone [FIK03, Sie13].

By the Morse theory, the level sets of $f$ near infinity of a Ricci
nonnegative expanding gradient soliton are diffeomorphic to
$\mathbb{S}^{n-1}$, which is simply connected for $n\geq 3$. Hence by
Corollary 1.16, for any $\alpha\in(0,1]$, we have that
$(C(\mathbb{RP}^{2}),dt^{2}+\alpha g_{\mathbb{RP}^{2}})$ is not a
$C^{2}$-asymptotic cone of any expander in the sense of Definition 1.13, where
$g_{\mathbb{RP}^{2}}$ is the standard metric on $\mathbb{RP}^{2}$.

As another application of Theorem 1.11, we remove the positive curvature
condition in Theorem 1.15 in dimension three.

######  Corollary 1.19.

Suppose that $(M^{3},g,f)$ is a $3$-dimensional complete and noncompact
expanding gradient Ricci soliton which is $C^{2}$-asymptotic to the cone
$\left(C(\mathbb{S}^{2}),dt^{2}+\alpha t^{2}g_{\mathbb{S}^{2}}\right)$, where
$\alpha\in(0,1]$ is a constant, and $g_{\mathbb{S}^{2}}$ is the standard
metric on $\mathbb{S}^{2}$. Then M is rotationally symmetric.

######  Remark 1.20.

As pointed out by Chen-Deruelle [CD15, Remark 1.5] and Deruelle [Der16,
Definition 1.1], the canonical form of a conical expander (with asymptotic
cone $C(X)$) is a Ricci flow with singular initial data (in the Gromov-
Hausdorff sense) given by $C(X)$. Hence Corollaries 1.16 and 1.19 can be
respectively interpreted as the preservation of nonnegative curvature and the
uniqueness of $3$-dimensional Ricci flow coming out of the cone $C(X)$, when
the flow is the canonical form of an expander. Singular Ricci flow on closed
$3$-manifold was introduced by Kleiner-Lott [KL17, KL18] and its uniqueness
and stability in dimension three were established by Bamler-Kleiner [BK17].

######  Remark 1.21.

The definition of conical expander in [Cho14, Definition 1.1] (see also
Definition 1.13) is slightly different from the ones in [Der16, Der17]. More
precisely, the condition in (1.9)

$\frac{2}{t}\frac{\partial\phi}{\partial t}=-\frac{\nabla f}{|\nabla f|^{2}}$
(1.11)

is required in [Cho14] but not in [Der16, Der17]. Indeed, Condition (1.11) is
only used when we apply the result [Cho14] in Corollary 1.19, and hence
Corollary 1.16 as well as Remark 1.17 also holds on those expanders satisfying
all conditions in Definition 1.13 except for possibly (1.11).

We shall sketch the idea of proof very briefly. Let $T_{ij}:=Rg_{ij}-2R_{ij}$.
The tensor $T$ was used extensively to study the curvature pinching in
$3$-dimensional Ricci flow (e.g. [Ham82, Ham95, CCGGIIKLLN08, Che09, CXZ13]).
In general, sectional curvature being nonnegative implies $T\geq 0$. When
$n=3$, we have $\operatorname{Rm}\geq 0$ at $p\in M$ if and only if $T\geq 0$
at $p\in M$ [Ham82, Corollary 8.2]. The key ingredient of the proof is to look
at the differential equation satisfied by $T$ and estimate the smallest
eigenvalue of $T$ via the maximum principle argument. When $n=3$, let
$\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}$ be the eigenvalues of
$\operatorname{Ric}$ and $\nu_{1}\leq\nu_{2}\leq\nu_{3}$ be the eigenvalues of
$T$. In the case of expanding soliton, $\nu_{1}$ satisfied the following nice
inequality in the barrier sense (see (2.16))

$\Delta_{f}\nu_{1}\leq-\nu_{1}-\nu_{1}^{2}-\nu_{2}\nu_{3}.$

The decay condition (1.7) then allows us to apply the asymptotic estimate in
[Der17] and the maximum principle argument in [MW15] to conclude that
$\nu_{1}$ is nonnegative in the $3$-dimensional expanding soliton case. For a
$4$-dimensional steady soliton, we invoke the dimension reduction trick via
the level set of $f$ by [MW15] and bound the smallest eigenvalue from below of
a tensor $U$ approximating $T$ (see (3.24)). Theorem 1.6 follows from a
similar argument as in the $3$-dimensional case together with some delicate
estimates of the error terms caused by the extra dimension.

This paper is organized as follows. We start with the $3$-dimensional
expanding soliton case to show Theorem 1.11, Corollary 1.16, and Corollary
1.19 in Section $2$. Next, we move on to the $4$-dimensional steady case in
Section $3$ and present the proofs of Theorem 1.6 and Corollary 1.9.

Acknowledgement. The authors would like to thank Professor Bennett Chow for
suggesting the problem of curvature of steady soliton singularity model. The
authors would also like to thank Professor Yuxing Deng for helpful
communications of ideas on steady solitons.

##  2 Positive curvature in 3-dimensional expanding soliton case

In this section, we shall consider a $3$-dimensional expanding gradient Ricci
soliton $(M^{3},g,f)$ satisfying (1.1) with $\kappa=-1$. Recall that we have
defined $T_{ij}:=Rg_{ij}-2R_{ij}$. As in the introduction, we denote the
eigenvalues of $\operatorname{Ric}$ by
$\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}$ and the eigenvalues of $T$ by
$\nu_{1}\leq\nu_{2}\leq\nu_{3}$. Then it can be seen from the definition of
$T$ that $R=\lambda_{1}+\lambda_{2}+\lambda_{3}=\nu_{1}+\nu_{2}+\nu_{3}$ and

$\begin{split}\nu_{1}&=\lambda_{1}+\lambda_{2}-\lambda_{3}=R-2\lambda_{3}\\\
\nu_{2}&=\lambda_{1}+\lambda_{3}-\lambda_{2}=R-2\lambda_{2}\\\
\nu_{3}&=\lambda_{2}+\lambda_{3}-\lambda_{1}=R-2\lambda_{1}.\end{split}$ (2.1)

To prove that $M$ has nonnegative curvature, It suffices to show that
$\nu_{1}\geq 0$ everywhere. We shall use the following well-known formula for
$\operatorname{Rm}$ in dimension three (c.f. [CLN06, (1.62)]):

$R_{ijkl}=R_{il}g_{jk}+R_{jk}g_{il}-R_{ik}g_{jl}-R_{jl}g_{ik}-\frac{R}{2}\left(g_{il}g_{jk}-g_{ik}g_{jl}\right).$
(2.2)

It is a result by Hamilton [Ham95] that $\nabla\left(|\nabla
f|^{2}+R+f\right)=0$, by adding a constant to $f$, we have

$|\nabla f|^{2}+R=-f$ (2.3)

On the other hand, it was proved by Pigola–Rimoldi–Setti [PRS11] and S.-J.
Zhang [Zhs11] that the scalar curvature of any complete expanding gradient
Ricci soliton satisfies

$R\geq-\frac{n}{2}.$ (2.4)

Throughout this article, for any smooth function $\gamma$, we define the
weighted laplacian $\Delta_{\gamma}$ as $\Delta-\nabla_{\nabla\gamma}$, where
$\Delta$ is the standard laplacian operator. Let $v=\frac{n}{2}-f$. By taking
the trace of the soliton equation (1.1), (2.4), and (2.3), we see that $\Delta
v=R+\frac{n}{2}$, and

$\displaystyle v$ $\displaystyle\geq$ $\displaystyle|\nabla v|^{2};$ (2.5)
$\displaystyle\Delta_{f}v$ $\displaystyle=$ $\displaystyle v.$

Integrating (2.5), we have that on any complete gradient Ricci expander of any
dimension, it hods that $v\leq r^{2}/4+Cr+C$ for some constant $C>0$ (see
[CCGGIIKLLN15, Corollary 27.10]). If, in addition,
$\liminf_{x\to\infty}\operatorname{Ric}>-1/2$, then we have that
$\operatorname{Ric}\geq(-1/2+\varepsilon_{0})g$ near infinity for some
constant $\varepsilon_{0}>0$, and hence $\nabla^{2}v\geq\varepsilon_{0}g$
outside a compact set. By integrating the inequality along minimizing
geodesics, we have

$v\geq\varepsilon_{0}r^{2}/2-C_{1}r-C_{1}$ (2.6)

for some positive constant $C_{1}$, and consequentially $v\sim r^{2}$.

The scalar curvature $R$ of an expanding gradient Ricci soliton satisfies
[CCGGIIKLLN07]

$\begin{split}\Delta_{f}R=-R-2|{\operatorname{Ric}}|^{2}\leq-R-2R^{2}/n=-(1+2R/n)R.\end{split}$
(2.7)

By the strong maximum principle [GT01], if $M$ has nonnegative scalar
curvature, then $R>0$ everywhere unless $R\equiv 0$, and in the latter case,
by (2.7) and (1.1), $\operatorname{Ric}\equiv 0$ and $M$ is flat [PRS11,
Theorem 1]. This also follows from the parabolic minimum principle applied to
the scalar curvature of the canonical form (1.2).

######  Lemma 2.1.

Suppose that $(M^{3},g,f)$ is a $3$-dimensional complete noncompact expanding
gradient Ricci soliton with the following scalar curvature decay,

$|R|\leq o(r^{-1}).$

Then the following estimate is satisfied:

$|{\operatorname{Rm}}|+|\nabla{\operatorname{Rm}}|\leq o(r^{-1}).$

###### Proof.

In view of (2.2), to prove the estimate on $\operatorname{Rm}$, we only need
to bound the Ricci tensor. By [Cha20, Theorem 7] and by applying Shi’s
estimate [CLN06] to the canonical form of the soliton, it holds that
$|{\operatorname{Rm}}|$ is bounded and that there exists a finite positive
constant $C_{k}$ for any $k\in\mathbb{N}$, such that

$|\nabla^{k}{\operatorname{Rm}}|\leq C_{k}\quad\text{ on }\quad M.$ (2.8)

It can be seen from (2.2) that in an orthonormal frame
$\left\\{\textbf{e}_{1},\textbf{e}_{2},\textbf{n}:=\frac{\nabla f}{|\nabla
f|}\right\\}$, we have

$R_{ij}=R_{\textbf{n}ij\textbf{n}}+\frac{R}{2}\left(g_{ij}-g_{i\textbf{n}}g_{j\textbf{n}}\right)-R_{\textbf{n}\textbf{n}}g_{ij}+R_{i\textbf{n}}g_{j\textbf{n}}+R_{j\textbf{n}}g_{i\textbf{n}},$
(2.9)

where $i,j=1,2$, and we are not summing over n. Moreover, from the Ricci
identity and the soliton equation we have

$R_{ij,k}-R_{ik,j}=R_{kjil}f_{l}.$ (2.10)

Thanks to [Cha20, Corollary 3] and (2.3), we have

$-f\sim r^{2}$ (2.11)

and there is a positive constant $C$ such that

$C^{-1}r^{2}\leq-f-R=|\nabla f|^{2}\leq Cr^{2}$ (2.12)

outside a compact set of $M$. By (2.8), (2.10), and
$2\operatorname{Ric}(\nabla f)=\nabla R$, we see that

$R_{\textbf{n}ij\textbf{n}}=O\left(\frac{|{\nabla\operatorname{Ric}}|}{|\nabla
f|}\right)=O(r^{-1})\quad\text{ and }\quad
R_{i\textbf{n}}=O\left(\frac{|\nabla R|}{|\nabla f|}\right)=O(r^{-1}).$

By virtue of (2.9), $|{\operatorname{Ric}}|$ and hence $|\operatorname{Rm}|$
are of $O(r^{-1})$. By the local Shi’s estimate [Der17, Lemma 2.6], there
exists a positive constant $C$ such that for all $p\in$ $M$ and $s\geq 1$

$|\nabla{\operatorname{Rm}}|(p)\leq
C\sup_{B_{s}(p)}|{\operatorname{Rm}}|\left[1+\sup_{B_{s}(p)}|{\operatorname{Rm}}|+\frac{\sup_{B_{s}(p)\setminus
B_{s/2}(p)}|\nabla f|}{s}\right]^{\frac{1}{2}}.$ (2.13)

Hence $|\nabla{\operatorname{Rm}}|=O(r^{-1})$, and both
$R_{\textbf{n}ij\textbf{n}}$ and $R_{i\textbf{n}}$ are of $O(r^{-2})$.
$|{\operatorname{Rm}}|=o(r^{-1})$ then follows from (2.2) and (2.9). The
estimate on $\nabla{\operatorname{Rm}}$ is now a consequence of the bound on
$|{\operatorname{Rm}}|$ and (2.13).

∎

######  Lemma 2.2.

Let $(M^{3},g,f)$ be a $3$-dimensional complete noncompact expanding gradient
Ricci soliton with positive scalar curvature. Assume that the tensor $T$
satisfies either one of the following conditions:

  1. 1.

T/R is asymptotically nonnegative, i.e.,

$\liminf_{x\to\infty}\frac{T}{R}\geq 0;$

  2. 2.

$\nu_{1}/R$ attains its minimum, where $\nu_{1}$ is the smallest eigenvalue of
$T$ as in (2.1).

Then M has nonnegative sectional curvature.

######  Remark 2.3.

The above lemma implies that any $3$-dimensional complete noncompact expanding
gradient Ricci soliton with nonnegative sectional curvature outside a compact
set has nonnegative sectional curvature everywhere.

###### Proof.

By the differential equation of $\operatorname{Ric}$ [PW10, Lemma 2.1] and
(2.2), we may compute in an orthonormal frame

$\begin{split}\Delta_{f}R_{il}&=-R_{il}-2R_{ijkl}R_{jk}\\\
&=-R_{il}-2RR_{il}-2|{\operatorname{Ric}}|^{2}g_{il}+4R_{ik}R_{kl}+R^{2}g_{il}-RR_{il}\\\
&=-R_{il}-3RR_{il}+4R_{ik}R_{kl}+R^{2}g_{il}-2|{\operatorname{Ric}}|^{2}g_{il}.\end{split}$
(2.14)

Hence, in the barrier sense, we have

$\Delta_{f}2\lambda_{3}\geq-2\lambda_{3}-6R\lambda_{3}+8\lambda_{3}^{2}+2R^{2}-4|{\operatorname{Ric}}|^{2}.$

By virtue of the formula $\Delta_{f}R=-R-2|{\operatorname{Ric}}|^{2}$, it
holds that

$\Delta_{f}\nu_{1}=\Delta_{f}(R-2\lambda_{3})\leq-\nu_{1}+6R\lambda_{3}-2R^{2}-8\lambda_{3}^{2}+2|{\operatorname{Ric}}|^{2}.$
(2.15)

Using (2.1), we may express the R.H.S. of (2.15) in terms of the eigenvalues
of $T$

$\displaystyle
6R\lambda_{3}-2R^{2}-8\lambda_{3}^{2}+2|{\operatorname{Ric}}|^{2}$
$\displaystyle=$ $\displaystyle
3(\nu_{1}+\nu_{2}+\nu_{3})(\nu_{2}+\nu_{3})-2(\nu_{1}+\nu_{2}+\nu_{3})^{2}-2(\nu_{2}+\nu_{3})^{2}$
$\displaystyle+\frac{1}{2}\left[(\nu_{2}+\nu_{3})^{2}+(\nu_{1}+\nu_{3})^{2}+(\nu_{1}+\nu_{2})^{2}\right]$
$\displaystyle=$
$\displaystyle(\nu_{2}+\nu_{3})^{2}+3(\nu_{1}\nu_{3}+\nu_{1}\nu_{2})-4(\nu_{1}\nu_{2}+\nu_{1}\nu_{3}+\nu_{2}\nu_{3})$
$\displaystyle-2(\nu_{1}^{2}+\nu_{2}^{2}+\nu_{3}^{2})+\left[\nu_{1}^{2}+\nu_{2}^{2}+\nu_{3}^{2}+\nu_{1}\nu_{2}+\nu_{1}\nu_{3}+\nu_{2}\nu_{3}\right]$
$\displaystyle=$ $\displaystyle-\nu_{1}^{2}-\nu_{2}\nu_{3}.$

Hence (2.15) can be rewritten as

$\Delta_{f}\nu_{1}\leq-\nu_{1}-\nu_{1}^{2}-\nu_{2}\nu_{3}.$ (2.16)

On the other hand, we have

$\begin{split}\Delta_{f}R^{-1}&=-R^{-2}\Delta_{f}R+2R^{-3}|\nabla R|^{2}\\\
&=-R^{-2}(-R-2|{\operatorname{Ric}}|^{2})+2R^{-3}|\nabla R|^{2}\\\
&=R^{-1}+2R^{-2}|{\operatorname{Ric}}|^{2}+2R^{-3}|\nabla R|^{2}.\end{split}$
(2.17)

$\displaystyle\Delta_{f}\left(R^{-1}\nu_{1}\right)$ $\displaystyle=$
$\displaystyle R^{-1}\Delta_{f}\nu_{1}+\nu_{1}\Delta_{f}R^{-1}+2\langle\nabla
R^{-1},\nabla\left(R^{-1}\nu_{1}R\right)\rangle$ $\displaystyle\leq$
$\displaystyle-R^{-1}\nu_{1}-R^{-1}\left(\nu_{1}^{2}+\nu_{2}\nu_{3}\right)+R^{-1}\nu_{1}+2R^{-2}|{\operatorname{Ric}}|^{2}\nu_{1}+2R^{-3}|\nabla
R|^{2}\nu_{1}$ $\displaystyle-2R^{-1}\langle\nabla
R,\nabla\left(R^{-1}\nu_{1}\right)\rangle-2R^{-3}|\nabla R|^{2}\nu_{1}$
$\displaystyle=$
$\displaystyle-R^{-1}\left(\nu_{1}^{2}+\nu_{2}\nu_{3}\right)+2R^{-2}|{\operatorname{Ric}}|^{2}\nu_{1}-2R^{-1}\langle\nabla
R,\nabla\left(R^{-1}\nu_{1}\right)\rangle.$

Hence we can rewrite the above equation as

$\begin{split}\Delta_{f-2\ln
R}\left(R^{-1}\nu_{1}\right)&\leq-R^{-1}\left(\nu_{1}^{2}+\nu_{2}\nu_{3}\right)+2R^{-2}|{\operatorname{Ric}}|^{2}\nu_{1}\\\
&=-R^{-2}\left(R(\nu_{1}^{2}+\nu_{2}\nu_{3})-2|{\operatorname{Ric}}|^{2}\nu_{1}\right).\end{split}$
(2.18)

Straightforward computation yields

$\displaystyle
R(\nu_{1}^{2}+\nu_{2}\nu_{3})-2|{\operatorname{Ric}}|^{2}\nu_{1}$
$\displaystyle=$
$\displaystyle(\nu_{1}+\nu_{2}+\nu_{3})(\nu_{1}^{2}+\nu_{2}\nu_{3})-\frac{\nu_{1}}{2}\left[(\nu_{1}+\nu_{2})^{2}+(\nu_{1}+\nu_{3})^{2}+(\nu_{2}+\nu_{3})^{2}\right]$
$\displaystyle=$
$\displaystyle\nu_{1}^{3}+\nu_{1}^{2}\nu_{2}+\nu_{1}^{2}\nu_{3}+\nu_{2}^{2}\nu_{3}+\nu_{2}\nu_{3}^{2}+\nu_{1}\nu_{2}\nu_{3}$
$\displaystyle-\left[\nu_{1}^{3}+\nu_{1}^{2}\nu_{2}+\nu_{1}\nu_{2}^{2}+\nu_{1}^{2}\nu_{3}+\nu_{1}\nu_{3}^{2}+\nu_{1}\nu_{2}\nu_{3}\right]$
$\displaystyle=$
$\displaystyle\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1}).$

Hence

$\Delta_{f-2\ln
R}\left(R^{-1}\nu_{1}\right)\leq-R^{-2}\left[\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1})\right]\leq
0.$ (2.19)

Note that so far the above computation works well for any $3$-dimensional
gradient Ricci expander with positive scalar curvature. In particular, (2.16)
and (2.19) also hold on these solitons without additional assumption.

If $\nu_{1}$ is negative somewhere, then either one of our assumptions will
imply that $R^{-1}\nu_{1}$ attains its negative minimum, say, at a point $q\in
M$ and $R^{-1}(q)\nu_{1}(q)<0$. Then we have

$0\leq\Delta_{f-2\ln
R}\left(R^{-1}\nu_{1}\right)(q)\leq-R^{-2}(q)\left[\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1})\right](q)$

in the barrier sense. As $\nu_{1}\leq\nu_{2}\leq\nu_{3}$, we have
$\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1})=0$ at $q$. Since
$3\nu_{3}(q)\geq R(q)>0$, we see that $\nu_{1}(q)=\nu_{2}(q)<0$. However,

$0<\nu_{2}^{2}(q)(\nu_{3}(q)-\nu_{1}(q))\leq 0,$

which is impossible. This shows that $\nu_{1}\geq 0$ on $M$ and completes the
proof of the lemma. ∎

The same line of argument of Lemma 2.2 also gives the following proposition
which shall not be used in the rest of the article.

######  Proposition 2.4.

If $(M^{3},g,f)$ is a $3$-dimensional complete noncompact expanding gradient
Ricci soliton with positive scalar curvature $R>0$ and $\operatorname{Ric}$
satisfies either one of the following conditions:

  1. 1.

$\operatorname{Ric}/R$ is asymptotically nonnegative, i.e.

$\liminf_{x\to\infty}\frac{\operatorname{Ric}}{R}\geq 0;$

  2. 2.

$\lambda_{1}/R$ attains its minimum, where $\lambda_{1}$ is the smallest
eigenvalue of $\operatorname{Ric}$,

then M has nonnegative Ricci curvature everywhere.

######  Remark 2.5.

The above proposition implies that any $3$-dimensional complete noncompact
expanding gradient Ricci soliton with nonnegative Ricci curvature outside a
compact set has nonnegative Ricci curvature everywhere.

###### Proof.

We sketch the proof and point out the essential differences from Lemma 2.2. It
follows from (2.14) that in the sense of barrier

$\Delta_{f}2\lambda_{1}\leq-2\lambda_{1}-6R\lambda_{1}+8\lambda_{1}^{2}+2R^{2}-4|{\operatorname{Ric}}|^{2}.$

Using $\nu_{3}=R-2\lambda_{1}$ and the computation showing (2.16), we see that

$\begin{split}\Delta_{f}\nu_{3}&\geq-\nu_{3}+6R\lambda_{1}-2R^{2}-8\lambda_{1}^{2}+2|{\operatorname{Ric}}|^{2}\\\
&=-\nu_{3}-\nu_{3}^{2}-\nu_{1}\nu_{2}.\end{split}$

By exchanging the roles of $\nu_{1}$ and $\nu_{3}$, we may compute as in
(2.17) and (2.18) to get

$\begin{split}\Delta_{f-2\ln
R}\left(R^{-1}\nu_{3}\right)&\geq-R^{-2}\left(R(\nu_{3}^{2}+\nu_{1}\nu_{2})-2|{\operatorname{Ric}}|^{2}\nu_{3}\right)\\\
&=R^{-2}\left(\nu_{1}^{2}(\nu_{3}-\nu_{2})+\nu_{2}^{2}(\nu_{3}-\nu_{1})\right)\geq
0.\end{split}$

Since $\nu_{3}/R=1-2\lambda_{1}/R$, it holds that

$\Delta_{f-2\ln
R}\left(R^{-1}\lambda_{1}\right)\leq-R^{-2}\left(\nu_{1}^{2}(\nu_{3}-\nu_{2})+\nu_{2}^{2}(\nu_{3}-\nu_{1})\right)/2\leq
0.$ (2.20)

If $\lambda_{1}$ is negative somewhere, then either one of our assumptions
will imply that $R^{-1}\lambda_{1}$ attains its negative minimum, say, at a
point $q\in M$ and $R^{-1}(q)\lambda_{1}(q)<0$. We may further suppose that
$\nu_{1}(q)<0$, otherwise
$\lambda_{1}(q)=(\nu_{1}(q)+\nu_{2}(q))/2\geq\nu_{1}(q)\geq 0$. By (2.20), we
have at $q$

$\nu_{1}^{2}(\nu_{3}-\nu_{2})+\nu_{2}^{2}(\nu_{3}-\nu_{1})=0.$

Hence $\nu_{2}=\nu_{3}>0$ and $0<R/3\leq\nu_{3}=\nu_{1}<0$, which is
impossible. The completes the proof of the proposition. ∎

######  Lemma 2.6.

Under the same assumptions of Theorem 1.11, it holds that

$|{\operatorname{Rm}}|\leq CR\quad\text{ on }\quad M,$ (2.21)

for some positive constant $C>0$.

######  Remark 2.7.

The proof of Lemma 2.6 also shows that (2.21) is true on any $3$-dimensional
complete noncompact gradient Ricci expander with
$\liminf_{x\to\infty}r^{2}\operatorname{Rm}\geq 0$ and
$|{\operatorname{Rm}}|\leq Cr^{-\delta}$ near infinity for some $\delta>0$.

###### Proof.

As $\lim_{x\to\infty}R=0$ and that $R\geq-o(r^{-2})$, the scalar curvature $R$
is nonnegative [Cha20, Theorem 6]. By the strong minimum principle, we may
assume that $R>0$, since otherwise the expander is flat. It follows from (1.7)
that $|R|\leq o(r^{-1})$ near infinity. Hence, by Lemma 2.1, (2.10), and
(2.12), we have

$R_{\textbf{n}ij\textbf{n}}=O\left(\frac{|\nabla{\operatorname{Ric}}|}{|\nabla
f|}\right)=o(r^{-2})\quad\text{ and }\quad
R_{i\textbf{n}}=O\left(\frac{|\nabla R|}{|\nabla f|}\right)=o(r^{-2}).$ (2.22)

where $\textbf{n}:=\frac{\nabla f}{|\nabla f|}$. Using (2.9), (2.22) and the
fact that $R>0$ near infinity, we have

$\begin{split}T_{ij}=Rg_{ij}-2R_{ij}&=-2R_{\textbf{n}ij\textbf{n}}+Rg_{i\textbf{n}}g_{j\textbf{n}}+2R_{\textbf{n}\textbf{n}}g_{ij}-2R_{i\textbf{n}}g_{j\textbf{n}}-2R_{j\textbf{n}}g_{i\textbf{n}}\\\
&\geq-2R_{\textbf{n}ij\textbf{n}}+2R_{\textbf{n}\textbf{n}}g_{ij}-2R_{i\textbf{n}}g_{j\textbf{n}}-2R_{j\textbf{n}}g_{i\textbf{n}}\\\
&\geq-o(r^{-2})g_{ij},\end{split}$ (2.23)

Hence $\nu_{1}\geq-o(r^{-2})$ and

$\limsup_{x\to\infty}-\nu_{1}r^{2}\leq 0.$ (2.24)

It is not difficult to see that (2.16) is still valid. As
$\nu_{1}+\nu_{2}+\nu_{3}=R\geq 0$ and $\nu_{1}\leq\nu_{2}\leq\nu_{3}$, we have
that $3\nu_{3}\geq R\geq 0$ and that the following holds in the barrier sense.

$\begin{split}\Delta_{f}\nu_{1}&\leq-\nu_{1}-\nu_{1}^{2}-\nu_{2}\nu_{3}\\\
&=-\nu_{1}-\nu_{1}^{2}-\nu_{1}\nu_{3}-(\nu_{2}-\nu_{1})\nu_{3}\\\
&\leq-\nu_{1}-\nu_{1}^{2}-\nu_{1}\nu_{3}.\end{split}$ (2.25)

Moreover by (2.24), $0\leq\nu_{3}\leq R-2\nu_{1}\leq R+Cr^{-2}\leq Cr^{-1}$
near infinity. Let $u=-\nu_{1}$, from (2.11), we may rewrite (2.25) on the set
$\\{x:u(x)\geq 0\\}=\\{x:\nu_{1}(x)\leq 0\\}$ as

$\Delta_{f}u\geq-u-c_{0}v^{-1/2}u,$

where $v=n/2-f\sim r^{2}$ and $c_{0}$ is some positive constant. Together with
the boundary condition at infinity $\limsup_{x\to\infty}vu\leq 0$, the maximum
principle argument by Deruelle [Der17, Lemma 2.9(1)] shows that there are
positive constants $C$ and $C_{1}$ such that on $M$

$\begin{split}-\nu_{1}=u&\leq Cv^{1-3/2}e^{-v}\\\ &\leq C_{1}R,\end{split}$
(2.26)

where we also used the scalar curvature lower bound [Cha20, Theorem 3] and the
fact that $M$ is not flat. Note that the argument in [Der17] requires $u\geq
0$. However, (2.26) becomes trivial if $u<0$ at the point under consideration.
Hence we can always assume that the maximum principle is applied on $\\{u\geq
0\\}$ and no extra sign assumption on $u$ is needed. From (2.26) we see that
$-C_{1}R\leq\nu_{1}\leq\nu_{2}\leq\nu_{3}\leq R-2\nu_{1}\leq(1+2C_{1})R$ and
$|T|\leq cR$. Since $T=Rg-2\operatorname{Ric}$, we have

$|{\operatorname{Ric}}|\leq c(|T|+R)\leq CR.$

The estimate on $|{\operatorname{Rm}}|$ follows from (2.2). ∎

###### Proof of Theorem 1.11.

By our assumption (1.7) and the argument in the proof of Lemma 2.6, we may
assume that $R>0$ on $M$. Using Lemma 2.1, (2.11), and (2.23), we see that
$v\sim r^{2}$ and

$R^{-1}T\geq-o(1)v^{-1}R^{-1}g.$ (2.27)

We claim that there is a large positive constant $a$, such that outside a
compact set $K_{0}\subset M$, it holds that

$\Delta_{f-2\ln
R}\left(e^{av^{-1/2}}v^{-1}R^{-1}\right)\leq-v^{-3/2}e^{av^{-1/2}}R^{-1}<0.$
(2.28)

We first assume (2.28) and prove the theorem. Fixing any positive integer $k$,
we define

$Q_{k}:=R^{-1}\nu_{1}+e^{av^{-1/2}}v^{-1}R^{-1}k^{-1}.$

By (2.27), we have $\liminf_{x\to\infty}Q_{k}\geq 0$. If $Q_{k}$ is
nonnegative everywhere for infinitely many $k$, then $\nu_{1}\geq 0$ follows
by letting $k\to\infty$. Arguing by contradiction, we may assume, without loss
of generality, that for all $k$, $Q_{k}$ is negative somewhere and hence it
attains its negative minimum at a point $x_{k}$. We must have $x_{k}\in K_{0}$
for each $k$, otherwise, by the minimum principle, (2.19), and (2.28), we have

$0\leq\Delta_{f-2\ln R}\,Q_{k}(x_{k})\leq k^{-1}\Delta_{f-2\ln
R}\left(e^{av^{-1/2}}v^{-1}R^{-1}\right)(x_{k})\leq-k^{-1}v^{-3/2}e^{av^{-1/2}}R^{-1}<0,$

which is impossible. Hence $\inf_{M}Q_{k}=Q_{k}(x_{k})=\min_{K_{0}}Q_{k}$. By
letting $k\to\infty$, we conclude that
$\inf_{M}R^{-1}\nu_{1}=\min_{K_{0}}R^{-1}\nu_{1}$ which implies that
$R^{-1}\nu_{1}$ attains its minimum. By Lemma 2.2, $M$ has nonnegative
sectional curvature.

Next we justify $\operatorname{Rm}>0$. Let $\pi:\widetilde{M}\longrightarrow
M$ be the universal covering map of $M$. It is clear that
$(\widetilde{M},\pi^{*}g,\pi^{*}f)$ is a complete nonflat expanding gradient
Ricci soliton. If the sectional curvature is not striclty positive, then by
the strong maximum principle of the Ricci flow [CLN06, Section 6.7] and the De
Rham splitting Theorem, $\widetilde{M}$ splits isometrically as
$\mathbb{R}\times N$, where $N$ is a $2$-dimensional expanding gradient Ricci
soliton. As $\lim_{x\to\infty}R_{g}=0$ and $M$ is not flat, $R_{g}$ attains
its maximum, say at $q_{0}\in M$. Let $(a,b)\in\mathbb{R}\times N$ such that
$\pi(a,b)=q_{0}$. For any $(s,b)\in\mathbb{R}\times\\{b\\}$, it is evident
that

$R_{\pi^{*}g}(s,b)=R_{\pi^{*}g}(a,b)=R_{g}(q_{0}).$

As $R_{g}\to 0$ at $\infty$, $\pi\left(\mathbb{R}\times\\{b\\}\right)\subseteq
K_{1}$ for some compact set $K_{1}$. Furthermore, the soliton equation
$\frac{\partial^{2}\pi^{*}f}{\partial s^{2}}=-1/2$ implies that

$\pi^{*}f(s,b)=-s^{2}/4+C_{1}s+C_{2}$

for some constants $C_{1}$ and $C_{2}$ independent on $s$. We have that for
any $s\in\mathbb{R}$

$-s^{2}/4+C_{1}s+C_{2}\geq\min_{K_{1}}f>-\infty,$

which is impossible. This completes the proof of Theorem 1.11 modulo formula
(2.28). It remains to establish formula (2.28).

###### Proof of formula (2.28):.

By virtues of (2.5) and $\Delta_{f}R=-R-2|{\operatorname{Ric}}|^{2}$, it holds
that

$\begin{split}\Delta_{f}v^{-1}&=-v^{-2}\Delta_{f}v+2|\nabla
v|^{2}v^{-3}=-v^{-1}+2|\nabla v|^{2}v^{-3},\\\
\Delta_{f}R^{-1}&=-R^{-2}\Delta_{f}R+2R^{-3}|\nabla
R|^{2}=R^{-1}(1+2R^{-1}|{\operatorname{Ric}}|^{2})+2R^{-3}|\nabla
R|^{2}.\end{split}$

By the product rule, we have

$\begin{split}\Delta_{f}\left(v^{-1}R^{-1}\right)&=R^{-1}\Delta_{f}v^{-1}+v^{-1}\Delta_{f}R^{-1}-2R^{-2}\langle\nabla
R,\nabla\left(v^{-1}R^{-1}R\right)\rangle\\\ &=-v^{-1}R^{-1}+2|\nabla
v|^{2}v^{-3}R^{-1}+v^{-1}R^{-1}(1+2R^{-1}|{\operatorname{Ric}}|^{2})+2v^{-1}R^{-3}|\nabla
R|^{2}\\\ &\,\,\,\,\,\,\,-2\langle\nabla\ln
R,\nabla\left(v^{-1}R^{-1}\right)\rangle-2v^{-1}R^{-3}|\nabla R|^{2}\\\
&=2|\nabla
v|^{2}v^{-3}R^{-1}+2v^{-1}R^{-2}|{\operatorname{Ric}}|^{2}-2\langle\nabla\ln
R,\nabla\left(v^{-1}R^{-1}\right)\rangle.\end{split}$

Hence from (1.7), (2.5), and Lemma 2.6, we have

$\begin{split}\Delta_{f-2\ln R}\left(v^{-1}R^{-1}\right)&\leq\left(2|\nabla
v|^{2}v^{-2}+2R^{-1}|{\operatorname{Ric}}|^{2}\right)v^{-1}R^{-1}\\\
&\leq\left(2v^{-1}+CR\right)v^{-1}R^{-1}\\\ &\leq
C_{0}v^{-3/2}R^{-1}.\end{split}$ (2.29)

Let $a$ be a large positive constant to be determined. By (2.5) again, we have

$\begin{split}\Delta_{f}e^{av^{-1/2}}=-\frac{a}{2}v^{-1/2}e^{av^{-1/2}}+\left(\frac{a^{2}}{4}v^{-3}+\frac{3a}{4}v^{-5/2}\right)|\nabla
v|^{2}e^{av^{-1/2}},\end{split}$ $\begin{split}2\langle\nabla\ln R,\nabla
e^{av^{-1/2}}\rangle=-aR^{-1}\langle\nabla R,\nabla v\rangle
v^{-3/2}e^{av^{-1/2}}.\end{split}$

Hence

$\Delta_{f-2\ln
R}\,e^{av^{-1/2}}\leq-\frac{a}{2}v^{-1/2}e^{av^{-1/2}}+C(a+a^{2})v^{-3/2}e^{av^{-1/2}}-aR^{-1}\langle\nabla
R,\nabla v\rangle v^{-3/2}e^{av^{-1/2}},$ (2.30)

where $C$ is some constant independent on $a$.

On the other hand, we compute

$\begin{split}2\langle\nabla
e^{av^{-1/2}},\nabla\left(v^{-1}R^{-1}\right)\rangle&=av^{-3/2}e^{av^{-1/2}}|\nabla
v|^{2}v^{-2}R^{-1}+av^{-3/2}e^{av^{-1/2}}\langle\nabla v,\nabla R\rangle
v^{-1}R^{-2}\\\ &=av^{-7/2}e^{av^{-1/2}}|\nabla
v|^{2}R^{-1}+av^{-5/2}e^{av^{-1/2}}\langle\nabla v,\nabla R\rangle
R^{-2}.\end{split}$ (2.31)

Thanks to (2.29), (2.30) and (2.31), we have

$\begin{split}\Delta_{f-2\ln
R}\left(e^{av^{-1/2}}v^{-1}R^{-1}\right)&=e^{av^{-1/2}}\Delta_{f-2\ln
R}\left(v^{-1}R^{-1}\right)+v^{-1}R^{-1}\Delta_{f-2\ln
R}\left(e^{av^{-1/2}}\right)\\\ &\,\,\,\,\,\,\,+2\langle\nabla
e^{av^{-1/2}},\nabla\left(v^{-1}R^{-1}\right)\rangle\\\
&\leq(C_{0}-a/2)v^{-3/2}e^{av^{-1/2}}R^{-1}+C(a+a^{2})v^{-5/2}e^{av^{-1/2}}R^{-1}\\\
&\,\,\,\,\,\,\,-a\langle\nabla R,\nabla v\rangle
v^{-5/2}e^{av^{-1/2}}R^{-2}+av^{-7/2}e^{av^{-1/2}}|\nabla v|^{2}R^{-1}\\\
&\,\,\,\,\,\,\,+av^{-5/2}e^{av^{-1/2}}\langle\nabla v,\nabla R\rangle
R^{-2}\\\
&\leq(C_{0}-a/2)v^{-3/2}e^{av^{-1/2}}R^{-1}+C(a+a^{2})v^{-5/2}e^{av^{-1/2}}R^{-1}.\end{split}$

Hence, fixing $a\gg C_{0}$, outside a compact set $K_{0}$ (possibly depending
on $a$), we have

$\Delta_{f-2\ln
R}\left(e^{av^{-1/2}}v^{-1}R^{-1}\right)\leq-v^{-3/2}e^{av^{-1/2}}R^{-1}<0.$

This finishes the proofs of formula (2.28) and Theorem 1.11. ∎

∎

Before proceeding to the proof of Corollary 1.16, we investigate some basic
geometric properties of a $C^{2}$ conical expanding soliton.

######  Proposition 2.8.

Let $(X^{n-1},g_{X})$ be a closed Riemannian manifold of dimension $n-1$,
where $n\geq 3$. Suppose that a complete noncompact expanding gradient Ricci
soliton $(M^{n},g,f)$ is $C^{2}$ asymptotic to $(C(X),g_{C})$, where
$g_{C}=dt^{2}+t^{2}g_{X}$. Then the following conditions hold:

  1. (a)

$f$ is proper, i.e. $\lim_{x\to\infty}f=-\infty$;

  2. (b)

$\operatorname{Ric}\geq(\varepsilon_{0}-1/2)g$ near infinity for some positive
constant $\varepsilon_{0}$;

  3. (c)

$\operatorname{Ric}\to 0$ at infinity (or $R\to 0$ at infinity if $n=3$).

###### Proof.

It suffices to show that $(c)\implies(b)\implies(a)$ and lastly $(c)$ holds.
$(c)\implies(b):$ We only need to consider the case when $n=3$ and
$\lim_{x\to\infty}R=0$. Under these assumptions, we may apply [Cha20, Theorems
11 and Theorem 12] to see that $\operatorname{Ric}\to 0$ as $x\to\infty$.
$(b)$ follows.
$(b)\implies(a):$ $\operatorname{Ric}\geq(\varepsilon_{0}-1/2)\,g$ and the
soliton equation imply that $-\nabla^{2}f\geq\varepsilon_{0}\,g$ outside a
compact subset. Integrating the inequality along minimizing geodesics as in
(2.6) yields

$-f\geq\varepsilon_{0}r^{2}/2-cr-c\quad\text{ on }\quad M$

for some positive constant $c$. It is then evident that
$\lim_{x\to\infty}f=-\infty$.

To see that $(c)$ holds, we recall that (1.8) implies

$\left|\phi^{*}g-g_{C}\right|_{g_{C}}(t,\omega)\leq
C_{0}\,S_{0}^{-3\varepsilon}$

and hence for any $S>S_{0}$, the set
$\phi\big{(}\\{(t,\omega):\,t\in(S_{0},S],\,\omega\,\in X\\}\big{)}$ is
bounded in $(M,g)$, where $\varepsilon$ is the positive number in Definition
1.13. As a result, we see that $\lim_{x\to\infty}t=\infty$. By the conical
estimate (1.8) with $l=2$, it holds that

$\begin{split}|{\operatorname{Ric}(g)}|_{g}&\leq
c|{\operatorname{Ric}(g_{C})}|_{g_{C}}+O(t^{-2-3\varepsilon/2})\\\ &\leq
O(t^{-2})\to 0\text{ as }x\to\infty.\end{split}$

This complete the proofs of $(c)$ and Proposition 2.8.

∎

With the above preparation, we are about to prove Corollary 1.16. For reader’s
convenience, we recall the statement of the corollary:

######  Corollary 2.9.

Suppose that $(M^{3},g,f)$ is a $3$-dimensional complete noncompact expanding
gradient Ricci soliton which is $C^{2}$ asymptotic to a cone
$\left(C(X^{2}),dt^{2}+t^{2}g_{X}\right)$, where $(X^{2},g_{X})$ is a
connected closed $2$ dimensional Riemannian manifold. Then the following are
equivalent:

  1. (1)

$M$ has nonnegative sectional curvature;

  2. (2)

$M$ has nonnegative scalar curvature;

  3. (3)

$C(X)$ has nonnegative scalar curvature;

  4. (4)

$\operatorname{Rm}(g_{X})\geq{\rm id}_{\Lambda^{2}TX}$

###### Proof.

$(1)\implies(2)$ is immediate.
$(2)\implies(3):$ Let $\phi$ be the diffeomorphism in Definition 1.13. By
$R_{g}\geq 0$ and (1.8), for any $\omega\in X$, we have

$0\leq\lim_{t\to\infty}4v\circ\phi(t,\omega)R_{g}\circ\phi(t,\omega)=R_{g_{C}}(1,\omega).$

Since $g_{C}=dt^{2}+t^{2}g_{X}$ is a warped product metric with warping
function $t$, the scalar curvature satisfies [Li12, Appendix A]

$R_{g_{C}}(t,\omega)=R_{g_{C}}(1,\omega)/t^{2}\geq 0.$

This proves $(3)$.
$(3)\implies(4):$ Since $X$ is of dimension $2$, $(4)$ is equivalent to
$R_{g_{X}}\geq 2$. Thanks to $(3)$ and the properties of warped product [Li12,
Appendix A], we have for any $(t,\omega)\in C(X)$,

$R_{g_{C}}(t,\omega)=\frac{R_{g_{X}}(\omega)-2}{t^{2}}.$ (2.32)

In particular, $0\leq R_{g_{C}}(1,\omega)=R_{g_{X}}(\omega)-2$ for any
$\omega\in X$ and hence $(4)$ holds.
$(4)\implies(1):$ By our assumptions and $(4)$, $(M,g,f)$ is $C^{2}$
asymptotic to $(C(X),g_{C})$, where $g_{C}=dt^{2}+t^{2}g_{X}$ and
$\operatorname{Rm}(g_{X})\geq 1$. Using Proposition 2.8 (b) and (2.6), we have
$-f\sim r^{2}$ near infinity. Moreover, we can find a diffeomorphism $\phi$
satisfying the conditions in Definition 1.13. By virtues of (1.8), (1.9) and
(2.32), we have for all sufficiently large $t$,

$\begin{split}R_{g}\circ\phi(t,\omega)=R_{\phi^{*}g}(t,\omega)&=R_{g_{C}}(t,\omega)+O(t^{-3\varepsilon-2})\\\
&=\left(R_{g_{X}}(\omega)-2\right)t^{-2}+O(t^{-3\varepsilon-2})\\\ &\leq
c\,t^{-2},\end{split}$ (2.33)

where $c$ is some positive constant independent on large $t$ and $\varepsilon$
is the positive number in Definition 1.13. We then use (1.9), (2.33), and the
fact $-f\sim r^{2}$ to conclude that outside some compact subset of $M$, it
holds that

$t\sim r\,;$ (2.34) $R_{g}\leq Cr^{-2}=o(r^{-1}),$ (2.35)

where $C$ is a positive constant. To apply Theorem 1.11, It remains to justify
the lower bound in (1.7), i.e. $R_{g}\geq-o(r^{-2})$. Indeed, by (1.8), It
holds that

$R_{g}\geq\left(R_{g_{X}}-2\right)t^{-2}-O(t^{-3\varepsilon-2})\geq-O(t^{-3\varepsilon-2}),$

where we used $R_{g_{X}}\geq 2$ from assumption $(4)$ in the last inequality.
Hence by (2.34) and (2.35), we have $r^{2}R_{g}\geq-Cr^{-3\varepsilon}\to 0$
as $x\to\infty$ and (1.7) is satisfied. We may then invoke Theorem 1.11 and
conclude that $M$ has nonnegative sectional curvature. This establishes $(1)$
and completes the proof of Corollary 1.16. ∎

We shall end this section by proving another application of Theorem 1.11:

###### Proof of Corollary 1.19.

As $\operatorname{Rm}(\alpha g_{\mathbb{S}^{2}})\geq 1/\alpha\geq 1$ for any
$\alpha\in(0,1]$, it follows from Corollary 1.16 that $M$ is nonnegatively
curved. By the strong maximum principle argument as in the proof of Theorem
1.11, $M$ is either flat or has positive sectional curvature. Corollary 1.19
becomes obvious in the former case. Hence we may assume that
$\operatorname{Rm}>0$ on $M^{3}$. The rotational symmetry of $M$ then follows
from the result by Chodosh [Cho14, Theorem 1.2] (see also Theorem 1.15). ∎

##  3 Curvature estimates in 4D steady soliton case

Let $(M^{4},g,f)$ be a $4$-dimensional complete steady gradient Ricci soliton,
i.e.,

$\operatorname{Ric}+\nabla^{2}f=0.$ (3.1)

By a result of Hamilton, it is well known that on a complete non-Ricci-flat
(and hence noncompact) steady gradient Ricci soliton, upon scaling the metric
if necessary, the following identity holds,

$|\nabla f|^{2}+R=1.$ (3.2)

Chen [Che09] showed that the scalar curvature of any complete ancient Ricci
flow must be nonnegative. Consequently, by the strong minimum principle, the
scalar curvature of a steady gradient soliton is positive everywhere unless
$M$ is Ricci flat. Recall that as in Section $2$, for any smooth function
$\gamma$, the weighted laplacian $\Delta_{\gamma}$ is defined as

$\Delta_{\gamma}:=\Delta-\nabla_{\nabla\gamma},$

where $\Delta$ is the standard laplace operator. In dimension four, the
curvature operator becomes more complicated than the $3$-dimensional case. We
shall apply the techniques by Munteanu-Wang [MW15] and carry out the dimension
reduction via the level sets of the potential function $f$. The crucial
observation is that the curvature tensor in $\nabla f$ direction can be
written as $\nabla{\operatorname{Ric}}$, namely, for any tangent vectors $X,Y$
and $Z$,

$\nabla_{Z}\operatorname{Ric}(X,Y)-\nabla_{Y}\operatorname{Ric}(X,Z)=R(Z,Y,X,\nabla
f).$ (3.3)

The above equation is a consequence of the soliton equation (3.1) and the
Ricci identity. It leads to the following lemma due to Munteanu-Wang [MW15,
Proposition 2.1] which allows us to control the Riemann curvature tensor by
the Ricci tensor and its derivatives.

######  Lemma 3.1.

[MW15] Suppose that $(M^{4},g,f)$ is a $4$-dimensional gradient Ricci soliton.
Then there exists a universal positive constant $A_{0}$, such that, if $\nabla
f\neq 0$ at $q$ $\in M$, then

$|{\operatorname{Rm}}|(q)\leq
A_{0}\left(|{\operatorname{Ric}}|(q)+\frac{|\nabla{\operatorname{Ric}}|}{|\nabla
f|}(q)\right).$ (3.4)

######  Lemma 3.2.

Let $(M^{4},g,f)$ be a $4$-dimensional complete noncompact non-Ricci-flat
steady gradient Ricci soliton with bounded Ricci curvature. Then there exists
a positive constant $c_{1}$ such that

$|{\operatorname{Ric}}|\leq c_{1}R\quad\text{ on }\quad M.$ (3.5)

###### Proof.

By the boundedness of $|{\operatorname{Ric}}|$, we can find a finite positive
constant $L$ such that

$|{\operatorname{Ric}}|\leq L\quad\text{ on }\quad M.$ (3.6)

Using the computation in [Cha19, p. 9001], we see that wherever $\nabla f\neq
0$,

$\begin{split}\Delta_{f}(|{\operatorname{Ric}}|+|{\operatorname{Ric}}|^{2})&\geq-\frac{A_{0}^{2}|{\operatorname{Ric}}|^{2}}{|\nabla
f|^{2}}-2A_{0}|{\operatorname{Ric}}|^{2}-4A_{0}|{\operatorname{Ric}}|^{3}-\frac{4A_{0}^{2}|{\operatorname{Ric}}|^{4}}{|\nabla
f|^{2}}.\end{split}$

Hence, by (3.6), one can find a constant $Q_{0}=Q_{0}(A_{0},L)>0$ such that

$\begin{split}\Delta_{f}(|{\operatorname{Ric}}|+|{\operatorname{Ric}}|^{2})&\geq-
Q_{0}|{\operatorname{Ric}}|^{2}\quad\text{ on }\quad\left\\{x\in M:|\nabla
f|^{2}(x)>\tfrac{1}{2}\right\\}.\end{split}$

Let $Q_{1}$ be another large positive constant such that

$\begin{split}Q_{1}&\geq\frac{Q_{0}}{2}+1\text{ and }\\\ \frac{Q_{1}}{2}&\geq
L^{2}+L+1.\end{split}$ (3.7)

Letting $u:=|{\operatorname{Ric}}|+|{\operatorname{Ric}}|^{2}-Q_{1}R$, from
[CCGGIIKLLN07, (1.33)], we have

$\Delta_{f}R=-2|{\operatorname{Ric}}|^{2}$ (3.8)

and

$\Delta_{f}u\geq 2|{\operatorname{Ric}}|^{2}\quad\text{ on
}\quad\left\\{|\nabla f|^{2}>\tfrac{1}{2}\right\\}.$ (3.9)

If we can prove that $u\leq 0$ on $M$, then we are done with the lemma. Since
the Ricci curvature is bounded, $u$ is also bounded. By a result of Pigola-
Rimoldi-Setti [PRS11, Corollary 10], the weak maximum principle of
$\Delta_{f}$ holds on $M$ and there is a sequence $\\{x_{k}\\}_{k=1}^{\infty}$
on $M$ such that

$u(x_{k})\geq\sup_{M}u-\frac{1}{k}\quad\text{ and
}\quad\Delta_{f}u(x_{k})\leq\frac{1}{k}.$ (3.10)

Since the Ricci curvature is bounded from below and $|\nabla f|$ is uniformly
bounded by (3.2), the classical Omori-Yau maximum principle can also be used
to obtain a sequence $\\{x_{k}\\}_{k=1}^{\infty}$ satisfying (3.10) (see
[AMP16, Theorem 2.3]).

Case 1: if $|\nabla f|^{2}(x_{k_{j}})\leq\frac{1}{2}$ for some subsequence
$\\{x_{k_{j}}\\}_{j=1}^{\infty}$, then by (3.2), (3.6), and (3.7), we see that
$R(x_{k_{j}})\geq\frac{1}{2}$ and

$\begin{split}\sup_{M}u&\leq u(x_{k_{j}})+\frac{1}{k_{j}}\\\ &\leq
L+L^{2}-\frac{Q_{1}}{2}+\frac{1}{k_{j}}\\\
&\leq-1+\frac{1}{k_{j}}.\end{split}$

Letting $j\to\infty$, we have $u\leq 0$ on $M$.

Case 2: If $|\nabla f|^{2}(x_{k})>\frac{1}{2}$ for all large $k$, then by the
differential inequality (3.9) and (3.10)

$2|{\operatorname{Ric}}|^{2}(x_{k})\leq\Delta_{f}u(x_{k})\leq\frac{1}{k}\to 0$

and hence

$\sup_{M}u=\lim_{k\to\infty}u(x_{k})=0.$

This finishes the proof of the lemma. ∎

With the above preparation, we are going to prove Theorem 1.3.

###### Proof of Theorem 1.3:.

By our assumption, there is a finite constant $L_{1}>0$ such that

$|{\operatorname{Rm}}|\leq L_{1}\quad\text{ on }\quad M.$ (3.11)

It is well known that in any gradient Ricci soliton, the gradient of the
scalar curvature $\nabla R$ can be expressed as (see [CCGGIIKLLN07, (1.27)])

$\nabla R=2\operatorname{Ric}(\nabla f).$ (3.12)

Then by the calculation in [Cha19, p.9002], (3.12), and Lemma 3.2, there exist
constants $c_{3}$ and $c^{\prime}_{3}$ depending on $A_{0}$, $L$, $\lambda$,
and $c_{1}$ in (3.5) such that

$\begin{split}\Delta_{f-2\ln
R}\,\frac{|{\operatorname{Ric}}|^{2}}{R^{2}}&\geq\frac{|{\nabla\operatorname{Ric}}|^{2}}{2R^{2}}-\frac{4A_{0}|{\operatorname{Ric}}|^{3}}{R^{2}}-\frac{4A_{0}^{2}|{\operatorname{Ric}}|^{4}}{R^{2}|\nabla
f|^{2}}-\frac{6|\nabla\ln R|^{2}|{\operatorname{Ric}}|^{2}}{R^{2}}\\\
&\geq\frac{|\nabla{\operatorname{Ric}}|^{2}}{2R^{2}}-c_{3}\\\ &\geq
c^{\prime}\left(\frac{|{\operatorname{Rm}}|}{R}+\lambda\frac{|{\operatorname{Ric}}|^{2}}{R^{2}}\right)^{2}-c_{3}^{\prime}\quad\text{
on }\quad\left\\{|\nabla f|^{2}>\tfrac{1}{2}\right\\}\end{split}$ (3.13)

for all large $\lambda$, where $c^{\prime}=c^{\prime}(A_{0})$ is a positive
constant. We also used (3.4) in the last inequality. By [Cha19, (48)], we have

$\Delta_{f-2\ln
R}\,\frac{|{\operatorname{Rm}}|}{R}\geq-5\frac{|{\operatorname{Rm}}|^{2}}{R}.$
(3.14)

Combining (3.13) and (3.14), we see that for all sufficiently large $\lambda$,
we have

$\Delta_{f-2\ln R}\,W\geq W^{2}-c^{\prime\prime}\quad\text{ on
}\quad\left\\{|\nabla f|^{2}>\tfrac{1}{2}\right\\},$ (3.15)

where $W:=R^{-1}|{\operatorname{Rm}}|+\lambda
R^{-2}|{\operatorname{Ric}}|^{2}$ and
$c^{\prime\prime}=c^{\prime\prime}(A_{0},L,,\lambda,c_{1})$ is some positive
constant. We localize the function $W$ by considering

$G:=\phi^{2}W,$

where $\phi=\psi\left(\frac{d(x,p_{0})}{\rho}\right)$, $\rho$ is a large
positive number, $\psi:[0,\infty)\longrightarrow[0,1]$ is a cut off function
satisfying

$\psi(t)=\begin{cases}1&\text{ if }t\leq 1;\\\ 0&\text{ if }t>2,\end{cases}$

$\psi^{\prime}\leq 0$, and $|\psi^{\prime}|+|\psi^{\prime\prime}|\leq c$ for
some constant $c$. Hence we have

$|\nabla\phi|\leq\frac{c}{\rho}.$

Thanks to the Laplacian comparison theorem for smooth metric measure space
[WW09, Theorem 1.1], it holds that

$\Delta_{f}r\leq\frac{3}{r}+1.$

Then by (3.12) and Lemma 3.2 , we have the estimate on the weighted Laplacian
of $\phi$

$\begin{split}\Delta_{f-2\ln R}\,\phi&=\Delta_{f}\phi+2\langle\nabla\ln
R,\nabla\phi\rangle\\\
&\geq\frac{\psi^{\prime}}{\rho}\Delta_{f}r+\frac{\psi^{\prime\prime}}{\rho^{2}}|\nabla
r|^{2}-\frac{4cc_{1}}{\rho}\\\
&\geq-\frac{c}{\rho}\left(\frac{3}{\rho}+1\right)-\frac{c}{\rho^{2}}-\frac{4cc_{1}}{\rho}\\\
&\geq-c\end{split}$

for all large $\rho$, see also [Per02, WW09, Cha19]. It follows from (3.15)
that

$\begin{split}\phi^{2}\Delta_{f-2\ln R}\,G=&\text{ }\phi^{4}\Delta_{f-2\ln
R}\,W+\left(2\phi\Delta_{f-2\ln R}\,\phi+2|\nabla\phi|^{2}\right)G\\\ &\text{
}+4\phi^{3}\langle\nabla\phi,\nabla(\phi^{-2}G)\rangle\\\ =&\text{
}\phi^{4}\Delta_{f-2\ln R}\,W+(2\phi\Delta_{f-2\ln
R}\,\phi-6|\nabla\phi|^{2})G\\\ &\text{ }+4\phi\langle\nabla\phi,\nabla
G\rangle\\\ \geq&\text{ }\phi^{4}W^{2}+(2\phi\Delta_{f-2\ln
R}\,\phi-6|\nabla\phi|^{2})G\\\ &\text{ }+4\phi\langle\nabla\phi,\nabla
G\rangle-c^{\prime\prime}\\\ \geq&\text{
}G^{2}-c_{4}G-c_{4}+4\phi\langle\nabla\phi,\nabla G\rangle\quad\text{ on
}\quad\left\\{|\nabla f|^{2}>\tfrac{1}{2}\right\\}\end{split}$ (3.16)

for some positive constant $c_{4}$ independent on $\rho$. If the maximum of
$G$ is attained on $\\{|\nabla f|^{2}>\frac{1}{2}\\}$, then by the maximum
principle and (3.16), at the point where $G$ attains its maximum we have

$0\geq G^{2}-c_{4}G-c_{4}$

and hence $G\leq c_{5}$ for some positive constant $c_{5}$ independent on
$\rho$. If instead $G$ attains its maximum on $\\{|\nabla
f|^{2}\leq\frac{1}{2}\\}$, then it follows from (3.2), (3.11), and the
definition of $G$ that $R\geq\frac{1}{2}$ and

$G\leq 2L_{1}+4L^{2}\lambda.$

By letting $\rho\to\infty$, we have $|{\operatorname{Rm}}|\leq c_{2}R$ for
some constant $c_{2}>0$. This completes the proofs of Theorem 1.3.

∎

Under the conditions that the scalar curvature decays at least linearly, i.e.
$R\leq C/(r+1)$ on $M$, we have, by [CC20, Theorem 4.1], that
$\operatorname{Rm}$ is bounded. It then follows from Theorem 1.3 that

$|{\operatorname{Rm}}|\leq c_{2}R\leq c_{2}C/(r+1).$ (3.17)

Therefore we may invoke Shi’s derivative estimates [CLN06] to obtain the
derivative estimates for $\operatorname{Rm}$, namely, for any nonnegative
integer $k$, we have

$|\nabla^{k}{\operatorname{Rm}}|\leq\frac{C_{k}}{(r+1)^{(k+2)/2}}.$ (3.18)

In view of (3.3) and the derivative estimate (3.18), we see that the
$\operatorname{Rm}$ in $\nabla f$ direction is of the order $r^{-3/2}$ and
hence we can restrict our consideration of $\operatorname{Rm}$ on the
orthogonal complement of $\nabla f$, i.e. the tangent space of level set of
$f$ which is of dimension three. This allows the use of Hamilton-Ivey estimate
as in the previous section with error terms due to the curvature in $\nabla f$
direction. By our assumption in Theorem 1.6, $R\to 0$ as $x\to\infty$ and $f$
is proper. Hence outside a compact set, $|\nabla f|^{2}=1-R\geq 1/2$. Thanks
to [Cha19, Lemma 2] and the properness of $f$, there is constant $c>0$ such
that outside a compact subset, it holds that

$c^{-1}r\leq v=-f\leq cr.$ (3.19)

Moreover, for all large $\tau\gg 1$, the level set of the potential
$\Sigma:=\\{f=-\tau\\}$ is a compact hypersurface; this level set is also
connected by a result of Munteanu-Wang [MW11]. Hence there exist a compact set
$K_{0}$ and a positive constant $\tau_{0}\gg 1$ such that $M\setminus K_{0}$
is foliated by the smooth and closed level sets, namely

$M\setminus K_{0}=\bigcup_{\tau>\tau_{0}}\\{f=-\tau\\}$ (3.20)

and $|\nabla f|^{2}\geq 1/2$ on $M\setminus K_{0}$. The second fundamental
form $A_{\Sigma}$ of $\Sigma$ with respect to the normal $\nabla f/|\nabla f|$
is given by

$A_{\Sigma}=\nabla^{2}f/|\nabla f|=-\operatorname{Ric}(g)/|\nabla
f|=O(r^{-1}).$

Let $\tilde{g}$ be the metric on $\Sigma$ induced by $g$. Let
$\\{e_{i}\\}_{i=1}^{4}$ be an orthonormal frame such that $e_{4}=\nabla
f/|\nabla f|$, which is well defined near infinity. As in [MW15], throughout
this section, $a,b,c,d$ and $i,j,k,l$ shall denote the indices in
$\\{1,2,3\\}$ and $\\{1,2,3,4\\}$, respectively. We begin with a
$4$-dimensional analog of (2.2); its shrinker version was proved by Munteanu-
Wang [MW15, Lemma 4.1].

######  Lemma 3.3.

Under the assumptions of Theorem 1.6, outside a compact subset of $M$, it
holds that

$\begin{split}\operatorname{Rm}(g)_{abcd}&=\operatorname{Ric}(g)_{ad}g_{bc}+\operatorname{Ric}(g)_{bc}g_{ad}-\operatorname{Ric}(g)_{ac}g_{bd}-\operatorname{Ric}(g)_{bd}g_{ac}\\\
&\,\,\,-R(g)\left(g_{ad}g_{bc}-g_{ac}g_{bd}\right)/2+O(r^{-3/2}),\end{split}$

where $\\{e_{i}\\}_{i=1}^{4}$ is any orthonormal frame such that $e_{4}=\nabla
f/|\nabla f|$ and $a,b,c,d\in\\{1,2,3\\}$.

###### Proof.

Let $K_{0}$ be the compact set as in (3.20). For any $x\in M\setminus K_{0}$,
$x$ belongs to $\Sigma:=\\{f=-\tau\\}$ for some large $\tau$. Let $\tilde{g}$
denote the induced metric on $\Sigma$. By the Gauss equation, we have

$\begin{split}\operatorname{Rm}(g)_{abcd}&=\operatorname{Rm}(\tilde{g})_{abcd}-f_{ad}f_{bc}/|\nabla
f|^{2}+f_{ac}f_{bd}/|\nabla f|^{2}\\\
&=\operatorname{Rm}(\tilde{g})_{abcd}+O(r^{-2}).\end{split}$ (3.21)

Similarly, using (3.3) and (3.18), we also have

$\begin{split}\operatorname{Ric}(g)_{ab}&=\operatorname{Ric}(\tilde{g})_{ab}+\operatorname{Rm}(g)_{a44b}+O(r^{-2})=\operatorname{Ric}(\tilde{g})_{ab}+O(r^{-3/2}),\\\
R(g)&=R(\tilde{g})+O(r^{-3/2}).\end{split}$ (3.22)

Moreover, $\Sigma$ is of dimension three. Hence, by (2.2), (3.21), and (3.22),
we have

$\begin{split}\operatorname{Rm}(g)_{abcd}&=\operatorname{Rm}(\tilde{g})_{abcd}+O(r^{-2})\\\
&=\operatorname{Ric}(\tilde{g})_{ad}\tilde{g}_{bc}+\operatorname{Ric}(\tilde{g})_{bc}\tilde{g}_{ad}-\operatorname{Ric}(\tilde{g})_{ac}\tilde{g}_{bd}-\operatorname{Ric}(\tilde{g})_{bd}\tilde{g}_{ac}\\\
&\,\,\,-R(\tilde{g})\left(\tilde{g}_{ad}\tilde{g}_{bc}-\tilde{g}_{ac}\tilde{g}_{bd}\right)/2+O(r^{-2})\\\
&=\operatorname{Ric}(g)_{ad}g_{bc}+\operatorname{Ric}(g)_{bc}g_{ad}-\operatorname{Ric}(g)_{ac}g_{bd}-\operatorname{Ric}(g)_{bd}g_{ac}\\\
&\,\,\,-R(g)\left(g_{ad}g_{bc}-g_{ac}g_{bd}\right)/2+O(r^{-3/2}).\end{split}$

This finishes the proof of the lemma. ∎

In view of (3.21), to bound $\operatorname{Rm}(g)$ from below, it suffices to
provide a quantitative lower bound of $\operatorname{Rm}(\tilde{g})$ by
considering the tensor
$T(\tilde{g}):=R(\tilde{g})\tilde{g}-2\operatorname{Ric}(\tilde{g})$
introduced in the previous section. Let

$\begin{split}\tilde{\lambda}_{1}\leq\tilde{\lambda}_{2}\leq\tilde{\lambda}_{3}&\text{
be the eigenvalues of }\operatorname{Ric}(\tilde{g}),\\\
\tilde{\nu}_{1}\leq\tilde{\nu}_{2}\leq\tilde{\nu}_{3}&\text{ be the
eigenvalues of }T(\tilde{g}).\end{split}$

Then it clearly follows from (2.1) that

$\tilde{\nu}_{1}=R(\tilde{g})-2\tilde{\lambda}_{3}.$ (3.23)

We further let $U$ be the tensor which approximates $T(\tilde{g})$

$U:=\big{(}R(g)-\operatorname{Ric}(g)_{44}\big{)}g-2\operatorname{Ric}(g);$
(3.24) $\begin{split}\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}&\text{ be the
eigenvalues of }\operatorname{Ric}(g)\text{ when restricted on }T\Sigma,\\\
\nu_{1}\leq\nu_{2}\leq\nu_{3}&\text{ be the eigenvalues of }U\text{ when
restricted on }T\Sigma,\end{split}$ (3.25)

where $T\Sigma$ denotes the tangent space of the level set $\Sigma$ of $f$. We
also consider the trace of $U$ with respect to $\tilde{g}$

$\overline{U}:=U_{aa}=\nu_{1}+\nu_{2}+\nu_{3}.$

It follows from the definition of $\lambda_{i}$ that
$\overline{U}=R(g)-\operatorname{Ric}(g)_{44}=\lambda_{1}+\lambda_{2}+\lambda_{3}$
is smooth as long as $\nabla f\neq 0$. By an argument similar to (2.1), we
have

$\begin{split}\nu_{1}&=\lambda_{1}+\lambda_{2}-\lambda_{3}=\overline{U}-2\lambda_{3}\\\
\nu_{2}&=\lambda_{1}+\lambda_{3}-\lambda_{2}=\overline{U}-2\lambda_{2}\\\
\nu_{3}&=\lambda_{2}+\lambda_{3}-\lambda_{1}=\overline{U}-2\lambda_{1}.\end{split}$
(3.26)

By virtues of (3.18), (3.22), and (3.23), it can be seen that
$\operatorname{Ric}(g)_{44}=O(r^{-3/2})$,
$\tilde{\lambda}_{3}\leq\lambda_{3}+O(r^{-3/2})$ and

$\begin{split}\tilde{\nu}_{1}&\geq R(g)-2\tilde{\lambda}_{3}-O(r^{-3/2})\\\
&\geq R(g)-2\lambda_{3}-O(r^{-3/2})\\\ &\geq\nu_{1}-O(r^{-3/2}).\end{split}$
(3.27)

Our goal is to bound $\nu_{1}$ from below near infinity by the maximum
principle. We first compute the differential inequality satisfied by
$\nu_{1}$. The computations involved in the proof are parallel to those in the
proof of Lemma 2.6 except for the error terms due to the extra dimension.

######  Lemma 3.4.

$\nu_{1}$ satisfies the following differential equation in the barrier sense,

$\Delta_{f}\nu_{1}\leq-\nu_{1}^{2}-\nu_{2}\nu_{3}+O(r^{-5/2}).$

###### Proof.

For the sake of simplicity, in the rest of this section, the connections,
norms, and various curvature quantities are taken with respect to the soliton
metric $g$ unless specified. As metioned above, $\\{e_{i}\\}_{i=1}^{4}$ is an
orthonormal frame with $e_{4}=\nabla f/|\nabla f|$. Let
$a,b,c,d\in\\{1,2,3\\}$ and $i,j,k,l\in\\{1,2,3,4\\}$. $\\{e_{a}\\}_{a=1}^{3}$
can be further chosen such that at the point under consideration,
$R_{ab}=\lambda_{a}\delta_{ab}$. By the differential equation of
$\operatorname{Ric}$ [PW10, Lemma 2.1], (3.3), a computation similar to
(2.14), and Lemma 3.3, it holds that

$\begin{split}\Delta_{f}R_{ab}&=-2R_{aijb}R_{ij}\\\
&=-2R_{acdb}R_{cd}+O(r^{-5/2})\\\
&=-2\overline{U}R_{ab}-2R_{cd}R_{cd}g_{ab}+4R_{ac}R_{bc}+\overline{U}^{2}g_{ab}-\overline{U}R_{ab}+O(r^{-5/2})\\\
&=-3\overline{U}R_{ab}-2R_{cd}R_{cd}g_{ab}+4R_{ac}R_{bc}+\overline{U}^{2}g_{ab}+O(r^{-5/2}),\end{split}$

where $\nu_{i}$ and $\lambda_{i}$ are given by (3.25). We first claim that in
the barrier sense, it holds that

$\Delta_{f}\lambda_{3}\geq-3\overline{U}\lambda_{3}-2R_{cd}R_{cd}+4\lambda_{3}^{2}+\overline{U}^{2}+O(r^{-5/2})$
(3.28)

outside a compact set. We include the details here, since $\lambda_{3}$ is an
eigenvalue of $\operatorname{Ric}$ restricted on $T\Sigma$, and extra care
needs to be taken.

###### Proof of Claim (3.28):.

At any point where $\nabla f\neq 0$, let $W$ be a tangent vector of $M$. We
denote the orthogonal projection of $W$ onto $\\{\nabla f\\}^{\perp}$ by
$W^{T}$, i.e.

$W^{T}:=W-\langle W,\nabla f\rangle\nabla f/|\nabla f|^{2}.$

Let $q$ be any point near infinity with $\nabla f\neq 0$. We choose a smooth
vector field $Q$ of $M$ near $q$ as in [CCGGIIKLLN08] such that $Q(q)\in
T_{q}\Sigma$, $Q(q)=e_{3}(q)$ and thus
$\left[\operatorname{Ric}(Q(q))\right]^{T}=\lambda_{3}Q(q)$. Moreover, fixing
any $j\in\\{1,2,3,4\\}$, we have

$\left(\nabla_{j}Q\right)(q)=0\quad\text{ and
}\quad\left(\nabla_{j}\nabla_{j}Q\right)(q)=0.$ (3.29)

Since $Q$ is not necessarily a section of $T\Sigma$ near $q$,
$\operatorname{Ric}(Q,Q)$ may not be a barrier function required. Instead, we
let $\Psi$ be the smooth function
$\operatorname{Ric}(Q^{T},Q^{T})/|Q^{T}|^{2}$ near $q$. By the choice of $Q$,
we have $Q^{T}(q)=Q(q)$ and $\Psi\leq\lambda_{3}$ near $q$ with equality
holding at $q$. Furthermore, it follows from (3.12), (3.18), and (3.29) that
at the point $q$ we have

$\nabla_{j}\left(Q^{T}\right)=\operatorname{Ric}(Q,e_{j})\nabla f/|\nabla
f|^{2}=\begin{cases}O(r^{-1})&\text{ if }j=1,2,3;\\\ O(r^{-3/2})&\text{ if
}j=4.\end{cases}$

and for $j=1,2,3,4$,

$\nabla_{j}\nabla_{j}\left(Q^{T}\right)=O(r^{-3/2})$

One may check using the above derivative estimates of $Q^{T}$ to see that at
the point $q$ we have

$\Delta_{f}\Psi\geq-3\overline{U}\lambda_{3}-2R_{cd}R_{cd}+4\lambda_{3}^{2}+\overline{U}^{2}+O(r^{-5/2}).$

This justifies Assertion (3.28). ∎

Next We claim that

$\Delta_{f}\overline{U}=-2R_{cd}R_{cd}+O(r^{-5/2})$ (3.30)

###### Proof of Claim (3.30):.

Since $\overline{U}=R-R_{44}$, it can be seen from (3.8), (3.12) and (3.18)
that

$\displaystyle\Delta_{f}\overline{U}$ $\displaystyle=$
$\displaystyle\Delta_{f}R-\Delta_{f}(R_{44})$ $\displaystyle=$
$\displaystyle-2|{\operatorname{Ric}}|^{2}-\Delta_{f}(R_{44})$
$\displaystyle=$ $\displaystyle-2R_{cd}R_{cd}-\Delta_{f}(R_{44})+O(r^{-5/2}).$

Using the facts that $2{\operatorname{Ric}}(\nabla f)=\nabla R$ (3.12) and
$2{\operatorname{Ric}}(\nabla f,\nabla f)=\langle\nabla R,\nabla
f\rangle=\Delta R+2|{\operatorname{Ric}}|^{2}$ (3.8), we have

$\displaystyle\Delta_{f}(R_{44})$ $\displaystyle=$
$\displaystyle\Delta_{f}\left(\frac{\operatorname{Ric}(\nabla f,\nabla
f)}{|\nabla f|^{2}}\right)$ $\displaystyle=$ $\displaystyle|\nabla
f|^{-2}\Delta_{f}\left(\operatorname{Ric}(\nabla f,\nabla
f)\right)+\operatorname{Ric}(\nabla f,\nabla f)|\nabla f|^{-4}\Delta_{f}R$
$\displaystyle+2\operatorname{Ric}(\nabla f,\nabla f)|\nabla f|^{-6}|\nabla
R|^{2}+2\langle\nabla\left(\operatorname{Ric}(\nabla f,\nabla f)\right),\nabla
R\rangle|\nabla f|^{-4}$ $\displaystyle=$ $\displaystyle|\nabla
f|^{-2}\Delta_{f}\left(\operatorname{Ric}(\nabla f,\nabla f)\right)-|\nabla
f|^{-4}|{\operatorname{Ric}}|^{2}\left(\Delta
R+2|{\operatorname{Ric}}|^{2}\right)$ $\displaystyle+|\nabla f|^{-6}|\nabla
R|^{2}\left(\Delta R+2|{\operatorname{Ric}}|^{2}\right)+|\nabla
f|^{-4}\langle\nabla\Delta
R+4\langle\nabla\operatorname{Ric},\operatorname{Ric}\rangle,\nabla R\rangle$
$\displaystyle=$ $\displaystyle|\nabla
f|^{-2}\Delta_{f}\left(\operatorname{Ric}(\nabla f,\nabla
f)\right)+O(r^{-4}).$

By a computation of Petersen-Wylie [PW10, Lemma 2.4], (3.18), and (3.3), we
have

$\displaystyle\Delta_{f}\left(\operatorname{Ric}(\nabla f,\nabla f)\right)$
$\displaystyle=$ $\displaystyle-4\langle\nabla_{\nabla
f}\operatorname{Ric},\operatorname{Ric}\rangle+2\operatorname{Ric}(\nabla_{i}\nabla
f,\nabla_{i}\nabla f)+2R_{4ij4}R_{ij}|\nabla f|^{2}$ $\displaystyle=$
$\displaystyle
O(r^{-5/2})+2R_{jk}R_{ij}R_{ik}+2(R_{ij,4}-R_{j4,i})R_{ij}|\nabla f|$
$\displaystyle=$ $\displaystyle O(r^{-5/2}).$

Thus $\Delta_{f}(R_{44})=O(r^{-5/2})$ and Claim (3.30) follows. ∎

Hence by Claims (3.28) and (3.30), in the barrier sense

$\Delta_{f}\nu_{1}=\Delta_{f}(\overline{U}-2\lambda_{3})\leq
6\overline{U}\lambda_{3}-2\overline{U}^{2}-8\lambda_{3}^{2}+2R_{cd}R_{cd}+O(r^{-5/2}).$
(3.31)

Thanks to (3.26) and a computation similar to the proof of (2.16), we have

$\begin{split}6\overline{U}\lambda_{3}-2\overline{U}^{2}-8\lambda_{3}^{2}+2R_{cd}R_{cd}&=6(\nu_{1}+\nu_{2}+\nu_{3})(\nu_{2}+\nu_{3})/2-2(\nu_{1}+\nu_{2}+\nu_{3})^{2}\\\
&\,\,\,-2(\nu_{2}+\nu_{3})^{2}+\left[(\nu_{1}+\nu_{2})^{2}+(\nu_{2}+\nu_{3})^{2}+(\nu_{1}+\nu_{3})^{2}\right]/2\\\
&=-\nu_{1}^{2}-\nu_{2}\nu_{3}.\end{split}$

We then rewrite (3.31) as the required inequality

$\Delta_{f}\nu_{1}\leq-\nu_{1}^{2}-\nu_{2}\nu_{3}+O(r^{-5/2}).$

∎

Hence by (3.21) and (3.27), Theorem 1.6 is a consequence of the following
proposition.

######  Proposition 3.5.

Under the above notations and the same assumptions in Theorem 1.6, the
following inequality holds outside a compact set, we have

$\nu_{1}\geq-Cr^{-3/2},$

where $C$ is a positive constant.

###### Proof.

Using (3.8), we see that
$\Delta_{f}R^{-1}=2R^{-2}|{\operatorname{Ric}}|^{2}+2R^{-1}|\nabla\ln R|^{2}$
and

$\begin{split}\Delta_{f}\left(R^{-1}\nu_{1}\right)&=R^{-1}\Delta_{f}\nu_{1}+\nu_{1}\Delta_{f}R^{-1}+2\langle\nabla
R^{-1},\nabla\left(RR^{-1}\nu_{1}\right)\rangle\\\
&=R^{-1}\Delta_{f}\nu_{1}+\nu_{1}\Delta_{f}R^{-1}-2\langle\nabla\ln
R,\nabla\left(R^{-1}\nu_{1}\right)\rangle-2R^{-1}|\nabla\ln R|^{2}\nu_{1}\\\
&\leq-R^{-1}(\nu_{1}^{2}+\nu_{2}\nu_{3})+2R^{-2}|{\operatorname{Ric}}|^{2}\nu_{1}-2\langle\nabla\ln
R,\nabla\left(R^{-1}\nu_{1}\right)\rangle+O(R^{-1}r^{-5/2}).\end{split}$
(3.32)

Note that by (3.12) and (3.18),
$|{\operatorname{Ric}}|^{2}=R_{cd}R_{cd}+2R_{i4}R_{i4}-R_{44}^{2}=R_{cd}R_{cd}+O(r^{-3})$.
Due to a computation similar to the proof of (2.19) and the facts that
$U=O(r^{-1})$ and $\overline{U}=R-R_{44}$, we have

$\begin{split}R(\nu_{1}^{2}+\nu_{2}\nu_{3})-2|{\operatorname{Ric}}|^{2}\nu_{1}&=\overline{U}(\nu_{1}^{2}+\nu_{2}\nu_{3})-2R_{cd}R_{cd}\nu_{1}+O(r^{-7/2})\\\
&=(\nu_{1}+\nu_{2}+\nu_{3})(\nu_{1}^{2}+\nu_{2}\nu_{3})-\frac{\nu_{1}}{2}\left[(\nu_{1}+\nu_{2})^{2}+(\nu_{1}+\nu_{3})^{2}+(\nu_{2}+\nu_{3})^{2}\right]\\\
&\,\,\,\,+O(r^{-7/2})\\\
&=\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1})+O(r^{-7/2}).\end{split}$
(3.33)

As before, we define the weighted operator $\Delta_{f-2\ln
R}:=\Delta_{f}+2\langle\nabla\ln R,\,\nabla\,\rangle$. Then, by (3.32) and
(3.33), we obtain a $4$-dimensional analog of (2.19)

$\Delta_{f-2\ln
R}\left(R^{-1}\nu_{1}\right)\leq-R^{-2}\left[\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1})\right]+O(R^{-2}r^{-7/2})+O(R^{-1}r^{-5/2}).$
(3.34)

To deal with the error terms in (3.34), we need the auxiliary function $v=-f$.
Since $f$ is assumed to be proper (3.19), by adding a constant if necessary,
we have $v\geq 1$ on $M$ and $v\sim r$ near infinity. Furthermore, by taking
the trace of (3.1) and (3.2), we have $\Delta_{f}v=1$. For any $\alpha>0$,
using $\langle\nabla R,\nabla f\rangle=\Delta R+2|{\operatorname{Ric}}|^{2}$
and (3.18), we have

$\begin{split}\Delta_{f-2\ln R}\,v^{-\alpha}&=-\alpha
v^{-\alpha-1}\Delta_{f-2\ln R}\,v+\alpha(\alpha+1)v^{-\alpha-2}|\nabla
v|^{2}\\\ &=-\alpha v^{-\alpha-1}-2\alpha v^{-\alpha-1}\langle\nabla\ln
R,\nabla v\rangle+\alpha(\alpha+1)v^{-\alpha-2}|\nabla v|^{2}\\\ &=-\alpha
v^{-\alpha-1}+2\alpha v^{-\alpha-1}R^{-1}(\Delta
R+2|{\operatorname{Ric}}|^{2})+\alpha(\alpha+1)v^{-\alpha-2}|\nabla v|^{2}\\\
&\leq-\alpha v^{-\alpha-1}/2+O(R^{-1}v^{-\alpha-3}),\end{split}$ (3.35)

outside a compact set of $M$. By (3.17) and [CZh21, Theorem 1.3], both
$R^{-1}U$ and $R^{-1}\nu_{1}$ are bounded, and there is a positive constant
$C$ such that _either one_ of the following holds near infinity

$C^{-1}r^{-1}\leq R\leq Cr^{-1};$ (3.36) $C^{-1}e^{-r}\leq R\leq Ce^{-r}.$
(3.37)

If $R$ decays exponentially, namely, if (3.37) is true, then $|\nu_{1}|\leq
cR\leq Ce^{-r}$ and Proposition 3.5 is done. Therefore we may suppose that
(3.36) holds. Using (3.19) and (3.34), one can find a constant $C_{1}>0$, such
that

$\Delta_{f-2\ln
R}\left(R^{-1}\nu_{1}\right)\leq-R^{-2}\left[\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1})\right]+C_{1}v^{-3/2}.$

Substitution $\alpha=1/2$ in (3.35), we have $\Delta_{f-2\ln
R}\,8v^{-1/2}\leq-v^{-3/2}$ near infinity and

$\Delta_{f-2\ln
R}\left(R^{-1}\nu_{1}+8C_{1}v^{-1/2}\right)\leq-R^{-2}\left[\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1})\right].$
(3.38)

Since $R^{-1}\nu_{1}$ is bounded near infinity, by the properness of $v=-f$
(3.19), there is a large positive constant $\tau_{0}$ such that (3.38) holds
on $\\{x:f(x)\leq-\tau_{0}\\}$, and we can choose a large constant $C_{1}$
such that

$R^{-1}\nu_{1}+8C_{1}v^{-1/2}>0\quad\text{ on }\quad\\{x:f(x)=-\tau_{0}\\}.$

By (3.18), $\langle\nabla R,\nabla f\rangle=\Delta
R+2|{\operatorname{Ric}}|^{2}$, and the assumption that (3.36) is true, it is
clear that

$\displaystyle\Delta_{f-2\ln R}\ln v$ $\displaystyle=$ $\displaystyle
v^{-1}\Delta_{F}v-v^{-2}|\nabla v|^{2}$ $\displaystyle=$ $\displaystyle
v^{-1}\left(1+2\langle\nabla\ln R,\nabla v\rangle\right)-v^{-2}|\nabla v|^{2}$
$\displaystyle=$ $\displaystyle v^{-1}\left(1+O(v^{-1})\right)-v^{-2}|\nabla
v|^{2}\leq 2v^{-1}.$

For all positive integer $k$, let
$G_{k}:=R^{-1}\nu_{1}+8C_{1}v^{-1/2}+\frac{1}{k}\ln v$. It can be seen that
$G_{k}>0$ on $\\{x:f(x)=-\tau_{0}\\}$. By the properness of $f$ (3.19), we see
that $\lim_{x\to\infty}G_{k}=\infty$. If $G_{k}$ is negative somewhere in
$\\{x:f(x)\leq-\tau_{0}\\}$, then it attains a negative minimum, say, at
$q_{k}\in\\{x:f(x)<-\tau_{0}\\}$. It is evident that $\nu_{1}(q_{k})<0$ and
$G_{k}$ satisfies the following inequality at $q_{k}$

$0\leq\Delta_{f-2\ln R}\left(R^{-1}\nu_{1}+8C_{1}v^{-1/2}+\tfrac{1}{k}\ln
v\right)\leq-R^{-2}\left[\nu_{2}^{2}(\nu_{3}-\nu_{1})+\nu_{3}^{2}(\nu_{2}-\nu_{1})\right]+2v^{-1}/k.$
(3.39)

Moreover, by (3.12), (3.18), and (3.36), we have
$3\nu_{3}\geq\overline{U}=R-R_{44}\geq R/2$ for large $\tau_{0}$. From (3.39)
and the linear scalar curvature decay condition we have, at $q_{k}$ it holds
that

$\displaystyle\nu_{2}^{2}\leq 6R^{-1}\nu_{2}^{2}\nu_{3}\leq
6R^{-1}\nu_{2}^{2}(\nu_{3}-\nu_{1})\leq Cv^{-2}/k$

and thus $|\nu_{2}|\leq v^{-1}\sqrt{C/k}$. Hence again by (3.36), (3.19),
(3.39), and the fact that $|R^{-1}U|\leq c_{1}$ near infinity, we have

$|\nu_{1}|/36\leq R^{-2}\nu_{3}^{2}|\nu_{1}|\leq
2v^{-1}/k+c_{1}^{2}v^{-1}\sqrt{C/k}$

and $R^{-1}|\nu_{1}|(q_{k})\leq C_{2}k^{-1/2}$ for some positive constant
$C_{2}$ independent on $q_{k}$ and $k$. In conclusion

$G_{k}\geq G_{k}(q_{k})\geq-C_{2}k^{-1/2}.$

The above inequality is obvious when $G_{k}$ is nonnegative on
$\\{x:f(x)\leq-\tau_{0}\\}$. By letting $k\to\infty$, we have
$R^{-1}\nu_{1}+8C_{1}v^{-1/2}\geq 0$ on $\\{x:f(x)\leq-\tau_{0}\\}$ and
Proposition 3.5 follows. ∎

###### Proof of Theorem 1.6:.

The lower bound of $\operatorname{Rm}$ in (1.5) is a consequence of (3.3),
(3.18), (3.21), (3.27), and the estimate on $\nu_{1}$ in Proposition 3.5. The
upper estimate of $\operatorname{Rm}$ in (1.5) follows from (3.17). This
completes the proof of Theorem 1.6. ∎

Next, we move on to the proof of Corollary 1.9.

###### Proof of Corollary 1.9:.

By [CZh21, Theorem 1.3, Proposition 6.1], if the curvature of $M$ decays
exponentially, then Corollary 1.9(2) holds. Therefore, we may assume that the
scalar curvature satisfies (3.36) and show that Corollary 1.9(1) is true. For
any sequence $p_{i}\to\infty$ in $M$, let $\Sigma_{i}:=\\{x:f(x)=f(p_{i})\\}$
and $\tau_{i}:=-f(p_{i})\to\infty$ by the properness of $f$ (3.19). From the
discussion before Lemma 3.3 and (3.19), $\Sigma_{i}$ is a smooth compact
connected hypersurface in $M$ and

$c^{-1}r\leq\tau_{i}\leq cr\quad\text{ on }\quad\Sigma_{i}.$ (3.40)

$\tilde{g}_{i}$ denotes the induced metric on $\Sigma_{i}$ by $M$ and
$h_{i}:=R(p_{i})\tilde{g}_{i}$ is the scaled metric by the scalar curvature at
$p_{i}$. In view of the proof of [CZh21, Proposition 6.1], the curvature and
intrinsic diameter of $(\Sigma_{i},h_{i})$ are uniformly bounded in $i$. Thus
by Gromov’s Compactness theorem [BBI01, Theorem 10.7.2], the sequence sub-
converges to certain compact Alexandrov space $(Y,d_{Y})$ of dimension $\leq
3$. Moreover by Theorem 1.6, the Gauss equation, (3.36), and (3.40), we see
that $(\Sigma_{i},h_{i})$ has almost nonnegative curvature in the following
sense

$\begin{split}\operatorname{Rm}(h_{i})=R(p_{i})\operatorname{Rm}(\tilde{g}_{i})&=R(p_{i})\left(\operatorname{Rm}(g)+\operatorname{Ric}\ast\operatorname{Ric}/|\nabla
f|^{2}\right)\\\ &\geq-R(p_{i})\left(C\tau_{i}^{-3/2}g\odot
g+C\tau_{i}^{-2}g\odot g\right)\\\
&\geq-2R(p_{i})^{-1}\left(C\tau_{i}^{-3/2}h_{i}\odot h_{i}\right)\\\
&\geq-C^{\prime}\tau_{i}^{-1/2}h_{i}\odot h_{i},\end{split}$ (3.41)

where $(g\odot g)_{abcd}:=2g_{ad}g_{bc}-2g_{ac}g_{bd}$ is the Kulkarni-Nomizu
product. Hence we have $(Y,d_{Y})$, being a limit of the sequence
$(\Sigma_{i},h_{i})$, is an Alexandrov space of nonnegative curvature. The
pointed Gromov-Hausdorff convergence to $\mathbb{R}\times Y$ then follows from
the same argument as in the proof of [CZh21, Proposition 6.1].

∎

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Department of Mathematics, University of California, San Diego, CA, 92093
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Department of Mathematics, University of California, San Diego, CA, 92093
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School of Mathematics, University of Minnesota, Twin Cities, MN, 55414
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