#  $4d$ steady gradient Ricci solitons with nonnegative curvature away from a
compact set

Ziyi $\text{Zhao}^{{\dagger}}$ and Xiaohua $\text{Zhu}^{{\ddagger}}$ BICMR and
SMS, Peking University, Beijing 100871, China<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

In the paper, we analysis the asymptotic behavior of noncompact
$\kappa$-noncollapsed steady gradient Ricci soliton $(M,g)$ with nonnegative
curvature operator away from a compact set $K$ of $M$. In particular, we
prove: any $4d$ noncompact $\kappa$-noncollapsed steady gradient Ricci soliton
$(M^{4},g)$ with nonnegative sectional curvature must be a Bryant Ricci
soliton up to scaling if it admits a sequence of rescaled flows of
$(M^{4},g)$, which converges subsequently to a family of shrinking quotient
cylinders.

###### Key words and phrases:

Steady Ricci soliton, Ricci flow, ancient $\kappa$-solution, Bryant Ricci
soliton

###### 2000 Mathematics Subject Classification:

Primary: 53E20; Secondary: 53C20, 53C25, 58J05

${\ddagger}$ partially supported by National Key R&D Program of China
2020YFA0712800 and NSFC 12271009.

##  0\. Introduction

Steady gradient Ricci soliton, as a singular model of type II of Ricci flow,
has been extensively studied. In dimensions 2, Hamilton [20, 13] proved that
cigar solution is the only 2d steady Ricci soliton up to scaling. In
dimensions 3, Perelman conjectured that Bryant Ricci soliton is the only 3d
$\kappa$-noncollapsed steady Ricci soliton up to scaling [26]. The conjecture
has been proved by Brendle [4]. The cigar solution and Bryant Ricci soliton
are both rotationally symmetric. In higher dimension $n\geq 4$, besides the
Bryant soliton [10], Lai recently constructed a family of ${\rm
SO}(n-1)$-symmetry solutions with positive curvature operator [22]. Thus it is
interesting to classify steady Ricci solitons under suitable conditions of
symmetry and curvature.

To character the Bryant Ricci soliton, Brendle introduced the following notion
[5].

######  Definition 0.1.

A complete steady gradient Ricci soliton $(M^{n},g,f)$ is called
asymptotically cylindrical if the following holds:

(i) The scalar curvature satisfies $\frac{C_{1}}{\rho(x)}\leq
R(x)\leq\frac{C_{2}}{\rho(x)}$ as $\rho(x)>>1$, where $C_{1}$, $C_{2}$ are two
positive constants.

(ii) Let $p_{i}$ be an arbitrary sequence of marked points going to infinity.
Consider the rescaled metrics

(0.1) $\displaystyle g_{p_{i}}(t)=r_{i}^{-1}\phi^{*}_{r_{i}t}(g),$

where $r_{i}R(p_{i})=1+o(1)$ as $i\rightarrow\infty$ and $\phi_{t}$ is a
family of transformations generated by $-\nabla f$, the flow
$(M,g_{p_{i}}(t);p_{i})$ converges in the Cheeger-Gromov sense to a family of
shrinking cylinders $(\mathbb{S}^{n-1}\times\mathbb{R},~{}\bar{g}(t))$,
$t\in(0,1)$. The metric $\bar{g}(t)$ is given by

(0.2) $\displaystyle\bar{g}(t)=(1-t)g_{\mathbb{S}^{n-1}(1)}+ds^{2},$

where $\mathbb{S}^{n-1}(1)$ is the unit sphere in the Euclidean space.

In [5], Brendle proved that any steady (gradient) Ricci soliton with positive
sectional curvature must be isometric to the Bryant Ricci soliton up to
scaling if it is asymptotically cylindrical. Latterly, Deng-Zhu found that the
Brendle’s result still holds if one of two conditions in Definition 0.1 is
satisfied for $\kappa$-noncollapsed steady Ricci solitons with nonnegative
curvature operator [16, 17]. In a very recent paper [29], the above results of
Brendle and Deng-Zhu have been improved instead of curvature condition by the
nonnegativity of curvature operator away from a compact set of $M$ together
with $P$-curvature pinching condition. Thus it is a natural question to ask:
is the Brendle’s result true if there is only one sequence in the condition
(ii) satisfied? In this paper, we give a positive answer for 4d
$\kappa$-noncollapsed steady Ricci solitons with nonnegative sectional
curvature.

Let $(M^{n},g)$ be a noncompact $\kappa$-noncollapsed steady Ricci soliton
with curvature operator $\rm{Rm}\geq 0$ (sectional curvature $\rm{Km}\geq 0$
for $n=4$) and $\rm{Ric}>0$ away from a compact set $K$ of $M$. Let
$p_{i}\rightarrow\infty$ be any sequence in $M$ and $g_{p_{i}}(t)$ the
rescaled flow of Ricci soliton $g$ as in Definition 0.1. Then
$(M,g_{p_{i}}(t);p_{i})$ converges to a splitting flow in the Cheeger-Gromov
sense,

(0.3) $\displaystyle\bar{g}(t)=h(t)+ds^{2},~{}{\rm on}~{}N\times\mathbb{R},$

where $h(t)$ ($t\in(-\infty,0]$) is an ancient $\kappa$-solution on an
$(n-1)$-dimensional $N$, see Proposition 1.2.

The following alternative principle is the main result in this paper.

######  Theorem 0.2.

Let $(M^{4},g)$ be a $4d$ noncompact $\kappa$-noncollapsed steady gradient
Ricci soliton with $\rm{Km}\geq 0$ and $\rm{Ric}>0$ away from a compact set
$K$ of $M$. Let $p_{i}\rightarrow\infty$ be any sequence in $M$ and
$\bar{g}(t)=h(t)+ds^{2}$ the splitting limit flow of $(M,g_{p_{i}}(t);p_{i})$
as in (0.3). Then either all $h(t)$ is a family of $3d$ shrinking quotient
spheres, or all $h(t)$ is a $3d$ noncompact ancient $\kappa$-solution.

We note that both of cases will happen in Theorem 0.2 with following examples.
For any $2n\geq 4$ and each $Z_{k}$-group, Appleton [2] has constructed an
example of noncompact $\kappa$-noncollapsed steady gradient Ricci soliton with
$\rm{Rm}>0$ on $M\setminus K$, where all split ancient $\kappa$-solution
$h(t)$ is a family of shrinking quotient spheres of dimension $(2n-1)$ with
$Z_{k}$-group. In each of Lai’s examples [22] of noncompact
$\kappa$-noncollapsed steady gradient Ricci solitons with $\rm{Rm}>0$ on $M$,
all split ancient $\kappa$-solutions $h(t)$ are noncompact.

We also note that $3d$ noncompact ancient $\kappa$-solution has been recently
classified by Brendle [6] and Bamlar-Kleiner [3], independently. Namely, it is
isometric to either a family of shrinking quotient cylinders, or a Bryant
Ricci soliton flow.

As an application of Theorem 0.2, we prove

######  Corollary 0.3.

Let $(M^{4},g)$ be a $4d$ noncompact $\kappa$-noncollapsed steady gradient
Ricci soliton with nonnegative sectional curvature. Suppose that there exists
a sequence of rescaled flows $(M,g_{p_{i}}(t);p_{i})$ of $(M,g)$ which
converges subsequently to a family of shrinking quotient cylinders. Then
$(M,g)$ is isometric to the $4d$ Bryant Ricci soliton up to scaling. Moreover,
if $(M,g)$ has nonnegative sectional curvature and positive Ricci curvature,
the result is still true just under assumption of the existence of $3d$
compact split limit flow $(N,h(t))$ as in (0.3).

Our proof of Theorem 0.2 depends on a deep classification result for $3d$
compact $\kappa$-solutions proved by Brendle-Daskalopoulos-Sesum [7] (also see
Theorem 3.2). But we guess that Theorem 0.2 and Corollary 0.3 are both true
for any dimension.

The paper is organized as follows. In Section 1, we prove a splitting result
for any limit flow of rescaled flows sequence from a $\kappa$-noncollapsed
steady gradient Ricci soliton $(M,g)$ with $\rm{Rm}\geq 0$ on $M\setminus K$,
see Proposition 1.2. In Section 2, we first get a decay estimate of curvature
and then study the level set geometry of $(M,g)$ by assuming the existence of
compact split ancient $\kappa$-solution $(N,h(t))$, see Lemma 2.2, Proposition
2.7, etc. All results in this section holds for any dimension. In Section 3,
we focus on a $4d$ steady Ricci soliton to get a diameter estimate of
$(N,h(0))$ for all 3d split flows $(N,h(t))$, see Proposition 3.6. Theorem 0.2
and Corollary 0.3 will be proved in Section 4.

##  1\. A splitting theorem

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

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. These three types of Ricci solitons correspond to three
different blow-up solutions of Ricci flow [20].

In case of steady Ricci solitons, we can rewrite (1.1) as

(1.2) $\displaystyle 2\operatorname{Ric}(g)=\mathscr{L}_{X}g,$

where $\mathscr{L}_{X}$ is the Lie operator along the gradient vector field
(VF) $X=\nabla f$ generalized by $f$. Let
$\\{\phi^{*}_{t}\\}_{t\in(-\infty,\infty)}$ be a 1-ps of transformations
generated by $-X$. Then $g(t)=\phi^{*}_{t}(g)$ ($t\in(-\infty,\infty))$ is a
solution of Ricci flow. Namely, $g(t)$ satisfies

(1.3) $\displaystyle\frac{\partial g}{\partial t}=-2{\rm Ric}(g),~{}g(0)=g.$

For simplicity, we call $g(t)$ the soliton Ricci flow of $(M,g)$.

By (1.2), we have

(1.4) $\displaystyle\langle\nabla R,\nabla
f\rangle=-2\operatorname{Ric}(\nabla f,\nabla f),$

where $R$ is the scalar curvature of $g$. It follows

$R+|\nabla f|^{2}={\rm Const.}$

Since $R$ is alway positive ([28, 11]), the above equation can be normalized
by

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

We recall that an ancient $\kappa$-solution is a $\kappa$-noncollapsed
solution of Ricci flow (1.3) with ${\rm R_{m}}(\cdot,t)\geq 0$ defined for any
$t\in(-\infty,T_{0}]$. The following result is a version of Perelman’s
compactness theorem for higher dimensional ancient $\kappa$-solutions.

######  Theorem 1.1.

Let $\left(M,g_{i}(t);p_{i}\right)$ be any sequence of $n$-dimensional ancient
$\kappa$-solutions on a noncompact manifold $M$ with
$R\left(p_{i},0\right)=1$. Then $(M,g_{i}(t);~{}$ $p_{i})$ subsequently
converge to a splitting flow $(N\times\mathbb{R},\bar{g}(t);p_{\infty})$ in
Cheeger-Gromov sense. Here

(1.6) $\displaystyle\bar{g}(t)=h(t)+ds^{2},$

and $(N,h(t))$ is an $(n-1)$-dimensional ancient $\kappa$-solution.

The convergence of $\left(M,g_{i}(t);p_{i}\right)$ comes from [15, Theorem
3.3]. The splitting property in (1.6) can be also obtained by Hamilton’s
argument [20, Lemma 22.2] with help of Perelman’s asymptotic volume ratio
estimate for $\kappa$-solutions [21, Proposition 41.13]. In fact, for a
sequence of rescaling Ricci flows arising from a steady Ricci soliton, we can
improve Theorem 1.1 under a weaker curvature condition as follows.

######  Proposition 1.2.

Let $(M^{n},g)$ be a noncompact $\kappa$-noncollapsed steady gradient Ricci
soliton with $\rm{Rm}\geq 0$ away from $K$. Let $p_{i}\rightarrow\infty$ and
$(M,g_{p_{i}}(t);p_{i})$ a sequence of rescaling flows with
$R_{p_{i}}\left(p_{i},0\right)=1$ as in (0.1) . Then $(M,g_{p_{i}}(t);~{}$
$p_{i})$ subsequently converge to a splitting flow
$(N\times\mathbb{R},\bar{g}(t);p_{\infty})$ as in Theorem 1.1. Moreover, for
$n=4$, $\rm{Rm}\geq 0$ can be weakened to $\rm{Km}\geq 0$ away from $K$.

###### Proof.

Since $\rm{Km}\geq 0$ on $M\setminus K$, we have the Harnack estimate by
(1.4),

(1.7) $\displaystyle\frac{d}{dt}R(x,t)\geq 0,~{}{\rm on}~{}M\setminus K.$

Then according to the proof of Theorem 1.1 (to see Lemma 3.5-3.7 for details
there), for the convergence part in the proposition, we need only to show that
the following asymptotic scalar curvature estimate,

(1.8) $\displaystyle{\rm limsup}_{x\rightarrow\infty}R(x)d^{2}(o,x)=\infty,$

where $o\in M$ is a fixed point. As a consequence, the rescaled flow
$(M,g_{p_{i}}(t);p_{i})$ has locally uniformly curvature estimate, and so
$(M,g_{p_{i}}(t);p_{i})$ subsequently converges to a limit ancient
$\kappa$-solution $(M_{\infty},\bar{g}(t);p_{\infty})$.

We note that (1.8) is true for any ancient $\kappa$-solution by the Perelman’s
result of asymptotic zero volume ratio [26, 21] (cf. [15, Corollary 2.4]). In
our case, we have only $\rm{Rm}\geq 0$ away from $K$. We will use a different
argument to prove (1.8) below.

On contrary, we suppose that (1.8) is not true. Then there exists a constant
$C>0$, such that

(1.9) $\displaystyle R(x)\leq\frac{C}{d^{2}(o,x)}=o(\frac{1}{d(o,x)}).$

In particular, the scalar curvature decays to zero uniformly. Due to a result
in [12, Theorem 2.1], we know that there are two constants $c_{1},c_{2}>0$
such that

(1.10) $\displaystyle c_{1}\rho(x)\leq f(x)\leq c_{2}\rho(x).$

Thus by [16, Theorem 6.1] with the help of (1.9) and (1.10), we get

$R(x)\geq\frac{C_{0}}{d(o,x)},$

for some constant $C_{0}$. But this is a contradiction with
$(\ref{contradiction-1})$. Hence (1.8) is true.

In the following, our goal is to show that $\bar{g}(t)$ is of form (1.6).
First we prove the volume ratio estimate,

(1.11)
$\displaystyle\rm{AVR}(g)=\lim_{r\rightarrow\infty}\frac{\rm{Vol}(B(p,r))}{r^{n}}=0.$

By (1.8), we can use the Hamilton’s argument in [20, Lemma 22.2] to find
sequences of points $q_{i}\rightarrow\infty$ and number $s_{i}>0$ such that
$\frac{s_{i}}{d(q_{i},o)}\rightarrow 0$,

(1.12) $\displaystyle R(q_{i})s_{i}^{2}\rightarrow\infty,$

and

(1.13) $\displaystyle R(x)\leq 2R(q_{i}),~{}\forall~{}x\in B(q_{i},s_{i}).$

Consider a sequence of the rescaled flows $(M,g_{q_{i}}(t);q_{i})$,
$t\in(-s_{i},0]$, such that $R_{q_{i}}\left(q_{i},0\right)=1$, where
$R_{q_{i}}(\cdot,t)$ is the scalar curvature of $g_{q_{i}}(t)$. Then by (1.7),
$R_{q_{i}}(x,t)\leq 2$ whenever $t\in(-s_{i},0]$ and $d_{q_{i}}(q_{i},x)\leq
R(q_{i})^{\frac{1}{2}}s_{i}$, where $d_{q_{i}}(q_{i},\cdot)$ is the distance
function from $q_{i}$ w.r.t $g_{q_{i}}(t)$. It follows that
$(M,g_{q_{i}}(t);q_{i})$ with $t\in(-s_{i},0]$ converges subsequently to a
limit ancient $\kappa$-solution $(M_{\infty},g_{\infty}(t);q_{\infty})$.
Moreover, by (1.12) and the curvature condition $\rm{Km}\geq 0$ on $M\setminus
K$, one can construct a geodesic line on
$(M_{\infty},g_{\infty}(0);q_{\infty})$ (cf. [23, Theorem 5.35]). Thus, by
Cheeger-Gromoll splitting theorem, $(M_{\infty},g_{\infty}(t);q_{\infty})$ is
in fact a splitting ancient flow
$(N^{\prime}\times\mathbb{R},h^{\prime}(t)+ds^{2};q_{\infty})$, where
$(N^{\prime},h^{\prime}(t);q_{\infty})$ is an $(n-1)$-dimensional
$\kappa$-noncollapsed ancient solution. Clearly,
$(N^{\prime},h^{\prime}(0);q_{\infty})$ can not be flat since
$R_{\infty}(q_{\infty},0)=1$, and so $(M_{\infty},g_{\infty}(t);q_{\infty})$
is a non-flat ancient solution. Hence, by [21, Proposition 41.13], the
asymptotic volume ratio of $(M_{\infty},g_{\infty}(t);q_{\infty})$ must be
zero. This will imply (1.11) by the volume monotone since the (1.11) is
invariant under the rescaling.

Next we let

(1.14) $\displaystyle r(p_{i})=\sup\\{\rho|~{}{\rm
Vol}(B(p_{i},\rho))\geq\frac{\omega}{2}\rho^{n}\\}.$

We prove

(1.15) $\displaystyle C_{0}^{-1}r(p_{i})\leq R^{-\frac{1}{2}}(p_{i})\leq
C_{0}r(p_{i}).$

In fact, for the first inequality in (1.15), by the volume comparison, there
is $C_{1}(D)>0$ for any $D>0$ such that

$\operatorname{Vol}(B(x,r(p_{i})))\geq
C_{1}^{-1}r(p_{i})^{n},~{}\forall~{}x\in B(p_{i},Dr(p_{i})).$

Then by [15, Lemma 3.5], there is $C_{0}(D)>0$ such that

(1.16) $\displaystyle R\leq C_{0}^{2}r(p_{i})^{-2},\forall~{}x\in
B(p_{i},\frac{D}{2}r(p_{i})).$

Thus we only need to prove the second inequality.

We use the above argument in the proof of (1.11). On contrary, there is a
sequence $p_{i}\rightarrow\infty$ (still denoted by $\\{p_{i}\\}$) such that

(1.17)
$\displaystyle\lim_{i\rightarrow\infty}\frac{R^{-1/2}(p_{i})}{r(p_{i})}=\infty.$

On the other hand, by $(\ref{R-bound-volume})$ and $(\ref{harnack-esti})$, we
have

$R(x,t)\leq C_{0}r(p_{i})^{-2},~{}\forall~{}x\in
B(p_{i},\frac{D}{2}r(p_{i})),t\in(-\frac{D}{2},0].$

Then the rescaled flow $\left(M,r(p_{i})^{-2}g(r(p_{i})^{2}t);p_{i}\right)$
converges subsequently to a limit ancient solution
$(M_{\infty}^{\prime},g_{\infty}^{\prime}(t);p_{\infty}^{\prime})$. Note that
$r(p_{i})<\infty$ for each $p_{i}$ by $(\ref{AVR})$. Moreover, by the volume
comparison, it follows

(1.18) $\displaystyle\lim_{i\rightarrow\infty}\frac{r(p_{i})}{d(p_{i},o)}=0.$

Hence, by (1.18) and the curvature condition $\rm{Km}\geq 0$ on $M\setminus
K$, one can construct a geodesic line on
$(M_{\infty}^{\prime},g_{\infty}^{\prime}(0);p_{\infty}^{\prime})$ (cf. [23,
Theorem 5.35]), and so
$(M_{\infty}^{\prime},g_{\infty}^{\prime}(t);p_{\infty}^{\prime})$ is a
splitting ancient flow
$(\hat{N}\times\mathbb{R},\hat{h}(t)+ds^{2};p_{\infty})$, where $\hat{h}(t)$
is an $(n-1)$ dimensional ancient $\kappa$-solution. As a consequence, by
$(\ref{curvature-less-volume})$, we have

(1.19) $\displaystyle R_{\infty}(p_{\infty}^{\prime},0)=0.$

By the strong maximum principle and (1.19), $(N,\hat{h}(0))$ is flat and so as
$(M_{\infty}^{\prime},g_{\infty}^{\prime}(0))$. Then by the injective radius
estimate (cf. [15, Lemma 3.6]), one can show that
$(M_{\infty}^{\prime},g_{\infty}^{\prime}(0))$ must be isometric to the
Euclidean space. In particular,
$\rm{Vol}(B_{g_{\infty}^{\prime}(0)}(p_{\infty}^{\prime},1))=\omega$. But this
is impossible by (1.14). Hence we finish the proof of (1.15).

At last, by $(\ref{r-d-order})$ and (1.15), we have

(1.20)
$\displaystyle\lim_{i\rightarrow\infty}\frac{R(p_{i})^{-1}}{d^{2}(p_{i},o)}=0.$

Then instead of the rescaled flow
$\left(M,r(p_{i})^{-2}g(r(p_{i})^{2}t);p_{i}\right)$ by
$(M,g_{p_{i}}(t);p_{i})$, the limit ancient solution
$(M_{\infty},\bar{g}(t);p_{\infty})$ will split off a line as
$(M_{\infty}^{\prime},g_{\infty}^{\prime}(0);p_{\infty}^{\prime})$. Thus
$\bar{g}(t)$ is of form (1.18).

In case of $n=4$, we note that both of split $3d$ $\kappa$-noncollapsed
ancient flows $h^{\prime}(t)$ and $\hat{h}(t)$ in the above arguments are non-
negatively curved under ${\rm K_{m}}\geq 0$ away from $K$. Thus both of
$h^{\prime}(t)$ and $\hat{h}(t)$ are same as ancient $\kappa$-solutions. Hence
the proofs above work for $4d$ steady Ricci solitons when the assumption ${\rm
R_{m}}\geq 0$ is replaced by ${\rm K_{m}}\geq 0$ away from $K$.

∎

According to the proof in Proposition 1.2, we also get the following curvature
comparison.

######  Lemma 1.3.

Let $(M^{n},g)$ be a noncompact $\kappa$-noncollapsed steady Ricci soliton as
in Proposition 1.2. Let $\\{p_{i}\\}\to\infty$ be any sequence of $(M^{n},g)$.
Then for any $q_{i}\in B_{g_{p_{i}}}(p_{i},D)$, there is a $C_{0}(D)>0$ such
that

(1.21) $\displaystyle C_{0}^{-1}R(p_{i})\leq R(q_{i})\leq C_{0}R(p_{i}).$

###### Proof.

We note that the rescaling flow $(M,g_{p_{i}}(t);p_{i})$ will converges to a
splitting of ancient solution
$(M_{\infty},\bar{g}(t)=h(t)+ds^{2};p_{\infty})$. Then by (1.15) and (1.16) in
the proof of Proposition 1.2, we get the second inequality of (1.21)
immediately. Thus we only need to prove the first inequality.

By contradiction, there exists a sequence of points $q_{i}\in
B_{g_{p_{i}}}(p_{i},D)$ for some $D>0$ such that

(1.22) $\displaystyle\frac{R(q_{i})}{R(p_{i})}\rightarrow 0,~{}{\rm
as}~{}i\to\infty.$

Then

$R_{\bar{g}(0)}(q_{\infty})=0,$

where $q_{\infty}$ is a limit of $\\{q_{i}\\}$ from the convergence of
$(M,g_{p_{i}}(t);p_{i})$. By the strong maximum principle, it follows that
$\bar{g}(0)$ is a flat metric, which contradicts to
$R_{\bar{g}(0)}(p_{\infty})=1$. Thus (1.21) is proved. ∎

##  2\.  Compact case of $(N,h(t))$

In this section, we assume that there exists a sequence of
$p_{i}\rightarrow\infty$ on an $n$-dimensional steady Ricci soliton
$(M^{n},g)$ such that the corresponding split ancient $\kappa$-solution $h(t)$
of $(n-1)$-dimension in Proposition 1.2 satisfies

(2.1) $\displaystyle\rm{Diam}(h(0))\leq C.$

We will use the method in [16, 17] to study level set geometry of $(M^{n},g)$
under the condition (2.1). All results in this section holds for any
dimension.

Firstly we show that $(M,g)$ has a convexity property in sense of geodesics.

######  Lemma 2.1.

Suppose that there exists a sequence of $p_{i}\rightarrow\infty$ such that the
split $(n-1)$-dimensional ancient $\kappa$-solution $(N,{h}(t))$ in
Proposition 1.2 satisfies (2.1). Then there exists a compact set $K^{\prime}$
$(K\subset K^{\prime}$) such that for $x_{1},x_{2}\in M\setminus K^{\prime}$
the minimal geodesic curve connecting $x_{1}$ and $x_{2}$, $\sigma(s)\subset
M\setminus K$, where $K$ is the compact set in Proposition 1.2.

###### Proof.

By the convergence of $(M,g_{p_{i}}(t);p_{i})$ together with (2.1), it is easy
to see that one can choose a point $p\in\\{p_{i}\\}$ such that
$B_{g}(p_{i},10CR(p_{i})^{-\frac{1}{2}})$ divides $M$ into three parts with a
compact part $\Sigma_{p}$ which contains $K$ as follows,

(2.2) $\displaystyle
M=B_{g}(p_{i},10CR(p_{i})^{-\frac{1}{2}})\cup\Sigma_{p}\cup M^{\prime},$

where $B_{g}(p_{i},10CR(p_{i})^{-\frac{1}{2}})\cap K=\emptyset$ and
$M^{\prime}=M\setminus(B_{g}(p_{i},10CR(p_{i})^{-\frac{1}{2}}\cup\Sigma_{p})$
is a noncompact set of $M$. Set

$K^{\prime}=\Sigma_{p}\cup B_{g}(p,10CR(p)^{-\frac{1}{2}}).$

We need to verify $K^{\prime}$ chosen as required in the lemma.

On contrary, there will exist two points $x_{1},x_{2}\in M\setminus
K^{\prime}$ and another point $x\in\sigma(s)\cap K$, where $\sigma(s)$ is the
minimal geodesic curve connecting $x_{1}$ and $x_{2}$. Then $\sigma(s)$ will
pass through $B_{g}(p,10CR(p)^{-\frac{1}{2}})$ at least twice. Denote $q_{1}$
to be the first point and $q_{2}$ to be the last point in
$B_{g}(p,10CR(p)^{-\frac{1}{2}})$ respectively, which intersects with
$\sigma(s)$. Let $\sigma^{\prime}$ be the part of $\sigma(s)$ between $q_{1}$
and $q_{2}$. Thus by the triangle inequality, we have

$\displaystyle d_{g}(q_{1},q_{2})$
$\displaystyle=\rm{Length}(\sigma^{\prime})=d_{g}(q_{1},x)+d_{g}(x,q_{2})$
$\displaystyle\geq d_{g}(q_{1},o)+d_{g}(o,q_{2})-2d_{g}(o,x)$
$\displaystyle\geq 2d_{g}(o,p)-d_{g}(q_{1},p)-d_{g}(q_{2},p)-2C^{\prime}$
(2.3) $\displaystyle\geq 2d_{g}(o,p)-20R(p)^{-\frac{1}{2}}-2C^{\prime}.$

On the other hand, by the estimate $(\ref{r-decay-upper})$, we see that for
any small $\delta$ it holds

(2.4) $\displaystyle R(p_{i})^{-\frac{1}{2}}\leq\delta d_{g}(p_{i},o),$

as long as $i>>1.$ By (2), it follows

$d_{g}(q_{1},q_{2})\geq 2d_{g}(o,p)-20\delta d_{g}(o,p)-2C^{\prime}\geq
d_{g}(o,p).$

However,

$\displaystyle d_{g}(q_{1},q_{2})$ $\displaystyle\leq
d_{g}(q_{1},p)+d_{g}(p,q_{2})\leq 20R(p)^{-\frac{1}{2}}$ $\displaystyle\leq
20\delta d_{g}(o,p)\leq\frac{1}{2}d_{g}(o,p).$

Thus we get a contradiction! The lemma is proved. ∎

###  2.1. Curvature decay estimate

By Lemma 1.3 and Lemma 2.1, we prove

######  Lemma 2.2.

Let $(M^{n},g)$ be the steady Ricci soliton in Proposition 1.2 with
$\rm{Ric}>0$ away from $K$. Suppose that there exists a sequence of
$p_{i}\rightarrow\infty$ such that the split $(n-1)$-dimensional ancient
$\kappa$-solution $(N,{h}(t))$ in Proposition 1.2 satisfies (2.1). Then the
curvature of $(M^{n},g)$ decays to zero uniformly. Namely,

(2.5) $\displaystyle\lim_{x\rightarrow\infty}R(x)=0.$

###### Proof.

First we prove

(2.6) $\displaystyle\lim_{p_{i}\rightarrow\infty}R(p_{i})=0.$

On contrary, we assume that $R(p_{i})\geq c$ for some constant $c>0$. We
consider a sequence of functions $f_{p_{i}}=f-f(p_{i})$ on Riemannian
manifolds $(M,g_{p_{i}}(0);p_{i})$. By $(\ref{scalar-equ})$, it is easy to see

$\displaystyle|\nabla f_{p_{i}}|_{g_{p_{i}}}\leq c^{-\frac{1}{2}}.$

Thus for any $D>0$ it holds

$\displaystyle|f_{p_{i}}(x)|\leq 2c^{-\frac{1}{2}}D,~{}\forall x\in
B_{g_{p_{i}}(p_{i},D)}.$

By the regularity of Laplace equation,

$\Delta_{g_{p_{i}}}f_{p_{i}}=R(g_{p_{i}(0)}),$

$f_{i}$ converges subsequently to a smooth function $f_{\infty}$ on
$N\times\mathbb{R}$ which satisfies the gradient steady Ricci soliton
equation,

$\rm{Ric}(\bar{g}(0))=\nabla^{2}f_{\infty}.$

Note that $\bar{g}(0)=h(0)+ds^{2}$ is a product metric. Hence $(N,h(0))$ is
also a steady gradient Ricci soliton,

On the other hand, by the maximum principle, $(N,h(0);p_{\infty})$ should be
Ricci-flat. However, by the normalization of

$R(g_{p_{i}}(0))(p_{i})=1,$

$R(h(0))(p_{\infty})$ is also $1$. This is a contradiction! (2.6) is proved.

By (2.6) and Lemma 1.3, we get

(2.7)
$\displaystyle\lim_{i\to\infty}\sup_{B_{g}(p_{i},10CR(p_{i})^{-\frac{1}{2}})}R(x)=0.$

Next we use (2.7) to derive (2.5).

Recall that the set of equilibrium points of $(M,g,f)$ is given by

$\displaystyle S:=\\{x\big{|}|~{}\nabla f|(x)=0\\}.$

In general, $S$ may be not empty. But we have

$\mathbf{Claim1}$: There is no any equilibrium point away from a compact set
$\hat{K}$ of $M$ which containing $K^{\prime}$. Here $K^{\prime}$ is the set
of $M$ determined in Lemma 2.1.

If $S$ is not empty and $\mathbf{Claim1}$ is not true, there will be two
equilibrium points $x_{1}$ and $x_{2}$ and a compact set $\hat{K}$ which
containing $K^{\prime}$ such that $x_{1},x_{2}\in M\setminus\hat{K}$. Then by
Lemma 2.1, there is a minimal geodesic curve $\sigma(s)$ connecting $x_{1}$
and $x_{2}$ such that $\sigma(0)=x_{1}$ and $\sigma(T)=x_{2}$ and
$\sigma(s)\subset M\setminus K$. Note

$\displaystyle\frac{d}{ds}(\langle\nabla
f,\sigma^{\prime})\rangle(\sigma(s)))=\nabla^{2}f(\sigma^{\prime},\sigma^{\prime})(s)=\rm{Ric}(\sigma^{\prime},\sigma^{\prime}).$

Thus we get

$\displaystyle 0$ $\displaystyle=\langle\nabla
f,\sigma^{\prime})\rangle(\sigma(t))-\langle\nabla
f,\sigma^{\prime})\rangle(\sigma(0))$
$\displaystyle=\int_{0}^{T}\rm{Ric}(\sigma^{\prime},\sigma^{\prime})ds>0,$

which is a contradiction! Hence, $\mathbf{Claim1}$ is true.

By (1.20), we can choose a subsequence of $\\{p_{i}\\}$, still denoted by
$\\{p_{i}\\}$ such that

(2.8) $\displaystyle B_{g}(p_{i},10CR(p_{i})^{-\frac{1}{2}})\cap
B_{g}(p_{j},10CR(p_{i+1})^{-\frac{1}{2}})=\emptyset,~{}\forall~{}i,j>>1.$

Then as in (2.2), there are a compact set $\bar{K}$ and a sequence of compact
set $\\{K_{i}\\}$ $(i\geq i_{0})$ of $M$ such that $\hat{K}\subset\bar{K}$ and

$\partial K_{i}\subset\partial
B_{g}(p_{i},10CR(p_{i})^{-\frac{1}{2}})\cup\partial
B_{g}(p_{i+1},10CR(p_{i+1})^{-\frac{1}{2}}),$

and $M$ is decomposed as

(2.9) $\displaystyle M=\bar{K}\cup_{i\geq
i_{0}}(K_{i}\cup(B_{g}(p_{i+1},10CR(p_{i+1})^{-\frac{1}{2}})),$

$\mathbf{Claim2}$: For any $q_{i}\in K_{i}$, there exists $t_{i}>0$ such that

(2.10) $\displaystyle q_{i}^{t_{i}}=\phi_{t_{i}}(q_{i})\in
B_{g}(p_{i},10R(p_{i})^{-\frac{1}{2}}C)\cup
B_{g}(p_{i+1},10R(p_{i+1})^{-\frac{1}{2}}C).$

On contrary, we see that $\phi_{t}(q_{i})\subset K_{i}$ for all $t\geq 0$.
Since $K_{i}$ is compact, there exists a $c^{\prime}>0$ by $\mathbf{Claim1}$
such that

$\rm{Ric}\geq c^{\prime}g,c^{\prime-1}\leq|\nabla f|\leq c^{\prime}.$

It follows

$\displaystyle\frac{d}{dt}R(\phi_{t}(q_{i}))$ $\displaystyle=-\langle\nabla
R,\nabla f\rangle(\phi_{t}(q_{i}))=2\rm{Ric}(\nabla f,\nabla f)$ (2.11)
$\displaystyle\geq 2c^{\prime-1}>0,~{}\forall~{}t\geq 0.$

As a consequence,

$R(\phi_{t}(q_{i}))\geq 2c^{\prime-1}t\to\infty,~{}{\rm as}~{}t\to\infty.$

This is impossible since $R(\cdot)$ is uniformly bounded. Hence,
$\mathbf{Claim2}$ is true.

By $\mathbf{Claim2}$ and (2.1), for any $q_{i}\in K_{i}$ we see

$\displaystyle R(q_{i})\leq R(q_{i}^{t_{i}})$
$\displaystyle\leq\max\\{R(x)|~{}x\in
B_{g}(p_{i},10R(p_{i})^{-\frac{1}{2}}C)\cup
B_{g}(p_{i+1},10R(p_{i+1})^{-\frac{1}{2}}C)\\}.$

Thus we get (2.5) from (2.7) and (2.9) immediately.

∎

######  Remark 2.3.

The steady Ricci soliton in Lemma 2.2 has a uniform curvature decay to zero.
Then $|\nabla f(x)|\to 1$ as $\rho(x)\to\infty$ by (1.5). Moreover, by [17,
Lemma 2.2] (or [12, Theorem 2.1]), $f$ satisfies (1.10). Hence, the integral
curve $\gamma(s)$ generated by $\nabla f$ extends to the infinity as
$s\rightarrow\infty$.

###  2.2. Estimate of level sets

By Lemma 2.2 and (1.5), there exists a point $p_{0}\in M$ such that

$R_{max}=\sup_{p\in M}R(p)=R(p_{0})=1.$

For any positive $c<1$, we set

$S(c)=\\{p\in M|~{}R(p)\geq R_{max}-c\\}.$

Then $S(c)$ is a compact set. Moreover, by Remark 2.3 there exists a $c_{0}$
such that $\hat{K}\subset S(c_{0})$ and $\nabla f\neq 0$ on
$S(c_{0})\setminus\hat{K}$. Thus VF $\hat{X}=\frac{\nabla f}{|\nabla f|}$ is
well-defined on $S(c)\setminus\hat{K}$ for any $c\geq c_{0}$.

By [12, Lemma 2.2, 2.3], it is known that there exists a $t_{q}$ such that
$\phi_{t_{q}}(q)\in S(c_{0})$ for any $q\in M\setminus S(c_{0})$.
Consequently, for any integral curve of $\hat{X}=\frac{\nabla f}{|\nabla f|}$,
$\Gamma(s):[0,\infty)\rightarrow M$, we can reparametrize $s$ such that
$\Gamma(0)=p\in S(c_{0})\setminus\hat{K}$, and so $\Gamma(s)\subset
M\setminus\hat{K}$ is a smooth curve for any $s>0$.

######  Lemma 2.4.

Let $(M^{n},g)$ be an $n$-dimensional steady soliton as in Lemma 2.2 and
$\Gamma(s)$ any integral curve of $\hat{X}$ with $\Gamma(0)=p\in
S(c_{0})\setminus\hat{K}$. Then for any $\epsilon$, there exists a uniform
constant $C=C(\epsilon)>0$ such that

(2.12) $\displaystyle(1-\epsilon)(s_{2}-s_{1})\leq
d(\Gamma(s_{2}),\Gamma(s_{1}))\leq(s_{2}-s_{1}),~{}\forall s_{2}>s_{1}>C.$

In particular,

(2.13) $\displaystyle(1-\epsilon)s\leq d(\Gamma(s),p)\leq s,~{}\forall~{}s>C.$

###### Proof.

Firstly by Remark 2.3, we note that for any $\epsilon>0$ there exists a
compact set $S^{\prime}$ such that

(2.14) $\displaystyle|\nabla f|(x)>1-\epsilon,~{}\forall x\in M\setminus
S^{\prime}.$

Moreover, $(\ref{gradient-f-almost-1})$ holds whenever $f(x)>L$. Since
$\Gamma(s)\subset M\setminus\hat{K}$, $|\nabla f|(\Gamma(s))\geq c_{0}>0$ by
$(\ref{R-monotonicity})$ for all $s\geq 0$. It follows

$\displaystyle
f(\Gamma(s))-f(\Gamma(0))=\int_{0}^{s}\frac{d}{dt}f(\Gamma(t))dt=\int_{0}^{s}|\nabla
f|(\Gamma(t))dt\geq cs.$

Thus there exists a uniform constant $C=\frac{L}{c}+1$ such that
$(\ref{gradient-f-almost-1})$ holds as long as $s>C$.

Let $\gamma:[0,D]\rightarrow M$ be a minimal geodesic from $\Gamma(s_{1})$ to
$\Gamma(s_{2})$, where $D=d(\Gamma(s_{1}),\Gamma(s_{2}))$. Then by

$\frac{d}{dr}\langle\nabla
f,\gamma^{\prime}(r)\rangle=\nabla^{2}f(\gamma^{\prime}(r),\gamma^{\prime}(r))\geq
0,$

we obtain

$f(\Gamma(s_{2}))-f(\Gamma(s_{1}))=\int_{0}^{D}\langle\nabla
f,\gamma^{\prime}(r)\rangle dr\leq D\langle\nabla
f,\gamma^{\prime}(D)\rangle.$

This implies

(2.15) $\displaystyle f(\Gamma(s_{2}))-f(\Gamma(s_{1}))\leq
d(\Gamma(s_{1}),\Gamma(s_{2})).$

On the other hand, by $(\ref{gradient-f-almost-1})$, we have

$\displaystyle f(\Gamma(s_{2}))-f(\Gamma(s_{1}))$
$\displaystyle=\int_{s_{1}}^{s_{2}}\langle\nabla f,\Gamma^{\prime}(r)\rangle
dr$ (2.16) $\displaystyle=\int_{s_{1}}^{s_{2}}|\nabla
f|(\Gamma(r))dr\geq(1-\epsilon)(s_{2}-s_{1}).$

Thus the first inequality in $(\ref{potential-distance})$ follows from
$(\ref{potential-less-geodesic})$ and $(\ref{potential-large-geodesic})$
immediately. Note that

(2.17) $\displaystyle
d(\Gamma(s_{1}),\Gamma(s_{2}))\leq\rm{Length}(\Gamma(s))|_{s_{1}}^{s_{2}}=s_{2}-s_{1}.$

Hence, the second inequality in $(\ref{potential-distance})$ also holds.
$(\ref{potential-distance-special})$ is a direct consequence of
$(\ref{potential-distance})$ by the triangle inequality.

∎

As in Lemma 2.4, we let $\Gamma_{i}(s)$ be an integral curve of $\hat{X}$
through $p_{i}$ with $\Gamma_{i}(0)\in S(c_{0})$ and
$\Gamma_{i}(s_{i})=p_{i}$. For any $D>0$, we set

$\hat{\Gamma}_{i}(s)=\Gamma_{i}(R(p_{i})^{-\frac{1}{2}}s+s_{i}),~{}s\in[-D,D].$

Then it is easy to see

$|\frac{d\hat{\Gamma}_{i}(s)}{ds}|_{g_{p_{i}}(0)}=1,~{}s\in[-D,D].$

Thus $\hat{\Gamma}_{i}(s)$ is an integral curve of
$\hat{X}_{i}=\frac{\nabla_{i}f}{|\nabla_{i}f|}$ through $p_{i}$, where
$\nabla_{i}$ is the gradient operator w.r.t. the metric
$(M,g_{p_{i}}(t);p_{i})$.

With help of Lemma 2.4, we prove that the splitting line obtained by
Proposition 1.2 is actually a limit of a family of integral curves of
$\hat{X}_{i}$ under the condition in Lemma 2.2.

######  Lemma 2.5.

Let $(M^{n},g)$ be the steady soliton in Lemma 2.2 and
$(N\times\mathbb{R},h(t)+ds^{2};p_{\infty})$ the splitting limit flow of
$(M,g_{p_{i}}(t);p_{i})$. Then $\hat{\Gamma}_{i}(s)$ converges locally to a
geodesic line on $N\times\mathbb{R}$ w.r.t. the metric
$(M,g_{p_{i}}(t);p_{i})$.

###### Proof.

Since $X_{i}=\nabla_{i}f$ is convergent w.r,t. the metrics
$(M,g_{p_{i}}(t);p_{i})$ (cf. [16, Lemma 4.6]), $\hat{X}_{i}$ also converges
subsequently to a VF $\hat{X}_{\infty}$ on
$(N\times\mathbb{R},h(t)+ds^{2};p_{\infty})$. Thus $\hat{\Gamma}_{i}(s)$
converges to an integral curve $\hat{\Gamma}_{\infty}(s)$ of
$\hat{X}_{\infty}$ on $N\times\mathbb{R}$, where $s\in(-\infty,\infty)$. It
remains to show that $\hat{\Gamma}_{\infty}(s)$ is a line.

Since $p_{i}\to\infty$, we have $s_{i}\to\infty$. Then by $(\ref{AVR})$ and
$(\ref{r-R-comparison})$ , for any number $D>0$, it holds

$s_{i}-\operatorname{D}R(p_{i})^{-\frac{1}{2}}\rightarrow\infty.$

By applying (2.12) to each $\hat{\Gamma}_{i}(s^{\prime})$, we get

$(1-\epsilon)DR(p_{i})^{-\frac{1}{2}}\leq
d(\hat{\Gamma}_{i}(-D),\hat{\Gamma}_{i}(0))\leq DR(p_{i})^{-\frac{1}{2}}$

and

$(1-\epsilon)DR(p_{i})^{-\frac{1}{2}}\leq
d(\hat{\Gamma}_{i}(D),\hat{\Gamma}_{i}(0))\leq DR(p_{i})^{-\frac{1}{2}}.$

It follows

$2(1-\epsilon)DR(p_{i})^{-\frac{1}{2}}\leq
d(\hat{\Gamma}_{i}(-D),\hat{\Gamma}_{i}(D))\leq 2DR(p_{i})^{-\frac{1}{2}},$

and consequently,

$2(1-\epsilon)D\leq
d_{g_{p_{i}}}(\hat{\Gamma}_{i}(-D),\hat{\Gamma}_{i}(D))\leq 2D.$

Thus by taking the limit of $\hat{\Gamma}_{i}(s)$ as well as $\epsilon\to 0$,
we obtain

(2.18) $\displaystyle
d_{g_{\infty}}\left(\hat{\Gamma}_{\infty}(-D),\hat{\Gamma}_{\infty}(D)\right)=2D.$

Note that $2D$ is the number of length of $\hat{\Gamma}_{\infty}(s)$ between
$\hat{\Gamma}_{\infty}(-D)$ and $\hat{\Gamma}_{\infty}(D)$. Hence,
$\hat{\Gamma}_{\infty}(s)$ must be a minimal geodesic connecting
$\hat{\Gamma}_{\infty}(-D)$ and $\hat{\Gamma}_{\infty}(D)$. Since $D$ is
arbitrary, $\hat{\Gamma}_{\infty}(s)$ can be extended to a geodesic line. ∎

Now we begin to prove main results in this subsection.

######  Lemma 2.6.

Let $(M^{n},g)$ be the steady soliton in Lemma 2.2 and
$(N\times\mathbb{R},h(t)+ds^{2};p_{\infty})$ the splitting limit flow of
$(M,g_{p_{i}}(t);p_{i})$, which satisfies (2.1). Then
$f^{-1}(f(p_{i}))\subseteq B_{g_{p_{i}}}(p_{i},200C)$ when $i>>1$.

###### Proof.

On contrary, there will exist a $q_{i}^{\prime}\in\partial
B_{g_{p_{i}}}(p_{i},100C)\cap f^{-1}(f(p_{i}))$ and a minimal geodesic
$\bar{\gamma}_{i}\subset f^{-1}(f(p_{i}))$ connecting $p_{i}$ and
$q_{i}^{\prime}$ w.r.t. the induced metric $\bar{g}_{p_{i}}$ on
$f^{-1}(f(p_{i}))$ such that

$\bar{\gamma}_{i}\subset B_{g_{p_{i}}}(p_{i},100C).$

Then

(2.19) $\displaystyle\rm{Length}_{\bar{g}_{p_{i}}}(\bar{\gamma}_{i})\geq
d_{g_{p_{i}}}(p_{i},q_{i}^{\prime})=100C.$

On the other hand, according to the proofs in [16, Lemma 4.3-Proposition 4.5],
the part $\Sigma_{i}=f^{-1}(f(p_{i}))\cap B_{g_{p_{i}}}(p_{i},100C)$ of level
set $f^{-1}(f(p_{i}))$, which contains $\bar{\gamma}_{i}$, converges
subsequently to an $(n-1)$-dimensional open manifold
$(\Sigma_{\infty},h^{\prime};p_{\infty})$ w.r.t. the induced metric
$\bar{g}_{p_{i}}$. As a consequence, the minimal geodesic $\bar{\gamma}_{i}$
converges subsequently to a minimal geodesic $\bar{\gamma}$ in
$\Sigma_{\infty}$. Thus by $(\ref{length-bar-gamma})$, we get

(2.20) $\displaystyle\rm{Length}_{h^{\prime}}(\bar{\gamma})\geq 100C.$

Next we show that $(\Sigma_{\infty},h^{\prime})$ is an open set of $(N,h(0))$.
Then it follows

${\rm{Diam}}(N,h(0))\geq{\rm{Diam}}(\Sigma_{\infty},h^{\prime})\geq 100C,$

which contradicts to (2.1). The lemma will be proved.

Let $\hat{X}_{i}=\frac{\nabla_{i}f}{|\nabla_{i}f|}$. By Lemma 2.2,
$(\ref{gradient-f-almost-1})$ and Shi’s estimates we can calculate that

$\displaystyle\sup_{B(p_{i},2D)_{g_{p_{i}}}}|\nabla_{i}\hat{X}_{i}|_{g_{p_{i}}}$
$\displaystyle=\sup_{B(p_{i},2D)_{g_{p_{i}}}}R(p_{i})^{-\frac{1}{2}}(\frac{|\rm{Ric}|}{|\nabla
f|}+\frac{|\rm{Ric}(\nabla f,\nabla f)|}{|\nabla f|^{3}})$ $\displaystyle\leq
CR(p_{i})^{\frac{1}{2}}\to 0,$

and

$\displaystyle\sup_{B(p_{i},2D)_{g_{p_{i}}}}|\nabla^{m}_{i}\hat{X}_{i}|_{g_{p_{i}}}$
$\displaystyle\leq
C(m)\sup_{B(p_{i},2D)_{g_{p_{i}}}}|\nabla^{m-1}_{i}\rm{Ric(g_{p_{i}})}|_{g_{p_{i}}}\leq
C^{\prime}.$

Thus $\hat{X}_{i}$ converges subsequently to a parallel vector field
$\hat{X}_{\infty}$ on $(N\times\mathbb{R},h(t)+ds^{2};p_{\infty})$. Moreover,

$\displaystyle\sup_{B(p_{i},2D)_{g_{p_{i}}}}|\hat{X}_{i}|_{g_{p_{i}}}=1.$

Hence, $\hat{X}_{\infty}$ is a non-trivial parallel vector field on
$N\times\mathbb{R}$.

$\hat{X}_{\infty}$ is also perpendicular to $(\Sigma_{\infty},h^{\prime})$. In
fact, for any $V\in T\Sigma_{\infty}$ with $|V|_{h^{\prime}}=1$, by [16,
Proposition 4.5], there is a sequence of $V_{i}\in T\Sigma_{i}$ such that
$R(p_{i})^{-\frac{1}{2}}V_{i}\to V$. Thus

$\displaystyle
h^{\prime}(V,\hat{X}_{\infty})=\lim_{i\to\infty}g_{p_{i}}(R(p_{i})^{-\frac{1}{2}}V_{i},\hat{X}_{i})=\lim_{i\to\infty}g(V_{i},\frac{\nabla
f}{|\nabla f|})=0.$

By Lemma 2.5, we have already known that $\hat{X}_{\infty}$ generates a
geodesic line $\hat{\Gamma}_{\infty}$ through $p_{\infty}$ on
$N\times\mathbb{R}$. Note that $(N,h(0))$ is compact by $(\ref{bound-h})$.
$\hat{X}_{\infty}$ must be tangent to the splitting line direction of
$N\times\mathbb{R}$, and consequently,
$(\Sigma_{\infty},h^{\prime};p_{\infty})\subset(N,h(0);p_{\infty})$. Namely,
$(\Sigma_{\infty},h^{\prime})$ is an open set of $(N,h(0))$. The proof is
complete.

∎

By Lemma 2.6, we prove

######  Proposition 2.7.

Let $(M^{n},g)$ be the steady soliton in Lemma 2.2 and
$(N\times\mathbb{R},h(t)+ds^{2};p_{\infty})$ the splitting limit flow of
$(M,g_{p_{i}}(t);p_{i})$, which satisfies (2.1). Then there exists
$C_{0}(C)>0$ such that for any $q_{i}\in f^{-1}(f(p_{i}))$ the splitting limit
flow $(h^{\prime}(t)+ds^{2},N^{\prime}\times\mathbb{R};q_{\infty})$ of
rescaled flow $(M,g_{q_{i}}(t);q_{i})$ satisfies

(2.21) $\displaystyle{\rm Diam}(h^{\prime}(0))\leq C_{0}.$

###### Proof.

The convergence part comes from Proposition 1.2. We need to check (2.21). In
fact, by Lemma 2.6 and Lemma 1.3, there are $C_{1},C_{2}>0$ such that for any
$D>0$ such that

$\displaystyle B_{g_{q_{i}}}(q_{i},D)\subset
B_{g_{p_{i}}}(q_{i},C_{1}D)\subset B_{g_{p_{i}}}(p_{i},C_{1}D+C_{2}),$

where $g_{q_{i}}=R(q_{i})g$. Similarly, we have

$\displaystyle B_{g_{q_{i}}}(q_{i},D)\supset
B_{g_{p_{i}}}(q_{i},C_{1}^{-1}D)\supset
B_{g_{p_{i}}}(p_{i},C_{1}^{-1}D-2C_{2}).$

Then it is easy to see that the splitting Ricci flow
$(h^{\prime}(t)+ds^{2},N^{\prime}\times\mathbb{R};q_{\infty})$ of
$(M,g_{q_{i}}(t);q_{i})$ is isometric to
$(h(t)+ds^{2},N\times\mathbb{R};p_{\infty})$ up to scaling. As a consequence,
we get

$\displaystyle\rm{Diam}(h^{\prime}(0))\leq(C_{1}+10)\rm{Diam}(h(0))\leq(C_{1}+10)C.$

The proposition is proved.

∎

######  Remark 2.8.

By Lemma 2.6 and Proposition 2.7, the argument in the proof Lemma 2.6 also
implies that both of $N$ and $N^{\prime}$ are diffeomorphic to each level set
$f^{-1}(f(p_{i}))$ when $i>>1$. In fact, the submanifolds
$(f^{-1}(f(p_{i})),\bar{g}_{p_{i}})$ converge to $(N,h(0))$ (also to
$(N^{\prime},h^{\prime}(0))$) in the Cheeger-Gromov sense.

##  3\. $4d$ steady Ricci solitons

In this section, we first recall recent works on compact $3d$ ancient
$\kappa$-solitons by Angenent-Brendle-Daskalopoulos-Sesum and Brendle-Daskalo-
poulos-Sesum [1, 7], then we estimate the diameter of $(N,h(0))$ for all split
limit flow $(N,h(t))$ of rescaled flow sequence.

As we know, Perelman model ancient solution is of type II, which is defined on
$S^{3}$ with $Z_{2}\times O(2)$-symmetry for any $t\in(-\infty,0)$ [27].
According to [20], we have the definition,

######  Definition 3.1.

An ancient solution with ${\rm K_{m}}\geq 0$ is called type I if it satisfies

$\displaystyle\sup_{M\times(-\infty,0]}(-t)R(x,t)<\infty.$

Otherwise, it is called type II, i.e., it satisfies

$\displaystyle\sup_{M\times(-\infty,0]}(-t)R(x,t)=\infty.$

Fix $p_{0}\in S^{3}$. We normalize the Perelman solution by

(3.1) $\displaystyle R_{max}(-1)=R(p_{0},-1)=1.$

For simplicity, we denote it by $(S^{3},g_{Pel}(t);p_{0})$, $t\in(-\infty,0)$.

The asymptotic behavior of $(S^{3},g_{Pel}(t);p_{0})$ has been computed in [1]
as follows,

$\displaystyle\rm{Diam}({g_{Pel}(t)})\geq 2.1\sqrt{(-t)\log(-t)},$
$\displaystyle R_{max}\leq 1.1\frac{\log(-t)}{-t},$ (3.2) $\displaystyle
R_{min}\geq\frac{C}{-t}.$

Here $-t\geq L$ for some large $L>10000C>10000$. Thus

$\rm{Diam}(g_{Pel}(t))R^{\frac{1}{2}}(q,t)$

is strictly increasing as $t\rightarrow-\infty$, and

(3.3)
$\displaystyle\lim_{t\rightarrow-\infty}\rm{Diam}(g_{Pel}(t))R^{\frac{1}{2}}(q,t)=\infty,~{}\forall~{}q\in
S^{3}.$

In particular, there exists a constant $C_{Diam}$ such that
$\rm{Diam}(g_{Pel}(t))\geq C_{Diam}$, when $-t\geq 2L$. Usually, we call all
$3d$ ancient $\kappa$-solitons of type II on $S^{3}$ as Perelman (ancient)
solutions.

The following classification result of 3d compact ancient $\kappa$-solutions
of type II was proved in [7].

######  Theorem 3.2.

Any 3d compact simply connected ancient $\kappa$-solution of type II coincides
with a reparametrization in space, a translation in time, and a parabolic
rescaling of Perelman solution $(S^{3},g_{Pel}(t);p_{0})$.

By Theorem 3.2, for any simply connected compact $3d$ $\kappa$-solution
$(M,h(t);q)$ of type II and a point $q\in M$, there exist constant $\lambda$,
a time $T$, $p\in S^{3}$ and a diffeomorphism $\Psi$ from $S^{3}$ to $M$ such
that $\Psi(p)=q$ and

(3.4)
$\displaystyle(\Psi^{-1}(M),\lambda\Psi^{*}(h(\lambda^{-1}t)),\Psi^{-1}(q))=(S^{3},g_{Pel}(t-T);p_{0}).$

We note that the Perelman’s solution $(S^{3},g_{Pel}(t);p_{0})$ is
$Z_{2}\times O(2)$-symmetric. Then the isometric subgroup of
$(S^{3},g_{Pel}(t);p_{0})$ must be as $Z_{2}\times G$, where $G$ is a subgroup
of $O(2)$. Thus $G$ fixes the minimal geodesic connecting two tips of the
Perelman solution. It follows that any quotient of Perelman solution, which is
also an ancient $\kappa$-solutions, satisfies the above asymptotic behavior
$(\ref{asymptotic-behaviour})$. Hence, by the classification Theorem 3.2, we
get

######  Proposition 3.3.

Let $(M,h(t))$ be a 3d compact ancient $\kappa$-solutions of type II and $p\in
M$, which satisfies

(3.5) $\displaystyle R(p,0)=1$

and

(3.6) $\displaystyle\rm{Diam}(h(0))=C^{\prime}>10C_{Diam}.$

Then for any $q\in M$, $\rm{Diam}(h(t))R^{\frac{1}{2}}(q,t)$ is strictly
decreasing for $t\leq 0$. Moreover, there exists a $T(C^{\prime})$ such that

(3.7)
$\displaystyle\rm{Diam}(h(T(C^{\prime})))R^{\frac{1}{2}}(q,T(C^{\prime}))=2C^{\prime}.$

By Theorem 3.2 and Proposition 3.3, we are able to classify the split ancient
$\kappa$-solutions of dimension 3 when the $4d$ noncompact
$\kappa$-noncollapsed steady Ricci soliton in Theorem 0.2 admits a split
noncompact ancient $\kappa$-solution $(N,h(t))$.

We need the following definition introduced by Perelman (cf. [27]).

######  Definition 3.4.

For any $\epsilon>0$, we say a pointed Ricci flow
$\left(M_{1},g_{1}(t);p_{1}\right),t\in$ $[-T,0]$, is $\epsilon$-close to
another pointed Ricci flow $\left(M_{2},g_{2}(t);p_{2}\right),t\in[-T,0]$, if
there is a diffeomorphism onto its image
$\bar{\phi}:B_{g_{2}(0)}\left(p_{2},\epsilon^{-1}\right)\rightarrow M_{1}$,
such that $\bar{\phi}\left(p_{2}\right)=p_{1}$ and
$\left\|\bar{\phi}^{*}g_{1}(t)-g_{2}(t)\right\|_{C^{\left[\epsilon^{-1}\right]}}<\epsilon$
for all $t\in\left[-\min\left\\{T,\epsilon^{-1}\right\\},0\right]$, where the
norms and derivatives are taken with respect to $g_{2}(0)$.

By Proposition 1.2 together with the above definition, we get immediately,

######  Proposition 3.5.

Let $(M^{n},g)$ be the steady Ricci soliton in Proposition 1.2. Then for any
$\epsilon>0$, There exists a compact set $D(\epsilon)>0$, such that for any
$p\in M\setminus D$, $(M,g_{p}(t);p)$ is $\epsilon$-close to a splitting flow
$(h_{p}(t)+ds^{2};p)$, where $h_{p}(t)$ is an $(n-1)$-dimensional ancient
$\kappa$-solution.

We note that for a given $p$ and a number $\epsilon>0$ the $\epsilon$-close
splitting flow $(h_{p}(t)+ds^{2};p)$ may not be unique in Proposition 3.5. Due
to [22], we introduce a function on $M$ for each $\epsilon$ by

(3.8) $\displaystyle
F_{\epsilon}(p)=\inf_{h_{p}}\\{{\rm{Diam}}(h_{p}(0))\in(0,\infty)\\}.$

For simplicity, we always omit the subscribe $\epsilon$ in $F_{\epsilon}(p)$
below.

By estimating (3.8), we prove

######  Proposition 3.6.

Let $(M^{4},g)$ be a noncompact $\kappa$-noncollapsed steady Ricci soliton in
Theorem 0.2. Suppose that there exists a sequence of pointed rescaled Ricci
flows $(M,g_{p_{i}}(t);p_{i})$ converges subsequently to a splitting Ricci
flow $(h(t)+ds^{2};p_{\infty})$ for some noncompact ancient $\kappa$-solution
$h(t)$. Then for any limit flow $(h^{\prime}(t)+ds^{2};q_{\infty})$ of
rescaled Ricci flows $(M,g_{q_{i}}(t);q_{i})$, $h^{\prime}(t)$ is a noncompact
ancient $\kappa$-solution.

###### Proof.

We argue by contradiction. Suppose that there exists a limit flow
$(h^{\prime}(t)+ds^{2};q_{\infty})$ converged by rescaled flows
$(M,g_{q_{i}}(t);q_{i})$, which satisfies $(\ref{bound-h})$. Then by
Proposition 2.7, there exists a uniform constant $C_{3}(C)>0$, such that

(3.9) $\displaystyle F(p_{i}^{\prime})\leq C_{3}$

for all $i$, and all $p_{i}^{\prime}\in f^{-1}(f(q_{i}))$.

Fix $C^{\prime}=\max\\{100C_{Diam},10C_{3}\\}$ and $T(C^{\prime})$ as in
Proposition 3.3. We choose an $\epsilon>0$ such that
$\epsilon^{-1}>\max\\{10T(C^{\prime}),100C^{\prime}\\}$. Thus for the sequence
of $(M,g_{p_{i}}(t);p_{i})$ in Proposition 3.6, we can choose a point
$p_{i_{0}}\in\\{p_{i}\\}$ such that

(3.10) $\displaystyle F(p_{i_{0}})>C^{\prime}\geq 100C_{Diam}.$

Let $\Gamma_{1}$ be the integral curve of $\hat{X}$ passing through
$p_{i_{0}}$ with $\Gamma_{1}(0)=p_{i_{0}}$, which tends to the infinity by
Lemma 2.4. We claim:

(3.11) $\displaystyle
F(\Gamma_{1}(s))>\frac{1}{2}C^{\prime},~{}\forall~{}s\geq 0.$

Define

$s_{0}=\sup\\{s\geq 0|F(s^{\prime})\geq C^{\prime}~{}\text{for
all}~{}s^{\prime}\in[0,s]\\}.$

If $s_{0}=\infty$, $F(s)>C^{\prime}$ for all $s\geq 0$. Then (3.11) is obvious
true in this case. Thus we may consider the case $s_{0}<\infty$, i.e.,
$F(s_{0})=C^{\prime}$ for some $s^{\prime}=s_{0}$. It follows that there
exists a $3d$ compact ancient $\kappa$-solution $h_{\Gamma_{1}(s_{0})}(t)$
such that

$\displaystyle(M,R(\Gamma_{1}(s_{0}))g(R(\Gamma_{1}(s_{0}))^{-1}t);\Gamma_{1}(s_{0}))$
(3.12)
$\displaystyle\overset{\epsilon-\text{close}}{\sim}(N\times\mathbb{R},h_{\Gamma_{1}(s_{0})}(t)+ds^{2};\Gamma_{1}(s_{0})).$

Since the diameter of $h_{\Gamma_{1}(s_{0})}(0)$ is large,
$h_{\Gamma_{1}(s_{0})}(t)$ can not be a family of shrinking quotient spheres.
Hence, by Theorem 3.2, it must be a quotient of Perelman solution after a
reparametrization.

By Proposition 3.3, we see that
$\rm{Diam}(h_{\Gamma_{1}(s_{0})}(t))R_{h}^{\frac{1}{2}}(\Gamma_{1}(s_{0}),t)$
is strictly decreasing for $t\in(-\epsilon^{-1},0]$. By $(\ref{F_0})$, it
follows

(3.13)
$\displaystyle\rm{Diam}(h_{\Gamma_{1}(s_{0})}(t))R_{h}^{\frac{1}{2}}(\Gamma_{1}(s_{0}),t)>C^{\prime},~{}t\in(-\epsilon^{-1},0].$

Moreover, by the choice of $T(C^{\prime})$, we have

$\displaystyle\rm{Diam}(h(T(C^{\prime}))R_{h}^{\frac{1}{2}}(\Gamma_{1}(s_{0}),-T(C^{\prime}))=2C^{\prime}.$

Let $t_{1}=\min\\{-1000,-T(C^{\prime})\\}\geq-\frac{\epsilon^{-1}}{2}$. Thus

(3.14)
$\displaystyle\rm{Diam}(h_{\Gamma_{1}(s_{0})}(t_{1}))R_{h}^{\frac{1}{2}}(\Gamma_{1}(s_{0}),t_{1})\geq
2C^{\prime}.$

Recall that $\left\\{\phi_{t}\right\\}_{t\in(-\infty,\infty)}$ is the flow of
$-\nabla f$ with $\phi_{0}$ the identity and $(g(t),\Gamma_{1}(s))$ is
isometric to $\left(g,\phi_{t}(\Gamma_{1}(s))\right)$. Then

$\displaystyle\phi_{t}(\Gamma_{1}(s))=\Gamma_{1}\left(s-\int_{0}^{t}|\nabla
f|\left(\phi_{\mu}(\Gamma_{1}(s))\right)d\mu\right)$

Let $T=tR\left(\Gamma_{1}\left(s_{0}\right)\right)^{-1}<0$ and

(3.15) $\displaystyle s=s_{0}-\int_{0}^{T}|\nabla
f|\left(\phi_{\mu}\left(\Gamma_{1}\left(s_{0}\right)\right)\right)d\mu.$

Set

$s_{1}=s_{0}-\int_{0}^{T_{1}}|\nabla
f|\left(\phi_{\mu}\left(\Gamma_{1}\left(s_{0}\right)\right)\right)d\mu,$

where $T_{1}=t_{1}R\left(\Gamma_{1}\left(s_{0}\right)\right)^{-1}.$ Since the
scalar curvature $R$ of $(M,g)$ decays to $0$ uniformly by Proposition 2.2, we
may assume $|\nabla f|\geq\frac{1}{2}$ along $\Gamma_{1}$. Thus

(3.16) $\displaystyle s_{1}-s_{0}\geq
500R^{-1}\left(\Gamma_{1}\left(s_{0}\right)\right)\geq 500R^{-1}_{max}=500.$

Note that
$\phi_{T}\left(\Gamma_{1}\left(s_{0}\right)\right)=\Gamma\left(s\right)$ and
$\left(g\left(T\right),\Gamma_{1}\left(s_{0}\right)\right)$ is isometric to
$\left(g,\Gamma_{1}\left(s\right)\right)$ for all $s\in[s_{0},s_{1}]$. Then

$\displaystyle(M,R(\Gamma_{1}(s))g;\Gamma_{1}(s))$
$\displaystyle\cong(M,R(\Gamma_{1}(s_{0}),T)g(T);\Gamma_{1}(s_{0}))$ (3.17)
$\displaystyle\cong(M,\frac{R(\Gamma_{1}(s_{0}),T)}{R(\Gamma_{1}(s_{0}))}R(\Gamma_{1}(s_{0}))g(T);\Gamma_{1}(s_{0})).$

Since $R(\Gamma_{1}(s_{0}),T)\leq R(\Gamma_{1}(s_{0}))$ by
$(\ref{R-monotonicity})$, we get from (3),

$\displaystyle(M,R(\Gamma_{1}(s))g;\Gamma_{1}(s))$ (3.18)
$\displaystyle\overset{\epsilon-\text{close}}{\sim}(N\times\mathbb{R},\frac{R(\Gamma_{1}(s_{0}),T)}{R(\Gamma_{1}(s_{0}))}h_{\Gamma_{1}(s_{0})}(t)+ds^{2};\Gamma_{1}(s_{0})).$

On the other hand, there is another $3d$ compact ancient $\kappa$-solution
$h_{\Gamma_{1}(s_{1})}(t)$ corresponding to the point $\Gamma_{1}(s)$ such
that

(3.19)
$\displaystyle(M,R(\Gamma_{1}(s))g;\Gamma_{1}(s))\overset{\epsilon-\text{close}}{\sim}(h_{\Gamma_{1}(s)}(0)+ds^{2},\Gamma_{1}(s)).$

Hence, combining $(\ref{s1-close1})$ and $(\ref{s1-close2})$, we derive

(3.20) $\displaystyle
h_{\Gamma_{1}(s)}(0)\overset{\epsilon-\text{close}}{\sim}\frac{R(\Gamma_{1}(s_{0}),T)}{R(\Gamma_{1}(s_{0}))}h_{\Gamma_{1}(s_{0})}(t).$

By the convergence of $(M,g_{p}(t);p)$, we have

$\frac{R(\Gamma_{1}(s_{0}),T)}{R(\Gamma_{1}(s_{0}))}\overset{\epsilon-\text{close}}{\sim}R_{h}(\Gamma_{1}(s_{0}),t),~{}\forall~{}t\in[t_{1},0],$

and so,

$\displaystyle\rm{Diam}(\frac{R(\Gamma_{1}(s_{0}),T)}{R(\Gamma_{1}(s_{0}))}h_{\Gamma_{1}(s_{0})}(t))\overset{\epsilon-\text{close}}{\sim}\rm{Diam}(R_{h}(\Gamma_{1}(s_{0}),t)h_{\Gamma_{1}(s_{0})}(t))).$

Then by (3.20), the monotonicity $(\ref{t0-t1-diameter})$ implies that

(3.21) $\displaystyle F(\Gamma_{1}(s))\geq
C^{\prime}-2\epsilon>\frac{1}{2}C^{\prime},~{}\forall~{}s\in\left[s_{0},s_{1}\right].$

Moreover, by $(\ref{t1-diameter})$,

(3.22) $\displaystyle
F\left(\Gamma_{1}(s_{1})\right)>2C^{\prime}-2\epsilon>C^{\prime}.$

By (3.22) together with (3.21) and (3.16), we can repeat the above argument to
obtain (3.11). On the other hand, the curve $\Gamma_{1}(s)$ passes through
level sets $f^{-1}(f(q_{i}))$ because of
$\lim_{s\rightarrow\infty}f(\Gamma_{1}(s))=\infty$. Thus for each $q_{i}$
($i>>1$) there exists $p_{i}^{\prime}\in f^{-1}(f(q_{i}))$ such that
$p_{i}^{\prime}=\Gamma_{1}(s_{i})$ for some $s_{i}$. By $(\ref{compact-
diameter})$, $F(p_{i}^{\prime})\leq C_{3}$, which contradicts with
$(\ref{noncompact-diameter})$. Hence, the proposition is proved. ∎

##  4\. Proofs of the main results

In this section, we prove Theorem 0.2 and Corollary 0.3. Firstly, we consider
a special case: there is a uniform constant $C$ such that all split ancient
$\kappa$-solution $h(t)$ in Proposition 1.2 satisfies $(\ref{bound-h})$. By
generalizing the argument in Section 3 we prove

######  Proposition 4.1.

Let $(M^{4},g)$ be a $4d$ noncompact $\kappa$-noncollapsed gradient steady
Ricci soliton with $\rm{Km}\geq 0$ and $\rm{Ric}>0$ away from a compact set
$K$ of $M$. Suppose that there is a uniform constant $C$ such that all split
ancient $\kappa$-solution $h(t)$ in Proposition 1.2 satisfies
$(\ref{bound-h})$. Then all $h(t)$ must be a family of shrinking quotient
spheres.

By a result of Ni [25], it suffices to prove all $h(t)$ is a compact
$\kappa$-noncollapsed ancient solution of type I. In other words, we shall
exclude the existence of $\kappa$-noncollapsed ancient solutions of compact
type II. The proof is based on two lemmas [1, Lemma 2.1, Lemma 2.2]. For the
reader’s convenience, we give a sketch proof of those lemmas below.

######  Lemma 4.2.

Let $(N^{3},h(t))$ be a $3d$ compact $\kappa$-solution of type II.111By
Theorem 3.2, $(N,h(t))$ coincides with a quotient of Perelman solution
$(S^{3},g_{Pel}(t);p_{0})$. Fix $p\in N$, we consider
$t_{k}\rightarrow-\infty$ and a sequence of points $x_{k}\in N$ such that
$\ell(x_{k},t_{k})<\frac{3}{2}$, where $\ell$ denotes the reduced distance
from $(p,0)$. Then the rescaled manifold by dilating the manifold
$(N,h(t_{k}))$ around the point $x_{k}$ by the factor
$\frac{1}{\sqrt{-t_{k}}}$ converges to a $3d$ noncompact shrinking gradient
Ricci soliton.

###### Proof.

By Perelman’s argument in [26, Section 11], the rescaled manifold converges in
the Cheeger-Gromov sense to a $3d$ $\kappa$-noncollapsed shrinking gradient
Ricci soliton $(N^{\prime},h^{\prime}(t))$ with nonnegative curvature
operator. If $(N^{\prime},h^{\prime}(t))$ is compact, it must be a quotient of
$3d$ round sphere by a result of Perelman [27, Section 1] (also see [24,
Corollary 4]). In particular, the sectional curvatures of $(N,h(t_{k}))$ must
lie in the interval
$[\frac{c-\epsilon_{k}}{-t_{k}},\frac{c+\epsilon_{k}}{-t_{k}}]$, where
$\epsilon_{k}\rightarrow 0$ as $k\rightarrow\infty$. Then by curvature
pinching estimates, $(N,h(t))$ has constant sectional curvature for each $t$
[18] (see also [9, 8]). It follows that $(N,h(t))$ is also a family of
shrinking round quotient spheres, which contradicts with the type II
condition. Hence, $(N^{\prime},h^{\prime}(t))$ must be non-compact. The lemma
is proved. ∎

######  Lemma 4.3.

Let $(N^{3},h(t))$ be a $3d$ compact $\kappa$-solution of type II. Then for
any sequence of times $t_{k}\rightarrow-\infty$, it holds

$\displaystyle\rm{R}_{max}(t_{k})\rm{Diam}(h(t_{k}))^{2}\rightarrow\infty,$

where $\rm{R}_{max}(t)=\max\\{R(h(\cdot,t)\\}$. In particular,

(4.1)
$\displaystyle\lim_{t\rightarrow-\infty}\rm{R}_{max}(t)\rm{Diam}(h(t))^{2}\rightarrow\infty.$

###### Proof.

(4.1) is true according to (3.3) and the classification Theorem 3.2. In the
following, we give a direct proof of (4.1) by Lemma 4.2. In fact, by a result
of Perelman [26, Section 11], for any sequence of times
$t_{k}\rightarrow-\infty$, we can always find a sequence of points $x_{k}\in
N$ such that $\ell(x_{k},t_{k})\leq\frac{3}{2}$ for each $k$. Then by Lemma
4.2, the rescaled flow $(N,(-t_{k})^{-1}h((-t_{k})t);x_{k})$ converges to a
noncompact shrinking Ricci soliton with non-negative curvature operator. It
follows

(4.2)
$\displaystyle\rm{Diam}((-t_{k})^{-1}h(t_{k}))=(-t_{k})^{-\frac{1}{2}}\rm{Diam}(h(t_{k}))\to\infty.$

Moreover, such a limit soliton is non-flat [21, Proposition 39.1]. Thus there
exists a uniform constant $\delta>0$, such that
$(-t_{k})R(x_{k},t_{k})\geq\delta$ for all $k>>1$. Hence, by (4.2), we derive

$\displaystyle\rm{R}_{max}(t_{k})\rm{Diam}(h(t_{k}))^{2}$
$\displaystyle\geq\rm{R}(x_{k},t_{k})\rm{Diam}(h(t_{k}))^{2}$
$\displaystyle\geq\delta(-t_{k})^{-1}\rm{Diam}(h(t_{k}))^{2}$
$\displaystyle\to\infty.$

∎

###### Proof of Proposition 4.1.

If the proposition is false, by the Ni’s result [25] (also see [7]), there
will exist a sequence of rescaled flow $(M,g_{p_{i}}(t),p_{i})$ converges
subsequently to a splitting Ricci flow
$(N\times\mathbb{R},h(t)+ds^{2};p_{\infty})$, where $(N,h(t))$ is a $3d$
compact ancient $\kappa$-solution of type II.

Choose $t_{k}\to-\infty$. Then for each $t_{k}$ there exists $q_{ik}\in
f^{-1}(f(p_{i}))$ such that

$R(q_{ik},R(p_{i})^{-1}t_{k})={\rm max}\\{R(x,R(p_{i})^{-1}t_{k})|~{}x\in
f^{-1}(f(p_{i}))\\}.$

It follows that for $x\in f^{-1}(f(p_{i}))$ it holds

(4.3) $\displaystyle R(q_{ik})^{-1}R(q_{ik},R(p_{i})^{-1}t_{k})\geq
R(q_{ik})^{-1}R(x,R(p_{i})^{-1}t_{k}).$

For each $k$, we know that rescaled flow $(M,g_{q_{ik}}(t);q_{ik})$ converges
subsequently to a splitting Ricci flow
$(N_{k}\times\mathbb{R},h_{k}(t)+ds^{2};q_{\infty k})$. Moreover, by Remark
2.8, submanifold $(f^{-1}(f(p_{i})),\bar{g}_{q_{ik}};q_{ik})$ converges
subsequently to $(N_{k},h_{k}(0),q_{\infty k})$ w.r.t. the induced metric,
where each $(N_{k},h_{k}(0))$ is isometric $(N,h(0))$ up to rescaling.

We note that $\phi_{t}(q_{i}),\phi_{t}(p_{i})\in f^{-1}(f(\phi_{t}(p_{i})))$
for any $t\leq 0$ when $i>>1$. Then by Lemma 1.3, there is a constant
$C_{0}(C)>1$, where the constant $C$ is determined in $(\ref{bound-h})$ such
that

(4.4)
$\displaystyle\frac{R(\phi_{t}(q_{ik}))}{R(\phi_{t}(p_{i}))}\in[C_{0}^{-1},C_{0}],~{}\forall
t\leq 0.$

In particular, $\frac{R(q_{ik})}{R(p_{i})}$ converge subsequently to a
constant $\lambda_{k}\in[C_{0}^{-1},C_{0}]$. Thus by $(\ref{sequence-max-
point})$ with help of the convergence of $(M,g_{q_{ik}}(t);q_{ik})$, we get

(4.5) $\displaystyle R_{h_{k}}(q_{\infty k},\lambda_{k}t_{k})\geq
R_{h_{k}}(x,\lambda_{k}t_{k}),~{}\forall x\in N.$

Since each $h_{k}(\lambda_{k}t_{k})$ is isometric to $h(t_{k})$ up to scaling,
we derive

$R_{h}(q_{\infty k},t_{k})={\rm max}\\{R_{h}(x,t_{k})|~{}x\in N\\}.$

Hence, by Lemma 4.3,

$\lim_{k\rightarrow\infty}\rm{Diam}(h(t_{k}))R_{h}^{\frac{1}{2}}(q_{\infty
k},t_{k})=\lim_{k\rightarrow\infty}\rm{Diam}(h(t_{k}))R_{h,max}^{\frac{1}{2}}(t_{k})=\infty.$

As a consequence, there is $t_{0}\in\\{t_{k}\\}$ for some $k_{0}$ such that

(4.6) $\displaystyle\rm{Diam}(h(t_{0}))R_{h}^{\frac{1}{2}}(q_{\infty
k_{0}},t_{0})>100CC_{0}^{\frac{1}{2}}.$

For simplicity, we let $q_{i}=q_{ik_{0}}$ and $q_{\infty}=q_{\infty k_{0}}$.
Set $T=t_{0}R^{-1}(p_{i})$ and choose $\epsilon<-\frac{1}{100t_{0}}$. Then
$(M,g_{p_{i}}(t);q_{i})$ is $\epsilon$-close to
$(N\times\mathbb{R},h(t)+ds^{2};q_{\infty})$ when $i>>1$. Thus by (4.6), it
holds

(4.7) $\displaystyle{\rm Diam}(\bar{g}(T))R^{\frac{1}{2}}(q_{i},T)={\rm
Diam}(\bar{g}_{p_{i}}(t_{0}))R_{p_{i}}^{\frac{1}{2}}(q_{i},t_{0})>90CC_{0}^{\frac{1}{2}},$

as long as $i>>1$, where $\bar{g}(T)$ is the induced metric of $g$ on
$f^{-1}(f(\phi_{T}(p_{i})))$ and $\bar{g}_{p_{i}}(t)$ is the induced metric of
$g_{p_{i}}(t)$ on $f^{-1}(f(p_{i}))$, respectively. On the other hand, since

$\frac{R(\phi_{T}(p_{i}))}{R(\phi_{T}(q_{i}))}\in[C_{0}^{-1},C_{0}],$

we have

(4.8) $\displaystyle\frac{R(p_{i},T)}{R(q_{i},T)}\in[C_{0}^{-1},C_{0}].$

Hence combining $(\ref{qi-sequence-large})$ and $(\ref{pi-qi-bound})$, we
obtain

$\displaystyle{\rm Diam}(\bar{g}(T))R^{\frac{1}{2}}(p_{i},T)$
$\displaystyle=(\frac{R(p_{i},T)}{R(q_{i},T)})^{\frac{1}{2}}{\rm
Diam}(\bar{g}(T))R^{\frac{1}{2}}(q_{i},T)$ $\displaystyle>90C,~{}i>>1.$

This means

$\displaystyle{\rm Diam}(R(\phi_{T}(p_{i}))\bar{g}(T))$ $\displaystyle={\rm
Diam}(\bar{g}(T))R^{\frac{1}{2}}(\phi_{T}(p_{i}))$ $\displaystyle={\rm
Diam}(\bar{g}(T))R^{\frac{1}{2}}(p_{i},T)$ (4.9) $\displaystyle>90C,~{}i>>1.$

Let $(\tilde{N},\tilde{h}(t))$ be a $3d$ split ancient flow of limit of
$(M,g_{(\phi_{T}(p_{i}))};\phi_{T}(p_{i}))$. Then $(\tilde{N},\tilde{h}(0))$
is a limit of submanifolds
$(f^{-1}(f(\phi_{T}(p_{i}))),R(\phi_{T}(p_{i}))\bar{g}(T);\phi_{T}(p_{i}))$
(see Remark 2.8). Thus by (4), we get

${\rm Diam}(\tilde{N},\tilde{h}(0))\geq 90C.$

It follows

$F(\phi_{T}(p_{i}))>50C,~{}i>>1,$

and so

(4.10) $\displaystyle\limsup_{i\rightarrow\infty}F(\phi_{T}(p_{i}))\geq 50C.$

On the other hand, by the condition of proposition, we see

(4.11) $\displaystyle\limsup_{p\rightarrow\infty}F(p)<2C.$

Hence, we get a contradiction between $(\ref{max-diameter})$ and $(\ref{max-
diameter-T1})$. The proposition is proved. ∎

Now we are able to prove Theorem 0.2 by Proposition 4.1 together with
Proposition 3.6.

###### Proof of Theorem 0.2.

Case 1:

$\limsup_{p\rightarrow\infty}F_{\epsilon}(p)<C.$

for any $\epsilon<1$. Then $(\ref{bound-h})$ holds for all split ancient
$\kappa$-solution $h(t)$. Thus by Proposition 4.1, all $h(t)$ must be a family
of shrinking quotient spheres.

Case 2:

$\limsup_{\epsilon\to 0}\limsup_{p\rightarrow\infty}F_{\epsilon}(p)=\infty.$

In this case, by taking a diagonal subsequence, there is a sequence of pointed
flows $(M,g_{q_{i}}(t);q_{i})$, which converges subsequently to a splitting
Ricci flow $(N^{\prime}\times\mathbb{R},h^{\prime}(t)+ds^{2};q_{\infty})$ for
some noncompact ancient $\kappa$-solution $h^{\prime}(t)$. Then by Proposition
3.6, $h(t)$ is a noncompact $\kappa$-solution for any splitting limit flow
$(N\times\mathbb{R},h(t)+ds^{2};p_{\infty})$. ∎

###### Proof of Corollary 0.3.

By the assumption, the split $3$-dimensional ancient flows $(N,h(t))$ of limit
of $(M,g_{p_{i}}(t),p_{i})$ is a family of shrinking quotient spheres. Namely,
$(N,h(0))$ is a quotient of round sphere. We claim: $(M,g)$ has positive Ricci
curvature on $M$.

On contrary, ${\rm Ric}(g)$ is not strictly positive. We note that $(\ref{pi-
decay})$ is still true in the proof of Lemma 2.2 without ${\rm Ric}(g)>0$ away
from a compact set of $M$. Then as in the proof of [16, Lemma 4.6], we see
that $X_{i}=R(p_{i})^{-\frac{1}{2}}\nabla f\to X_{\infty}$ w.r.t.
$(M,g_{p_{i}}(t),p_{i})$, where $X_{\infty}$ is a non-trivial parallel vector
field. Thus according to the argument in the proof of [16, Theorem 1.3], the
universal cover of $(N,h(t))$ must split off a flat factor $\mathbb{R}^{d}$
($d\geq 1$). However, the universal cover of $N$ is $S^{3}$. This is a
contradiction! Hence, we prove ${\rm Ric}(g)>0$ on $M$.

Now we can apply Theorem 0.2 to see that any split $3$-dimensional ancient
flow $(N^{\prime},h^{\prime}(t))$ of limit of $(M,g_{q_{i}}(t),q_{i})$ is a
family of shrinking quotient spheres. We claim: $(N^{\prime},h^{\prime}(t))$
is in fact a family of shrinking spheres.

By Lemma 2.2, the scalar curvature of $(M,g)$ decays to zero uniformly. Then
$(M,g)$ has unique equilibrium point $o$ by the fact ${\rm Ric}(g)>0$. Thus
the level set $\Sigma_{r}=\\{f(x)=r\\}$ is a closed manifold for any $r>0$,
and it is diffeomorphic to $S^{3}$ (cf. [16, Lemma 2.1]).

On the other hand, as in the proof of Lemma 2.6, the level set
$(\Sigma_{f(q_{i})},\bar{g}_{q_{i}};q_{i})$ converges subsequently to
$(N^{\prime},h^{\prime}(0);q_{\infty})$ w.r.t. the induced metric
$\bar{g}_{q_{i}}$ on $\Sigma_{f(q_{i})}$ by $g_{q_{i}}$. Since each
$\Sigma_{f(q_{i})}$ is diffeomorphic to $S^{3}$, $N^{\prime}$ is also
diffeomorphic to $S^{3}$. Thus $(N^{\prime},h^{\prime}(t))$ is a family of
shrinking spheres.

By the above claim, the condition (ii) in Definition 0.1 is satisfied. Thus by
[17, Lemma 6.5], $(M,g)$ is asymptotically cylindrical. It follows that
$(M,g)$ is isometric to the Bryant Ricci soliton up to scaling by [5].

In addition of ${\rm Ric}(g)>0$, we see that all level set
$\Sigma_{r}=\\{f(x)=r\\}$ is diffeomorphic to $S^{3}$ and so as $N$. Then by
Theorem 0.2 together with the existence of $3d$ compact split limit flow, all
$3d$ split limit flow $(N^{\prime},h^{\prime}(t))$ must be a family of
shrinking spheres. Thus $(M,g)$ is also isometric to the Bryant Ricci soliton
up to scaling. The proof of corollary is complete.

∎

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11institutetext: Department of Computer Science, Posts and Telecommunications
Institute of Technology, Hanoi, Vietnam 22institutetext: FPT Technology
Research Institute, FPT University, Hanoi, Vietnam 22email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Analyzing Vietnamese Legal Questions using Deep Neural Networks with
Biaffine Classifiers

Nguyen Anh Tu 1122 Hoang Thi Thu Uyen 11
Tu Minh Phuong 11 Ngo Xuan Bach(✉) 11

###### Abstract

In this paper, we propose using deep neural networks to extract important
information from Vietnamese legal questions, a fundamental task towards
building a question answering system in the legal domain. Given a legal
question in natural language, the goal is to extract all the segments that
contain the needed information to answer the question. We introduce a deep
model that solves the task in three stages. First, our model leverages recent
advanced autoencoding language models to produce contextual word embeddings,
which are then combined with character-level and POS-tag information to form
word representations. Next, bidirectional long short-term memory networks are
employed to capture the relations among words and generate sentence-level
representations. At the third stage, borrowing ideas from graph-based
dependency parsing methods which provide a global view on the input sentence,
we use biaffine classifiers to estimate the probability of each pair of start-
end words to be an important segment. Experimental results on a public
Vietnamese legal dataset show that our model outperforms the previous work by
a large margin, achieving 94.79% in the F1 score. The results also prove the
effectiveness of using contextual features extracted from pre-trained language
models combined with other types of features such as character-level and POS-
tag features when training on a limited dataset.

###### Keywords:

Question Answering Legal Domain Deep Neural Network Biaffine Classifier BERT
BiLSTM.

##  1 Introduction

Question answering (QA) [7, 12, 22, 28, 29, 30], a sub-field of natural
language processing (NLP) and information retrieval (IR), aims to build
computer systems that can automatically answer questions in natural languages.
There are two main approaches for building QA systems: IR-based and knowledge-
based. The former approach finds and extracts answers from a collection of
unstructured text, while the latter utilizes structured text or knowledge
bases. Although two approaches exploit different types of resources, they
share the first step, question analysis, which extracts the needed information
to answer the input question. Such information is then used to form a query in
various ways which serves as the input for the next step. Question analysis is
therefore a crucial task for both IR-based and knowledge-based question
answering.

In this paper, we target at the question analysis task in the legal domain,
which is undoubtedly important but has received less attention. Legal and
regulatory documents are ubiquitous and have a great impact on our life.
Figure 1 shows two examples of Vietnamese legal questions annotated with key
information. The goal is to correctly extract two types of information in the
first question (Traffic Light-TL and Question Type-QT), and four types of
information in the second question (Type of Vehicle-TV, Alcohol Concentration-
AC, Value-V, and Question Type-QT). We call the segments that contain
important information important segments.

Figure 1: Examples of two legal questions and their important information.

Traditional methods often frame the task as a sequence labeling problem and
exploit probabilistic graphical models like conditional random fields (CRFs)
[14] to solve it. The main advantage of those methods is that they can build
an accurate model using a relatively small annotated corpus with a handcrafted
feature set. Recently, however, deep neural networks have made tremendous
breakthroughs in various NLP tasks and applications, including sentence
classification [13, 26], sequence labeling [2, 9, 33], syntactic and
dependency parsing [21, 34], natural language inference [5, 10], machine
translation [27, 35], as well as question answering [10, 12]. Furthermore, the
well-known limitation of deep models, i.e. data hungry, could be mitigated by
using advanced pre-trained models and fine-tuning [5, 10]. All of these make
deep neural networks the preferred choice for NLP tasks in general and QA in
particular.

Here, we propose to use deep neural networks for Vietnamese question analysis
in the legal domain. Our method combines several recent advanced techniques in
the NLP and deep learning research communities: pre-trained language models
[5] for contextual word embeddings, convolutional neural networks (CNNs) [16]
for extracting character-level features, and bidirectional long-short term
memory networks (BiLSTM) [8] for sentence-level representations. Furthermore,
instead of formulating it as a sequence labeling problem, we employ biaffine
classifiers to estimate directly the possibility that a pair of words becomes
an important segment. The main advantage of biaffine classifiers is that they
provide a global view on the input sentence, which has been shown to be
effective in dependency paring [6, 17], and named entity and relation
extraction [23, 36]. Experimental results on a Vietnamese corpus consisting of
1678 legal questions show that our model outperforms a SOTA method by a large
margin, showing the F1 score of 94.79%. The effectiveness of these components
of the model is also validated by an ablation study.

The remainder is organized as follows. Section 2 reviews related work. Section
3 presents our model for extracting important information from Vietnamese
legal questions. The model architecture is presented first, and its key
components are then described in more detail. Experimental results and error
analysis are introduced in Section 4. Finally, Section 5 concludes the paper
and shows some future work.

##  2 Related Work

Several studies have been performed on Vietnamese QA in various domains,
including travel, education, as well as legal. Tran et al. [28] introduce a
Vietnamese QA system, which can answer simple questions in the travel domain
by mining information from the Web. Bach et al. [3] focus on the task of
analyzing Vietnamese question in education. Using deep neural networks
(BiLSTM-CRFs) with a rich set of features, their model can accurately extract
14 types of vital information from education utterances. In the legal domain,
Duong and Ho [7] develop a QA system to answer Vietnamese questions about
provisions, procedures, processes, etc. in enterprise laws. Kien et al. [12]
introduce a retrieval-based method for answering Vietnamese legal questions by
learning text representation. Their model leverages CNNs and attention
mechanisms to extract and align important information between a question and a
legal article. Other works on Vietnamese question answering include Nguyen et
al. [22], Tran et al. [29], and Le-Hong and Bui [15].

Perhaps the most closely work to ours is the one of Bach et al. [1], which
also focuses on analyzing Vietnamese legal questions. Our method, however, is
distinguished from theirs in two aspects. First, we formulate the task as a
multi-class classification problem instead of sequence labeling. Second, we
utilize deep neural networks instead of using traditional models like CRFs.

##  3 Method

###  3.1 Model Overview

Our goal is to extract all important segments from an input legal question,
where each segment is a triple of start/end positions and a label for the
information type. Figure 2 illustrates our proposed architecture. First, we
create word representations by concatenating different types of features:
contextual word embeddings, character-level features, and part-of-speech (POS)
tag embeddings. The outputs of the word representation layer are then fed into
two stacked BiLSTMs to obtain the sentence-level representations. After that,
we use two feed forward neural networks (FFN) to generate different
representations for the start/end of segments. Finally, we employ a biaffine
classifier to create a $n\times n\times c$ scoring tensor $R$, where $n$
denotes the length of the input question, and $c$ is the number of labels
(including Null for non important information). Tensor $R$ provides scores for
all possible segments that could contain key information.

In the up-coming sections, we show the model components in detail. For
notation, we denote vectors, matrices, and scalars with bold lower-case (e.g.,
$\textbf{x}_{t}$, $\textbf{h}_{t}$, b), bold upper-case (e.g., H,
$\textbf{W}_{i}$, $\textbf{V}_{i}$), and italic lower-case (e.g., $n$, $c$),
respectively.

Figure 2: Model architecture.

###  3.2 Word Representations

Because words are basic elements to form written languages, a good word
representation method is the first and crucial step to build successful NLP
systems. In this work, we create rich information word representations by
integrating multiple information sources, including contextual word
embeddings, character-level features, and POS-tag embeddings.

Contextual word embeddings. Traditional NLP approaches usually represent words
by one-hot vectors with one value 1 and the rest 0. These high-dimensional
sparse vectors are memory consuming and cannot capture the word semantics.
Distributed word representation methods, which use low-dimensional continuous
vectors, have been introduced to handle these issues. Word2vec [20], Fasttext
[4], and Glove [25] are successful examples of such methods that represent
similar words with similar vectors. Although these methods have made many
breakthroughs in NLP research, they represent words by fix vectors which are
context independent. Static word embedding methods like word2vec are therefore
limited in representing polysemous words.

Recently, contextual word embedding methods have been shown to be the key
component of many SOTA NLP systems [5, 18]. The main advantage of these
methods it that they can learn different representations for polysemous words
by considering the sequence of all words in sentences/documents. Perhaps BERT
proposed by Devlin et al. [5] is the most famous and popular contextual word
embedding method. The key technical innovation of BERT is applying the
bidirectional training of Transformer [31] to language modeling with two
strategies: masked-language modeling and next sentence prediction.

In this work, we use PhoBERT [24], a monolingual variant of RoBERTa [18] pre-
trained on a 20GB word-level Vietnamese dataset. Like BERT and RoBERTa,
PhoBERT segments the input sentence into sub-words, which brings the balance
between character- and word-level hybrid representations and enables the
encoding of rare words with appropriate sub-words. We represent each sub-word
by concatenating embeddings of the last four encoding layers (9 to 12) of
PhoBERT-base, and the contextual embedding of a word is the embedding of its
first sub-word.

Character-level features. Beside contextual word embeddings, we also utilize
morphological information from characters. Additional character embeddings are
derived from character-level convolutional (charCNN) networks. As shown in
Figure 3, charCNN consists of 1D operations: convolution and max pooling.
Feature maps are then concatenated to produce a character-level representation
for the word.

Figure 3: CNN-based character-level features.

POS-tag embeddings. We suppose POS tags are another useful source of
information. Therefore, we also use POS tag embeddings to represent words.
These embedding vectors are initialized randomly in range
$(-\sqrt{3/dim},\sqrt{3/dim})$, where $dim$ denotes their dimension. We use
VnCoreNLP [32] for word segmentation and POS tagging for input questions.

Finally, all feature vectors are concatenated into a single embedding for
representing a word. Word vectors are then fed into BiLSTM networks to create
sentence-level representations.

###  3.3 BiLSTM

Long short-term memory (LSTM) networks [11] are designed for sequence data
modeling problem. Let
$\textbf{X}=(\textbf{x}_{1},\textbf{x}_{2},\ldots,\textbf{x}_{n})$ denote the
input question where $\textbf{x}_{i}$ is the embedding of the $i^{th}$ word.
At each position $t$, the LSTM computes an intermediate representation using a
hidden state h:

$\textbf{h}_{t}=f(\textbf{h}_{t-1},\textbf{x}_{t})$

where $f$ includes an input gate, a forget gate, an output gate, and a memory
cell (denoted by $\textbf{i}_{t}$, $\textbf{f}_{t}$, $\textbf{o}_{t}$,
$\textbf{c}_{t}$, respectively) to update $\textbf{h}_{t}$:

$\textbf{i}_{t}=\sigma(\textbf{W}_{i}\textbf{x}_{t}+\textbf{V}_{i}\textbf{h}_{t-1}+\textbf{b}_{i}),$
$\textbf{f}_{t}=\sigma(\textbf{W}_{f}\textbf{x}_{t}+\textbf{V}_{f}\textbf{h}_{t-1}+\textbf{b}_{f}),$
$\textbf{o}_{t}=\sigma(\textbf{W}_{o}\textbf{x}_{t}+\textbf{V}_{o}\textbf{h}_{t-1}+\textbf{b}_{o}),$
$\textbf{c}_{t}=\textbf{f}_{t}\odot\textbf{c}_{t-1}+\textbf{i}_{t}\odot\text{tanh}(\textbf{W}_{c}\textbf{x}_{t}+\textbf{V}_{c}\textbf{h}_{t-1}+\textbf{b}_{c}),$
$\textbf{h}_{t}=\textbf{o}_{t}\odot\text{tanh}(\textbf{c}_{t}),$

and the output $\textbf{y}_{t}$ can be produced based on $\textbf{h}_{t}$:

$\textbf{y}_{t}=\sigma(\textbf{W}_{y}\textbf{h}_{t}+\textbf{b}_{y}),$

where $\odot$ indicates the multiplication operator function, $\sigma$ is the
element-wise softmax, and $\textbf{W}_{*}$, $\textbf{V}_{*}$, and
$\textbf{b}_{*}$ ($*$ denotes $i$,$f$,$o$,$c$,$y$) are weight matrices and
vectors to be learned during the training process.

Bidirectional long short-term memory (BiLSTM) [8] combine two LSTMs: one
network moves from the left to the right and the other network moves from the
right to the left of the sequence. Two BiLSTMs are exploited to learn a higher
level of semantics within the sentence.

###  3.4 Biaffine Layer

We employ two feed forward neural networks (FFNs) to create different
representations for start/end positions. The outputs of two FFNs at position
$t$ are denoted by $\textbf{g}_{t}^{start}$ and $\textbf{g}_{t}^{end}$:

$\textbf{g}_{t}^{start}=\text{FFN}^{start}(\textbf{y}_{t})$
$\textbf{g}_{t}^{end}=\text{FFN}^{end}(\textbf{y}_{t})$

For each start-end candidate pair $(i,j)$, $1\leq i\leq j\leq n$, we apply the
biaffine classifier:

$\textbf{r}_{i,j}=\text{Biaffine}(\textbf{g}_{i}^{start},\textbf{g}_{j}^{end})=(\textbf{g}_{i}^{start})^{\top}\textbf{U}\textbf{g}_{j}^{end}+\textbf{W}(\textbf{g}_{i}^{start}\oplus\textbf{g}_{j}^{end})+\textbf{b},$

where U, W, b are a $d\times c\times d$ tensor, a $c\times 2d$ matrix, and a
bias vector, respectively, and $d$ is the size of the output layers of both
FFNstart and FFNend.

Vector $\textbf{r}_{i,j}$ is then fed into a softmax layer to produce
probability scores $\textbf{s}_{i,j}$:

$\textbf{s}_{i,j}(k)=\frac{exp(\textbf{r}_{i,j}(k))}{\sum_{k^{\prime}=1}^{c}{exp(\textbf{r}_{i,j}(k^{{}^{\prime}}))}}$

The label of segment $(i,j)$ can be determined as:
$\widehat{l}=\arg\max_{k}\textbf{s}_{i,j}(k).$

The question analysis task now becomes a multi-class classification problem
and model parameters are learned to minimize the cross-entropy loss function.

##  4 Experiments

###  4.1 Data and Evaluation Method

In our experiments we used the Vietnamese dataset of legal questions
introduced by Bach et al. [1]. This dataset consists of 1678 legal questions
about the traffic law in Vietnam. Questions were annotated with 16 labels
reflecting different aspects of the domain listed in Table 1.

Table 1: Types of important information and their occurrence numbers Label | Meaning | # | Label | Meaning | #
---|---|---|---|---|---
A | Action of vehicle | 1087 | L | Location | 426
AC | Alcohol concentration | 44 | QT | Question type | 1678
ANO | Annotation | 75 | SP | Speed | 115
DL | Driving license | 119 | TI | Traffic instructor | 93
IF1 | Add. info. about vehicle | 196 | TL | Traffic light | 31
IF2 | Add. info. about traffic light | 12 | TP | Traffic participant | 20
IF3 | Add. info. about traffic participant | 287 | TV | Type of vehicle | 1245
IF4 | Add. info. | 227 | V | Value | 231

We performed cross-validation tests with the same training/test data splits of
Bach et al. [1]. For each fold, we used $10$% of the training set as a
validation set. The performance of extraction models was measured using
popular metrics such as precision, recall and the F1 score.

###  4.2 Network Training

Our models were implemented in PyTorch111https://pytorch.org/ using
Huggingface’s Transformers222https://huggingface.co/transformers/. In all
experiments, we set the batch size to $64$. The max character length was set
to $15$, and the max sequence length was tuned in [$50,60,80,100$] for all
models, and the best value was $60$. We set the dimensions of character-level
features and POS-tag embeddings to $256$ and $100$, respectively. We used
dimension of $300$ for FFNs, and kernel sizes of 3 and 4 for charCNN. To
mitigate overfitting, we used a dropout rate of $0.3$ for each hidden layer.
Our models were trained using the AdamW optimizer [19]. We set the epsilon and
weight decay to default values in PyTorch, i.e. 1$e$-8. The learning rate was
tuned in [3$e$-5, 4$e$-5, 5$e$-5] and the best learning rate value was 5$e$-5.
For each model, we trained for $30$ epochs and selected the version that
obtained the highest F1 score on the validation set to apply to the test set.

###  4.3 Experimental Results

####  4.3.1 Our Model vs. Baseline

We first conducted experiments to compare our model with the previous work of
Bach et al. [1] (our baseline). Table 2 shows experimental results on each
type of information and the overall scores of two models. Our model
outperformed the baseline for 15 out of 16 types of information. Types with
the biggest improvements in the F1 score include TL (Traffic light: 19.79%),
ANO (Annotation: 9.61%), TP (Traffic participant: 7.48%), A (Action: 4.47%),
and SP (Speed: 4.33%). Overall, our model achieved a micro F1 score of 94.79%,
which improved 1.85% (26.20% error rate reduction) in comparison with the
baseline. Experimental results demonstrated the effectiveness of deep neural
networks compared to traditional methods like CRFs.

Table 2: Experimental results of our model compared with the baseline (the improvements are indicated in bold) |  Baseline (Bach et al. [1]) | Our Model
---|---|---
Type | Prec.(%) | Rec.(%) | F1(%) | Prec.(%) | Rec.(%) | F1(%)
A | 88.20 | 89.14 | 88.66 | 92.91 | 93.35 | 93.13 (4.47$\uparrow$)
AC | 95.78 | 95.78 | 95.78 | 96.12 | 97.28 | 96.70 (0.92$\uparrow$)
ANO | 85.58 | 60.57 | 68.82 | 73.17 | 84.51 | 78.43 (9.61$\uparrow$)
DL | 97.97 | 99.20 | 98.54 | 100.00 | 98.26 | 99.12 (0.58$\uparrow$)
IF1 | 94.64 | 87.35 | 90.67 | 90.34 | 91.48 | 90.91 (0.24$\uparrow$)
IF2 | 100.00 | 73.33 | 82.67 | 86.72 | 70.44 | 77.74 (4.93$\downarrow$)
IF3 | 88.08 | 75.06 | 80.91 | 85.77 | 83.45 | 84.59 (3.68$\uparrow$)
IF4 | 85.77 | 74.34 | 79.51 | 80.14 | 82.97 | 81.53 (2.02$\uparrow$)
L | 92.23 | 92.71 | 92.44 | 93.18 | 96.31 | 94.72 (2.28$\uparrow$)
QT | 96.03 | 94.84 | 95.42 | 95.89 | 96.07 | 95.98 (0.56$\uparrow$)
SP | 95.61 | 91.44 | 93.23 | 100.00 | 95.24 | 97.56 (4.33$\uparrow$)
TI | 99.05 | 95.53 | 97.24 | 100.00 | 100.00 | 100.00 (2.76$\uparrow$)
TL | 86.95 | 60.35 | 68.59 | 87.33 | 89.46 | 88.38 (19.79$\uparrow$)
TP | 80.00 | 60.67 | 67.21 | 76.35 | 73.11 | 74.69 (7.48$\uparrow$)
TV | 97.32 | 98.78 | 98.04 | 99.57 | 97.99 | 98.77 (0.73$\uparrow$)
V | 98.18 | 99.26 | 98.71 | 100.00 | 98.23 | 99.11 (0.40$\uparrow$)
Overall | 93.84 | 92.05 | 92.94 | 94.30 | 95.28 | 94.79 (1.85$\uparrow$)

####  4.3.2 Ablation Study

Next we evaluated the contribution of individual components of our model by
performing an ablation study as follows:

  * •

Biaffine classifier: We replaced the biaffine classifier with a CRF/softmax
layer and reformulated the task as a sequence labeling problem

  * •

Character-level features, POS-tag embeddings, BiLSTMs: We removed each
component in turn.

  * •

PhoBERT Embeddings: We replaced contextual word embeddings with static word
embeddings. In our experiments, we used Fasttext [4], a variant of Word2Vec
[20] which deals with unknown words and sparsity in languages by using sub-
word models.

Experimental results in Table 3 proved that all the components contributed to
the success of our model. Our first observation is that the performance of the
system degraded when we modified or removed a component. While the full model
got 94.79%, the F1 score reduced to 94.62% and 93.59% when we replaced the
biaffine classifier with CRF/softmax function and reformulated the task as a
sequence labeling problem. The score was only 94.46%, 94.33%, and 93.77%, when
we removed POS-tag features, character-level features, and the BiLSTM layers,
respectively. The results showed that the BiLSTM layers have a high impact on
the performance of our model. The second observation is that replacing
contextual word embeddings by static word embeddings leads to the biggest
decrease of 1.98%. This indicated that contextual word embeddings from pre-
trained language models like PhoBERT played a critical role in our model.

Table 3: Ablation study (the decrease in the F1 score of the modified models is indicated in bold) Component | Modification | Prec.(%) | Rec.(%) | F1(%)
---|---|---|---|---
Full | None | 94.30 | 95.28 | 94.79
Biaffine classifier | Seq. labeling, CRF | 93.92 | 95.33 | 94.62 (0.17$\downarrow$)
Seq. labeling, Softmax | 94.02 | 93.17 | 93.59 (1.20$\downarrow$)
POS-tag embeddings | Removal | 94.54 | 94.39 | 94.46 (0.33$\downarrow$)
Character features | Removal | 93.51 | 95.16 | 94.33 (0.46$\downarrow$)
BiLSTM | Removal | 92.95 | 94.60 | 93.77 (1.02$\downarrow$)
PhoBERT | Fasttext | 92.00 | 93.63 | 92.81 (1.98$\downarrow$)

###  4.4 Error Analysis

This section discusses the cases in which our model failed to extract
important information. By analyzing the model’s predictions on the test sets,
we found that most errors belong to one of two following types:

  * •

Type I: Incorrect segments. Our model identified segments that are shorter or
longer than the gold segments.

  * •

Type II: Incorrect information types. Our model recognized segments (start and
end positions) correctly but assigned wrong information types (labels) to the
segments.

Figure 4 shows four examples of error cases. Among them, the first two
examples belong to Type I and the others belong to Type II. While our model
identified a longer segment in the first case, it detected a shorter segment
in the second case. In the third and fourth examples, our model made a mistake
on labels (TP instead of ANO, and IF4 instead of A).

Figure 4: Examples of error cases.

##  5 Conclusion

We have introduced a deep neural network model for analyzing Vietnamese legal
questions, a key step towards building an automatically question answering
system in the legal domain. By utilizing recent advanced techniques in the NLP
and deep learning research communities, our model can correctly extract 16
types of important information. For future work, we plan to develop a question
answering system for Vietnamese legal questions. Studying deep neural network
models for other NLP tasks in Vietnamese language is another direction for
future work.

## Acknowledgements

We would like to thank FPT Technology Research Institute, FPT University for
financial support which made this work possible.

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