# Soliton Solutions to the Curve Shortening Flow on the 2-dimensional
hyperbolic plane

Fábio Nunes da Silva111Universidade de Brasília, Department of Mathematics,
70910-900, Brasília-DF, Brazil<EMAIL_ADDRESS>Keti Tenenblat222
Universidade de Brasília, Department of Mathematics, 70910-900, Brasília-DF,
Brazil<EMAIL_ADDRESS>Partially supported by CNPq Proc.
312462/2014-0, Ministry of Science and Technology, Brazil and FAPDF/Brazil
grant 0193.001346/2016.

###### Abstract

We show that a curve is a soliton solution to the curve shortening flow if and
only if its geodesic curvature can be written as the inner product between its
tangent vector field and a fixed vector of the 3-dimensional Minkowski space.
We use this characterization to provide a qualitative study of the solitons.
We show that for each fixed vector there is a 2-parameter family of soliton
solution to the curve shortening flow on the 2-dimensional hyperbolic space.
Moreover, we prove that each soliton is defined on the entire real line, it is
embedded and its geodesic curvature converges to a constant at each end.

_Keywords_ : Curve shortening flow; solitons solutions.

##  1 Introduction

A family of curves $\hat{X}^{t}:I\longrightarrow M$, $t\in[0,T)$, on a
2-dimensional Riemannian manifold $M^{2}$ is said to be a solution to the
Curve Shortening Flow (CSF) with initial condition
$\hat{X}^{0}(\cdot)=X(\cdot)$, if it satisfies the following equation

$\left\\{\begin{array}[]{ll}\displaystyle{\frac{\partial}{\partial
t}\hat{X}^{t}(\cdot)}=\hat{k}^{t}(\cdot)\hat{N}^{t}(\cdot)\\\
\hat{X}^{0}(\cdot)=X(\cdot),\end{array}\right.$ (1.1)

where $\hat{k}^{t}(\cdot)$ is the geodesic curvature and $\hat{N}^{t}(\cdot)$
is the unit vector field normal to $\hat{X}^{t}(\cdot)$ for each $t\in[0,T)$.

Epstein and Gage [6] showed that when $M^{2}=\mathbb{R}^{2}$, the CSF is
geometrically the same if tangential components are added to the right hand
side of the differential equation (1.1). Therefore, one can define that a
1-parameter family of curves $\hat{X}^{t}:I\rightarrow\mathbb{R}^{2}$,
$t\in[0,T)$ is a solution to the CSF in $\mathbb{R}^{2}$ with initial
condition $\hat{X}^{0}(\cdot)=X(\cdot)$, if it satisfies

$\left\langle\displaystyle\frac{\partial}{\partial
t}\hat{X}^{t}(\cdot),\hat{N}^{t}(\cdot)\right\rangle=\hat{k}^{t}(\cdot),$
(1.2)

where $\langle\cdot,\cdot\rangle$ is the cannonical inner product on
$\mathbb{R}^{2}$, $\hat{k}^{t}(\cdot)$ is the curvature and
$\hat{N}^{t}(\cdot)$ is the unit vector field normal to $\hat{X}^{t}(\cdot)$
for each $t\in[0,T)$. The name curve shortening flow is justified by the fact
that when the curves of the family $\hat{X}^{t}$ are closed, then the length
of the curves decreases along the flow, i.e., it is a gradient type of flow
for the arc length functional. Grayson [12] observed that the CSF is also
known as the curvature flow or the heat flow for isometric immersions.

According to Epstein and Gage [6], the original motivation for studying
equation (1.1) was to find a new and maybe more natural proof of the existence
of closed geodesics on Riemannian manifolds. However, the first results in
this direction were obtained by Grayson [13] em 1989. But equation (1.1) for
the Euclidean plane was investigated earlier by several authors [2], [7], [8],
[9], [10] e [12].

An important class of solutions to the CSF are those that evolve by isometries
or homotheties. Such solutions are called self-similar solutions  and solitons
if they evolve just by isometries. On the Euclidean plane, the straight lines
are not affected by the flow and they are considered to be trivial solutions.
Circles evolve homothetically to a point in finite time. The Grim Reaper curve
given by the graph of the function $f(s)=ln(cos(s))$ evolves by a flow of
translations. Giga [11] proved that this is the unique curve on the plane that
evolves by translations. An example of a plane curve that evolves by
isometries of the plane is the yin-yang spiral. Abresch-Langer [2] and
Epstein-Weinstein [7] investigated the closed curves, not necessarily simple,
that evolve by homotheties. Halldorsson [14] concluded the description of all
self-similar solutions on the plane.

Some authors studied classes of solutions to the CSF on the plane and proved
that after a certain time the flow evolves into a self-similar one. First,
Gage [8] [9] showed that closed convex curves on $\mathbb{R}^{2}$ evolve into
circular curves after a certain time. Then Gage and Hamilton [10] showed that
convex closed curves collapse into a point. Grayson [12] proved that closed
embedded curves evolve to circular curves and then they collapse into a point
at a finite time. Moreover, Angenent [3], under more general conditions,
proved that the CSF evolves in a sense into a self-similar flow, showing the
importance of self-similar solutions.

One should point out that, when the ambient space is not the Euclidean plane,
there are very few results on self-similar solutions to the CSF. In 2015,
Halldorsson [15] classified all the self-similar solutions on the Minkowski
plane. Dos Reis and Tenenblat [5] caracterized and described all the soliton
solutions on the sphere $\mathbb{S}^{2}$. Some results on the CSF for
Riemannian manifolds different from the plane can be found in [10], [13], [16]
and [20], among others. Moreover, Angenent, [4] studied the topology of the
closed geodesics on compact surfaces by using the CSF.

In this paper, we will study the soliton solutions of curve shortening flow on
the 2-dimensional hyperbolic space $\mathbb{H}^{2}\subset\mathbb{R}_{1}^{3}$,
where $\mathbb{R}_{1}^{3}$ is the 3-dimensional Minkowski space.

##  2 Main Results

We consider the 3-dimensional Minkowki space as
$\mathbb{R}_{1}^{3}=(\mathbb{R}^{3},\langle,\rangle)$, where $\mathbb{R}^{3}$
is the 3-dimensional vector space and $\langle,\rangle$ is the Minkowski
metric defined by

$\langle u,v\rangle=-u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}.$

Let $\displaystyle
X:I\subset\mathbb{R}\rightarrow\mathbb{H}^{2}\subset\mathbb{R}_{1}^{3}$ be a
regular curve parametrized by arc length $s$. We denote by
$T(s)=X^{\prime}(s)$ the tangent vector field, $N(s)=X(s)\times T(s)$ the unit
normal vector field and $k(s)=\langle T^{\prime}(s),N(s)\rangle$ the geodesic
curvature of $X$. A one parameter family of curves
$\displaystyle\hat{X}:I\times J\rightarrow\mathbb{H}^{2}$ is called a curve
shortening flow (CSF) with initial condition $X$, if

$\left\\{\begin{array}[]{ll}\displaystyle\left\langle\frac{\partial}{\partial
t}\hat{X}(s,t),\hat{N}(s,t)\right\rangle=\hat{k}(s,t),\\\
\hat{X}(s,0)=X(s),\end{array}\right.$ (2.1)

where $\hat{k}^{t}(\cdot)=\hat{k}(\cdot,t)$ is the geodesic curvature and
$\hat{N}^{t}(\cdot)=\hat{N}(\cdot,t)$ is the unit normal vector field of
$\hat{X}^{t}(\cdot)=\hat{X}(\cdot,t)$. When $X$ is a geodesic i.e. $k=0$, then
the family $\hat{X}^{t}(s)=X(s)$ gives a trivial solution to the CSF. Our goal
is to study the case when $\hat{X}^{t}(s)$ evolves by a 1-parameter family of
isometries of $\mathbb{H}^{2}$.

######  Definition 2.1.

Let $\displaystyle\hat{X}:I\times
J\rightarrow\mathbb{H}^{2}\subset\mathbb{R}_{1}^{3}$ be a solution to the
curve shortening flow (2.1) on $\mathbb{H}^{2}$, with initial condition
$\displaystyle X:I\rightarrow\mathbb{H}^{2}$. We say that $X$ is a soliton
solution to the curve shortening flow if there is a 1-parameter family of
isometries $M(t):\mathbb{H}^{2}\rightarrow\mathbb{H}^{2}$ such that $M(0)=Id$
and

$\hat{X}^{t}(s)=M(t)X(s)$

for all $t\in J$, where $Id$ is the identity map.

We remark that an isometry of $\mathbb{H}^{2}$ is an element of the Lie group
$O_{1}(3)=\\{M\in GL(3,\mathbb{R}):M^{T}\epsilon M=\epsilon\\}$ that preserves
$\mathbb{H}^{2}$, where $M^{T}$ is the transpose of $M$ and

$\epsilon=\left(\begin{array}[]{lll}-1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right).$

######  Theorem 2.2.

Let $\displaystyle X:I\rightarrow\mathbb{H}^{2}$ be a regular curve
parametrized by arc length. Then $X(s)$, $s\in I$, is a soliton solution to
the curve shortening flow if, and only if, there is a vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ such that

$\langle T(s),v\rangle=k(s),$ (2.2)

where $T(s)$ is the unit tangent vector field and $k(s)$ is the geodesic
curvature of $X$.

We observe that when $X$ is a geodesic of
$\mathbb{H}^{2}\subset\mathbb{R}^{3}_{1}$, then it is a planar curve and hence
there exists a vector $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$ such that
$\displaystyle\langle T,v\rangle=0.$

The following theorem describes the qualitative behaviour of the soliton
solutions to the CSF in $\mathbb{H}^{2}.$

######  Theorem 2.3.

For any $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$, there is a 2-parameter
family of non-trivial soliton solutions to the curve shortening flow on the
2-dimensional hyperbolic space. Each soliton solution is an embedded curve
$X(s)$ on $\mathbb{H}^{2}$, defined for all $s\in\mathbb{R}$. Moreover, at
each end, the curvature function $k(s)$ tends to one of the following
constants $\\{-1,0,1\\}$.

##  3 Proofs of the main results

In this section we prove our main results.

Proof of Theorem 2.2. Suppose that $X(s)$ is parametrized by arc length $s$.
If $X$ is a soliton solution to the CSF, then
$\displaystyle\hat{X}^{t}(s)=M(t)X(s)$ is solution to (2.1), where $M(t)$ is a
family of isometries of $\mathbb{H}^{2}$.Taking the derivative of
$\displaystyle\hat{X}(s,t)$ at $t$, we have

$\frac{\partial}{\partial t}\hat{X}(s,t)=M^{\prime}(t)X(s).$

It follows from definition of the CSF that

$\hat{k}(s,t)=\left\langle\frac{\partial}{\partial
t}\hat{X}(s,t),\hat{N}(s,t)\right\rangle=\left\langle
M^{\prime}(t)X(s),M(t)N(s)\right\rangle.$

In particular, for $t=0$, we have

$k(s)=\langle M^{\prime}(0)X(s),N(s)\rangle.$

$M^{\prime}(0)$ is an element of the Lie algebra $\mathfrak{o}_{1}(3)$ of the
Lie group $O_{1}(3)$. Let ${A_{1},A_{2},A_{3}}$ be a basis of
$\mathfrak{o}_{1}(3)$, where

$\displaystyle A_{1}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&-1&0\end{array}\right),\,\,\,A_{2}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&1\\\
1&-1&0\end{array}\right)\,\,\,\,\text{e}\,\,\,\,\,A_{3}=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&0\end{array}\right).$

Then $M^{\prime}(0)=c_{1}A_{1}+c_{2}A_{2}+c_{3}A_{3}$, for real numbers
$c_{i}$.

By simple computations, we can prove that $\displaystyle\langle
M^{\prime}(0)X(s),N(s)\rangle=\langle T(s),v\rangle$, where $X(s)\times
N(s)=-T(s)$ and $v=(c_{1}+c_{2},c_{2},-c_{3})$. Therefore,

$k(s)=\langle T(s),v\rangle.$

Conversely, let $X(s)$ be a curve in $\mathbb{H}^{2}\subset\mathbb{R}^{3}_{1}$
parametrized by arc length $s$, such that $\langle T(s),v\rangle=k(s)$ for a
vector $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$. Without loss of generality,
up to isometries of $\mathbb{H}^{2}$, we can consider $v$ to be a multiple of
$w_{1}=(1,0,0)$ if $v$ is a timelike vector, a multiple of $w_{2}=(1,1,0)$ if
$v$ is a lightlike vector and a multiple of $w_{3}=(0,0,-1)$ if $v$ is a
spacelike vector. Thus, depending on the type of the vector $v$, we can write
the curvature as $k_{i}(s)=\langle T(s),v_{i}\rangle$ where $v_{i}=aw_{i}$,
$a>0$ and $i=1,2,3$.

Now, we define the evolution of $X$ in $\mathbb{H}^{2}$ to be
$\hat{X}_{i}(s,t)=M_{i}(t)X(s)$, where

$\displaystyle M_{1}(t)$ $\displaystyle:=$
$\displaystyle\left(\begin{array}[]{lll}1&0&0\\\
0&cos(\varphi_{1}(t))&sen(\varphi_{1}(t))\\\
0&-sen(\varphi_{1}(t))&cos(\varphi_{1}(t))\end{array}\right),\,\,\,M_{2}(t):=\left(\begin{array}[]{ccc}1+\frac{(\varphi_{2}(t))^{2}}{2}&-\frac{(\varphi_{2}(t))^{2}}{2}&\varphi_{2}(t)\\\
\frac{(\varphi_{2}(t))^{2}}{2}&1-\frac{(\varphi_{2}(t))^{2}}{2}&\varphi_{2}(t)\\\
\varphi_{2}(t)&-\varphi_{2}(t)&1\end{array}\right),$ (3.8) $\displaystyle
M_{3}(t)$ $\displaystyle:=$
$\displaystyle\left(\begin{array}[]{lll}cosh(\varphi_{3}(t))&senh(\varphi_{3}(t))&0\\\
senh(\varphi_{3}(t))&cosh(\varphi_{3}(t))&0\\\ 0&0&1\end{array}\right),$
(3.12)

and $\varphi_{i}(t)=at$ for each $i=1,2,3.$

A straightforward computation shows that

$\displaystyle\left\langle
M^{\prime}_{i}(t)X(s),M_{i}(t)N(s)\right\rangle=-\varphi^{\prime}_{i}(t)\langle
X(s)\times
N(s),w_{i}\rangle=-\varphi^{\prime}_{i}(t)\langle-T(s),w_{i}\rangle.$

Thus,

$\displaystyle\left\langle\frac{\partial}{\partial
t}\hat{X}_{i}(s,t),\hat{N}_{i}(s,t)\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle M^{\prime}_{i}(t)X(s),M_{i}(t)N(s)\right\rangle$
$\displaystyle=$
$\displaystyle-\varphi^{\prime}_{i}(t)\langle-T(s),w_{i}\rangle\;=\;\langle
T(s),v_{i}\rangle$ $\displaystyle=$ $\displaystyle
k_{i}(s)\;=\;\hat{k}_{i}(s,t),$

where the last equality follows from the fact that isometries preserve
geodesic curvature. Therefore, $X$ is a soliton solution to the CSF.

$\square$

It follows from Theorem 2.2 that the study of the solitons solutions to the
CSF on the 2-dimensional hyperbolic space is reduced to describing the curves
that satisfy Equation (2.2) for some vector
$v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$. Up to isometries of $\mathbb{H}^{2}$
we consider $v$ as being $v_{i}=ae_{i}$, where $a\in\mathbb{R}^{+}$,
$e_{1}=(-1,0,0)$ if $v$ is a timelike vector, $e_{1}=(-1,1,0)$ if $v$ is a
lightlike vector and $e_{1}=(0,0,1)$ if $v$ is a spacelike vector. Our next
result characterizes (2.2) in terms of a system of differential equations.

######  Proposition 3.1.

Let $\displaystyle X:I\rightarrow\mathbb{H}^{2}$ be a regular curve
parametrized by arc length $s$. Consider the vectors

$e_{1}=(-1,0,0),\qquad e_{2}=(-1,1,0)\qquad e_{3}=(0,0,1).$ (3.13)

For each $i\in\\{1,2,3\\}$, define the functions

$\alpha_{i}(s)=\langle X(s),e_{i}\rangle,\qquad\tau_{i}(s)=\langle
T(s),e_{i}\rangle\qquad\eta_{i}(s)=\langle N(s),e_{i}\rangle,$

where $T$ and $N$ are the unit vector fields tangent and normal to $X$,
respectively. For a fixed $a>0$,

$k_{i}(s)=a\tau_{i}(s)$

is satisfied, for all $s\in I$ if, and only if, the functions $\alpha_{i}(s)$,
$\tau_{i}(s)$ and $\eta_{i}(s)$ satisfy the system

$\left\\{\begin{array}[]{lll}\alpha^{\prime}_{i}(s)=\tau_{i}(s),\\\
\tau^{\prime}_{i}(s)=a\tau_{i}(s)\eta_{i}(s)+\alpha_{i}(s),\\\
\eta^{\prime}_{i}(s)=-a\tau^{2}_{i}(s),\end{array}\right.$ (3.14)

with initial condition $(\alpha_{i}(0),\tau_{i}(0),\eta_{i}(0))$ satisfying

$-\alpha^{2}_{i}(0)+\tau^{2}_{i}(0)+\eta^{2}_{i}(0)=\left\\{\begin{array}[]{cll}-1,&\text{if}&i=1,\\\
0,&\text{if}&i=2,\\\ 1,&\text{if}&i=3.\end{array}\right.$ (3.15)

For such functions, the expression
$-\alpha^{2}_{i}(s)+\tau^{2}_{i}(s)+\eta^{2}_{i}(s)$ is equal to the right
hand side of (3.15), for all $s\in I$. Moreover, $\eta_{i}(s)$ is a decreasing
function.

###### Proof.

The vector fields $X$, $T$ and $N$ satisfy the following system of equations

$\left\\{\begin{array}[]{lll}X^{\prime}(s)=T(s),\\\
T^{\prime}(s)=k(s)N(s)+X(s),\\\ N^{\prime}(s)=-k(s)T(s).\end{array}\right.$
(3.16)

Taking the inner product with $e_{i}$, we get that $\alpha_{i}(s)$,
$\tau_{i}(s)$ and $\eta_{i}(s)$ satisfy the system of equations

$\left\\{\begin{array}[]{lll}\alpha^{\prime}_{i}(s)=\tau_{i}(s),\\\
\tau^{\prime}_{i}(s)=k_{i}(s)\eta_{i}(s)+\alpha_{i}(s),\\\
\eta^{\prime}_{i}(s)=-k_{i}(s)\tau_{i}(s),\end{array}\right.$ (3.17)

Suppose that $k_{i}(s)=a\tau_{i}(s)$ for all $s\in I$. Then substituting into
(3.17), we obtain (3.14). Note that,

$e_{i}=\alpha_{i}(s)X(s)+\tau_{i}(s)T(s)+\eta_{i}N(s).$

Therefore, $\langle
e_{i},e_{i}\rangle=-\alpha^{2}_{i}(s)+\tau^{2}_{i}(s)+\eta^{2}_{i}(s)$ is
constant for all $s\in I$. In particular, for $s=0$, we obtain (3.17).
Moreover, it follows from the third equation of the system (3.14) that the
function $\eta_{i}(s)$ is decreasing.

Conversely, suppose that the functions $\alpha_{i}(s)$, $\tau_{i}(s)$ and
$\eta_{i}(s)$ satisfy (3.14) and (3.15) for each $i\in\\{1,2,3\\}$. Since
(3.17) holds, we have

$\displaystyle\left\\{\begin{array}[]{l}a\tau_{i}(s)\eta_{i}(s)+\alpha_{i}(s)=k_{i}(s)\eta_{i}(s)+\alpha_{i}(s),\\\
-(a\tau_{i}(s))\tau_{i}(s)=-k_{i}(s)\tau_{i}(s),\end{array}\right.$

i.e.,

$\displaystyle\left\\{\begin{array}[]{l}\left[a\tau_{i}(s)-k_{i}(s)\right]\eta_{i}(s)=0,\\\
\left[a\tau_{i}(s)-k_{i}(s)\right]\tau_{i}(s)=0,\end{array}\right.$

for all $s\in I$. For eaxch $i$, in order to conclude that
$k_{i}(s)=a\tau_{i}(s)$, for all $s$, we will assume that $k_{i}(s)\neq
a\tau_{i}(s)$ at some point $s_{0}$. Then this will occur on some interval
$J\subset I$ around $s_{0}$. Hence $\eta_{i}(s)=\tau_{i}(s)=0$ for $s\in J$.
Therefore, $e_{i}$ will be orthogonal to $T(s)$ and $N(s)$ for all $s\in J$.
Thus, $e_{i}$ will be parallel to $X(s)$ for all $s\in J$. But $e_{i}$ is a
constant vector for each $i$, so this can only happen at some isolated points
of a curve $X$ in $\mathbb{H}^{2}$, which is a contradiction. Therefore,
$k_{i}(s)=a\tau_{i}(s)$ for all $s\in I$ and for each $i\in\\{1,2,3\\}$. ∎

Our next proposition shows how a solution of the system (3.14), with initial
conditions satisfying (3.15), is related to a soliton solution to the CSF.

######  Proposition 3.2.

Given a solution $(\alpha(s),\tau(s),\eta(s))$ to the system (3.14) on some
interval $J$ with fixed $a>0$ and initial conditions
$(\alpha(0),\tau(0),\eta(0))$ satisfying
$-\alpha^{2}(0)+\tau^{2}(0)+\eta^{2}(0)=-1$ (resp. $0$ and $1$), there exists
a smooth curve $\displaystyle X:I\rightarrow\mathbb{H}^{2}$ parametrized by
arc length $s$, such that its tangent and normal unit vector fields $T$ and
$N$ satisfy

$\alpha(s)=\langle X(s),e\rangle,\hskip 14.22636pt\tau(s)=\langle
T(s),e\rangle\hskip 14.22636pt\text{and}\hskip 14.22636pt\eta(s)=\langle
N(s),e\rangle,$ (3.20)

where $e=(-1,0,0)$ (resp. $e=(-1,1,0)$ and $e=(0,0,1)$).

###### Proof.

Define $k(s)=a\tau(s)$. Thus, up to isometries of $\mathbb{H}^{2}$, there
exists an unique curve $\displaystyle X:I\rightarrow\mathbb{H}^{2}$, whose
curvature is $k(s)$ i.e. $X(s)$ and its tangent and normal unit vector fields
$T(s)$ and $N(s)$ satisfy the system (3.16). The curve $X(s)$ is uniquely
determined by the initial conditions $X(0)$, $T(0)$ and $N(0)$, that can be
chosen such that

$-\alpha(0)X(0)+\tau(0)T(0)+\eta(0)N(0)=e,$

where $e=(-1,0,0)$ (resp. $e=(-1,1,0)$ and $e=(0,0,1)$). A straightforward
computations shows that (3.15) and (3.16) imply

$\displaystyle\frac{d}{ds}\left[-\alpha(s)X(s)+\tau(s)T(s)+\eta(s)N(s)\right]=0.$

Therefore, (3.20) is satisfied. ∎

######  Remark 3.3.

Let $\displaystyle X:I\rightarrow\mathbb{H}^{2}$ be a regular curve
parametrized by arc length $s$ given by $X(s)=(x_{1}(s),x_{2}(s),x_{3}(s))$.
The function $\alpha(s)$ defined by (3.20) has the following geometric
interpretation.

  * •

If $e=(-1,0,0)$ (timelike vector), then $\alpha(s)=x_{1}(s)>0$ for all $s\in
I$. Moreover, $\alpha(s)$ is the height function with respect to the vector
$(1,0,0).$

  * •

If $e=(-1,1,0)$ (lightlike vector), then $\alpha(s)=x_{1}(s)+x_{2}(s)>0$ for
all $s\in I$. Moreover, $\alpha(s)$ is the height function with respect to the
vector $(1,1,0).$

  * •

If $e=(0,0,1)$ (spacelike vector), then $\alpha(s)=x_{3}(s)$ for all $s\in I$.
Moreover, $\alpha(s)$ is the height function (with sign) with respect to the
vector $(0,0,1).$

Figure 1a (resp. 1b and 1c) provides a geometric illustration of the function
$\alpha(s)$ as a height function with respect to vector $(-1,0,0)$ (resp.
$e=(-1,1,0)$ and $e=(0,0,1)$.)

(a)

(b)

(c)

Figure 1: Geometric interpretation of the functions $\alpha(s)$.

As we have seen in Propositions 3.1, 3.2 and Remark 3.3, the investigation of
the soliton solutions to the CSF on the 2-dimensional hyperbolic space is
equivalent to studying the solutions $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of
the system

$\left\\{\begin{array}[]{lll}\alpha^{\prime}(s)=\tau(s)\\\
\tau^{\prime}(s)=a\tau(s)\eta(s)+\alpha(s)\\\
\eta^{\prime}(s)=-a\tau^{2}(s),\end{array}\right.$ (3.21)

for each constant $a>0$ and initial condition $\displaystyle\psi(0)\in H\cup
C\cup S\subset\mathbb{R}^{3},$ where

$\begin{array}[]{l}H:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}:-\alpha^{2}+\tau^{2}+\eta^{2}=-1,\alpha>0\\},\\\
C:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}\setminus\\{0\\}:-\alpha^{2}+\tau^{2}+\eta^{2}=0,\alpha>0\\},\\\
S:=\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}:-\alpha^{2}+\tau^{2}+\eta^{2}=1\\}.\end{array}$
(3.22)

These are disjoint sets and if the initial condition $\psi(0)\in H$ (resp. $C$
or $S$) then the solution $\psi(s)$ defined on the maximal interval $I$ will
be contained in $H$ (resp. $C$ or $S$) for all $s\in I$.

From now on, using (3.21), we will prove a series of lemmas that will provide
the proof of the main result (Theorem 2.3). Namely, we will prove that for any
initial condition, the solutions $\psi(s)$ of (3.21) and hence the associated
soliton solutions to the hyperbolic space are defined on the whole
$\mathbb{R}$. Moreover, we will analize the behaviour of the curvature
function of the solitons at each end.

In the first lemma we will study the solution of (3.21) such that the function
$\tau(s)$ is constant. As we will see such solutions (that will be called
trivial) only exit on $S$.

######  Lemma 3.4.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non null solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in H\cup C\cup S$, where $H$, $C$ and $S$ are given by
(3.22). Then the function $\tau(s)=b$, $s\in I$, where $b$ is a real constant
if, and only if, $b\in\\{-1,0,1\\}$, $I=\mathbb{R}$ and $\psi(s)\in S$ for all
$s\in\mathbb{R}.$ Moreover,

  * i)

If $b=0$, then $\psi(s)=(0,0,\pm 1)$ are singular solutions of (3.21) in $S$.

  * ii)

If $b^{2}=1$, then $a=1$ and $\psi(s)=(\pm s+\alpha(0),\,\pm
1,\,-s\pm\alpha(0)).$

###### Proof.

If $b=0$, it follows from (3.21) that $\alpha(s)=0$ for all $s\in I$. Using
the equation $-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\gamma$, where
$\gamma\in\\{-1,0,1\\}$, we obtain $\eta^{2}(s)=1$ for all $s\in I$. Hence,
$\psi(s)=(0,0,\pm 1)$, $\forall\;s\in\mathbb{R}$ are singular solutions of
(3.21) in $S$.

If $\psi(s)$ is not a singular solution, then $b\neq 0$ and it follows from
(3.21) that

$a\,b\,\eta(s)=-\alpha(s)\hskip 28.45274pt\text{and}\hskip
28.45274pt\eta(s)=-ab^{2}s+\eta(0),$

for all $s\in\mathbb{R}$. Using the relation
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\gamma$, where
$\gamma\in\\{-1,0,1\\}$, we conclude that

$[-a^{2}\,b^{2}+1]\eta^{2}(s)=\gamma-b^{2},$

and hence the function $[-a^{2}b^{2}+1]\alpha^{2}(s)$ is also constant. Since
$\psi(s)$ is not a singular solution, it follows that $\displaystyle
a^{2}b^{2}=1$ and $b^{2}=\gamma$. Therefore, $\gamma=1$, $b=\pm 1$, $a=1$ and
$\alpha(s)=\mp\eta(s)$, for all $s\in\mathbb{R}$. This concludes the proof. ∎

It follows from Lemma 3.4 that when $\tau(s)$ is a constant function then the
functions $\alpha(s)$ and $\eta(s)$ are linear in $s$ and its corresponding
soliton solutions to the CSF are curves of constant curvature i.e. geodesics
when $k(s)=\tau(s)=0$ or planar curves with curvature $k(s)=\tau(s)=\pm 1$.

In this context, we define a trivial solution
$\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of (3.21), when $\tau(s)$ is a constant
function. From now on, we will study only non trivial solutions of (3.21). It
follows from Lemma 3.4 that there are no trivial solutions of (3.21) in $H\cup
C$.

On our next lemmas we will study the solutions $\psi(s)$ of (3.21) contained
in $H\cup C$ and those contained in $S$ separately.

######  Lemma 3.5.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22).

  * i)

If $\alpha(s)$ has a critical point then it is a global minimum point of
$\alpha$. Moreover, there exists always $\overline{s}\in I$ such that
$\alpha(s)$ is strictly monotone on the intervals $(\omega_{-},\overline{s}]$
and $[\overline{s},\omega_{+})$.

  * ii)

If $s_{0}$ is a critical point of $\tau(s)$, then $a^{2}\tau^{2}(s_{0})>1$ and
$s_{0}$ is a local minimum (resp. maximum) point of $\tau(s)$ if, and only if,
$\tau(s_{0})<0$ (resp. $\tau(s_{0})>0$).

###### Proof.

i) Let $s_{0}$ be a critical point of $\alpha(s)$. Note that $\alpha(s)>0$ for
all $s\in I$ whenever $\psi(0)\in H\cup C$. Taking the second derivative of
$\alpha(s)$ and using (3.21) at $s=s_{0}$, we have

$\alpha^{\prime\prime}(s_{0})=\tau^{\prime}(s_{0})=a\tau(s_{0})\eta(s_{0})+\alpha(s_{0})=\alpha(s_{0})>0.$

Hence, $s_{0}$ is a global minimum point of $\alpha(s)$. Therefore,
$\alpha(s)$ has at most one critical point. If there are no critical points
then $\alpha(s)$ is trictly monotone on $I$.

ii) Let $s_{0}$ be a critical point of $\tau(s)$. Then
$\tau^{\prime}(s_{0})=a\eta(s_{0})\tau(s_{0})+\alpha(s_{0})=0$ and
$\eta(s_{0})\tau(s_{0})\neq 0$ because $\alpha(s)>0$ for all $s\in I$. Since
$\psi(0)\in H\cup C$, it follows that
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\delta\leq 0$ for all $s\in I$, where
$\delta\in\\{-1,0\\}$. Thus,

$\delta=-a^{2}\tau^{2}(s_{0})\eta^{2}(s_{0})+\tau^{2}(s_{0})+\eta^{2}(s_{0})=\eta^{2}(s_{0})[-a^{2}\tau^{2}(s_{0})+1]+\tau^{2}(s_{0})\leq
0.$

Hence, $-a^{2}\tau^{2}(s_{0})+1<0$. Taking the second derivative of $\tau(s)$
and using (3.21) at $s=s_{0}$, we have

$\tau^{\prime\prime}(s_{0})=a\tau^{\prime}(s_{0})\eta(s_{0})+a\tau(s_{0})\eta^{\prime}(s_{0})+\alpha^{\prime}(s_{0})=\tau(s_{0})\left[-a^{2}\tau^{2}(s_{0})+1\right].$
(3.23)

This concludes the proof of ii). ∎

######  Lemma 3.6.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in C$, where $C$ is given by (3.22).

  * i)

If $\tau(s)>0$ in $I$, then $\alpha(s)$ is strictly increasing in $I$,
$\tau(s)$ is bounded and it has at most one critical point in $I$. Moreover,
$\omega_{-}=-\infty$, $\displaystyle\lim_{s\to-\infty}\psi(s)=(0,0,0)$ and
$\displaystyle\lim_{s\to\omega_{+}}-\eta(s)=\lim_{s\to\omega_{+}}\alpha(s)=+\infty.$

  * ii)

If $\tau(s)<0$ in $I$, then $\alpha(s)$ is strictly decreasing in $I$,
$\tau(s)$ is bounded and it has at most one critical point in $I$. Moreover,
$\omega_{+}=+\infty$, $\displaystyle\lim_{s\to+\infty}\psi(s)=(0,0,0)$ and
$\displaystyle\lim_{s\to\omega_{-}}\eta(s)=\lim_{s\to\omega_{-}}\alpha(s)=+\infty.$

###### Proof.

i) If $\tau(s)>0$, then it follows from Lemma 3.5 that $\tau(s)$ has only
local maximum points i.e. $\tau(s)$ has at most one critical point. The
positive function $\alpha(s)$ is bounded and there exists $\overline{s}\in I$
such that $\alpha$ is strictly increasing on $(\omega_{-},\overline{s})$.
Thus, it follows from equation $\alpha^{2}(s)=\tau^{2}(s)+\eta^{2}(s)$ that we
can take $\overline{s}$ such that $\tau(s)$ and $\eta(s)$ are bounded and
monotone on $(\omega_{-},\overline{s})$. The interval $I$ is maximal, hence
$\omega_{-}=-\infty.$ Since the limits
$\displaystyle\lim_{s\to-\infty}\alpha(s)$,
$\displaystyle\lim_{s\to-\infty}\alpha^{\prime}(s)=\lim_{s\to-\infty}\tau(s)$
and $\displaystyle\lim_{s\to-\infty}\tau^{\prime}(s)$ exist, we obtain that
$\displaystyle\lim_{s\to-\infty}\tau(s)=0$ and
$\displaystyle\lim_{s\to-\infty}\alpha(s)=\lim_{s\to-\infty}\eta(s)=0$. Using
the fact that the function $\eta(s)$ is decreasing, we get that $\eta(s)<0$
for all $s\in I.$

We claim that $\alpha(s)$ is ununbounded on $(\overline{s},\omega_{+})$. In
fact, assume by contradiction that the strictly increasing function
$\alpha(s)$ is bounded on $(\overline{s},\omega_{+})$. Thus, it follows from
equation $\alpha^{2}(s)=\tau^{2}(s)+\eta^{2}(s)$ that we can take
$\overline{s}$ such that $\tau(s)$ and $\eta(s)$ are bounded and monotone on
$(\overline{s},\omega_{+})$. Hence, there exists $p\in C$ such that
$\displaystyle\lim_{s\to\omega_{+}}(\alpha(s),\tau(s),\alpha(s))=p$ and $p$ is
a singular (trivial) solution in $C$, which contradicts Lemma 3.4. Therefore,
$\displaystyle\lim_{s\to\omega_{+}}\alpha(s)=+\infty.$

Now, assume by contradiction that the strictly decreasing and negative
function $\eta(s)$ is bounded on $(\overline{s},\omega_{+})$. Since
$\tau^{2}(s)+\eta^{2}(s)=\alpha^{2}(s)$, it follows that the function
$\tau(s)$ is unbounded and positive on $(\overline{s},\omega_{+})$, because we
showed that $\alpha(s)$ is unbounded on $(\overline{s},\omega_{+})$. Thus, we
can choose $\overline{s}$ such that $2\tau(s)<a\tau^{2}(s)$ for all
$s>\overline{s}$. Using the equations of (3.21), we obtain

$2\alpha(s)-2\alpha(\overline{s})=2\int_{\overline{s}}^{s}\tau(s)ds<\int_{\overline{s}}^{s}a\tau^{2}(s)ds=-\eta(s)+\eta(\overline{s}).$
(3.24)

Hence

$2\alpha(s)<-\eta(s)+\eta(\overline{s})+2\alpha(\overline{s}),$

for each $s\in(\overline{s},\omega_{+})$. But this contradicts the fact that
$\alpha(s)$ is unbounded. Therefore, $\eta(s)$ is unbounded and
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$.

Finally, if $\tau(s)$ is unbounded on $(\overline{s},\omega_{+})$,
$\overline{s}\in I$, then we can choose again $\overline{s}$ such that
$2\tau(s)<a\tau^{2}(s)$ for all $s>\overline{s}$. Using (3.24), we obtain

$\alpha(s)+\eta(s)<2\alpha(\overline{s})-\eta(\overline{s})-\alpha(s),$

this is a contradiction, because $\alpha^{2}(s)=\tau^{2}(s)+\eta^{2}(s)$ i.e.
$\alpha(s)>-\eta(s)$ and
$\displaystyle\lim_{s\to\omega_{+}}\alpha(s)=+\infty$.

ii) This proof is analogous to the proof of item i). ∎

######  Lemma 3.7.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H$, where $H$ is given by (3.22). Then there exists a unique
$s_{0}$ such that $\alpha^{\prime}(s_{0})=\tau(s_{0})=0$.

###### Proof.

Assume by contradiction that such an $s_{0}$ does not exist. Then, either
$\tau(s)<0$ or $\tau(s)>0$ for all $s\in I$. If $\tau(s)<0$, then $\alpha(s)$
is strictly decreasing. Taking $\overline{s}\in I$ we have
$1\leq\alpha(s)\leq\alpha(\overline{s})$ for all $s>\overline{s}$. Since
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=-1$, then
$\tau^{2}(s)+\eta^{2}(s)<\alpha^{2}(s)<\alpha^{2}(\overline{s})$ for all
$s>\overline{s}$. Hence, the functions $\alpha(s),\eta(s)$ are bounded and
monotone in $[\overline{s},\omega_{+})$, and
$\displaystyle\lim_{s\to\omega_{+}}\tau(s)$ exists, because
$\tau^{2}(s)=-1+\alpha^{2}(s)-\eta^{2}(s)$. Thus, there exists a point $p\in
H$ such that
$\displaystyle\lim_{s\to\omega_{+}}(\alpha(s),\tau(s),\alpha(s))=p$.
Therefore, $\displaystyle\omega_{+}=+\infty$ and $p$ is a singular (trivial)
solution of (3.21), which contradicts Lemma 3.4. In a similar way, we can
prove that $\tau(s)<0$ for all $s\in I$ cannot occur.

Therefore, there is an $s_{0}\in I$ such that
$\alpha^{\prime}(s_{0})=\tau(s_{0})=0$. It follows from Lemma 3.5, that
$s_{0}$ is a global minimum of the function $\alpha(s)$. Hence, $s_{0}$ is
unique. ∎

In the next three lemmas, we will suppose that $\psi(0)\in H\cup C$ and that
$\alpha(s)$ has only one critical point. Note that, this hypothesis only
excludes the case presented in Lemma 3.6 because it follows from Lemma 3.4
that $\alpha(s)$ has at most one critical point when $\psi(0)\in H\cup C$ and
Lemma 3.7 shows that $\alpha(s)$ has a unique critical point, when $\psi(0)\in
H$.

######  Lemma 3.8.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22). If $\alpha(s)$
has one critical point, then
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{+}}\alpha(s)=\infty$.

###### Proof.

Let $s_{0}$ be the global minimum point of $\alpha(s)$. Thus, $\alpha(s)$ is
monotone on the intervals $(\omega_{-},s_{0}]$ and $[s_{0},\omega_{+})$.
Assume by contradiction that $\alpha(s)$ is bounded on the intervals
$(\omega_{-},s_{0}]$ and $[s_{0},\omega_{+})$. Since the functions
$\alpha(s)$, $\tau(s)$ and $\eta(s)$ satisfy
$\tau^{2}(s)+\eta^{2}(s)=\delta+\alpha^{2}(s)\leq\alpha^{2}(s)$, where
$\delta\in\\{-1,0\\}$, then the functions $\alpha(s)$ and $\eta(s)$ are
bounded and monotone on $(\omega_{-},s_{0}]$ and $[s_{0},\omega_{+})$. The
limits $\displaystyle\lim_{s\to\omega_{-}}\tau(s)$ and
$\displaystyle\lim_{s\to\omega_{+}}\tau(s)$ exist, because
$\tau^{2}(s)=\delta+\alpha^{2}(s)-\eta^{2}(s)$. Thus, there are points $p_{1}$
and $p_{2}\in H\cup C$ such that
$\displaystyle\lim_{s\to\omega_{-}}(\alpha(s),\tau(s),\alpha(s))=p_{1}$ and
$\displaystyle\lim_{s\to\omega_{+}}(\alpha(s),\tau(s),\alpha(s))=p_{2}$.
Hence, $\omega_{-}=-\infty$, $\omega_{+}=+\infty$ and $\\{p_{1},p_{2}\\}$ is a
set of singular (trivial) solutions of (3.21), which contradicts Lemma 3.4.

Therefore, we conclude that the function $\alpha(s)$ is unbounded on the
intervals $(\omega_{-},s_{0}]$ and $[s_{0},\omega_{+})$, and moreover
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{+}}\alpha(s)=\infty$.
∎

######  Lemma 3.9.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22). If $\alpha(s)$
has one critical point, then
$\displaystyle\lim_{s\to\omega_{-}}\eta(s)=\infty$ and
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$.

###### Proof.

Let $s_{0}$ be the global minimum point of $\alpha(s)$. Then $\tau(s)<0$ for
all $s<s_{0}$ and $\tau(s)>0$ for all $s>s_{0}$. Moreover, $\alpha(s)$ is
unbounded and monotone on the intervals $(\omega_{-},s_{0}]$ and
$[s_{0},\omega_{+})$.

Assume by contradiction that the function $\eta(s)$ is bounded on
$(\omega_{-},s_{0}]$. Since $\tau^{2}(s)+\eta^{2}(s)=\delta+\alpha^{2}(s)$,
where $\delta\in\\{-1,0\\}$, then it follows from Lemma 3.7 that the function
$\tau(s)$ is unbounded and negative on $(\omega_{-},s_{0})$ i.e. there exists
$s_{1}\in(\omega_{-},s_{0}]$ such that $a\tau(s)<-1$ and
$-a\tau^{2}(s)<\tau(s)$ for all $s\in(\omega_{-},s_{1}]$. Thus, using (3.21)
for each $s\in(\omega_{-},s_{1}]$, we obtain

$\alpha(s)-\alpha(s_{1})=-\int_{s}^{s_{1}}\tau(s)ds<\int_{s}^{s_{1}}a\tau^{2}(s)ds=\eta(s)-\eta(s_{1}),$

that is,

$\alpha(s)<\eta(s)-\eta(s_{1})+\alpha(s_{1}),$

for each $s\in(\omega_{-},s_{1}]$ which contradicts Lemma 3.8. Therefore,
$\eta(s)$ is unbounded on $(\omega_{-},s_{1}]$.

In a similar way, we can prove that the function $\eta(s)$ is unbounded on
$[s_{0},\omega_{+})$.

Since $\eta(s)$ is decreasing on $(\omega_{-},\omega_{+})$, it follows that
$\displaystyle\lim_{s\to\omega_{-}}\eta(s)=\infty$ and
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$. ∎

######  Lemma 3.10.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21), with $a>0$,
defined on the maximal interval $I=(\omega_{-},\omega_{+})$ and initial
condition $\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22). If
$\alpha(s)$ has one critical point, then the function $\displaystyle\tau(s)$
is unbounded on $I$ and it has only two critical points.

###### Proof.

Let $s_{0}\in I$ be the global minimum point of $\alpha(s)$. The arguments
consist in studying the existence and the properties of the critical points of
$\tau(s)$.

Claim. If $\tau(s)$ does not have any critical point on $I$, then
$-1<a\tau(s)<0$ on $(\omega_{-},s_{0})$ and $0<a\tau(s)<1$ on
$(s_{0},\omega_{+})$. In fact, suppose that $\tau^{\prime}(s)\neq 0$ for all
$s\in I$. At $s_{0}$, $\tau(s_{0})=0$ and
$\tau^{\prime}(s_{0})=\alpha(s_{0})>0$. Moreover,
$\tau^{\prime}(s)=a\tau(s)\eta(s)+\alpha(s)>0$ i.e. $\tau(s)$ is an increasing
function on $I$, $\displaystyle\lim_{s\to\omega_{-}}\tau(s)\neq 0$ and
$\displaystyle\lim_{s\to\omega_{+}}\tau(s)\neq 0$. It follows from Lemma 3.9
that $\displaystyle\lim_{s\to\omega_{-}}\eta(s)=+\infty$ and
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$. We also know that
$\tau(s)$ is negative on $(\omega_{-},s_{0})$ and positive on
$(s_{0},\omega_{+})$. Thus, there are $s_{1}\in(\omega_{-},s_{0})$ and
$s_{2}\in(s_{0},\omega_{+})$ such that $-\alpha(s)<a\tau(s)\eta(s)<0$ for all
$s\in I\setminus[s_{1},s_{2}]$. Hence,
$-\alpha^{2}(s)<-a^{2}\tau^{2}(s)\eta^{2}(s)$ for all $s\in
I\setminus[s_{1},s_{2}]$ and

$\delta=-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)<-a^{2}\tau^{2}(s)\eta^{2}(s)+\tau^{2}(s)+\eta^{2}(s),$

i.e.,

$a^{2}\tau^{2}(s)\eta^{2}(s)<-\delta+\tau^{2}(s)+\eta^{2}(s).$

Therefore,

$1<\frac{-\delta}{a^{2}\eta^{2}(s)\tau^{2}(s)}+\frac{1}{a^{2}\eta^{2}(s)}+\frac{1}{a^{2}\tau^{2}(s)}$

for all $s\in I\setminus[s_{1},s_{2}]$.

Taking the limit when $s\to\omega_{+}$ and $s\to\omega_{-}$, using the fact
that $\tau(s)$ is increasing and Lemma 3.9, we obtain that

$\lim_{s\to\omega_{+}}\frac{1}{a^{2}\tau^{2}(s)}>1\hskip
28.45274pt\text{and}\hskip
28.45274pt\lim_{s\to\omega_{-}}\frac{1}{a^{2}\tau^{2}(s)}>1,$

i.e.,

$\lim_{s\to\omega_{+}}a^{2}\tau^{2}(s)<1\hskip 28.45274pt\text{and}\hskip
28.45274pt\lim_{s\to\omega_{-}}a^{2}\tau^{2}(s)<1.$

Thus, using that $\tau(s)$ is increasing, we conclude that $-1<a\tau(s)<0$ on
$(\omega_{-},s_{0})$ and $0<a\tau(s)<1$ on $(s_{0},\omega_{+})$. This proves
our Claim.

Still assuming that $\tau(s)$ does not have any critical point, we define the
positive functions $f(s)=\alpha(s)+\eta(s)$ and $g(s)=\alpha(s)-\eta(s)$
(observe that $-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\delta\leq 0$). Taking
the derivatives of $f$ and $g$ and using (3.21), we obtain
$f^{\prime}(s)=\tau(s)[1-a\tau(s)]$ and $g^{\prime}(s)=\tau(s)[1+a\tau(s)]$.
It follows from our Claim that the functions $f$ and $g$ are decreasing when
$\tau(s)<0$ and they are increasing when $\tau(s)>0$ and

$\displaystyle 0<f(s)\cdot g(s)=-\delta+\tau^{2}(s)<\frac{-\delta
a^{2}+1}{a^{2}}$

for all $s\in I$, where $\delta\in\\{-1,0\\}$. Hence, the functions $f$ and
$g$ are positive, monotone and bounded on the interval $(\omega_{-},s_{0})$.
Similarly one shows that the functions $f$ and $g$ are positive, monotone and
bounded on $(s_{0},\omega_{+})$. Thus, we conclude that there exist
$M_{1},M_{2}\in\mathbb{R}$ such that

$\left\\{\begin{array}[]{ll}\alpha(s)+\eta(s)\leq M_{1},\\\
\alpha(s)-\eta(s)\leq
M_{2},\end{array}\right.\qquad\forall\;s\in(\omega_{-},s_{0})\cup(s_{0},\omega_{+}).$

Hence,

$2\alpha(s)\leq
M_{1}+M_{2}\,\,\forall\,\,s\,\,\in\,\,(\omega_{-},s_{0})\,\,\text{and}\,\,2\alpha(s)\leq
M_{1}+M_{2}\,\,\forall\,\,s\,\,\in\,\,(s_{0},\omega_{+}).$

These inequalities contradict Lemma 3.8. Hence, the function $\tau(s)$ has at
least one critical point on each interval $(\omega_{-},s_{0})$ and
$(s_{0},\omega_{+})$. From item ii) of Lemma 3.5 we have that $\tau(s)$ has
only one local minimum $s_{1}\in(\omega_{-},s_{0})$ and it has only one local
maximum $s_{2}\in(s_{0},\omega_{+})$ i.e.
$\tau(s_{1})\leq\tau(s)\leq\tau(s_{2})$ for all $s\in I$ and $\tau(s)$ is
bounded on the interval $I$. This concludes the proof. ∎

######  Lemma 3.11.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in H\cup C$, where $H$ and $C$ are given by (3.22). Then
$I=\mathbb{R}$.

###### Proof.

From Lemmas 3.6 and 3.10 we have that $\tau(s)$ is bounded. Let $M>0$ be such
that $|\tau(s)|\leq M$, for all $s\in I$. Using (3.21), we obtain,

$|\alpha(s)-\alpha(s_{0})|=\left|\int_{s_{0}}^{s}\tau(s)ds\right|\leq
M|s-s_{0}|$ (3.25)

for each $s\in I$. If $\alpha(s)$ has a global minimum point, then it follows
from Lemma 3.8 that
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=\lim_{s\to\omega_{+}}\alpha(s)=\infty$.
Hence, $I=\mathbb{R}$.

If $\alpha(s)$ does not have any critical point and $\tau(s)<0$ (resp.
$\tau(s)>0$) for all $s\in I$, then it follows from Lemma 3.6 that
$\omega_{+}=+\infty$ and $\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=+\infty$
(resp. $\omega_{-}=-\infty$ and
$\displaystyle\lim_{s\to\omega_{+}}\alpha(s)=+\infty$). Using (3.25), we
conclude that $\omega_{-}=-\infty$ (resp. $\omega_{+}=+\infty$). Therefore,
$I=\mathbb{R}$. ∎

We will now study the solutions of the system (3.21), with initial conditions
on $S$. We will first classify the singular points of the system that are in
the set $S$.

######  Lemma 3.12.

Let $\Phi:S\rightarrow TS\subset\mathbb{R}^{3}$ be the differential vector
field given by

$\Phi(\alpha,\tau,\eta)=\left(\tau,a\tau\eta+\alpha,-a\tau^{2}\right),$

where $a>0$. Then $p=(0,0,1)$ and $-p=(0,0,-1)$ are the singular points of
$\Phi$ and the eigenvalues of $d\Phi_{p}$ and $d\Phi_{-p}$ are given
respectively by

$\lambda_{p}=\frac{a\pm\sqrt{a^{2}+4}}{2},\hskip
14.22636pt\lambda_{-p}=\frac{-a\pm\sqrt{a^{2}+4}}{2}.$ (3.26)

###### Proof.

Note that, if $\Phi(\alpha,\tau,\eta)=0$, then $\alpha=\tau=0$ and $\eta=\pm
1$. Hence, $p=(0,0,1)$ and $-p=(0,0,-1)$ are the singular points of $\Phi$.
The tangent plane in each singular point is defined by $T_{\pm
p}S\approx\\{(\alpha,\tau,\eta)\in\mathbb{R}^{3}:\eta=0\\}$. Thus,

$d\Phi=\left(\begin{array}[]{ccc}0&1&0\\\ 1&a\eta&a\tau\\\
0&-2a\tau&0\end{array}\right).$

Hence $\lambda$ is an eigenvalue of $d\Phi_{\pm p}$ if there is a non null
vector $w=(w_{1},w_{2},0)\in T_{\pm p}S$ such that $d\Phi_{\pm p}(w)=\lambda
w$ i.e.

$\left\\{\begin{array}[]{cc}w_{2}=\lambda w_{1},\\\ w_{1}\pm aw_{2}=\lambda
w_{2}.\end{array}\right.$

Hence, $\lambda$ satisfies $\lambda^{2}\mp a\lambda-1=0$, which gives (3.26).

∎

In Lema 3.12, we saw that $(0,0,\pm 1)$ are saddle points for the vector field
$\Phi$ on $S$, i.e., $\psi(s)=(0,0,\pm 1)$, $s\in I$ are singular solutions of
(3.21). If the functions $\alpha$ and $\tau$ are identically zero then the
corresponding curve $X(s)$ is the intersection of the upper half hyperboloid
with the plane going through the origin, orthogonal to $(0,0,1)$. Hence both
singular solutions of the system correspond to the same curve.

In order to study the non trivial solutions of the system (3.21), we consider
the singular point $p=(0,0,1)$ and $\psi(s,q)$ a solution of the system with
initial condition $q\in S$. Since the eigenvalues of the linearized system at
the singular point are not zero, it follows that the local behavior of the
system (3.21) is equivalent to the linearized one. Hence, there exist initial
conditions $q,\overline{q}\in S\setminus\\{p\\}$ such that
$\displaystyle\lim_{s\to-\infty}\psi(s,q)=p$ and
$\displaystyle\lim_{s\to+\infty}\psi(s,\overline{q})=p$. We define the
unstable and stable sets as

$\displaystyle W^{u}(p)=\\{q\in
S:\lim_{s\to-\infty}\psi(s,q)=p\\}\,\,\;\;\text{and }\,\,\;\;\displaystyle
W^{s}(p)=\\{q\in S:\lim_{s\to+\infty}\psi(s,q)=p\\}.$ (3.27)

From Lemma 3.4 we know that, if the function $\tau$ is a non zero
constant,i.e., $\tau(s)=b$, $b\in\mathbb{R}\setminus\\{0\\}$ for all $s\in I$,
then $b^{2}=a=1$. Our next result provides two non trivial solutions of the
system (3.21), $a=b^{2}=1$, defined on $\mathbb{R}$, with initial conditions
on the set $S$. They are particular cases of the solutions obtained in Lemma
3.4 with the constant of integration being zero. Moreover, we also obtain the
soliton solutions corresponding to these soltions.

######  Proposition 3.13.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I$, $a=1$ and $\psi(0)\in S$, where $S$ is given in
(3.22). Then, $\psi(s)=(-s,-1,-s)$ (resp. $\overline{\psi}(s)=(s,1,-s)$),
$s\in I=\mathbb{R}$ satisfy (3.21) with initial condition $(0,-1,0)$ (resp.
$(0,1,0)$). Moreover,

  * i)

The curve

$X(s)=\left(\frac{1+l^{2}+s^{2}}{2l},\frac{l^{2}-1-s^{2}}{2l},-s\right),$
(3.28)

where $l>0$ is the soliton solution to the CSF in $\mathbb{H}^{2}$ which
corresponds to the solution $\psi(s)=(-s,-1,-s)$ of (3.21);

  * ii)

The curve

$\overline{X}(s)=\left(\frac{1+l^{2}+s^{2}}{2l},\frac{l^{2}-1-s^{2}}{2l},s\right),$

where $l>0$ is the soliton solution to the CSF in $\mathbb{H}^{2}$ which
corresponds to the solution $\psi(s)=(s,1,-s)$ of (3.21)

###### Proof.

Straightforward computations show that $\psi(s)$ and $\overline{\psi}(s)$
satisfy (3.21) with initial conditions $(0,-1,0)$ and $(0,1,0)$ respectively.

i) Note that, if a curve $X(s)$ is given by (3.28), then

$\displaystyle T(s)$ $\displaystyle=$
$\displaystyle\left(\frac{s}{l},-\frac{s}{l},1\right),$ $\displaystyle N(s)$
$\displaystyle=$ $\displaystyle X(s)\times
T(s)=\left(\frac{-1+l^{2}+s^{2}}{2l},\frac{l^{2}+1-s^{2}}{2l},-s\right),$

$\alpha(s)=\langle X(s),(0,0,1)\rangle=s$, $\tau(s)=\langle
T(s),(0,0,1)\rangle=1$ and $\eta(s)=\langle N(s),(0,0,1)\rangle=-s$.

ii) The proof is similar to the case i). ∎

In Figure 2, we illustrate the curve $X(s)$ in $\mathbb{H}^{2}$ given by
(3.28) with $l=1$.

Figure 2: Soliton solution to the CSF with $a=1$, fixed vector $v=(0,0,1)$ and
constant curvature ($k(s)=1$).

In the following lemmas, we study the behavior of the functions $\alpha(s)$,
$\tau(s)$ and $\eta(s)$ when $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ is a non
trivial solution of the system (3.21) and initial condition $\psi(0)\in S$.

######  Lemma 3.14.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given in (3.22). If $s_{0}\in I$ is a
critical point of $\alpha(s)$, then $s_{0}$ is the global minimum (resp.
maximum) of $\alpha(s)$ if, and only if, $\alpha(s_{0})>0$ (resp.
$\alpha(s_{0})<0$). Moreover, there exists always $\overline{s}$ such that the
function $\displaystyle\alpha(s)$ is monotone on the intervals
$(\omega_{-},\overline{s})$ and $(\overline{s},\omega_{+}).$

###### Proof.

Let $s_{0}$ be a critical point of $\alpha(s)$, then $\tau(s_{0})=0$. It
follows from (3.21) that

$\alpha^{\prime\prime}(s_{0})=\alpha(s_{0}).$ (3.29)

Note that, if $\alpha(s_{0})=\tau(s_{0})=0$ and $\eta(s_{0})=1$, then
$\alpha(s)$ is constant, because $(0,0,1)$ is a singular (trivial) solution of
(3.21).

Therefore, it follows from (3.29) that $s_{0}$ is a local minimum point of
$\alpha(s)$ if $\alpha(s_{0})>0$. If there is another critical point $s_{1}$
of $\alpha(s)$ such that $s_{0}$ and $s_{1}$ are consecutive, then
$\alpha(s_{1})>\alpha(s_{0})>0$, because $s_{0}$ is a local minimum point.
Thus, $\alpha^{\prime\prime}(s_{1})=\alpha(s_{1})>0$ and $s_{1}$ is a local
minimum point $\alpha(s)$, this is a contradiction. Therefore, if
$\alpha(s_{0})>0$, then $s_{0}$ is a global minimum of $\alpha(s)$.

If $\alpha(s_{0})<0$, it follows from (3.29) that $s_{0}$ a local maximum
point of $\alpha(s)$. The proof that $s_{0}$ is a global maximum point of
$\alpha(s)$ is analogue to the previous case.

Since the function $\alpha(s)$ has at most one critical point, there exists
always $\overline{s}$ such that the function $\displaystyle\alpha(s)$ is
monotone on the intervals $(\omega_{-},\overline{s})$ and
$(\overline{s},\omega_{+}).$ ∎

We observe that if for a solution $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ of
(3.21) with $\psi(0)\in S$, the function $\alpha(s)$ does not have a crtitical
point, then $\alpha(s)$ is monotone on $I$. Moreover, it follows from Lemma
3.14, that for any solution of (3.21) with $\psi(0)\in W^{u}\cup
W^{s}\setminus\\{(0,0,1)\\}$, the functions $\alpha(s)$ and $\eta(s)$ do not
have critical points.

######  Lemma 3.15.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given by (3.22). Consider $W^{u}(p)$
and $W^{s}(p)$ given by (3.27). If $\psi(0)\in S\setminus W^{u}(p)$ (resp.
$S\setminus W^{s}(p)$), then
$\displaystyle\lim_{s\to\omega_{-}}|\alpha(s)|=+\infty$ (resp.
$\displaystyle\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$).

###### Proof.

It follows from Lemma 3.14 that there exits $\overline{s}\in I$ such that
$\alpha(s)$ is monotone on the intervals $(\omega_{-},\overline{s})$ and
$(\overline{s},\omega_{+})$. If $\psi(0)\in S\setminus W^{u}(p)$ assume by
contradiction that $\alpha(s)$ is bounded on $(\omega_{-},\overline{s})$.
Since $\tau^{2}(s)+\eta^{2}(s)=\alpha^{2}(s)+1$, it follows that the functions
$\alpha(s)$ and $\eta(s)$ are bounded and monotone on
$(\omega_{-},\overline{s})$ and the limit
$\displaystyle\lim_{s\to\omega_{-}}\tau(s)$ exists. Hence, there exists
$q\in\mathbb{R}_{1}^{3}$ such that
$\displaystyle\lim_{s\to\omega_{-}}(\alpha(s),\tau(s),\alpha(s))=q$,
$\omega_{-}=-\infty$ and $q$ is a singular solution of (3.21). But the system
(3.21) does not have any singular solution on the set $S\setminus W^{u}(p)$.
Therefore, $\displaystyle\lim_{s\to\omega_{-}}|\alpha(s)|=+\infty$. Similarly,
when $\psi(0)\in S\setminus W^{s}(p)$ one proves that
$\displaystyle\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$. ∎

######  Lemma 3.16.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given in (3.22).

  * i)

If $s_{0}$ is a critical point of $\tau(s)$. Then $\tau^{2}(s_{0})\neq 1$ and
$a^{2}\tau^{2}(s_{0})\neq 1$. If $\tau(s_{0})>0$, then $s_{0}$ is a local
minimum (resp. maximum) point of $\tau(s)$ if, and only if, $\tau(s_{0})>1$
(resp. $0<\tau(s_{0})<1$). If $\tau(s_{0})<0$, then $s_{0}$ is a local minimum
(resp. maximum) point of $\tau(s)$ if, and only if, $\tau(s_{0})<-1$ (resp.
$-1<\tau(s_{0})<0$).

  * ii)

The function $\tau(s)$ has at the most a finite number of critical points.

###### Proof.

i) Let $s_{0}$ be a critical point of $\tau(s)$. If $\tau^{2}(s_{0})=1$, it
follows from $-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=1$ that
$\alpha^{2}(s_{0})=\eta^{2}(s_{0})$. Thus, $0=\tau^{\prime}(s_{0})=\pm
a\eta(s_{0})+\alpha(s_{0})$ i.e. $a=1$. Hence, it follows from Lemma 3.4 that
the solution $\psi(s)$ of (3.21) with initial condition $\psi(s_{0})$ is a
trivial solution, which contradicts the hypothesis. Therefore,
$\tau(s_{0})\neq 1$. If $a^{2}\tau^{2}(s_{0})=1$, then
$0=\tau^{\prime}(s_{0})=\pm\eta(s_{0})+\alpha(s_{0})$ and from
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=1$ we have that $\tau(s_{0})=1$, which
also contradicts the hypothesis. Hence, $\tau^{2}(s_{0})\neq 1$ and
$a^{2}\tau^{2}(s_{0})\neq 1$. Moreover, taking the second derivative of
$\tau(s)$ at $s=s_{0}$, we obtain (3.23). This concludes the proof of the item
i).

ii) Note that, it follows from Lemma 3.14 that there exists always
$\overline{s}$ such that $\tau(s)$ does not change sign on each interval
$(\omega_{-},\overline{s})$ and $(\overline{s},\omega_{+})$. We will prove for
the interval $(\overline{s},\omega_{+})$, since similar arguments can be used
for the interval $(\omega_{-},\overline{s})$. If $\psi(0)\in W^{s}(p)$ i.e.
$\displaystyle\lim_{s\to+\infty}\tau(s)=0$, then it follows from item i) that
$\tau(s)$ has at most a finite number of critical points on
$(\overline{s},\omega_{+})$. If $\psi(0)\in S\setminus W^{s}(p)$, then it
follows from Lemma 3.15 that
$\displaystyle\lim_{s\to\omega_{+}}|\alpha(s)|=+\infty$.

Assume that $\tau(s)>0$ for all $s\in(\overline{s},\omega_{+})$, then there
exists $s_{1}$ such that $\alpha(s)>0$ for all $s>s_{1}$. If the function
$\eta(s)$, which is monotone, is always positive, then $\tau^{\prime}(s)>0$
for all $s>s_{1}$, i.e. $\tau$ has no critical on $(s_{1},\omega_{+})$. Now,
consider $s_{1}$ such that $\eta(s)<0$ for all $s>s_{1}$. Assume by
contradiction that there are $s_{2},s_{3}\in(s_{1},\omega_{+})$, two local
maximum points of $\tau(s)$. From item i), we obtain that there are
$b,d\in(s_{2},s_{3})$ such that $b<d$, $\tau(b)=\tau(d)=1$,
$\tau^{\prime}(b)<0$, $\tau^{\prime}(d)>0$. It follows from
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=1$ that $\eta^{2}(b)=\alpha^{2}(b)$
and $\eta^{2}(d)=\alpha^{2}(d)$. Thus, from $\tau^{\prime}(b)<0$ we obtain
$a\eta(b)<-\alpha(b)<0$ i.e. $a>1$ and from $\tau^{\prime}(d)>0$ we conclude
$\alpha(d)>-a\eta(d)>0$ i.e. $a<1$, this is a contradiction. Therefore, $\tau$
has at most one local maximum point on the interval $(s_{1},\omega_{+})$.

Analogously, assume that $\tau(s)<0$ for all $s\in(\overline{s},\omega_{+})$,
then there exists $s_{1}$ such that $\alpha(s)<0$ for all $s>s_{1}$. If the
monotone function $\eta(s)$ is always positive, then $\tau^{\prime}(s)<0$ for
all $s>s_{1}$, and hence $\tau$ has no critical point on $(s_{1},\omega_{+})$.
Now, we consider $s_{1}$ such that $\eta(s)<0$ for all $s>s_{1}$. Assume by
contradiction that there are $s_{2},s_{3}\in(s_{1},\omega_{+})$, two local
maximum points of $\tau(s)$. From item i) we obtain that there are
$b,d\in(s_{2},s_{3})$ such that $b<d$, $\tau(b)=\tau(d)=-1$,
$\tau^{\prime}(b)<0$, $\tau^{\prime}(d)>0$. It follows from
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=1$ that $\eta^{2}(b)=\alpha^{2}(b)$
and $\eta^{2}(d)=\alpha^{2}(d)$. Thus, from $\tau^{\prime}(b)<0$ we obtain
that $\alpha(b)<a\eta(b)<0$ i.e $a<1$ and from $\tau^{\prime}(d)>0$ we
conclude that $0>\alpha(d)>a\eta(d)$ i.e. $a>1$, this is a contradiction.
Therefore, $\tau$ has at most one local maximum point on the interval
$(s_{1},\omega_{+})$.

Similar arguments for the interval $(\omega_{-},\overline{s})$ imply that
$\tau(s)$ has at most a finite number of critical points on $I$. ∎

The following lemma shows that the function $\tau(s)$ is bounded and hence the
curvature of the soliton $X(s)$ on $\mathbb{H}^{2}$ is bounded.

######  Lemma 3.17.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ with
initial condition $\psi(0)\in S$, where $S$ is given by (3.22). Then the
function $\tau(s)$ is bounded on $I$.

###### Proof.

It follows from Lemma 3.14 that there exists $\overline{s}\in I$ such that
$\alpha(s)$ is monotone on the intervals $(\omega_{-},\overline{s})$ and
$(\overline{s},\omega_{+})$. Moreover, from Proposition 3.2 we have that
$\eta(s)$ is monotone.´

If $\psi(0)\in W^{s}(p)$, where $p$ is a singular point, then
$\displaystyle\lim_{s\to+\infty}\tau(s)=0$ and $\tau(s)$ is bounded on
$(\overline{s},+\infty)$ for any $\overline{s}\in I$ fixed. Similarly, if
$\psi(0)\in W^{u}(p)$, then $\displaystyle\lim_{s\to-\infty}\tau(s)=0$ and
$\tau(s)$ is bounded on $(-\infty,\overline{s})$, $\overline{s}\in I$ fixed.

We will now consider the cases when the initial condition $\psi(0)$ belongs to
$S\setminus W^{u}(p)$ or $S\setminus W^{s}(p)$. If $\psi(0)\in S\setminus
W^{u}(p)$ assume by contradiction that $\tau(s)$ is unbounded on
$(\omega_{-},\overline{s})$, then it follows from Lemma 3.16 that there exists
$s_{1}\in(\omega_{-},\overline{s})$ such that $|\tau(s)|>1$ and $a|\tau(s)|>2$
for all $s\in(\omega_{-},s_{1})$. Thus, $|\alpha(s)|>|\eta(s)|$, because
$\alpha^{2}(s)-\eta^{2}(s)=\tau^{2}(s)-1>0$ and $a\tau^{2}(s)>2|\tau(s)|$ for
all $s\in(\omega_{-},s_{1})$. From Lemma 3.14 we have that $\tau(s)$ does not
sign on $(\omega_{-},s_{1})$.

If $\alpha^{\prime}(s)=\tau(s)<0$ on $(\omega_{-},s_{1})$, then $\alpha(s)$ is
strictly decreasing on this interval. Thus, it follows from Lemma 3.15, that
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=+\infty$. Hence, $s_{1}$ can be
chosen so that $\alpha(s)$ is decreasing and positive for all $s<s_{1}$.
Therefore, using (3.21) and the fact that $a\tau^{2}(s)>2|\tau(s)|$, we obtain

$2\alpha(s)-2\alpha(s_{1})=-2\int^{s_{1}}_{s}\tau(s)ds<\int^{s_{1}}_{s}a\tau^{2}(s)ds=\eta(s)-\eta(s_{1}),$

i.e.,

$0<\alpha(s)-\eta(s)<2\alpha(s_{1})-\eta(s_{1})-\alpha(s),$

which contradicts Lemma 3.15, because
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=+\infty$. Hence, $\tau(s)$ is
bounded on $(\omega_{-},s_{1})$.

If $\alpha^{\prime}(s)=\tau(s)>0$ on $(\omega_{-},s_{1})$, then $\alpha(s)$ is
strictly increasing on this interval. Thus, it follows from Lemma 3.15 that
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=-\infty$. Hence, $s_{1}$ can be
chosen so that $\alpha(s)$ is increasing and negative for all $s<s_{1}$.
Therefore, using (3.21) and the fact that $a\tau^{2}(s)>2|\tau(s)|$, we obtain

$2\alpha(s_{1})-2\alpha(s)=2\int^{s_{1}}_{s}\tau(s)ds<\int^{s_{1}}_{s}a\tau^{2}(s)ds=-\eta(s_{1})+\eta(s),$

i.e. ,

$0<-\alpha(s)-\eta(s)<-2\alpha(s_{1})-\eta(s_{1})+\alpha(s),$

which contradicts Lemma 3.15, because
$\displaystyle\lim_{s\to\omega_{-}}\alpha(s)=-\infty$. Hence, $\tau(s)$ is
bounded on $(\omega_{-},\overline{s})$.

When $\psi(0)\in S\setminus W^{s}(p)$, the similar arguments show that
$\tau(s)$ is bounded on $(\overline{s},\omega_{+})$.

Therefore, the function $\tau(s)$ is bounded on $I$. ∎

Our next lemma provides the behavior of the function $\eta(s)$.

######  Lemma 3.18.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given by (3.22). Consider $W^{u}(p)$
and $W^{s}(p)$ given by (3.27). If $\psi(0)\in S\setminus W^{u}(p)$ (resp.
$S\setminus W^{s}(p)$), then
$\displaystyle\lim_{s\to\omega_{-}}\eta(s)=+\infty$ (resp.
$\displaystyle\lim_{s\to\omega_{+}}\eta(s)=-\infty$).

###### Proof.

The proof follows from Lemmas 3.15 and 3.17 and the fact that
$\tau^{2}(s)+\eta^{2}(s)=\alpha^{2}(s)+1$. ∎

######  Lemma 3.19.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21)
defined on the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial
condition $\psi(0)\in S$, where $S$ is given by (3.22). Then $I=\mathbb{R}$.

###### Proof.

It follows from Lemma 3.17 that $\tau(s)$ is bounded on the interval $I$. Let
$M>0$ be such that $|\tau(s)|\leq M$ for all $s\in I$. Using (3.21), we obtain
that

$|\alpha(s)-\alpha(s_{0})|=\left|\int_{s_{0}}^{s}\tau(s)ds\right|\leq
M|s-s_{0}|$ (3.30)

for all $s\in I$. We will now show that $\omega_{-}=-\infty$ and
$\omega_{+}=+\infty$.

If $\psi(0)\in S\setminus W^{u}(p)$, then from Lemma 3.15 we have that
$\alpha(s)$ is unbounded on $(\omega_{-},\overline{s})$ for any
$\overline{s}\in I$ fixed. Hence, it follows from (3.30) that
$\omega_{-}=-\infty$. It follows from the definition of $W^{u}(p)$ that
$\omega_{-}=-\infty$ when $\psi(0)\in W^{u}(p)$. Since $\displaystyle
S=W^{u}(p)\cup\left[S\setminus W^{u}(p)\right]$, we conclude that
$\omega_{-}=-\infty$.

If $\psi(0)\in S\setminus W^{s}(p)$, then from Lemma 3.15 we have that
$\alpha(s)$ is unbounded on $(\overline{s},\omega_{+})$ for any
$\overline{s}\in I$ fixed. Hence, it follows from (3.30) that
$\omega_{+}=+\infty$. It follows from the definition of $W^{s}(p)$ that
$\omega_{+}=+\infty$ when $\psi(0)\in W^{s}(p)$. Since $\displaystyle
S=W^{s}(p)\cup\left[S\setminus W^{s}(p)\right]$, then $\omega_{+}=+\infty$.

Therefore, $I=\mathbb{R}$. ∎

######  Lemma 3.20.

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a non trivial solution of (3.21),
with $a>0$ and initial condition $\psi(0)\in H\cup C\cup S$, where $H$, $C$
and $S$ are given by (3.22). Then $\psi(s)$ and the corresponding soliton
solution $X(s)$ to the CSF on $\mathbb{H}^{2}$ are defined for all
$s\in\mathbb{R}$. Moreover, at each end the curvature $k(s)$ of $X$ converges
to one of the following constants $\\{-1,0,1\\}$.

###### Proof.

Since $X(s)$ is a soliton solution to the CSF corresponding to $\psi(s)$, then
$k(s)=a\tau(s)$. Thus, Lemmas 3.6, 3.10, 3.14 and 3.17 imply that $k(s)$ is
bounded on $\mathbb{R}$ and it has at most a finite number of critical points.
Thus, the limits
$\displaystyle\lim_{s\to\pm\infty}k(s)=\displaystyle\lim_{s\to\pm\infty}a\tau(s)$
exist. In particular, when $\psi(0)\in W^{u}(p)$ then
$\displaystyle\lim_{s\to-\infty}\tau(s)=0$. Similarly, when $\psi(0)\in
W^{s}(p)$ then $\displaystyle\lim_{s\to+\infty}\tau(s)=0$. In these cases, the
curvature function converges to zero at $-\infty$ and $+\infty$, respectively.

If $\displaystyle\lim_{s\to\pm\infty}\tau(s)\neq 0$, then
$\displaystyle\lim_{s\to\pm\infty}|\alpha(s)|=+\infty$ and it follows from
$-\alpha^{2}(s)+\tau^{2}(s)+\eta^{2}(s)=\delta$, where $\delta\in\\{-1,0,1\\}$
that

$\displaystyle\lim_{s\to\pm\infty}\frac{\eta^{2}(s)}{\alpha^{2}(s)}=\displaystyle\lim_{s\to\pm\infty}\left(\frac{-\tau^{2}(s)+\delta}{\alpha^{2}(s)}+1\right)=1.$

Using (3.21), Lemmas 3.8, 3.9, 3.15, 3.18 and L’Hospital rule, we obtain

$\displaystyle\lim_{s\to\pm\infty}-\frac{\eta(s)}{\alpha(s)}=\lim_{s\to\pm\infty}a\tau(s)=\lim_{s\to\pm\infty}k(s).$

Therefore, $\displaystyle\lim_{s\to-\infty}k(s)=\pm 1$ and
$\displaystyle\lim_{s\to+\infty}k(s)=\pm 1$. ∎

Finally, we will prove our main theorem.

Proof of Theorem 2.3. For any vector $v\in\mathbb{R}^{3}_{1}\setminus\\{0\\}$,
without loss generality we can consider $v=ae$, where $a>0$ and

$\displaystyle e=\left\\{\begin{array}[]{clll}(-1,0,0)&\text{if}&v&\text{is a
timelike vector},\\\ (-1,1,0)&\text{if}&v&\text{is a lightlike vector },\\\
(0,0,1)&\text{if}&v&\text{is a spacelike vector.}\end{array}\right.$

Let $\psi(s)=(\alpha(s),\tau(s),\eta(s))$ be a solution of (3.21) defined on
the maximal interval $I=(\omega_{-},\omega_{+})$, $a>0$ and initial condition
$\psi(0)\in\mathbb{R}^{3}$ satisfying

$\displaystyle-\alpha^{2}(0)+\tau^{2}(0)+\eta^{2}(0)=\left\\{\begin{array}[]{clll}-1&\text{if}&v&\text{is
a timelike vector},\\\ 0&\text{if}&v&\text{is a lightlike vector},\\\
1&\text{if}&v&\text{is a spacelike vector,}\end{array}\right.$

i.e., $\psi(0)\in H\cup C\cup S$, where $H$, $C$ and $S$ are the disjoint sets
given by (3.22). Moreover, it follows from Proposition 3.2 that there is a
soliton solution $X(s)$ to the CSF, with curvature $k(s)=a\tau(s)$, such that
the relations

$\alpha(s)=\langle X(s),e\rangle,\hskip 8.5359pt\tau(s)=\langle
T(s),e\rangle\hskip 8.5359pt\text{and}\hskip 8.5359pt\eta(s)=\langle
N(s),e\rangle,$

are satisfied, where $T$ and $N$ are the unit vector fields tangent and normal
to $X$.

Thus, the initial conditions of (3.21), which are given by two constants,
determine the soliton solution in each case. Therefore, for each fixed vector
$v\in\mathbb{R}_{1}^{3}\setminus\\{0\\}$ there is a 2-parameter family of non
trivial soliton solutions to the CSF in $\mathbb{H}^{2}$. Moreover, it follows
from Lemmas 3.4, 3.11 and 3.19 that each soliton solution is defined for all
$s\in\mathbb{R}$, i.e. $I=\mathbb{R}$ and Lemma 3.20 shows that the curvature
at each end converges to one of the following constants $\\{-1,0,1\\}$.

Note that, from Lemmas 3.4, 3.5 and 3.14 we know that there exists
$\overline{s}\in\mathbb{R}$ such that $\alpha(s)$ is strictly monotone on the
intervals $(-\infty,\overline{s})$ and $(\overline{s},+\infty)$. Since
$\alpha(s)$ describes the Euclidean height of $X(s)$ with respect to a fixed
plane, then $X(s)$ does not have self-intersections in each one of the
intervals$(-\infty,\overline{s})$ and $(\overline{s},+\infty)$. Therefore,
$X(s)$ is embedded if $\alpha(s)$ is monotone in $\mathbb{R}$.

If $\alpha(s)$ is not monotone in $\mathbb{R}$ then $\alpha(s)$ has only one
critical point. Suppose that $X(s)$ has some self-intersection and consider
$\Sigma$ the simple region bounded by $X([s_{1},s_{2}])$ with
$X(s_{1})=X(s_{2})$, $s_{1}<\overline{s}<s_{2}$ and $\theta$ the external
angle between the tangent vectors $T(s_{1})$ and $T(s_{2})$, which is at the
most $\pi$. By Gauss-Bonnet’s theorem, we obtain

$\displaystyle 0<2\pi\chi(\Sigma)-\theta$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\kappa d\sigma+\int_{X([s_{1},s_{2}])}k(s)ds$
$\displaystyle=$
$\displaystyle-\int_{\Sigma}d\sigma+\int_{X([s_{1},s_{2}])}a\tau(s)ds$
$\displaystyle=$
$\displaystyle-\int_{\Sigma}d\sigma+a[\alpha(s_{2})-\alpha(s_{1})]$
$\displaystyle=$ $\displaystyle-\int_{\Sigma}d\sigma<0.$

This is a contradiction. Hence, the soliton solution $X(s)$ to the CSF in
$\mathbb{H}^{2}$ does not admit self-intersections. Note that, $X(s)$ is
already embedded on the intervals $(-\infty,\overline{s})$ and
$(\overline{s},+\infty)$ and from Lemmas 3.8 and 3.15 we have that the two
ends of the curve are unbounded. Therefore, $X(s)$ is an embedded curve.
$\square$

##  4 Visualizing some Soliton Solutions to the CSF on $\mathbb{H}^{2}$

In this setion, we visualize some examples of soliton solutions to the CSF on
the hyperbolic space. In order to do so, we use the following parametrization.
for $\mathbb{H}^{2}$

$\chi(u,w)=(\sqrt{1+u^{2}+w^{2}},u,w)$

If a curve of $\mathbb{H}^{2}$ $X(s)=\chi(u(s),w(s))$ is parametrized by arc
lentgh, then

$T(s)=\left(\frac{u(s)u^{\prime}(s)+w(s)w^{\prime}(s)}{\sqrt{1+u^{2}(s)+w^{2}(s)}},u^{\prime}(s),w^{\prime}(s)\right)$

the functions $u(s)$ and $w(s)$ satisfy the following system of ODEs

$\left\\{\begin{array}[]{ll}(u^{\prime})^{2}+(w^{\prime})^{2}+\left(u^{\prime}w-uw^{\prime}\right)^{2}=1+u^{2}+w^{2},\\\
w^{\prime\prime}u^{\prime}-u^{\prime\prime}w^{\prime}+uw^{\prime}-u^{\prime}w=k(s)\sqrt{1+u^{2}+w^{2}},\end{array}\right.$
(4.1)

where $k(s)$ is the curvature of $X(s)$. The first equation follows from the
fact that the curve is parametrized by arc lentgh and the second one from the
expression of the curvature of $X$.

In Theorem 2.2, we saw that the curvature of a soliton solution to the CSF on
$\mathbb{H}^{2}$ is determined by its tangent vector field and a non zero
fixed vector $v$. We use (4.1) and the software Maple to plot examples of such
solitons. In each example, we visualize the curve on the three models of the
2-dimensional hyperbolic space, namely the hyperboloid, the Poincaré disk and
the upper half space.

In Figure 3 a), the blue curve on the hyperboloid provides the visualization
of a soliton solution $X(s)$ to the CSF on $\mathbb{H}^{2}$ whose curvature is
given by $\displaystyle k(s)=\langle T(s),(-1,0,0)\rangle$ and $a=1$. The red
curve is the Euclidean orthogonal projection of $X(s)$ on the plane that
contains the origin and it is orthogonal to the vector $(-1,0,0)$. In Figures
3 b) and c) we visualize the same soliton on the Poincaré disk and on the half
space model respectively.

(a)

(b)

(c)

Figure 3: Soliton solution to the CSF on $\mathbb{H}^{2}$ with fixed vector
$v=(-1,0,0)$ and $a=1$.

In Figure 4 a), the blue curve provides the visualization of a soliton
solution $X(s)$ to the CSF on $\mathbb{H}^{2}$ whose curvature is given by
$\displaystyle k(s)=\langle T(s),(-1,1,0)\rangle$ and $a=1$. In Figures 4 b)
and c) we visualize the same soliton on the Poincaré disk and on the half
space model respectively.

(a)

(b)

(c)

Figure 4: Sóliton do fluxo FRC em $\mathbb{H}^{2}$ com vetor fixado $(-1,1,0)$
e $a=1$.

Finally, in Figure 5 a), the blue curve provides the visualization of a
soliton solution $X(s)$ to the CSF on $\mathbb{H}^{2}$ whose curvature is
given by $\displaystyle k(s)=\langle T(s),(0,0,1)\rangle$ and $a=1$. In
Figures 4 b) and c) we visualize the same soliton on the Poincaré disk and on
the half space model respectively. We point out that this soliton has non
constant curvature and hence it is different from the one given in Proposition
3.13. In fact, in order to obtain Figure 5, we used initial condition
$u(0)=w(0)=0$ and $\displaystyle
u^{\prime}(0)=-w^{\prime}(0)=-\,\frac{1}{\sqrt{2}}$ for the system (4.1),
i.e., $\displaystyle
T(0,0)=\left(0,-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$ and
$\displaystyle\tau(0)=\frac{1}{\sqrt{2}}\neq\pm 1$. Hence the curvature is not
constant. In Figures 5 b) and c) we visualize the soliton given in Figure 5 a)
on the Poincaré disk and on the half space model respectively.

(a)

(b)

(c)

Figure 5: Sóliton do fluxo FRC em $\mathbb{H}^{2}$ com vetor fixado $(0,0,1)$
e $a=1$.

Aknowlegment: The first author aknowledges the support given by the
Universidade Federal do Oeste da Bahia during his graduate program at the
Universidade de Brasília, when this research was undertaken.

## References

  * [1]
  * [2] Abresch, U.; Langer, J. The normalized curve shortening flow and homothetic solutions, Journal Differential Geometry 23, n. 2, p. 175–196 (1986).
  * [3] Angenent, S. B. On the formation of singularities in the curve shorteningow, Journal of Differential Geometry, v. 33, 601-633 (1991).
  * [4] Angenent, S. B. Curve shortening and the topology of closed geodesies on surfaces, Annals of Mathematics, v.162, 1187-1241 (2005).
  * [5] Dos Reis, H. F. S.; Tenenblat, K. Soliton solutions to the curve shortening flow on the sphere, Proc. Amer. Math. Soc., v. 147, 4955-4967 (2019).
  * [6] Epstein, C.L., Gage, M. The curve shortening flow. In: Chorin A.J., Majda A.J. (eds) Wave Motion: Theory, Modelling, and Computation. Mathematical Sciences Research Institute Publications, vol 7. Springer, (1987).
  * [7] Epstein, C.L.; Weinstein, M.I. A stable manifold theorem for the curve shortening equation, Comm. Pure Appl. Math., v. 40, 119-139 (1987).
  * [8] Gage, M. E., An isoperimetric inequality with applications to curve shortening, Duke Mathematical Journal, v. 50, n.4, p. 1225–1229 (1983).
  * [9] Gage, M. E., Curve shortening makes convex curves circular, Inventiones mathematicae, v. 76, n.2, p. 357–364 (1984).
  * [10] Gage, M. E.; Hamilton, R. S., The heat equation shrinking convex plane curves, Journal Differential Geometry, v.23, p. 69-96 (1986)
  * [11] Giga, Y., Surface evolutions equations. A level set approach, Monographs in Mathematics, vol. 99, Birkhauser, Basel, 2006.
  * [12] Grayson, M. A. The heat equation shrinks embedded plane curves to round a points, Journal Differential Geometry, v.26, p. 285-314 (1987)
  * [13] Grayson, M. A. Shortening embedded curves, Annals of Mathematics, v. 129, n.1, p. 71–111 (1989).
  * [14] Halldorsson, H. P. Self-similar solutions to the curve shortening flow, Transactions of the American Mathematical Society, v. 364, n. 10, p. 5285–5309 (2012).
  * [15] Halldorsson, H. P. Self-similar solutions to the mean curvature flow in the Minkowski plane $\mathbb{R}^{1,1}$, J. Reine Angew. Math., v. 704, 209–243 (2015).
  * [16] Ma, L.; Chen, D. Curve shortening in a Riemannian manifold, Annali di Matematica v. 186, p. 663-684 (2007)
  * [17] O’Neill, B. Semi-riemannian geometry with applications to relativity. Academic Press 103, (1983).
  * [18] Palis, J.; Melo, W. geometric theory of dynamical systems an introduction, Translated by A. K. Manning, Springer-Verlag New York (1982).
  * [19] Sotomayor, J. Equações diferenciais ordinárias, Editora Livraria da Física (2011).
  * [20] Zhou, H. Curve Shortening Flows in warped product manifolds, Transactions of the American Mathematical Society, Proc. Amer. Math. Soc., v. 145, 4505-4516 (2017).


[table]capposition=bottom capbtabboxtable[][]

††thanks: Xianbiao Qi is the corresponding author. If you have any questions
or concerns about the implementation details, please do not hesitate to
contact<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
# 1st Place Solution for ICDAR 2021 Competition on Mathematical Formula
Detection

Yuxiang Zhong  Xianbiao Qi  Shanjun Li  Dengyi Gu  Yihao Chen  Peiyang Ning
Rong Xiao

###### Abstract

In this technical report, we present our 1st place solution for the ICDAR 2021
competition on mathematical formula detection
(MFD)111http://transcriptorium.eu/~htrcontest/MathsICDAR2021/. The MFD task
has three key challenges, i.e. a large scale span, large variation of the
ratio between height and width, and rich character set and mathematical
expressions. Considering these challenges, we used Generalized Focal Loss
(GFL), an anchor-free method, instead of the anchor-based method, and prove
the Adaptive Training Sampling Strategy (ATSS) and proper Feature Pyramid
Network (FPN) can well solve the important issue of scale variation.
Meanwhile, we also found some tricks, e.g., Deformable Convolution Network
(DCN), SyncBN, and Weighted Box Fusion (WBF), were effective in MFD task. Our
method was ranked 1st place in the final 15 teams.

##  1 Introduction

Digitization of document images is vital for enabling searching in massive
collections of digitized printed scientific documents. Different from common
text content, general OCR software fails to process mathematical formulas.
Usually, a specific mathematical expression recognition is needed for
mathematical formulas. To do this, we have to detect the regions of
mathematical formulas. The ICDAR 2021 competition on mathematical formula
detection (MFD) is targeting this problem. In the MFD task, there are two
forms of formulas, i.e. embedded and isolated.

Different from general object detection [1, 2, 3, 4, 5], the MFD task [6, 7,
8] has three key characteristics, i.e. large scale variance especially between
isolated and embedded formulas, large variation of the ratio between width and
height, and rich character set and mathematical expressions. In this paper, we
fully consider the features of the MFD task, and develop an empirical solution
that is ranked 1st place in the ICDAR 2021 competition on mathematical formula
detection. In our solution, we used an anchor-free framework, Generalized
Focal Loss (GFL) [9], instead of the anchor-based model [4] because the
proposal generated by the latter is hard to cover different scales and
different ratios between width and height. Meanwhile, we employed an adaptive
training sample selection (ATSS) [10] to balance the sampled positive points
on each formula instance. In addition, we leveraged the feature pyramid
network (FPN) [11], the large input resolution, and the deformable
convolutional network (DCN) [12] because these techniques could benefit large
formula detection or small formula detection, or both. Finally, some other
tricks, such as ResNeSt [13], SyncBN [14], the large batch size and the
weighted box fusion (WBF) [15], Ranger [16] optimizer, were adopted in our
solution.

Figure 1: Comparison of random sampling strategy and ATSS sampling. Left.
positive samples with random sampling strategy; Right. positive samples with
ATSS.

##  2 Challenge

Compared to general object detection in natural images or scene text detection
in documents, mathematical formula detection in this task will encounter some
specific difficulties. As shown in Figure 1, these challenges include

  * •

a large scale span between embedded formula and isolated formula. The areas of
some large isolated formulas are hundreds of times larger than that of some
embedded formulas;

  * •

huge variation of the ratio between height and width. As shown in Figure 1,
the ratios range from around 0.5 to 12;

  * •

rich character set and mathematical expressions. There are hundreds of
mathematical characters, and their combinations are extremely rich.

The large scale span of formulas and the huge variation of the ratio between
height and width will lead to a classical scale problem in object detection.
To be specific, as shown in the left of Figure 1, isolated formulas with large
area will obtain large number of positive samples, but small embedded formulas
only have a few or even none of positive samples under the random sampling
strategy. In addition, large scale span will bring in a challenge to the head
of the detector. To deal with this issue, attention needs to be paid to the
choice of the pyramid of the detector head. In the following section, we will
describe methods targeting these two challenges.

##  3 Method

We treated this task as a pure object detection task, and adopted Generalized
Focal Loss (GFL [9]) framework as our baseline. GFL is built on a Fully
Convolutional One-Stage (FCOS) [17] detector that is a robust anchor-free
method. Based on the up-mentioned challenges, we reckon the anchor-free method
is better than the anchor-based methods (e.g., Faster R-CNN [4], Cascade R-CNN
[5]) in this task. In the following, we will describe two key modules in
detail that greatly improved our solution.

ATSS. Adaptive Training Sample Selection [10] is an effective sampling
strategy to alleviate the complex artificial design of sampling ratio between
positive and negative, and overcomes the sampling imbalance disadvantage of
random strategy. As shown in the left of the Figure 1, the results of random
sampling strategy obey an uniform distribution according to the area of
positive samples. It means that larger formula will obtain more positive
samples, while some tiny formulas will only get a few or even zero positive
samples. The sampling frequency of these tiny embedded formulas is much lower
than the large isolated formulas. Obviously, there is a sampling imbalance
between tiny and large formulas with random sampling strategy. On the
contrary, when using ATSS, the number of positive samples has no relationship
with the area of instance. Each instance will get the same number of positive
samples. As shown in the right of the Figure 1, ATSS can effectively eliminate
the impact of imbalanced sampling.

FPN. Feature Pyramid Network (FPN) [11] is to explicitly address the problem
of large scale span. The MFD task contains a large number of extremely small
formulas, which bring great challenges to our model. As illustrated in Figure
2, for such a single extremely small character formula, their short sides are
usually around 16 pixels. We observe that for any layer of FPN, the limit of
the detector is 3 pixels. This means that if we use the default FPN (3-7), the
short side of the formula needs to be at least 24 pixels to be detected.
Obviously, there are many small embedded formulas that do not satisfy this
condition. So we change the selection of FPN level to (2-6) so that our model
can overcome this defect.

Figure 2: Illustration of the importantce of the FPN on the MFD task. (a)
Input size; (b) Corresponding size in P2; (c) Corresponding size in P3. Under
some small scales, some positive samples will be missed.

##  4 Experiment

In this section, we will describe the implementation of our mathematical
formula detection system in detail.

Dataset. Our used data is the official IBEM dataset. The IBEM dataset consists
of 600 documents with a total number of 8,273 pages. There are 58,834 isolated
and 260,323 embedded expressions. Each document contains approximately an
average of 14 pages, 98 isolated mathematical expressions, and 434 embedded
mathematical expressions. Each document is separated into individual pages
with the corresponding ground truth. It is unknown how the organizers make a
change to the original IBEM data set. Finally, there are 7,244 images with
labels for training, and 4,028 images without a label for final evaluation.
Only the provide training data is used for training.

Implementation Details. Our baseline model is GFL with ResNet101 [18] as the
backbone. In the training of the baseline model, 8 Tesla V100 GPUs are used
with batch size 2 in each GPU. The input image is resized equally to
$1447\times 2048$ with a keeping ratio. An $800\times 800$ region is cropped
from the original image. We use FPN (3-7) in default. In the baseline model,
we use Adam [19] optimizer.

In the inference stage, we resize the test image to $1583\times 2048$ due to
the resolution distribution of the test dataset. Flip augment is turned on.
For the post-processing, Non-Maximum Suppression (NMS) [20] with 0.6 IoU
threshold is applied to filter the redundant box. All models are trained based
on MMDetection [21] toolbox.

###  4.1 Ablation Study

We have conducted many attempts in this competition. Some effective tricks are
listed below. We list these tricks according to the timeline we tried them.

Double Training Epoch. (DTE) Initially, We train 12 epochs. Then, we extend
the training epoch to 24, and get a substantial score gain. Therefore, we
further train 36 epochs, the result has no obvious change. It means that model
is under-fitting with 12 epochs, and suitable with 24 epochs.

Large Crop Size. Considering that a small crop size will cause the long-
isolated formula to be truncated sometimes. We decide to expand the crop size
to $1440\times 1600$.

Random Flip is used at the training phase with a 0.5 ratios. In the inference
phase, the image is also flipped as the input. The result is attained by
mapping back the flipped boxes. The final result will be obtained by NMS.

ResNeSt [13] is used as our backbone to replace ResNet. Here, we use
ResNeSt101 as the backbone.

Synchronized Batch Normalization (SyncBN) [14] is an effective batch
normalization approach that is more suitable for the situation that the batch
size is relatively small on each GPU graphics card.

Deformable Convolution Network (DCN) [12]. There are some discrete and large
isolated formulas in the data set. To detect such formulas, a large receipt
filed is needed. Deformable convolution is an effective way to extend the
receptive field. We replace all convolutions with deformable convolution
layers from c3 to c5 in ResNeSt101.

Larger Batch Size (LBS). We expand batch size from 2 to 3 each GPU by
utilizing the memory consume, such as mixed precision and replaced ReLU.

Ranger [16] is a synergistic optimizer combining RAdam (Rectified Adam) [22],
LookAhead [23], and GC (gradient centralization) [24].

RegMax is a hyperparameter of GFL. It represents the largest position
regression offset in each layer of FPN. In order to deal with the slim
isolated formula, we increase the default 16 to 24.

FPN Selection. We change the FPN (3-7) to FPN (2-6).

Weight Box Fusion (WBF) [15] is used for model ensemble. It adopts an
Expectation-Maximization algorithm to fuse the final bounding boxes. We select
two models with the best performance which are trained initially with
different random seeds. The IoU threshold of NMS is set to be 0.4 in WBF.

DTE | LCS | Flip | NeSt | SyBN | DCN | LBS | Ranger |  | Reg
---
24
| FPN
---
(2-6)
WBF |  | F1-score
---
Embedded
| F1-score
---
Isolated
| F1-score
---
Total
|  |  |  |  |  |  |  |  |  |  |  | 89.88
---
p:91.34 r:88.47
| 85.92
---
p:89.77 r:82.39
| 89.17
---
p:91.02 r:87.39
$\surd$ |  |  |  |  |  |  |  |  |  |  |  | 91.36
---
p:92.80 r:89.96
| 86.29
---
p:90.93 r:82.10
| 90.45
---
p:92.26 r:88.71
$\surd$ | $\surd$ |  |  |  |  |  |  |  |  |  |  | 92.95
---
p:94.43 r:91.52
| 87.84
---
p:92.04 r:84.01
| 92.03
---
p:93.58 r:90.53
$\surd$ | $\surd$ | $\surd$ |  |  |  |  |  |  |  |  |  | 93.58
---
p:94.79 r:92.41
| 88.38
---
p:92.29 r:84.78
| 92.66
---
p:94.35 r:91.03
$\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |  |  |  |  |  |  | 93.49
---
p:93.82 r:93.16
| 91.33
---
p:93.56 r:89.20
| 93.12
---
p:93.71 r:92.54
$\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |  |  |  |  |  | 93.81
---
p:94.00 r:93.63
| 91.60
---
p:93.82 r:89.48
| 93.42
---
p:93.97 r:92.89
$\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |  |  |  |  | 94.33
---
p:95.62 r:93.07
| 95.25
---
p:95.62 r:94.88
| 94.49
---
p:95.62 r:93.39
$\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |  |  |  | 94.58
---
p:95.16 r:94.00
| 95.60
---
p:95.97 r:95.23
| 94.76
---
p:95.31 r:94.22
$\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |  |  | 95.22
---
p:95.70 r:94.74
| 95.81
---
p:96.12 r:95.51
| 95.33
---
p:95.79 r:94.87
$\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |  | 95.01
---
p:95.86 r:94.16
| 97.28
---
p:97.18 r:97.38
| 95.41
---
p:96.10 r:94.73
$\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  | 95.67
---
p:96.34 r:95.00
| 97.67
---
p:97.46 r:97.88
| 96.03
---
p:96.54 r:95.53
$\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  | 96.01
---
p:95.79 r:96.23
| 98.14
---
p:97.32 r:98.98
| 96.33
---
p:96.81 r:95.85
Table 1: Experimental results of baseline, DTE (double training epoch), LCS
(larger crop size), Flip (training and testing time flip), SyBN (SyncBN), DCN
(deformable convolution network), LBS (larger batch size), Ranger optimizer,
Reg 24 (a hyperparameter regmax), FPN (2-6) (FPN selection), WBF (weighted box
fusion).

According to Table 1, we have the following observations.

  * •

The number of training epochs should be large to make the model learn the data
set fully.

  * •

Large input resolution is good for the MFD task. Random crops from the images
will cause the long-isolated formula to be truncated, and thus decrease the
performance.

  * •

ResNeSt is better than ResNet in this task. SyncBN can improve the performance
a little.

  * •

DCN is a powerful weapon in detection tasks. DCN largely improves the
performance of the isolated formulas. Such improvement can be attributed to
that DCN can extend the receipt field of the network. That is important for
large isolated formulas.

  * •

Ranger optimizer has outperformed Adam optimizer consistently. Similar
observation is also found in our another report [25] about ICDAR 2021
Competition on Scientific Table Image Recognition to LaTeX [26]. In our
evaluation of standard benchmarks, we also find that Ranger can improve the
average accuracy by around 0.6%.

  * •

The appropriate FPN level should be selected according to the scale of the
object in the task. It can largely improve the performance.

  * •

Weighted Box Fusion is useful in our task. It will improve the recall largely
although it will slightly decrease the precision. Overall, it can improve the
performance by 0.3%.

Remarks. Besides of the up-mentioned methods, we also tried many other
approaches including CutMix data augmentation, soft NMS instead of the
original NMS, IoU loss and GIoU loss instead of the smooth L1 loss, BFP and
PAFPN instead of the original FPN, and Swish and Mish activation functions
instead of ReLU. However, unfortunately, we did not observe obvious
improvement on our validation set.

###  4.2 Final Results

The final results are shown on Table 2 reported by the organizers, and our
team is PAPCIC (Visual Computing Group from Ping An Property & Casualty
Insurance Company, China). The types, E, I, S characters, stand for Embedded,
Isolated, and the Averaged, respectively. F1 score is used as the metric. Our
method is ranked 1st place among all 15 teams. From Table 2, we found that

  * •

almost all submissions obtained high performance on the isolated formula, but
some methods performed poor on the embedded formula. The gap among the highest
and the lowest on the isolated formula was around 3.0%, but this gap amounted
to around 10.6% on the embedded formula. It means most algorithms can detect
the isolated formulas well, and the main challenge of the MFD task lies on the
detection of the embedded formula.

  * •

compared to the “DLVCLab” and “TYAI” methods, our method slightly beat them on
the isolated formula, but outperformed them largely on the embedded formula.
We believe this improvement is attributed to the large input resolution, the
ATSS and the proper FPN choice.

Group ID | Type |  | F1
---
(Ts10+Ts11)
| F1
---
Task dependent(Ts11)
| F1
---
Task independent(Ts10)
PAPCIC |  | E
---
I
S
| 94.89
---
98.76
95.47
| 95.11
---
98.70
95.68
| 94.64
---
98.79
95.37
Lenovo |  | E
---
I
S
| 94.29
---
98.19
94.96
| 93.98
---
97.85
94.60
| 94.44
---
98.33
95.13
DLVCLab |  | E
---
I
S
| 93.79
---
98.54
94.60
| 93.88
---
98.61
94.64
| 93.75
---
98.51
94.59
TYAI |  | E
---
I
S
| 93.39
---
98.55
94.28
| 93.94
---
98.42
94.66
| 93.13
---
98.61
94.10
SPDBLab |  | E
---
I
S
| 92.80
---
98.06
93.70
| 92.14
---
97.76
93.03
| 93.12
---
98.19
94.01
YoudaoAI |  | E
---
I
S
| 92.73
---
98.34
93.70
| 92.71
---
98.38
93.63
| 92.74
---
98.32
93.74
PKUF-MFD |  | E
---
I
S
| 91.94
---
96.56
92.72
| 92.32
---
96.87
93.04
| 91.76
---
96.43
92.57
HW-L |  | E
---
I
S
| 90.53
---
98.94
91.97
| 90.57
---
98.61
91.86
| 90.51
---
99.08
92.02
Komachi |  | E
---
I
S
| 90.39
---
98.57
91.79
| 89.69
---
98.60
91.11
| 90.72
---
98.55
92.10
AIG |  | E
---
I
S
| 89.71
---
95.95
90.75
| 89.19
---
96.07
90.26
| 89.95
---
95.90
90.97
PKUSG |  | E
---
I
S
| 89.10
---
97.96
90.62
| 88.59
---
97.94
90.09
| 89.34
---
97.96
90.87
TAL |  | E
---
I
S
| 87.87
---
96.85
89.42
| 88.51
---
97.09
89.89
| 87.57
---
96.75
89.19
UIT |  | E
---
I
S
| 86.04
---
97.05
87.94
| 85.64
---
98.11
87.63
| 86.23
---
96.60
88.08
AV-DFKI |  | E
---
I
S
| 85.35
---
97.48
87.45
| 84.75
---
97.40
86.80
| 85.63
---
97.52
87.76
VH |  | E
---
I
S
| 84.25
---
98.59
86.67
| 84.39
---
98.51
86.61
| 84.18
---
98.62
86.70
Table 2: Final results of all 15 teams on ICDAR 2021 Competition on
Mathematical Formula Detection.

##  5 Conclusion

In this paper, we presented our 1st place method to ICDAR 2021 Competition on
Mathematical Formula Detection. We built our approach on GFL, with ResNeSt as
our backbone, to detect embedded and isolated formulas. We employed ATSS as
our sampling strategy instead of random sampling to eliminate the effects of
sample imbalance. Moreover, we observed and revealed the influence of
different FPN levels on the detection result. Finally, with a series of useful
tricks and model ensembles, our method was ranked 1st in the MFD task.

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