∎

11institutetext: Muhittin Evren Aydin 22institutetext: Department of
Mathematics
Faculty of Science, Firat University
Elazig, 23200, Turkey
22email<EMAIL_ADDRESS>33institutetext: Rafael López 44institutetext:
Departamento de Geometría y Topología
Universidad de Granada
18071 Granada, Spain 44email<EMAIL_ADDRESS>
# Translating solitons of translation and homothetical types††thanks: Rafael
López has been partially supported by the grant no. MTM2017-89677-P, MINECO/
AEI/FEDER, UE.

Muhittin Evren Aydin  Rafael López

(Received: date / Accepted: date)

###### Abstract

We prove that if a translating soliton can be expressed as the sum of two
curves and one of these curves is planar, then the other curve is also planar
and consequently the surface must be a plane or a grim reaper. We also
investigate translating solitons that can be locally written as the product of
two functions of one variable. We extend the results in Lorentz-Minkowski
space.

###### Keywords:

Mean curvature flowtranslating solitonsurfaces of translationhomothetical
surface

###### MSC:

53A10 53C21 53C42

##  1 Introduction

Let $\vec{v}\in\mathbb{R}^{3}$ be non-zero vector. A translating soliton in
Euclidean $3$-dimensional space $\mathbb{R}^{3}$ with respect to $\vec{v}$,
called the velocity of the flow, is a surface $M$ whose mean curvature $H$
satisfies

$H(p)=\langle N(p),\vec{v}\rangle,$ (1)

for all $p\in M$, where $N$ is the unit normal vector field on $M$.
Translating solitons appear in the theory of the mean curvature flow of
Huisken and Ilmanen as the solutions of the flow when $M$ evolves purely by
translations along the direction $\vec{v}$ (hsi1 ; il ). In particular,
$M+t\vec{v}$, $t\in\mathbb{R}$, satisfies that fixed $t$, the normal component
of the velocity vector $\vec{v}$ at each point is equal to the mean curvature
at that point. In non-parametric way $z=u(x,y)$, Equation (1) is

$(1+u_{y}^{2})u_{xx}-2u_{x}u_{y}u_{xy}+(1+u_{x}^{2})u_{yy}=2(1+u_{x}^{2}+u_{y}^{2})(-v_{1}u_{x}-v_{2}u_{y}+v_{3}),$
(2)

where the subindices indicate the corresponding partial differentiation and
$\vec{v}=(v_{1},v_{2},v_{3})$. This equation is a quasilinear elliptic PDE, so
the solvability is not assured. Some results of the solvability of the
Dirichlet problem can be proved assuming convexity in the initial data (Lopez5
). A way to reduce the complexity of (2) is assuming some type of symmetry on
the surface which makes that (2) converts into an ordinary differential
equation, where classical theory ensures local existence of solutions.
Following this strategy, we can assume that the surface is invariant under a
uniparametric group of translations (cylindrical surfaces) or rotations
(surfaces of revolution). Both families of surfaces are classified and play a
remarkable role in the theory of translating solitons. We now describe both
examples. Let $(x,y,z)$ be the canonical coordinates of $\mathbb{R}^{3}$.

  1. 1.

Cylindrical surfaces. The translating solitons are planes parallel to the
velocity vector $\vec{v}$ if the rulings are parallel to $\vec{v}$ or grim
reapers otherwise. See a detailed discussion in Lopez5 . For grim reapers, we
can assume without loss of generality that $\vec{v}=(0,0,1)$. Let $w$ be the
direction of the rulings. After a rotation about the $z$-axis, let
$w=\cos\theta e_{1}+\sin\theta e_{3}$, where $\\{e_{1},e_{2},e_{3}\\}$ is the
canonical basis of $\mathbb{R}^{3}$, $e_{3}=\vec{v}$ and $\cos\theta\not=0$.
The generating curve is included in the plane spanned by $\\{e_{2},e\\}$,
where $e=-\sin\theta e_{1}+\cos\theta e_{3}$. If we write this curve as
$\beta(s)=se_{2}+u(s)e$, then $u$ satisfies
$u^{\prime\prime}=\cos\theta(1+u^{\prime 2})$. This equation can be completely
integrated. For example, if $\theta=0$, $u(s)=-\log(\cos(s+a))+b$,
$a,b\in\mathbb{R}$. We point out that if a translating soliton is a ruled
surface, then it must be cylindrical, hence a plane or a grim reaper (Hieu ).

  2. 2.

Surfaces of revolution. The rotation axis is not arbitrary and must be
parallel to the velocity vector $\vec{v}$. There are two types of rotational
translating solitons depending on whether or not the surface meets the
rotation axis (aw ; css ). In the first case, the surface is known in the
literature as the bowl soliton and in the second one, the surfaces have
winglike shape.

Another way to address Equation (2) is by the technique of separation of
variables. We have two possibilities, $u(x,y)=f(x)+g(y)$ and
$u(x,y)=f(x)g(y)$, where $f$ and $g$ are smooth functions of one variable. In
both cases, Equation (2) is an ODE where the unknowns are the functions
$f=f(x)$ and $g=g(y)$. If $z=f(x)+g(y)$, the translating soliton equation (2)
is now

$(1+g^{\prime 2})f^{\prime\prime}+(1+f^{\prime
2})g^{\prime\prime}=(1+f^{\prime 2}+g^{\prime
2})(-v_{1}f^{\prime}-v_{2}g^{\prime}+v_{3}),$ (3)

where ′ indicates the derivative with respect to the corresponding variable.
In Lopez3 , the second author proved that if $\vec{v}=(0,0,1)$, grim reapers
are the only solutions of (3). Let us observe that the planes parallel to
$\vec{v}$ are not graphs on the $xy$-plane. A surface that is the graph of
$z=f(x)+g(y)$ can be expressed as the sum of two planar curves
$\alpha(x)+\beta(y)$, where $\alpha(x)=(x,0,f(x))$ and $\beta(y)=(0,y,g(y))$.
Let us observe that both curves are contained in orthogonal planes. More
generally, a surface is said to be a translation surface if it is the sum of
two curves called generating curves. The name of translation surface is due to
the surface can be viewed from the kinematic viewpoint as the translation of
the curve $\alpha$ (of $\beta$) by means of translations through $\beta$ (or
$\alpha$, respectively). Thus the result in Lopez3  is only a partial answer
to the following

> Problem 1. Classify all translating solitons that are translation surfaces.

This problem has its analogy in the classical theory of minimal surfaces of
$\mathbb{R}^{3}$. Scherk proved that besides the planes, the only minimal
surface that can be expressed as $z=f(x)+g(y)$ is

$u(x,y)=\frac{1}{c}\log\left|\frac{\cos(cy)}{\cos(cx)}\right|,$

where $c\not=0$ (sc ). More recently, Dillen et al. proved that if one of the
generating curves of a minimal surface of translation type is planar, then the
other generating curve is also planar (Dillen ) and the surface belongs to a
family of minimal surfaces discovered by Scherk (ni ). Surprisingly, very
recently the second author together Hasanis and Perdomo discovered many
minimal surfaces of translation type where both generating curves are not
planar (lh ; lp ).

In this paper we follow the same approach for translating solitons. However,
the presence of the vector $\vec{v}$ in Equation (1) makes a great difference
because $\vec{v}$ is an arbitrary vector in relation with the spatial
coordinates $(x,y,z)$ of $\mathbb{R}^{3}$. We give a partial answer to Problem
1 assuming that one of the generating curves is planar and proving that the
surface is a plane or a grim reaper (Theorem 2.2). As a previous step, we
prove this result in case that the two generating curves are planar curves but
not necessarily contained in orthogonal planes (Theorem 2.1). Both results are
analogous to the minimal surfaces obtained in Dillen . The goal of all both
theorems is that we do not presuppose any relation between the velocity vector
$\vec{v}$ and the surface. More precisely, the notion of translation surface
is affine because we use the sum of vectors of $\mathbb{R}^{3}$. However the
velocity vector $\vec{v}$ in the translating soliton equation (1) is assumed
in all its generality without any relation with the coordinates of
$\mathbb{R}^{3}$. This should be pointed out because one may be tempted to fix
$\vec{v}$ since Equation (1) is invariant after a rigid motion. However, such
rigid motion also changes the spatial coordinates of $\mathbb{R}^{3}$. This
seems to be subtle, but if one assumes that the surface is $z=f(x)+g(y)$, then
the vector $\vec{v}$ must be arbitrary. All this complicates the
demonstrations, which are not straightforward.

The second case of separation of variables that we investigate for the
translating soliton equation is $z=f(x)g(y)$. Then (2) is

$(1+f^{2}g^{\prime 2})gf^{\prime\prime}-2fgf^{\prime 2}g^{\prime
2}+(1+g^{2}f^{\prime 2})fg^{\prime\prime}-2(1+f^{\prime 2}g^{2}+f^{2}g^{\prime
2})(-v_{1}f^{\prime}g-v_{2}fg^{\prime}+v_{3})=0.$ (4)

Let us observe the symmetry of (4) in terms of $f$ or $g$, hence any
discussion on one of both functions also holds for the other one. As far as
the authors know, the first approach to this kind of surfaces in relation to
the study of the curvature of surfaces appeared in wo1 ; wo2 , where the
authors coined this type of surfaces as homothetical surfaces (see also GV ;
LM ). We have the analogous question.

> Problem 2. Classify all translating solitons of homothetical type.

It was proved in Lopez3 , and in the particular case $\vec{v}=(0,0,1)$, that
the only homothetical translating solitons are grim reapers. Grim reapers
appear when one of the functions $f$ or $g$ are constant. Indeed, if say
$f(x)=a$, $a\in\mathbb{R}$, then the parametrization of the surface is
$X(x,y)=(x,y,ag(y))$ deducing that the surface is cylindrical and the rulings
are parallel to the vector $e_{1}$ of the canonical basis. In contrast to
Equation (3), now Equation (4) is more difficult to work. The result that we
prove is assuming that $\vec{v}$ is one of the canonical directions of
$\mathbb{R}^{3}$ and proving that the surface is a plane or a grim reaper
(Theorem 3.1). Again we can make the same observation as before and although
this seems elementary analysis and would yield no non-trivial solutions
besides cylindrical surfaces, one can expect the existence of new examples.
For instance, in the family of minimal surfaces, the plane and the helicoid
(which is not cylindrical but ruled) are the only homothetical surfaces (wo1
). But if one replaces $z=f(x)g(y)$ by $h(z)=f(x)g(y)$, then there are many
minimal surfaces (ni ).

Finally in Section 4 we extend all the above results for translating solitons
in Lorentz-Minkowski space $\mathbb{R}_{1}^{3}$. Since the underlying affine
space for $\mathbb{R}_{1}^{3}$ coincides with the Euclidean space, the
concepts of translation surfaces and homothetical surfaces are equally valid
in the Lorentzian setting. The results are analogous to that of Euclidean
space.

##  2 Translating solitons of translation type

Consider a translation surface where the generating curves are planar curves.
If the planes containing the generating curves are orthogonal and $\vec{v}$ is
parallel to both planes, the second author proved that the only translating
solitons are grim reapers whose rulings are parallel to one of the above
planes (Lopez5 ). We now investigate Problem 1 in case that $\vec{v}$ is
arbitrary and the generating curves are planar but not necessarily containing
in orthogonal planes.

###### Theorem 2.1

Planes and grim reapers are the only translating solitons that are the sum of
two planar curves.

###### Proof

If the planes containing the curves are parallel then the sum of the two
curves is (part of) a plane. Suppose now that both planes are not parallel.
After renaming coordinates, we will assume that the $z$-axis is the common
straight line of the two planes, one of the generating curves is included in
the plane of equation $x=0$ and the other in the plane $cx+y=0$,
$c\in\mathbb{R}$. The cosine of angle between the two planes is
$c/\sqrt{1+c^{2}}$ and if $c=0$ then both planes become perpendicular. The
first curve parametrizes as $\beta(y)=(0,y,g(y))$ and the second one by
$\alpha(x)=(x,-cx,f(x))$, where $f$ and $g$ are two smooth functions defined
in intervals $I$ and $J$, respectively. Thus a parametrization of the surface
is

$X(x,y)=\alpha(x)+\beta(y)=(x,y-cx,f(x)+g(y)).$

Notice that if we name $\tilde{y}=y-cx$, then the surface is
$z=f(x)+g(\tilde{y}+cx)$. These surfaces are known in the literature as affine
translation surfaces (Liu2 ).

In case that $f$ or $g$ is a linear function, then the surface is cylindrical
and the surface must be a plane or a grim reaper, proving the result. Now we
discard this case. Then there are $x_{0}\in I$ and $y_{0}\in J$ such that
$f^{\prime\prime}(x_{0})\not=0$ and $g^{\prime\prime}(y_{0})\not=0$. Then
$f^{\prime\prime}\not=0$ and $g^{\prime\prime}\not=0$ in some subintervals
around $x_{0}$ and $y_{0}$ respectively, which we can assume to be $I$ and
$J$. In both intervals, there are points where $f^{\prime}\not=0$ and
$g^{\prime}\not=0$, otherwise $f$ or $g$ would be constant functions. Abusing
of notation, suppose $f^{\prime}f^{\prime\prime}(x_{0})\not=0$ and
$g^{\prime}g^{\prime\prime}(y_{0})\not=0$ and analogously,
$f^{\prime}f^{\prime\prime}\not=0$ in $I$ and
$g^{\prime}g^{\prime\prime}\not=0$ in $J$. If $\vec{v}=(v_{1},v_{2},v_{3})$,
Equation (2) writes as

$\left(1+g^{\prime 2}\right)f^{\prime\prime}+\left(1+c^{2}+f^{\prime
2}\right)g^{\prime\prime}=2\left(-v_{1}\left(f^{\prime}+cg^{\prime}\right)-v_{2}g^{\prime}+v_{3}\right)\left(1+g^{\prime}{}^{2}+\left(f^{\prime}+cg^{\prime}\right)^{2}\right).$

Divide by $(1+g^{\prime 2})(1+c^{2}+f^{\prime 2})$,

$\frac{f^{\prime\prime}}{1+c^{2}+f^{\prime
2}}+\frac{g^{\prime\prime}}{1+g^{\prime
2}}=2\left(-v_{1}\left(f^{\prime}+cg^{\prime}\right)-v_{2}g^{\prime}+v_{3}\right)\frac{1+g^{\prime}{}^{2}+\left(f^{\prime}+cg^{\prime}\right)^{2}}{(1+g^{\prime
2})(1+c^{2}+f^{\prime 2})}.$

Because the left hand side is the sum of a function on the variable $x$ and a
function on the variable $y$, when we differentiate with respect to $x$ and
next with respect to $y$, these terms are zero. The corresponding
differentiations on the right hand side give the expression

$\left(\sum_{n=0}^{4}P_{n}(y){f^{\prime}}^{n}\right)\frac{f^{\prime\prime}g^{\prime\prime}}{(1+g^{\prime
2})^{2}(1+c^{2}+f^{\prime 2})^{2}}=0,$

where $P_{n}$ are functions on the variable $y$. Thus
$\sum_{n=0}^{4}P_{n}(y)f^{\prime}(x)^{n}=0$ in $I\times J$. Since this is a
polynomial of the function $f^{\prime}=f^{\prime}(x)$, all coefficients
$P_{n}$ must vanish in $J$. The computation of $P_{4}$ yields
$P_{4}=-v_{1}g^{\prime}$, deducing $v_{1}=0$ because $g^{\prime}\not=0$.
Taking into account that $v_{1}=0$, the computation of $P_{2}$ gives

$P_{2}=c(-v_{3}{g^{\prime}}^{2}-2v_{2}g^{\prime}+v_{3}).$

We discuss two cases:

  1. 1.

Case $c=0$. Then all $P_{n}$ are trivially $0$ except $P_{1}$, which is
$P_{1}=-g^{\prime}(v_{2}g^{\prime 3}+3v_{2}g^{\prime}-2v_{3})$. From $P_{1}=0$
and because $g^{\prime}\not=0$, we have $v_{2}g^{\prime
3}+3v_{2}g^{\prime}-2v_{3}=0$. Since $g^{\prime\prime}\not=0$, the functions
$\\{1,g^{\prime},g^{\prime 3}\\}$ are linearly independent, concluding
$v_{2}=v_{3}=0$, so $\vec{v}=0$ obtaining a contradiction.

  2. 2.

Case $c\not=0$. Then $-v_{3}g^{\prime 2}-2v_{2}g^{\prime}+v_{3}=0$. Thus
$v_{2}=v_{3}=0$ again, which is contradictory.

We point out that in yoon  the authors obtained a partial result of Theorem
2.1 in case that $\vec{v}$ is one vector of the canonical basis.

Our next progress in Problem 1 is considering that one of the generating
curves is non-planar.

###### Theorem 2.2

Planes and grim reapers are the only translating solitons that are the sum of
two curves and where one of the generating curves is planar.

###### Proof

Suppose that the surface is parametrized by $X(s,t)=\alpha(s)+\beta(t)$, where
$\beta$ is a planar curve. Without loss of generality, we assume that $\beta$
is contained in the plane $\Pi$ of equation $x=0$ and that $\beta$
parametrizes as $\beta(y)=(0,y,g(y))$, where $g$ is a smooth function defined
in an interval $J$. The proof of theorem is by contradiction so by Theorem
2.1, we suppose that the curve $\alpha$ is not planar. Since $\alpha$ is a
space curve, then $\alpha$ is a graph on one of the coordinates axes. We can
assume that this axis is the $x$-axis because otherwise, the curve $\alpha$
would be contained in a plane parallel to $\Pi$ and the sum of $\alpha$ and
$\beta$ would be (part of) a plane. Definitively, $\alpha$ can be expressed as
$\alpha(x)=(x,f(x),h(x))$, where $f$ and $h$ are two smooth functions defined
in an interval $I\subset\mathbb{R}$. If we parametrize the surface by
$X(x,y)=(x,y+f(x),h(x)+g(y))$, the unit normal vector field is

$N=\frac{1}{\sqrt{1+g^{\prime
2}+\left(f^{\prime}g^{\prime}-h^{\prime}\right)^{2}}}\left(f^{\prime}g^{\prime}-h^{\prime},-g^{\prime},1\right)$

and the mean curvature $H$ is

$H=\frac{\left(h^{\prime\prime}-f^{\prime\prime}g^{\prime}\right)\left(1+g^{\prime
2}\right)+\left(1+f^{\prime 2}+h^{\prime
2}\right)g^{\prime\prime}}{2\left(1+g^{\prime
2}+\left(f^{\prime}g^{\prime}-h^{\prime}\right)^{2}\right)^{3/2}}.$

Let $\vec{v}=(v_{1},v_{2},v_{3})$. The translating soliton equation (2) is

$\left(h^{\prime\prime}-f^{\prime\prime}g^{\prime}\right)\left(1+g^{\prime
2}\right)+\left(1+f^{\prime 2}+h^{\prime
2}\right)g^{\prime\prime}=2\left(v_{1}(f^{\prime}g^{\prime}-h^{\prime})-v_{2}g^{\prime}+v_{3}\right)\left(1+g^{\prime
2}+\left(f^{\prime}g^{\prime}-h^{\prime}\right)^{2}\right).$ (5)

If $f$ or $h$ are linear functions, then the generating curve $\alpha$ is
planar, which is not possible. Therefore, with a similar argument as in the
beginning of the proof of Theorem 2.1, we can assume that in some subintervals
of $I$ and $J$, we have
$f^{\prime}f^{\prime\prime}h^{\prime}h^{\prime\prime}\not=0$ and
$g^{\prime}g^{\prime\prime}\not=0$. Without loss of generality we will assume
that these subintervals are $I$ and $J$ again. Our arguments will use the next
two claims

Claim 1. If there are $a,b,c\in\mathbb{R}$ such that
$a+bf^{\prime}(x)^{2}+ch^{\prime}(x)^{2}=0$ for all $x\in I$, then either
$abc\not=0$ or $a=b=c=0$. In the first case, we conclude that
$h^{\prime}(x)=\pm\sqrt{m_{0}+m_{1}f^{\prime}(x)^{2}}$, where
$m_{0},m_{1}\not=0$, $m_{0},m_{1}\in\mathbb{R}$.

The proof of the claim is as follows. According to the value of the constant
$b$, we have two cases. If $b=0$, then $a+ch^{\prime 2}=0$. In case that
$c=0$, then $a=0$ and the claim is proved. If $c\not=0$, we deduce that
$h^{\prime}h^{\prime\prime}=0$, which is not possible. The other case is
$b\not=0$. With a similar argument, we deduce $c\not=0$. If $a=0$, then
$bf^{\prime}(x)^{2}+ch^{\prime}(x)^{2}=0$ for all $x\in I$, in particular,
$bc<0$. Then $h^{\prime}(x)=\pm\sqrt{-b/c}f^{\prime}(x)$ so
$h(x)=\pm\sqrt{-b/c}f(x)+m$, $m\in\mathbb{R}$. Thus
$\alpha(x)=(x,f(x),\pm\sqrt{-b/c}f(x)+m)$ concluding that $\alpha$ is planar,
which is contradictory. Thus $a\not=0$. Hence $h^{\prime 2}=-a/c-b/cf^{\prime
2}$ and the result follows by taking $m_{0}=-a/c$ and $m_{1}=-b/c$.

Claim 2. Suppose $h^{\prime}=\pm\sqrt{m_{0}+m_{1}f^{\prime 2}}$, where
$m_{0},m_{1}\not=0$. Then the functions $\left\\{1,f^{\prime
2},f^{\prime}h^{\prime}\right\\}$ are linearly independent.

The proof is the following. Since $f^{\prime\prime}\neq 0$, let us introduce
$s=f^{\prime}$. Then the Wronskian of the set $\left\\{1,s^{2},\pm
s\sqrt{m_{0}+m_{1}s^{2}}\right\\}$ is
$\mp\frac{2m_{0}^{2}}{\left(m_{0}+m_{1}s^{2}\right)^{3/2}},$ and this proves
the claim.

We come back to the proof of theorem. Dividing (5) by $Q=1+f^{\prime
2}+h^{\prime 2}$ and differentiating with respect to $x$, we obtain a
polynomial equation on $g^{\prime}$

$\sum_{n=0}^{3}P_{n}(x)g^{\prime n}=0,$

where

$\begin{split}P_{0}(x)&=\left(\frac{h^{\prime\prime}-2(v_{3}-v_{1}h^{\prime})(1+h^{\prime
2})}{Q}\right)^{\prime}\\\
P_{1}(x)&=\left(\frac{-f^{\prime\prime}-2f^{\prime}\left(3v_{1}h^{\prime
2}-2v_{3}h^{\prime}+v_{1}\right)+2v_{2}h^{\prime
2}+2v_{2}}{Q}\right)^{\prime}\\\
P_{2}(x)&=\left(\frac{h^{\prime\prime}-2f^{\prime
2}\left(v_{3}-3v_{1}h^{\prime}\right)-4v_{2}f^{\prime}h^{\prime}+2v_{1}h^{\prime}-2v_{3}}{Q}\right)^{\prime}\\\
P_{3}(x)&=\left(\frac{-f^{\prime\prime}+2(v_{2}-v_{1}f^{\prime})(1+f^{\prime
2})}{Q}\right)^{\prime}.\end{split}$

Thus there are real constants $p_{n}\in\mathbb{R}$, $0\leq n\leq 3$, such that

$\begin{split}h^{\prime\prime}-2(v_{3}-v_{1}h^{\prime})(1+h^{\prime
2})&=p_{0}Q\\\ -f^{\prime\prime}-2f^{\prime}\left(3v_{1}h^{\prime
2}-2v_{3}h^{\prime}+v_{1}\right)+2v_{2}h^{\prime 2}+2v_{2}&=p_{1}Q\\\
h^{\prime\prime}-2f^{\prime
2}\left(v_{3}-3v_{1}h^{\prime}\right)-4v_{2}f^{\prime}h^{\prime}+2v_{1}h^{\prime}-2v_{3}&=p_{2}Q\\\
-f^{\prime\prime}+2(v_{2}-v_{1}f^{\prime})(1+f^{\prime
2})&=p_{3}Q.\end{split}$ (6)

In order to simplify the notation, set

$c_{1}=p_{0}-p_{2},\quad c_{2}=p_{1}-p_{3}.$

We distinguish two cases.

  1. 1.

Case $v_{3}=0$. There are two subcases.

    1. (a)

Subcase $v_{1}=0$. After a dilation of $\mathbb{R}^{3}$, we can assume
$\vec{v}=(0,1,0)$. Equations (6) are

$\displaystyle h^{\prime\prime}$ $\displaystyle=$ $\displaystyle p_{0}Q,$ (7)
$\displaystyle-f^{\prime\prime}+2(1+h^{\prime 2})$ $\displaystyle=$
$\displaystyle p_{1}Q,$ (8) $\displaystyle
h^{\prime\prime}-4f^{\prime}h^{\prime}$ $\displaystyle=$ $\displaystyle
p_{2}Q,$ (9) $\displaystyle-f^{\prime\prime}+2(1+f^{\prime 2})$
$\displaystyle=$ $\displaystyle p_{3}Q.$ (10)

Combining (8) and (10) and using the value of $Q$, we have

$c_{1}+(c_{1}+2)f^{\prime 2}+(c_{1}-2)h^{\prime 2}=0.$

Because the three coefficients are distinct from $0$, Claim 1 implies that
$h^{\prime}=\pm\sqrt{m_{0}+m_{1}f^{\prime 2}}$, where $m_{0},m_{1}\not=0$.
Using now (7) and (9), $4f^{\prime}h^{\prime}=c_{1}Q$, or equivalently,

$c_{2}+m_{0}c_{2}+(c_{2}+c_{2}m_{1})f^{\prime 2}\mp 4f^{\prime}h^{\prime}=0.$

From Claim 2, the functions $\\{1,f^{\prime 2},f^{\prime}h^{\prime}\\}$ are
linearly independent, hence the coefficients must vanish, obtaining a
contradiction.

    2. (b)

Subcase $v_{1}\not=0$. Since $\vec{v}=(v_{1},v_{2},0)$, after a dilation of
$\mathbb{R}^{3}$, we can assume that $\vec{v}=(1,v_{2},0)$. Now (6) is

$\displaystyle h^{\prime\prime}+2h^{\prime}(1+h^{\prime 2})$ $\displaystyle=$
$\displaystyle p_{0}Q,$ (11)
$\displaystyle-f^{\prime\prime}+2v_{2}(1+h^{\prime
2})-2f^{\prime}(1+3h^{\prime 2})$ $\displaystyle=$ $\displaystyle p_{1}Q,$
(12) $\displaystyle h^{\prime\prime}+(2-4v_{2}f^{\prime}+6f^{\prime
2})h^{\prime}$ $\displaystyle=$ $\displaystyle p_{2}Q,$ (13)
$\displaystyle-f^{\prime\prime}+2(v_{2}-f^{\prime})(1+f^{\prime 2})$
$\displaystyle=$ $\displaystyle p_{3}Q.$ (14)

Combining (11) and (13),

$2h^{\prime}(2v_{2}f^{\prime}-3f^{\prime 2}+h^{\prime
2})=c_{1}Q=c_{1}(1+f^{\prime 2}+h^{\prime 2}).$ (15)

Hence we can get the expression

$h^{\prime 2}=\frac{-c_{1}(1+f^{\prime
2})+4v_{2}f^{\prime}h^{\prime}-6h^{\prime}f^{\prime 2}}{c_{1}-2h^{\prime}}.$

From (12) and (14), we have

$-2v_{2}f^{\prime 2}+2f^{\prime 3}+(2v_{2}-6f^{\prime})h^{\prime
2}=c_{2}Q=c_{2}(1+f^{\prime 2}+h^{\prime 2}).$

Substituting the above value of $h^{\prime 2}$,

$(-4c_{1}f^{\prime 3}+2c_{1}v_{2}f^{\prime
2}-3c_{1}f^{\prime}+c_{1}v_{2})+h^{\prime}(2c_{2}v_{2}f^{\prime}-4c_{2}f^{\prime
2}-c_{2}-4v_{2}^{2}f^{\prime}+16v_{2}f^{\prime 2}-16f^{\prime 3})=0.$

In this polynomial equation on $h^{\prime}$ of degree $\leq 1$, if the
coefficient of $h^{\prime}$ is $0$, then $-4c_{1}f^{\prime
3}+2c_{1}v_{2}f^{\prime 2}-3c_{1}f^{\prime}+c_{1}v_{2}=0$. This is a
polynomial on $f^{\prime}$ and the leading coefficient of $f^{\prime 3}$ is
not $0$, which is not possible. Thus the coefficient of $h^{\prime}$ is not
$0$, obtaining

$h^{\prime}=-\frac{2c_{1}v_{2}f^{\prime 2}-4c_{1}f^{\prime
3}-3c_{1}f^{\prime}+c_{1}v_{2}}{2c_{2}v_{2}f^{\prime}-4c_{2}f^{\prime
2}-c_{2}-4v_{2}^{2}f^{\prime}+16v_{2}f^{\prime 2}-16f^{\prime 3}}.$

Substituting into (15), and after some manipulations, we have an expression of
type

$\sum_{n=0}^{9}A_{n}f^{\prime}(x)^{n}=0,$

where $A_{n}$ are real constants. Thus all coefficients $A_{n}$ must vanish.
However, the computation of $A_{9}$ gives $A_{9}=-512$. This completes the
proof for the case $v_{3}=0$.

  2. 2.

Case $v_{3}\not=0$. After a dilation, we suppose $\vec{v}=(v_{1},v_{2},1)$. We
distinguish four subcases.

    1. (a)

Subcase $v_{1}=v_{2}=0$. Then (6) is

$\displaystyle h^{\prime\prime}-2\left(1+h^{\prime 2}\right)$ $\displaystyle=$
$\displaystyle p_{0}Q$ (16)
$\displaystyle-f^{\prime\prime}+4f^{\prime}h^{\prime}$ $\displaystyle=$
$\displaystyle p_{1}Q$ (17) $\displaystyle h^{\prime\prime}-2f^{\prime 2}-2$
$\displaystyle=$ $\displaystyle p_{2}Q$ (18) $\displaystyle-f^{\prime\prime}$
$\displaystyle=$ $\displaystyle p_{3}Q.$ (19)

We deduce from (16) and (18) that $c_{1}Q=2f^{\prime 2}-2h^{\prime 2}$ or
equivalently, by $Q=1+f^{\prime 2}+h^{\prime 2}$,

$c_{1}+\left(c_{1}-2\right)f^{\prime 2}+\left(c_{1}+2\right)h^{\prime 2}=0.$

Clearly the coefficients are nonzero and Claim 1 implies
$h^{\prime}=\pm\sqrt{m_{0}+m_{1}f^{\prime 2}}$, $m_{0},m_{1}\not=0$. Moreover
from (17) and (19), we deduce $c_{2}Q=4f^{\prime}h^{\prime},$ and substituting
$h^{\prime 2}=m_{0}+m_{1}f^{\prime 2}$,

$c_{2}\left(1+m_{0}\right)+c_{2}\left(1+m_{1}\right)f^{\prime 2}\mp
4f^{\prime}h^{\prime}=0,$

which gives a contradiction from Claim 2.

    2. (b)

Subcase $v_{1}=0$ and $v_{2}\neq 0$. Then (6) is

$\displaystyle h^{\prime\prime}-2\left(1+h^{\prime 2}\right)$ $\displaystyle=$
$\displaystyle p_{0}Q$ (20)
$\displaystyle-f^{\prime\prime}+4f^{\prime}h^{\prime}+2v_{2}(1+h^{\prime 2})$
$\displaystyle=$ $\displaystyle p_{1}Q$ (21) $\displaystyle
h^{\prime\prime}-2f^{\prime 2}-4v_{2}f^{\prime}h^{\prime}-2$ $\displaystyle=$
$\displaystyle p_{2}Q$ (22)
$\displaystyle-f^{\prime\prime}+2v_{2}\left(1+f^{\prime 2}\right)$
$\displaystyle=$ $\displaystyle p_{3}Q.$ (23)

It follows from (20) and (22) that

$c_{1}Q=2f^{\prime 2}-2h^{\prime 2}+4v_{2}f^{\prime}h^{\prime}.$ (24)

Similarly from (21) and (23),

$c_{2}Q=4f^{\prime}h^{\prime}+2v_{2}(h^{\prime 2}-f^{\prime 2}).$ (25)

Combining (24) and (25), we deduce

$2(1+v_{2}^{2})(f^{\prime 2}-h^{\prime
2})=\left(c_{1}-c_{2}v_{2}\right)Q=\left(c_{1}-c_{2}v_{2}\right)(1+f^{\prime
2}+h^{\prime 2}).$

Let $c_{3}=p_{0}-p_{2}-v_{2}\left(p_{1}-p_{3}\right)$. This equation writes as

$c_{3}+(c_{3}-2(1+v_{2}^{2}))f^{\prime 2}+(c_{3}+2(1+v_{2}^{2}))h^{\prime
2}=0,$

where the coefficients of $\\{1,f^{\prime 2},h^{\prime 2}\\}$ are clearly
nonzero. Then Claim 1 implies $h^{\prime}=\pm\sqrt{m_{0}+m_{1}f^{\prime 2}}$,
$m_{0},m_{1}\not=0$. Coming back to (24), we derive

$4v_{2}f^{\prime}h^{\prime}=c_{1}\left(1+m_{0}\right)+2m_{0}+\left(c_{1}\left(1+m_{1}\right)+2m_{1}-2\right)f^{\prime
2}.$

Claim 2 concludes that this subcase is false because $v_{2}\not=0$

    3. (c)

Subcase $v_{1}\neq 0$ and $v_{2}=0$. Then (6) is

$\displaystyle
h^{\prime\prime}-2\left(1-v_{1}h^{\prime}\right)\left(1+h^{\prime 2}\right)$
$\displaystyle=$ $\displaystyle p_{0}Q$ (26)
$\displaystyle-f^{\prime\prime}-2f^{\prime}\left(3v_{1}h^{\prime
2}-2h^{\prime}+v_{1}\right)$ $\displaystyle=$ $\displaystyle p_{1}Q$ (27)
$\displaystyle h^{\prime\prime}-2f^{\prime
2}\left(1-3v_{1}h^{\prime}\right)+2\left(v_{1}h^{\prime}-1\right)$
$\displaystyle=$ $\displaystyle p_{2}Q$ (28)
$\displaystyle-f^{\prime\prime}-2v_{1}f^{\prime}\left(1+f^{\prime 2}\right)$
$\displaystyle=$ $\displaystyle p_{3}Q.$ (29)

From (26) and (28) we derive $f^{\prime 2}=A/B$, where

$A=\left(2v_{1}h^{\prime}-c_{1}-2\right)h^{\prime 2}-c_{1},\quad
B=c_{1}-2\left(1-3v_{1}h^{\prime}\right).$

Let us observe that $B\not=0$ because $h^{\prime\prime}\not=0$ and
$v_{1}\not=0$. Similarly, from (27) and (29), we have

$2f^{\prime}(v_{1}f^{\prime 2}+2h^{\prime 2}-3v_{1}h^{\prime
2})=c_{2}Q=c_{2}\left(1+f^{\prime 2}+h^{\prime 2}\right).$

Substituting $f^{\prime}$ by $\pm\sqrt{A/B}$,

$c_{2}\sqrt{B}\left(A+B\left(1+h^{\prime 2}\right)\right)=\pm
2\sqrt{A}(Av_{1}+Bh^{\prime}(2-3v_{1}h^{\prime})).$

After squaring both sides, we obtain

$\sum_{n=0}^{9}A_{n}h^{\prime}(x)^{n}=0,$

where $A_{n}$ are real constants. Being $A_{9}=2^{11}v_{1}^{5}\neq 0$, we
arrive to a contradiction.

    4. (d)

Subcase $v_{1}v_{2}\neq 0$. Then (6) writes

$\displaystyle
h^{\prime\prime}-2\left(1-v_{1}h^{\prime}\right)\left(1+h^{\prime 2}\right)$
$\displaystyle=$ $\displaystyle p_{0}Q$ (30)
$\displaystyle-f^{\prime\prime}-2f^{\prime}\left(3v_{1}h^{\prime
2}-2h^{\prime}+v_{1}\right)+2v_{2}\left(1+h^{\prime 2}\right)$
$\displaystyle=$ $\displaystyle p_{1}Q$ (31) $\displaystyle
h^{\prime\prime}-2f^{\prime
2}\left(1-3v_{1}h^{\prime}\right)-4v_{2}f^{\prime}h^{\prime}+2\left(v_{1}h^{\prime}-1\right)$
$\displaystyle=$ $\displaystyle p_{2}Q$ (32)
$\displaystyle-f^{\prime\prime}+2\left(v_{2}-v_{1}f^{\prime}\right)\left(1+f^{\prime
2}\right)$ $\displaystyle=$ $\displaystyle p_{3}Q.$ (33)

From (30) and (32),

$2\left(v_{1}h^{\prime}-1\right)h^{\prime 2}+2f^{\prime
2}\left(1-3v_{1}h^{\prime}\right)+4v_{2}f^{\prime}h^{\prime}=c_{1}Q=c_{1}\left(1+f^{\prime
2}+h^{\prime 2}\right),$

or equivalently

$Af^{\prime 2}+Bf^{\prime}+C=0,$

where

$\begin{split}A&=c_{1}-2+6v_{1}h^{\prime},\\\ B&=-4v_{2}h^{\prime},\\\
C&=c_{1}+\left(c_{1}+2-2v_{1}h^{\prime}\right)h^{\prime 2}.\end{split}$

Note that $A$ and $B$ cannot vanish because $v_{1}v_{2}\not=0$ and
$h^{\prime\prime}\neq 0$. Then

$f^{\prime}=\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}.$

On the other hand, (31) and (33) imply

$c_{2}Q+2\left(f^{\prime}(3v_{1}h^{\prime 2}-2h^{\prime})+f^{\prime
2}(v_{2}-v_{1}f^{\prime})-v_{2}h^{\prime 2}\right)=0.$ (34)

As in previous subcase, our purpose is to substitute the value of $f^{\prime}$
in order to obtain a polynomial on $h^{\prime}$. Let $D=B^{2}-4AC$. If we
write (34) again in terms of $\sqrt{D}$, then we have a polynomial equation of
type $a+b\sqrt{D}=0$, hence, $a^{2}-b^{2}D=0$. Substituting the value of $D$
as well as of $A$, $B$ and $C$, we obtain the desired polynomial equation on
$h^{\prime}$, namely,

$\sum_{n=0}^{12}A_{n}h^{\prime n}=0,$

where $A_{n}$ are real constants. Because $A_{12}=2^{16}3^{3}v_{1}^{8}\neq 0$,
we arrive to a contradiction, completing the proof of theorem.

##  3 Translating solitons of homothetical type

Let $u(x,y)=f(x)g(y)$ be a homothetical surface. Suppose that the surface is
also a translating soliton with respect to $\vec{v}$. There are three initial
cases that can be previously considered.

  1. 1.

Case that $f$ or $g$ is constant. Then the surface is ruled, so we know that
the surface is a grim reaper or a plane parallel to the vector $\vec{v}$.

  2. 2.

Case that $f$ (or $g$) is linear. Indeed, if $f(x)=ax+b$ with
$a,b\in\mathbb{R}$, $a\not=0$, then (4) is a polynomial equation
$\sum_{n=0}^{3}A_{n}(y)x^{n}=0$. In particular all coefficients $A_{n}$ must
vanish. The computation of $A_{3}$ yields $2a^{3}v_{2}g^{\prime 3}$. Since $g$
is not constant and $a\not=0$, we deduce $v_{2}=0$. Now the computation of
$A_{2}$ gives $A_{2}=2a^{3}(av_{1}g)g^{\prime 2}$. Hence, $v_{1}=v_{3}=0$,
obtaining a contradiction.

  3. 3.

Case that $f$ and $f^{\prime}$ (or $g$ and $g^{\prime}$) are linearly
dependent. Assume that $f^{\prime}=af,$ $a\in\mathbb{R}$, $a\not=0$. Then
$f^{\prime\prime}=a^{2}f$ and (4) is a polynomial equation on $f$ of degree
$3$, namely, $\sum_{n=0}^{3}A_{n}(y)f^{n}=0$. Then all coefficients $A_{n}$
must vanish. In particular, $A_{0}=-2v_{3}$, hence $v_{3}=0$. Now $A_{1}=0$
and $A_{3}$ lead to

$\displaystyle
A_{1}=a\left(a+2v_{1}\right)g+2v_{2}g^{\prime}+g^{\prime\prime}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle
A_{3}=\left(av_{1}g+v_{2}g^{\prime}\right)\left(ag^{2}+g^{\prime
2}\right)+a^{2}\left(g^{2}g^{\prime\prime}-gg^{\prime 2}\right)$
$\displaystyle=$ $\displaystyle 0.$

The linear combination $A_{1}-a^{2}gA_{3}=0$ writes
$-a^{4}g^{3}+(2av_{1}-a^{2})gg^{\prime 2}+2v_{2}g^{\prime 3}=0$, or
equivalently,

$-a^{3}+(2av_{1}-a^{2})\left(\frac{g^{\prime}}{g}\right)^{2}+2v_{2}\left(\frac{g^{\prime}}{g}\right)^{3}=0.$

Since all coefficients are not all zero, there is $b\in\mathbb{R}$ such that
$g^{\prime}/g=b$ with $b\not=0$. Taking into account that
$g^{\prime\prime}=b^{2}g$ and that $a^{2}+b^{2}\not=0$, then $A_{1}=0$ is

$a^{2}+(2(av_{1}+bv_{2})+b^{2})g=0,$

obtaining a contradiction.

After this discussion, we can assume that $f^{\prime}f^{\prime\prime}\not=0$
and $g^{\prime}g^{\prime\prime}\not=0$ in their domains and let us introduce
new variables. So, let $p=p(f)=f^{\prime}$, as well as, $q=q(g)=g^{\prime}$.
Then $p^{\prime}=f^{\prime\prime}/f^{\prime}$ and
$q^{\prime\prime}=g^{\prime\prime}/g^{\prime}$. Let us observe that
$pp^{\prime}qq^{\prime}\not=0$. Now the translating soliton equation (4) is

$(1+f^{2}q^{2})gpp^{\prime}-2fgp^{2}q^{2}+(1+g^{2}p^{2})fqq^{\prime}-2(1+p^{2}g^{2}+f^{2}q^{2})(-v_{1}pg-
v_{2}fq+v_{3})=0.$ (35)

We now give a partial result on Problem 2 in case that $\vec{v}$ is one of the
canonical basis of $\mathbb{R}^{3}$.

###### Theorem 3.1

Grim reapers are the only translating solitons of homothetical type when
$\vec{v}$ is one vector of the canonical basis.

###### Proof

We know that the case $\vec{v}=(0,0,1)$ was solved in Lopez3 . It remains the
case that $\vec{v}$ is $(1,0,0)$ or $(0,1,0)$. By the symmetry of Equation
(35) with respect to $v_{1}$ and $v_{2}$, it suffices to consider the case
that $\vec{v}=(0,1,0)$. Then

$(1+f^{2}q^{2})gpp^{\prime}-2fgp^{2}q^{2}+(1+g^{2}p^{2})fqq^{\prime}+2fq(1+p^{2}g^{2}+f^{2}q^{2})=0.$
(36)

We divide by $fgp^{2}q^{2}$,

$\frac{1}{q^{2}}\left(\frac{p^{\prime}}{fp}\right)+\frac{1}{p^{2}}\left(\frac{q^{\prime}}{gq}\right)+\frac{1}{q}\left(gq^{\prime}-q\right)+\frac{1}{p}\left(fp^{\prime}-p\right)+2\left(\frac{1}{p^{2}gq}+\frac{g}{q}+\frac{f^{2}q}{p^{2}g}\right)=0.$

Differentiating with respect to $f$ and $g$ successively, we obtain

$\left(\frac{1}{q^{2}}\right)^{\prime}\left(\frac{p^{\prime}}{fp}\right)^{\prime}+\left(\frac{1}{p^{2}}\right)^{\prime}\left(\frac{q^{\prime}}{gq}\right)^{\prime}+2\left[\left(\frac{1}{p^{2}}\right)^{\prime}\left(\frac{1}{gq}\right)^{\prime}+\left(\frac{f^{2}}{p^{2}}\right)^{\prime}\left(\frac{q}{g}\right)^{\prime}\right]=0.$

Notice that
$\left(\frac{1}{p^{2}}\right)^{\prime}\left(\frac{1}{q^{2}}\right)^{\prime}\neq
0$. Dividing by
$2\left(\frac{1}{p^{2}}\right)^{\prime}\left(\frac{1}{q^{2}}\right)^{\prime}$
and next differentiating with respect to $f$ and $g$ successively,

$\left(\dfrac{\left(\dfrac{f^{2}}{p^{2}}\right)^{\prime}}{\left(\dfrac{1}{p^{2}}\right)^{\prime}}\right)^{\prime}\left(\dfrac{\left(\dfrac{q}{g}\right)^{\prime}}{\left(\dfrac{1}{q^{2}}\right)^{\prime}}\right)^{\prime}=0.$
(37)

  1. 1.

Case

$\left(\frac{f^{2}}{p^{2}}\right)^{\prime}=a\left(\frac{1}{p^{2}}\right)^{\prime}$

for some $a\not=0$. Integrating we have $p^{2}=kf^{2}-ak$ for some $k\not=0$.
Differentiating with respect to $f$, we deduce $pp^{\prime}=kf$. Substituting
into (36), we have a polynomial equation $B_{1}(g)f+B_{3}(g)f^{3}=0$. Thus
$B_{1}=B_{3}=0$. The computation of these coefficients yield

$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-q\left(ag^{2}k-1\right)\left(q^{\prime}+2\right)+2agkq^{2}+gk,$
$\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle
q\left(g^{2}k\left(q^{\prime}+2\right)-gkq+2q^{2}\right).$

Equation $B_{3}=0$ can be solved explicitly. Suppose $k>0$ (an analogous
argument if $k<0$). Then

$q(g)=g\sqrt{k}\tan\left(m-\frac{2}{\sqrt{k}}\log(g)\right),\quad
m\in\mathbb{R}.$

Substituting into $B_{1}=0$, we conclude

$2\left(ag^{2}k-1\right)\tan^{3}\left(m-\frac{2\log(g)}{\sqrt{k}}\right)+\sqrt{k}\left(ag^{2}k+1\right)\tan^{2}\left(m-\frac{2\log(g)}{\sqrt{k}}\right)+\sqrt{k}=0,$

obtaining a contradiction.

  2. 2.

Case

$\left(\dfrac{q}{g}\right)^{\prime}=a\left(\dfrac{1}{q^{2}}\right)^{\prime}$

for some $a\not=0$. Integrating,

$g=\frac{q^{3}}{a+kq^{2}},$

for some constant $k\in\mathbb{R}$. Differentiating with respect to $g$,

$q^{\prime}=\frac{q^{2}\left(3a+kq^{2}\right)}{\left(a+kq^{2}\right)^{2}}.$

By substituting these values of $g$ and $q$ in (36), we obtain

$\frac{q}{(a+kq^{2})^{4}}\sum_{n=0}^{10}C_{n}(x)q^{n}=0.$

Then all coefficients $C_{n}$ must vanish. However, the computation of $C_{0}$
gives $C_{0}=2a^{4}f$ which is not possible.

##  4 Extension of the results to the Lorentzian setting

In this last section, we extend the results to the Lorentz-Minkowski 3-space
$\mathbb{R}_{1}^{3}$. Here $\mathbb{R}_{1}^{3}$ is the affine space
$\mathbb{R}^{3}$ endowed with the canonical Lorentzian metric
$dx^{2}+dy^{2}-dz^{2}$. Denote $\left\langle\cdot,\cdot\right\rangle_{L}$ and
$\times_{L}$ the Lorentzian inner and cross product, respectively.

We first consider a (non-planar) non-degenerate cylindrical surface
$X(s,t)=\alpha(s)+tw$ where $\alpha=\alpha(s)$ is parametrized by the arc
length $s$ and $w\in\mathbb{R}_{1}^{3}$. The unit normal vector $N$ is
parallel to $\alpha^{\prime}(s)\times_{L}w$ and hence (1) writes

$\left\langle
w,w\right\rangle_{L}\left\langle\alpha^{\prime}\times_{L}w,\alpha^{\prime\prime}\right\rangle_{L}=2\epsilon\left(\epsilon_{1}\left\langle
w,w\right\rangle_{L}-\langle\alpha^{\prime}(s),w\rangle_{L}^{2}\right)\left\langle\alpha^{\prime}(s)\times_{L}w,\vec{v}\right\rangle_{L},$
(38)

where $\epsilon$ is the sign of
$\left\langle\alpha^{\prime}(s)\times_{L}w,\alpha^{\prime}(s)\times_{L}w\right\rangle_{L}$
and
$\epsilon_{1}=\left\langle\alpha^{\prime}(s),\alpha^{\prime}(s)\right\rangle_{L}$.
In case that the rulings are lightlike, the surface is a translating soliton
if $\langle\alpha^{\prime}(s)\times_{L}w,\vec{v}\rangle_{L}=0$. In particular,
this equation holds if $\vec{v}$ is parallel to $w$ being $\alpha$ is an
arbitrary curve.

In all Lorentzian versions of the results, we will conclude that the surface
is a cylindrical surface. According to the causal character of the rulings,
the description of the translating solitons of $\mathbb{R}_{1}^{3}$ of
cylindrical type is the following (al ). After a rigid motion of
$\mathbb{R}_{1}^{3}$, we can fix $w$.

  1. 1.

Spacelike rulings. Let $w=(1,0,0)$. If $X(s,t)=(0,s,u(s))+tw$, then (1)

$u^{\prime\prime}=\left\\{\begin{array}[]{ll}2(1-u^{\prime
2})(v_{2}u^{\prime}-v_{3}),&1-u^{\prime 2}>0\\\ -2(1-u^{\prime
2})(v_{2}u^{\prime}-v_{3}),&1-u^{\prime 2}<0.\end{array}\right.$

For example, if $\vec{v}=(0,0,1)$, the rulings are orthogonal to $\vec{v}$ and
the integration of both equations give

$u(s)=\left\\{\begin{array}[]{ll}-\frac{1}{2}\log(\cosh(-2s+a))+b,&1-u^{\prime
2}>0\\\ \frac{1}{2}\log(\sinh(2s+a))+b,&1-u^{\prime 2}<0,\end{array}\right.$

where $a,b\in\mathbb{R}$. These two curves appeared in castro .

  2. 2.

Timelike rulings. Let $w=(0,0,1)$. If $X(s,t)=(s,u(s),0)+tw$, then the surface
is timelike and Equation (1) is

$u^{\prime\prime}=2(1+u^{\prime 2})(v_{1}u^{\prime}-v_{2}).$

If $\vec{v}=(0,1,0)$, the rulings are orthogonal to $\vec{v}$ and the solution
is $u(s)=\log(\cos(2s+a))/2+b$, $a,b\in\mathbb{R}$.

  3. 3.

Lightlike rulings. Then $H=0$, so the translating equation (1) is $\langle
N,\vec{v}\rangle=0$. Let $w=(1,0,1)$ and $X(s,t)=(u(s),s,-u(s))+tw$. The
surface is not degenerated if $u^{\prime}\not=0$. Then Equation (1) is

$v_{1}-2v_{2}u^{\prime}-v_{3}=0.$

If $v_{2}=0$, then $\vec{v}$ is parallel to $w$ and with arbitrary generating
curve. Otherwise the function $u$ is linear and $X(s,t)$ is a plane.

Summarizing, the cylindrical translating solitons in $\mathbb{R}_{1}^{3}$ are
planes (when the rulings are parallel to $\vec{v}$), Lorentzian grim reapers
and cylindrical surfaces whose rulings are lightlike and parallel to
$\vec{v}$.

As we have pointed out, the extensions of Theorems 2.1, 2.2 and 3.1 to the
Lorentzian setting are straightforward and the conclusion is that the surfaces
must be cylindrical surfaces.

###### Theorem 4.1

A translating soliton in $\mathbb{R}_{1}^{3}$ that is the sum of two curves
and where one of the generating curve is planar must be a cylindrical surface.

###### Theorem 4.2

A translating soliton in $\mathbb{R}_{1}^{3}$ of homothetical type when
$\vec{v}$ is one vector of the canonical basis must be a cylindrical surface.

## References

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## Declarations

Funding. Rafael López has been partially supported by the grant no.
MTM2017-89677-P, MINECO/ AEI/FEDER, UE.

Code availability. Not applicable.

Contributions. The authors equally conceived of the study, participated in its
design and coordination, drafted the manuscript, participated in the sequence
alignment, and read and approved the final manuscript.

Conflict of interest. The authors declare that there is no conflict of
interest.


[1]

1]organization=Hong Kong Space Museum, addressline=10 Salisbury Road, Tsim Sha
Tsui, Kowloon, city=Hong Kong, country=China

2]organization=Department of Physics, The Chinese University of Hong Kong,
addressline=Sha Tin, New Territories, city=Hong Kong, country=China

# Solar image quality assessment: a proof of concept using Variance of
Laplacian method and its application to optical atmospheric condition
monitoring

Chu Wing So<EMAIL_ADDRESS>Edwin Lok Hei Yuen<EMAIL_ADDRESS>Edgar
Heung Fat Leung<EMAIL_ADDRESS>Jason Chun Shing Pun<EMAIL_ADDRESS>[
[

###### Abstract

Here we present a proof of concept for the application of the Variance of
Laplacian (VL) method in quantifying the sharpness of optical solar images. We
conducted a comprehensive study using over 65,000 individual solar images
acquired on more than 160 days. Each image underwent processing using a VL
image processing algorithm, which assigns a ‘score’ based on the sharpness of
the solar disk’s edges. We studied the scores obtained from images acquired at
different conditions. Our findings demonstrate that the sharpness of the
images exhibits daily trends that are closely linked to the altitude of the
Sun at the observation site. We observed a significant degradation in image
quality only below a certain altitude threshold. Furthermore, we compared
airmass formulae from the literature with our sharpness observations and
concluded that the degradation could be modeled as an Image Sharpness Function
(ISF), which exhibits similarities to airmass variations. In addition to
assessing image quality, our method has the potential to evaluate the optical
atmospheric conditions during daytime observations. Moreover, this technique
can be easily and cost-effectively applied to archival or real-time images of
other celestial bodies, such as the Moon, bright planets and defocused stars.
Given that ISF is unique to each location and sensitive to sky conditions, the
development of an ISF is not only beneficial for routine observation
preparation but also essential for long-term site monitoring.

###### keywords:

Atmospheric effects The Sun Astronomy data analysis

##  1 Introduction

The quality of astronomical data has garnered significant attention due to its
impact on scientific discoveries. Higher-quality images with improved spatial
resolution have the ability to reveal finer details within celestial objects.
This has been exemplified in the resolved images of notable red giants such as
Betelgeuse (Burns et al., 1997) and W Hydrae (Vlemmings et al., 2021).
Additionally, the study of exoplanets has benefited from higher-quality
images, as demonstrated in the case of 2M1207b (Patience et al., 2010; Stone,
2012; Ricci et al., 2017).

In ground-based optical observations conducted under turbulent atmospheric
conditions, the quality of the acquired images can be significantly affected
by the attenuation of starlight due to scattering and absorption (Coulman,
1985). To mitigate the impact of atmospheric extinction, astronomers often
choose to observe their targets when the zenith angle is small, corresponding
to a lower airmass. For the same reason, observatories are typically
established in locations with dry or thin-air conditions to minimize
atmospheric interference (e.g. Giovanelli et al., 2001; Yoshii et al., 2016).
In parallel, remarkable progress has been made in developing techniques to
overcome the attenuation of starlight caused by the atmosphere. One
advancement is the implementation of adaptive optics (Davies and Kasper, 2012)
which enables real-time corrections for atmospheric distortions, resulting in
improved image quality and higher spatial resolution.

Astronomical seeing is a widely used metric for characterizing atmospheric
conditions during observations. It is commonly quantified by measuring the
stellar image’s Full Width at Half Maximum (FWHM), which represents the
angular size of the point spread function of a stellar image on the sensor
(Burns, 2022). A smaller FWHM corresponds to better seeing conditions,
indicating a smaller atmospheric turbulence. Conversely, a larger FWHM
indicates poorer seeing conditions with increased atmospheric distortion. In
the visual band, the typical size of the seeing disk at the world’s best
observational sites during photometric nights is approximately 0.6" to 0.8"
(Skidmore et al., 2009; Ramió et al., 2012; Ma et al., 2020). These sites are
renowned for their exceptional atmospheric conditions, enabling high-quality
observations. In contrast, suburban sites located near coastal metropolitan
cities typically experience significantly larger seeing disks, typically
around 10 times bigger (So, 2010). The larger seeing disks at such locations
are attributed to the increased atmospheric turbulence caused by local factors
like urbanization and proximity to the coast.

The traditional seeing measurement from starlight is not applicable to daytime
observations, such as solar observations. Attempting to represent diurnal
conditions using measurements derived from nocturnal seeing is far from
optimal, primarily due to the dynamic and rapidly changing nature of
atmospheric optical properties caused by various natural and artificial
factors (Perrone et al., 2022). While there are sophisticated solutions and
specialized devices available for daytime seeing measurements, they often
require substantial financial, temporal and labor resources. For instance, the
solar equivalent of a Differential Image Motion Monitor (DIMM) involves the
use of a specially designed telescope with two apertures and a prism to split
the light from the solar limb. The seeing condition is then quantified by
measuring the differential motion between images of the solar limb (e.g.
Beckers, 2001; Ozışık and Ak, 2004; Kawate et al., 2011; Irbah et al., 2016;
Song et al., 2018). Once the setup is installed, the telescope becomes
dedicated solely to seeing monitoring. This necessitates the investment in an
additional set of equipment, including the telescope, mount, detector and
other necessary components. Given the significant investment of resources
required by these dedicated devices, their widespread adoption for routine
daytime seeing measurements is not always feasible. As a result, alternative
approaches or methodologies that are more accessible and cost-effective are
sought for daytime seeing assessment in solar observations.

It is widely recognized that good atmospheric seeing conditions result in
sharper images, while poor seeing conditions lead to image blurring. Building
upon this relationship, we aim to assess image quality and variations in
daytime seeing by directly analyzing the sharpness or blurriness of solar
images. In the present study, we investigate the potential of applying a
sharpness detection algorithm to solar images for this purpose. Through
preliminary tests conducted on a dataset of over 65,000 individual solar
images, we demonstrate the practicality of our approach. We observe a
correlation between solar altitude and image sharpness, where higher solar
altitudes generally correspond to sharper images and vice versa. This trend
can be modeled in a manner similar to airmass variation, providing insights
into the variations in daytime seeing. Importantly, our method does not
require any additional resources beyond the existing solar image archive. It
leverages the available data and makes efficient use of the resources already
in place. Moreover, the potential of our method extends beyond solar images
and can be applied to other celestial objects such as the Moon, bright planets
and defocused stars.

This paper is structured as follows: Sect. 2 provides an introduction to the
sources of the solar images used in this study and outlines the methodology of
the sharpness detection algorithm. Subsequently, in Sect. 3, we present the
data quality control, statistics and results obtained from applying the
algorithm to the solar images and provide a detailed analysis of the findings.
Finally, in Sect. 4, we draw conclusions based on the results and engage in a
comprehensive discussion of the implications and potential applications of our
method in the assessment of image quality and variations in daytime seeing.

##  2 Methods

###  2.1 Solar Telescope and Images

We obtained pseudo white light solar images (with a peak at 430 nm) using the
Weatherproof Vacuum Solar Telescope System (STS) at the Hong Kong Space
Museum. STS is manufactured by Nishimura Co,. Ltd. in Japan. Permanently
installed on the museum’s roof (located at 22.29407∘N, 114.17146∘E) in the
urban harbor front of Hong Kong, STS provides a reliable platform for
conducting solar observations. Refer to Fig. 1 for a photograph of STS. In
addition to white light observations, STS has been utilized since 2017 to
observe the Sun in Hydrogen-alpha (with a peak at 656.3 nm) and Calcium-K
(with a peak at 393.3 nm) bands using separate telescopes. Furthermore, STS
includes a lunar cum planetary telescope and two auto-guiders. The six
refracting telescopes in a 2$\times$3 array are mounted on an equatorial fork
mount, as depicted in the technical drawing shown in Fig. 2.

Figure 1:  This photograph shows the Hong Kong Space Museum’s Weatherproof
Vacuum Solar Telescope System (STS) and provides a glimpse of its
surroundings. The Hong Kong Cultural Centre, located to the west of STS, is
partially visible in the background. As discussed in Sect. 3.1, the presence
of the cultural center has imposed certain limitations on the unobstructed
view of the Sun. Figure 2: Technical drawing of STS

Under favorable weather conditions, the auto-guider effectively maintains
precise tracking of the Sun by automatically tracing the solar disk’s edge.
The presence of occasional cloud passages within the field of view does not
significantly impact the quality of guiding. However, during prolonged periods
of cloudy weather, STS may temporarily lose track of the Sun, necessitating
manual re-centering.

Each sun-observing telescope of STS has an aperture of 100 mm and a focal
ratio of F/5. Each telescope is fitted with a color CCD camera that captures
an image of the solar disk every minute (every 5 minutes since 2023 January
19). These images are saved in JPG format for archival purposes. The image has
a resolution of 2,448$\times$2,050 and a dynamic range of 8 bits. Each pixel
sizes 4 $\mu$m$\times$4 $\mu$m. The effective field of view of the sensor is
enough to cover the entire solar disk throughout the year. In addition to
archival purposes, the images are live-streamed to the museum’s gallery and
the public
website111https://hk.space.museum/SolarTelescope/SolarTelescope.html.

###  2.2 Variance of Laplacian

The Variance of Laplacian (VL) method is a blur detection algorithm used in
computer vision. It was originally developed for digital image analysis
(Rosebrock, 2015). The method involves convolving the image with a 3$\times$3
Laplacian operator and calculating the variance, which is the squared standard
deviation. The Laplacian operator is commonly used for edge detection by
measuring the second derivative of the image’s intensity and highlighting
regions with rapid intensity changes (Wang, 2007). Blurred images tend to have
fewer edges, resulting in a lower variance. On the other hand, sharper or
well-focused images have a higher variance, indicating a wider spread of edge-
like and non-edge-like responses. The VL method is effective in determining
the level of blurriness in an image based on its variance. The Laplacian
operator has also been applied as an auto-focusing technique in microscopy due
to its reliability and speed (Pech-Pacheco et al., 2000).

Solar images, especially white light images, are ideal for edge detection. It
is because solar images often contain a featureless symmetric disk with a
clearly defined and high-contrast edge. Sunspots, which are captured in some
solar images, also create edges that can affect the VL of the image. However,
sunspots generally move slowly relative to the integration time of the image
and their impact on the VL is similar to the factors affecting the solar disk.
As a result, there is typically no systematic difference in the VL of images
with and without sunspots.

To compute the VL from batches of solar images, we developed a Python program.
We provide a pair of sample images in Fig. 3 and 4. In Fig. 3, the solar
disk’s edge and the sunspots appear blurrier, indicating a smaller VL value.
On the other hand, in Fig. 4, the same features appear sharper, indicating a
larger VL value. This demonstrates the relationship between the VL value and
the clarity of solar images.

Figure 3: Here is an example of a blurry solar image with a smaller VL value.
The zoomed-in sub-images highlight the sunspots and the edges of the solar
discs in detail. Figure 4: Same as Fig. 3 for a sharp image with a larger VL
value.

##  3 Results & Analysis

###  3.1 Data quality control & statistics

A total of 288,756 individual images captured between 2021 January and 2023
June were downloaded from the archive. To ensure data quality, a series of
selection criteria were applied in the following sequence, taking into account
the system operation and the Sun’s position:

  1. 1.

Exposure time: Images were selected if they were exposed after sunrise and
before sunset. This criterion was implemented to exclude images captured
during periods when the operator forgot to shut down the system after sunset.

  2. 2.

Unobstructed Sun: Images were filtered to ensure that the Sun was not blocked
by nearby buildings during exposure. A site survey was conducted and it was
determined that the telescopic view to the Sun’s path below 30∘ altitude in
the azimuthal direction between 205∘ and 300∘ may be obstructed by the Hong
Kong Cultural Centre (see Fig. 1).

  3. 3.

Exclusion of initial images: The first few images captured within the first 5
minutes of each day were excluded. These initial images were used for testing
purposes, during which the operator performed tasks such as image centering
and focusing. Manual checks on sample images revealed that these tasks were
typically completed within 5 minutes.

The previous selection criteria resulted in the exclusion of approximately 1.5
per cent of the archival data. The remaining images were exposed under varying
cloud conditions, ranging from overcast to clear. Some images appeared
completely black due to overcast conditions, while others had cloud patches
covering only a portion of the Sun. On the other hand, a subset of the images
showed a clear and distinct view of the Sun. To minimize the influence of
black and cloudy images effectively and to expedite the selection process
without relying solely on subjective and time-consuming manual tasks, two
additional objective selection criteria were implemented:

  1. 4.

Black pixel threshold: Images were selected if the total number of black
pixels was smaller than 80 per cent. The statistic was calculated using Python
codes. By setting a threshold on the amount of black pixels, it aimed to
exclude images that were excessively dark or had a significant portion of the
image obscured.

  2. 5.

Daily sunshine percentage: Images were further refined based on the sunshine
duration. The daily sunshine duration data, measured by the Hong Kong
Observatory (Hong Kong Observatory, 2021, 2022, 2023), were used to calculate
the percentage of time during the day that the Sun was visible. Images were
selected if the daily sunshine percentage exceeded 80 per cent. This criterion
aimed to ensure that the selected images were captured during periods of
favorable weather conditions with a significant amount of sunshine, providing
a sizable number of images per day for the calculation of DVL described in
Sect. 3.2.

It is understandable that setting stricter selection criteria may result in a
smaller number of available images for analysis. Conversely, including cloudy
images could potentially impact the accuracy of sharpness detection. The
determination of the cut-off values for these criteria is typically based on
tests conducted on real images, striking a balance between data quality and
sample size. In the end, 80.8 per cent of images passed criterion 4 while a
further 29.4 per cent of images passed criterion 5.

Based on the selection process, a total of 65,172 individual images were
chosen for further analysis. These images represent approximately 22.6 per
cent of the original dataset before applying any selection criteria. It is
worth noting that these images cover a span of 168 individual days. To gain
further insights into the selected images, statistical distributions were
analysed against various parameters. Figures 5, 6, 7 and 8 present the
distributions of the selected images in terms of month, time of day, solar
altitude and azimuth, respectively. To compute the solar positions for the
selected images, NASA Horizons System (NASA, Jet Propulsion Laboratory Solar
System Dynamics Group, 2021) was utilized.

Figure 5: The distribution of data acquisition dates as a function of the
month. This graph provides an overview of when the data was collected over the
course of years. Figure 6: The distribution of data acquisition as a function
of the time of day, binned to 5-minute intervals. This graph provides insights
into the timing patterns of data collection throughout the day. Figure 7: The
distribution of data acquisition as a function of solar altitude. This graph
provides an overview of the elevation angle of the Sun during the data
collection period. Figure 8: The distribution of data acquisition as a
function of solar azimuth. This graph provides an overview of the azimuthal
position of the Sun during the data collection period.

The statistical analysis reveals several interesting patterns in the data
distribution:

  * •

Month histogram (Fig. 5): The sampling is biased towards autumn and winter
(September to December), which are the dry seasons. This bias is likely due to
favorable weather conditions during these months. Furthermore, the change in
image acquisition interval from 1 minute to 5 minutes since 2023 January 19
has further reduced the sampling size before June.

  * •

Time histogram (Fig. 6): The sampling is relatively even between 10:00 and
16:30 (local time, UTC+8 hr, hereafter). However, the sample size drops
towards both ends of the day due to the filtering process based on selection
criteria 1 to 3. Besides, the shorter daylight duration in winter contributes
to the decrease in sample size after 16:30.

  * •

Altitude histogram (Fig. 7): The histogram displays a sharp peak at 45
degrees, which corresponds to the maximum solar altitude during the winter
season.

  * •

Azimuth histogram (Fig. 8): The data is concentrated on the southern sky
(azimuth range of $90^{\circ}$ to $270^{\circ}$), which is expected for a
location in the subtropical region. The four peaks observed at around
9$0^{\circ}$, $135^{\circ}$, $240^{\circ}$ and $270^{\circ}$ can be further
explained by referring to the polar plot in Fig. 10.

###  3.2 DVL of Solar Images

Upon analyzing the VL values obtained from each solar image, it was observed
that these values exhibit day-to-day variability. This variability can be
attributed to fluctuations in atmospheric conditions or potential
inconsistencies in the daily telescope’s focusing position made by the
operator. In order to accentuate the underlying trend of the VL values amidst
these daily variations, a novel metric termed DVL (Difference in VL) was
devised. DVL represents the disparity between the VL value of an individual
image and the corresponding daily median VL value. By calculating DVL, we
aimed to emphasize the relative changes in VL over time, thereby facilitating
the identification of significant patterns or trends within the dataset.

Fig. 9 showcases both the time series and histogram of DVL. Upon closer
examination, the time series reveals a discernible daily rhythm that aligns
with the movement of the Sun across the sky. Specifically, the DVL values are
relatively small in the morning, reach their peak around noon when the Sun is
at its highest point, and subsequently decrease towards the daily minimum near
sunset. This pattern reflects the expected seeing variation throughout the
day. Furthermore, the histogram of DVL demonstrates a prominent peak centered
around $2.5\times 10^{-4}$.

Figure 9: $(a)$ The time series and $(b)$ histogram of DVL

Fig. 10 illustrates the distribution of DVL values in polar coordinates. In
this representation, larger DVL values are depicted in a bluer shade, while
smaller DVL values appear in a yellower shade. Upon examination, it becomes
evident that the larger DVL values are primarily concentrated at higher
altitudes. On the other hand, smaller DVL values are scattered across lower
angles, indicating a correlation between DVL and solar altitude.

The presence of peaks observed in Fig. 8 can now be reasonably explained. The
peaks correspond to azimuth angles around 90∘, 135∘, 240∘ and 270∘, where the
‘line-of-sight’ from the horizon to the zenith intersects a greater number of
data points compared to other angles. This phenomenon aligns with the
observation of increased data density along these particular azimuth
directions.

No apparent azimuthal asymmetry is observed in the distribution of DVL,
suggesting that there are no significant directional biases in the DVL values.

Figure 10: This polar plot of DVL provides a visualization of the data
distribution and the corresponding DVL values across various solar positions,
offering insights into the relationship between solar position and DVL
magnitude.

###  3.3 DVL v.s. Solar Altitudes

Fig. 11 displays a scatter plot depicting the relationship between DVL and
solar altitudes. The plot reveals a weak trend of DVL increasing as solar
altitude rises. This observation suggests that solar images tend to exhibit
greater sharpness when the Sun is positioned higher in the sky, and
conversely, lower solar altitudes are associated with relatively reduced
sharpness.

Figure 11: Scatter plot of DVL as a function of solar altitudes. The plot
demonstrates how DVL values vary across different solar altitudes. There is a
discernible trend of increasing DVL as solar altitudes rise, suggesting that
solar images tend to exhibit greater sharpness when the Sun is positioned
higher in the sky. Conversely, lower solar altitudes are associated with
relatively lower DVL values, indicating reduced sharpness in the corresponding
images.

Beyond approximately $40^{\circ}$ solar altitude, the central data
distribution reaches a plateau, indicating that the variation in DVL becomes
relatively stable. This trend suggests that the image quality does not
significantly degrade when the solar altitude exceeds this angle in general.
In other words, approximately 65 per cent of the observation time is expected
to yield solar images of satisfactory quality, disregarding the potential
impact of cloud coverage. These findings imply that during a substantial
portion of the observation period, the quality of the captured solar images
remains reasonably acceptable, contributing to sound data acquisition for
analysis and research purposes.

###  3.4 Image Sharpness Function, ISF

Image degradation is attributed to starlight’s atmospheric attenuation. This
degradation is typically quantified using airmass ($X$), which is formulated
based on the thickness of the air column along the line of sight to the
target. Specifically, we express airmass as a function of the zenith angle
($z$) of the target, where $z$ is defined as $90^{\circ}$ minus the altitude.

In this section, our objective is to develop an Image Sharpness Function (ISF)
based on our observational data. ISF serves as a means to characterize the
relationship between image sharpness and airmass, providing a quantitative
measure of the degradation in image quality due to atmospheric attenuation. To
accomplish this, we referred to existing airmass formulae available in the
literature as a reference point.

Existing formulae expresses $z$ in radian, therefore we initially converted
the solar altitudes (in degrees) in our dataset to $z$ values (in radians).
Subsequently, we plotted the median DVL for each $0.25^{\circ}z$ interval, as
depicted in Fig. 12. It is important to note that the last three data points
with $z>1.381$ (corresponding to solar altitudes less than $10.88^{\circ}$)
were excluded from the analysis due to under-sampling. Each of these bins
contained fewer than 10 samples, rendering them insufficient for meaningful
statistical analysis.

We evaluated two distinct forms of ISF. The first form is as follows:

$DVL=a-[(1-b\sin^{2}(z)]^{c}$ (1)

which aligns with the airmass formula proposed by Krisciunas and Schaefer
(1991) (referred to as KS91 hereafter): $X=[1-0.96\sin^{2}(z)]^{-0.5}$.

By applying the Levenberg-Marquardt method (Press et al., 2007), we fitted the
median DVL observations to Eq. 1, resulting in a well-fitted curve with a high
coefficient of determination ($R^{2}$=0.962).

We also explored a second form of ISF, which can be represented as:

$DVL=\frac{d\cos(z)+e}{f\cos^{2}(z)+g\cos(z)+h}$ (2)

according to the low accuracy airmass formula in Young (1994) (referred to as
Y94 hereafter) which utilizes parameter values of 1.003198, 0.101632, 1,
0.090560 and 0.003198 for $d$, $e$, $f$, $g$ and $h$ respectively. We repeated
the fitting process using Eq. 2 and obtained a curve with a slightly improved
coefficient of determination ($R^{2}$=0.985). This indicates a stronger
correlation between the predicted sharpness values based on this form of ISF
and the actual median DVL observations.

Figure 12: The trend of image sharpness is evident from the median DVL observations (depicted as black squares) at various zenith angles $z$. The red curve, derived from the airmass formula in KS91, exhibits a reasonable fit to the data points. Similarly, the blue curve, which corresponds to the airmass formula in Y94, also demonstrates a good alignment with the observed median DVL values. Table 1: Best-fitted values of the parameters in two different forms of ISF ISF form | Parameter | Best-fitted value
---|---|---
| $a$ | 1.039
$a-[(1-b\sin^{2}(z)]^{c}$ | $b$ | 1.029
| $c$ | -0.081
| $d$ | -3.256
| $e$ | 2.412
$\frac{d\cos(z)+e}{f\cos^{2}(z)+g\cos(z)+h}$ | $f$ | -40.035
| $g$ | 5.209
| $h$ | -4.123

Table 1 provides the fitted parameters obtained from the regression analysis.
Fig. 12 illustrates the best-fitted curves derived from the ISF. Taken as a
whole, it becomes apparent that the overall trend of DVL aligns more closely
with the variation in airmass as described by the Y94 formula. However, it is
worth noting that for larger zenith angles ($z$) beyond 1.36 (equivalent to
$78^{\circ}$), the form of the KS91 airmass formula fits better. This
observation is consistent with the findings highlighted by Y94, who noted that
accurately formulating airmass near the horizon presents notable challenges
due to its sensitivity to various atmospheric conditions, such as surface
temperature, pressure and the vertical structure of the atmosphere.

##  4 Conclusions & Discussion

Our study has demonstrated the feasibility, efficiency and simplicity of using
the Variance of Laplacian (VL) method to analyse the quality of solar images.
This method assigns a ‘score’ to each image based on the sharpness of the
solar disk, providing a quantitative measure of image quality and,
consequently, atmospheric conditions, such as seeing.

While the measurement of seeing is typically achieved by estimating the
angular size of the point spread function of stellar images or using
specialized equipment like seeing monitors, our method utilizes standard solar
images. Our method can be employed to extract the diameter of the solar disk
from individual images. This approach enables a direct and expeditious
estimation of the site’s seeing parameter $r_{0}$ by analyzing variations in
the measured solar diameters, as illustrated in a recent paper by (Duan et
al., 2024). It eliminates the need for time-consuming data reduction
processes, extensive computational resources, or additional investments in
instrumentation. In the future, we plan to expand our method by testing it
with disk images of the Moon, bright planets and defocused stars. This will
allow us to develop real-time image quality monitors or seeing monitors that
can be executed on-the-spot.

Our analysis has reproduced a well-known trend indicating a relationship
between altitude and image quality: higher targets in the sky (with lower
airmass) exhibit better image quality, while lower targets have poorer image
quality. In particular, the quality of solar images taken above $40^{\circ}$
altitude remains reasonably acceptable in general. We have represented the
relationship through an Image Sharpness Function (ISF), which describes the
image quality as a function of the target’s zenith angle. Although we have
only tested two common forms of airmass formulas, our analysis strongly
suggests that ISF closely resembles the behavior of airmass, indicating a deep
relationship between the two. By studying image quality, we hope to facilitate
the challenging task of modeling airmass.

Each observing site possesses unique characteristics, including its own ISF.
Developing a site-specific ISF offers at least two benefits. Firstly, it can
be incorporated into calculation tools like the SKYCALC Sky Model
Calculator222https://www.eso.org/observing/etc/bin/gen/form?INS.MODE=swspectr+INS.NAME=SKYCALC
to enhance airmass or seeing estimations for observers’ reference during the
preparation of observing runs. Secondly, by regularly monitoring ISF rather
than relying on separate seeing monitors, one can continuously assess the
observing site conditions over an extended period. Furthermore, if archival
images are available, retrospective analysis becomes possible.

Future research should focus on enhancing the accuracy and robustness of the
VL algorithm. This can be achieved by testing alternative edge detection
algorithms (Song et al., 2018), noise filtering algorithms (Wang, 2007; Jain
et al., 2016) and employing machine learning techniques.

Lastly, this study highlights the value of the vast collection of archival
solar images, which can serve as a valuable data repository for research
beyond solar physics.

## Data Availability

Data and codes available on request.

## Declaration of Competing Interest

None.

## Acknowledgements

E. Yuen acknowledges the support received from the Department of Physics at
The University of Hong Kong. We would also like to express our gratitude to
the reviewers for their valuable time and effort in reviewing the manuscript.
Their feedback and suggestions have significantly contributed to the
improvement of this study.

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