1. Introduction
The theory of minimal surfaces has found many applications in differential geometry and also in physics. In [
1] and [
2], H. Liu gave some classification results for translation surfaces. A minimal translation hypersurface in a Euclidean space is either locally a hyperplane or an open part of a cylinder on Scherk’s surfaces, as proved in Dillen
et al. [
3]. In [
1] was generalized to translation surfaces with constant mean curvature and constant Gaussian curvature in \(\mathbb{E}^{3}.\) Saglam and Sabuncuoglu proved that every homothetical lightlike hypersurface in a semi-Euclidean \(\mathbb{E}_{q}^{m+2}\) space is minimal [
4]. Jiu and Sun studied \(n-\) dimensional minimal homothetical hypersurfces and gave their classification [
5]. R. Lopez [
6] studied translation surfaces in the 3-dimensional hyperbolic space
and classified minimal translation surfaces. Meng and Liu [
7] considered factorable surfaces along two lightlike directions and spacelike-lightlike directions in Minkowski 3-space \(\mathbb{E}_{1}^{3}\) and they also gave some classifcation theorems. In [
8], Yu and Liu studied the factorable minimal surfaces in \(\mathbb{E}_{1}^{3}\) and \(\mathbb{E}^{3}\), and gave some classification theorems. Guler
et al. [
9] defined by translation and homothetical TH-surfaces in the three dimensional Euclidean space.
2. Preliminaries
Let \(\mathbb{E}_{1}^{3}\) be a 3-dimensional Minkowski space with the scalar product of index \(1\) given by
\begin{equation*}
g_{L}=ds^{2}=-dx^{2}+dy^{2}+dz^{2},
\end{equation*}
where \((x,y,z)\) is a rectangular coordinate system of \(\mathbb{E}_{1}^{3}.\)
A vector \(V\) of \(\mathbb{E}_{1}^{3}\) is said to be timelike if \(g_{L}(V,V)0\) or \(V=0\) and lightlike or null if \(g_{L}(V,V)=0 \) and \(V\neq 0.\) A surface in \(\mathbb{E}_{1}^{3}\) is spacelike, timelike or lightlike if the tangent plane at any point is spacelike, timelike or lightlike, respectively.
The Lorentz scalar product of the vectors \(V\) and \(W\) is defined by \(g_{L}(V,W)=-v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3},\) where \(V=(v_{1},v_{2},v_{3}),\)
\(W=(w_{1},w_{2},w_{3})\in \) \(\mathbb{E}_{1}^{3}.\)
For any \(V,\) \(W\in \) \(\mathbb{E}_{1}^{3}\), the pseudo-vector product of \(V\)
and \(W\) is defined as follows:
\begin{equation*}
V\wedge _{L}W=\big(-v_{2}w_{3}+v_{3}w_{2},\text{ }v_{3}w_{1}-v_{1}w_{3}\text{
},v_{1}w_{2}-v_{2}w_{1}\big).
\end{equation*}
We denote a surface \(M^{2}\) in \(\mathbb{E}_{1}^{3}\) by
\begin{equation*}
r(u,v)=\big(r_{1}(u,v),\text{ }r_{2}(u,v),\text{ }r_{3}(u,v)\big).
\end{equation*}
Definition 1. [10]
A translation surface in Minkowski \(3\)-space is a surface that is
parameterized by either
\begin{eqnarray*}
r(u,v) &=&(u,\text{ }v,\text{ }f(u)+g(v))\;\; if\;\; L\;\; is \;\; timelike, \\
r(u,v) &=&(f(u)+g(v),\text{ }u,\text{ }v)\;\; if \;\; L\;\; is \;\; spacelike, \\
r(u,v) &=&(u+v,\text{ }g(v),\text{ }f(u)+v)\;\; if\;\; L\;\; is \;\; lightlike,
\end{eqnarray*}
with \(L\) the intersection of the two planes that contain the curves that
generate the surface.
Theorem 2. [11]
- The only translation surfaces with constant Gauss curvature \(K=0\) are cylindrical surfaces.
- There are no translation surfaces with constant Gauss curvature \(K\neq 0\) if one of the generating curves is planar.
Definition 3.
A homothetical (factorable) surface \(M^{2}\) in the 3-dimensional Lorentzian space \(\mathbb{E}_{1}^{3}\) is a surface that is a graph of a function
\begin{equation*}
z(u,\text{ }v)=f(u)g(v),
\end{equation*}
where \(f:I\subset \mathbb{R}\rightarrow \mathbb{R}\) and \(g:J\subset \mathbb{R}\rightarrow \mathbb{R}\) are two smooth functions.
Theorem 4. [11]
Planes and helicoids are the only minimal homothetical surfaces in Euclidean
space.
Accordingly, we define an extended surface in \(\mathbb{E}_{1}^{3}\) using
definitions as above and called it TH-type surface as follows [
9]:
Definition 5.
A surface \(M^{2}\) in the 3-dimensional Lorentzian space \(\mathbb{E}_{1}^{3}\)
is a TH-surface if it can be parameterized either by a patch
\begin{equation}
r(u,\text{ }v)=(u,\text{ }v,\text{ }A(f(u)+g(v))+Bf(u)g(v)) \label{TF-SUR}
\end{equation}
(1)
or
\begin{equation}
r(u,\text{ }v)=(A(f(u)+g(v))+Bf(u)g(v),\text{ }u,\text{ }v),
\label{TF-SUR 2}
\end{equation}
(2)
where \(A\) and \(B\) are non-zero real numbers.
Remark 1.
- If \(A\neq 0\) and \(B=0\) in (1), then surface is a translation surface.
- If \(A=0\) and \(B\neq 0\) in (1), then surface is a homothetical (factorable) surface.
Let \(\mathbf{N}\) denotes the unit normal vector field of \(M^{2}\) and put \( g_{L}(\mathbf{N},\mathbf{N})=\varepsilon =\pm 1\), so that \(\varepsilon =-1\)
or \(\varepsilon =1\) according to \(M^{2}\) is endowed with a Lorentzian or
Riemannian metric, respectively.
The mean curvature and the Gauss curvature are
\begin{equation*}
H=\frac{EN+GL-2FM}{2\left\vert EG-F^{2}\right\vert },\text{ }K=g_{L}(\mathbf{
N},\mathbf{N})\frac{LN-M^{2}}{EG-F^{2}},
\end{equation*}
where \(E,\) \(G,\) \(F\) are the coefficients of the first fundamental form, \(L,\)
\(M,\) \(N\) \ are the coefficients of the second fundamental form.
In this paper, we define TH-surfaces in the 3-dimensional Euclidean space \(
\mathbb{E}^{3}\) and Lorentzian-Minkowski space \(\mathbb{E}_{1}^{3},\) and
completely classify minimal or flat TH-surfaces.
3. Minimal TH-surfaces in \(\mathbb{E}_{1}^{3}\)
A surface \(M^{2}\) in the 3-dimensional Lorentzian space \(\mathbb{E}_{1}^{3}\)
is called minimal when locally each point on the surface has a neighborhood
which is the surface of least area with respect to its boundary [12]. In 1775, J. B. Meusnier showed that the condition of minimality
of a surface in \(\mathbb{E}^{3}\) is equivalent with the vanishing of its
mean curvature function, \(H=0.\)
Let \(z=f(x,y)\) define a graph \(M^{2}\) in the Euclidean 3-space \(\mathbb{E}
^{3}\). If \(M^{2}\) is minimal, the function \(f\) \ satisfies
\begin{equation}
(1+f_{y}{}^{2})f_{xx}-2f_{xy}f_{x}f_{y}+(1+f_{x}{}^{2})f_{yy}=0,
\label{surf-mini}
\end{equation}
(3)
which was obtained by J. L. Lagrange in \(1760\).
Let \(M^{2}\) be a TH-surface in \(\mathbb{E}_{1}^{3}\) parameterized by a patch
\begin{equation*}
r(u,\text{ }v)=(u,\text{ }v,\text{ }A(f(u)+g(v))+Bf(u)g(v)),
\end{equation*}
where \(A\) and \(B\) are non-zero real numbers.
So
\begin{equation*}
r_{u}=(1,\text{ }0,\text{ }f^{\prime }\alpha ),\;\; r_{v}=(0,\text{ }1,
\text{ }g^{\prime }\gamma ),
\end{equation*}
where \(\alpha =A+Bg\) and \(\gamma =A+Bf.\)
After eliminating \(f^{\prime }\) and \(g^{\prime }\) we find
\begin{equation*}
E=\frac{\gamma ^{\prime 2}\alpha ^{2}-B^{2}}{B^{2}},\;F=\frac{\alpha
\gamma \alpha ^{\prime }\gamma ^{\prime }}{B^{2}},\text{ }G=\frac{\gamma
^{2}\alpha ^{\prime 2}+B^{2}}{B^{2}}.
\end{equation*}
The unit normal vector is given by
\begin{equation*}
N =\frac{1}{WB}(\alpha \gamma ^{\prime },\text{ }-\gamma \alpha
^{\prime },\text{ }B),
\end{equation*}
where \(W^{2}=B^{-2}g_{L}(N,N)(\gamma ^{\prime
2}\alpha ^{2}-\alpha ^{\prime 2}\gamma ^{2}-B^{2})\) and
\begin{equation*}
g_{L}(N,N)=\varepsilon ,\;\varepsilon
=\left\{
\begin{array}{l}
1\;\; M^{2}\text{ is spacelike }(\gamma ^{\prime 2}\alpha
^{2}-\alpha ^{\prime 2}\gamma ^{2}-B^{2}>0), \\
-1\;\; M^{2}\text{ is timelike }(\gamma ^{\prime 2}\alpha ^{2}-\alpha
^{\prime 2}\gamma ^{2}-B^{2}< 0)\text{.}
\end{array}
\right.
\end{equation*}
The constant \(\varepsilon \) is called the sign of \(M^{2}\).
The coefficients of the second fundamental form are given by
\begin{equation*}
L=\frac{\alpha \gamma ^{\prime \prime }}{BW},\;M=\frac{\alpha
^{\prime }\gamma ^{\prime }}{BW},\;N=\frac{\gamma \alpha ^{\prime
\prime }}{BW}.
\end{equation*}
The expression of \(H\) is
\begin{eqnarray}
H &=&\frac{B^{2}(\alpha f^{\prime \prime }(1+g^{\prime 2}\gamma
^{2})-2B\alpha \gamma f^{\prime 2}g^{\prime 2}+\gamma g^{\prime \prime
}(f^{\prime 2}\alpha ^{2}-1))}{2W^{3}} \notag \\
&=&\frac{\alpha \gamma ^{\prime \prime }(B^{2}+\alpha ^{\prime 2}\gamma
^{2})-2\alpha \gamma \alpha ^{\prime 2}\gamma ^{\prime 2}+\gamma \alpha
^{\prime \prime }(\gamma ^{\prime 2}\alpha ^{2}-B^{2})}{2BW^{3}}.
\label{MEAN CURVA}
\end{eqnarray}
(4)
Then \(M^{2}\) is a minimal surface if and only if
\begin{equation}
\alpha \gamma ^{\prime \prime }(B^{2}+\alpha ^{\prime 2}\gamma ^{2})-2\alpha
\gamma \alpha ^{\prime 2}\gamma ^{\prime 2}+\gamma \alpha ^{\prime \prime
}(\gamma ^{\prime 2}\alpha ^{2}-B^{2})=0. \label{surface minimal}
\end{equation}
(5)
We distinguish the following cases.
Case 1. Let \(\gamma ^{\prime }=0\). In this case (5) gives \(\gamma \alpha ^{\prime \prime }=0.\)
- If \(\gamma =0\), then \(f(u)=-\frac{A}{B}\), \(M^{2}\) is the horizontal
plane of equation \(z=-\frac{A^{2}}{B}\).
- If \(\alpha ^{\prime \prime }=0\), then \(\alpha (v)=a_{1}v+b_{1},\) \(a_{1},\) \(b_{1}\in\mathbb{R}\), and \(\gamma (u)=c_{1},\) \(c_{1}\in\mathbb{R},\ M^{2}\) is the plane of equation \(z=c_{2}v+c_{3},\) \(c_{2},\) \(c_{3}\in\mathbb{R}\).
Case 2. Let \(\alpha ^{\prime }=0\). In this case (5) gives \(\gamma ^{\prime \prime }\alpha =0.\)
- ]If \(\alpha =0\), then \(g(v)=-\frac{A}{B}\), \(M^{2}\) is the horizontal
plane of equation \(z=-\frac{A^{2}}{B}\).
- If \(\gamma ^{\prime \prime }=0\), then \(\gamma (u)=a_{2}u+b_{2},\) \(
a_{2},\) \(b_{2}\in\mathbb{R}\), and \(\alpha (v)=c_{4},\) \(c_{4}\in\mathbb{R},\ M^{2}\) is the plane of equation \(z=c_{5}u+c_{6},\) \(c_{5},\) \(c_{6}\in\mathbb{R}\).
Case 3. Let \(\gamma ^{\prime \prime }=0\) and \(\gamma ^{\prime }\neq
0\). Then \(\gamma (u)=\lambda u+\delta ,\) \((\lambda ,\) \(\delta )\in\mathbb{R}\setminus \left\{ 0\right\} \times\mathbb{R}\) and \(\alpha \) is a solution of the following
ODE
\begin{equation}
-2\lambda ^{2}\alpha \alpha ^{\prime 2}+\alpha ^{\prime \prime }(\lambda
^{2}\alpha ^{2}-B^{2})=0. \label{surface minimal2}
\end{equation}
(6)
Then the general solution of (6) is given by
\begin{equation*}
\alpha (v)=-\frac{B}{\lambda }\coth (\lambda _{1}v+\lambda _{2}),\text{ }
\lambda _{1},\text{ }\lambda _{2}\in\mathbb{R}.
\end{equation*}
Hence
\begin{equation*}
g(v)=-\frac{1}{\lambda }\coth (\lambda _{1}v+\lambda _{2})-\frac{A}{B},\text{
}\lambda _{1},\text{ }\lambda _{2}\in\mathbb{R}.
\end{equation*}
Case 4. Let \(\alpha ^{\prime \prime }=0\) and \(\alpha ^{\prime }\neq
0\). Then \(\alpha (v)=\lambda v+\delta ,\) \((\lambda ,\) \(\delta )\in\mathbb{R}\setminus \left\{ 0\right\} \times\mathbb{R}\) and \(\gamma \) is a solution of the following
ODE
\begin{equation}
-2\lambda ^{2}\gamma \gamma ^{\prime 2}+\gamma ^{\prime \prime }(\lambda
^{2}\gamma ^{2}+B^{2})=0. \label{surface minimal33}
\end{equation}
(7)
Then the general solution of (7) is given by
\begin{equation*}
\gamma (u)=\frac{B}{\lambda }\tan (\lambda _{1}u+\lambda _{2}),\text{ }
\lambda _{1},\text{ }\lambda _{2}\in\mathbb{R}.
\end{equation*}
Hence
\begin{equation*}
f(u)=\frac{1}{\lambda }\tan (\lambda _{1}u+\lambda _{2})-\frac{A}{B},\text{ }
\lambda _{1},\text{ }\lambda _{2}\in\mathbb{R}.
\end{equation*}
Case 5. Let \(\gamma ^{\prime \prime }\neq 0.\) By symmetry in the
discussion of the case, we also suppose \(\alpha ^{\prime \prime }\neq 0\). If
we divide (5) by \(\alpha \gamma \alpha ^{\prime 2}\gamma
^{\prime 2},\) we obtain
\begin{equation*}
\frac{B^{2}\gamma ^{\prime \prime }}{\gamma \alpha ^{\prime 2}\gamma
^{\prime 2}}+\frac{\gamma \gamma ^{\prime \prime }}{\gamma ^{\prime 2}}-
\frac{B^{2}\alpha ^{\prime \prime }}{\alpha \alpha ^{\prime 2}\gamma
^{\prime 2}}+\frac{\alpha \alpha ^{\prime \prime }}{\alpha ^{\prime 2}}-2=0.
\end{equation*}
Thus, after a derivation with respect to \(u\), followed by a derivation with
respect to \(v,\) we obtain
\begin{equation*}
\left( \frac{\gamma ^{\prime \prime }}{\gamma \gamma ^{\prime 2}}\right)
_{,u}\left( \frac{1}{\alpha ^{\prime 2}}\right) _{,v}-\left( \frac{\alpha
^{\prime \prime }}{\alpha \alpha ^{\prime 2}}\right) _{,v}\left( \frac{1}{
\gamma ^{\prime 2}}\right) _{,u}=0.
\end{equation*}
Hence we deduce the existence of a real number \(k\in\mathbb{R}\) such that
\begin{equation}
\left\{
\begin{array}{c}
\left( \frac{\gamma ^{\prime \prime }}{\gamma \gamma ^{\prime 2}}\right)
_{,u}=k\left( \frac{1}{\gamma ^{\prime 2}}\right) _{,u} \\
\left( \frac{\alpha ^{\prime \prime }}{\alpha \alpha ^{\prime 2}}\right)
_{,v}=k\left( \frac{1}{\alpha ^{\prime 2}}\right) _{,v}.
\end{array}
\right. \label{surface minimal3}
\end{equation}
(8)
The first equation of (8) can integrate obtaining
\begin{equation}
\gamma ^{\prime \prime }=\gamma (k+c\gamma ^{\prime 2}). \label{Equi 1}
\end{equation}
(9)
From the second equation in (8), we obtain
\begin{equation}
\alpha ^{\prime \prime }=\alpha (k+b\alpha ^{\prime 2}). \label{Equi 2}
\end{equation}
(10)
Substituting the above in (5), we get
\begin{equation*}
\alpha \gamma ((k+c\gamma ^{\prime 2})(B^{2}+\alpha ^{\prime 2}\gamma
^{2})-2\alpha ^{\prime 2}\gamma ^{\prime 2}+(k+b\alpha ^{\prime 2})(\gamma
^{\prime 2}\alpha ^{2}-B^{2}))=0.
\end{equation*}
If we simplify by \(\alpha \gamma \) and then we divide by \(\alpha ^{\prime
2}\gamma ^{\prime 2}\), we get
\begin{equation*}
\frac{bB^{2}-k\gamma ^{2}}{\gamma ^{\prime 2}}-c\gamma ^{2}+2=\frac{
cB^{2}+k\alpha ^{2}}{\alpha ^{\prime 2}}+b\alpha ^{2}.
\end{equation*}
Hence, we deduce the existence of a real number \(\lambda \in\mathbb{R}\) such that
\begin{equation}
\left\{
\begin{array}{c}
\gamma ^{\prime 2}=\frac{bB^{2}-k\gamma ^{2}}{\lambda -2+c\gamma ^{2}} \\
\alpha ^{\prime 2}=\frac{cB^{2}+k\alpha ^{2}}{\lambda -b\alpha ^{2}}.
\end{array}
\right. \label{surface minimal4}
\end{equation}
(11)
Differentiating with respect to \(u\) and \(v\), respectively, we have
\begin{equation}
\left\{
\begin{array}{l}
\gamma ^{\prime \prime }=-\frac{\gamma ((\lambda -2)k+bcB^{2})}{(\lambda
-2+c\gamma ^{2})^{2}} \\
\alpha ^{\prime \prime }=\frac{\alpha (\lambda k+bcB^{2})}{(\lambda -b\alpha
^{2})^{2}}.
\end{array}
\right. \label{surface minimal5}
\end{equation}
(12)
Let us compare these expressions of \(\alpha ^{\prime \prime }\) and \(\gamma
^{\prime \prime }\) with those ones that appeared in (9) and (10) and replace the values of \(\gamma ^{\prime 2}\) and \(\alpha
^{\prime 2}\) obtained in (11).
We get
\begin{equation*}
\left\{
\begin{array}{l}
(\lambda k+bcB^{2})(\lambda -1-b\alpha ^{2})=0 \\
((\lambda -2)k+bcB^{2})(\lambda -1+c\gamma ^{2})=0.
\end{array}
\right.
\end{equation*}
We discuss all possibilities.
- If
\begin{equation*}
\left\{
\begin{array}{l}
\lambda k+bcB^{2}=0 \\
(\lambda -2)k+bcB^{2}=0,
\end{array}
\right.
\end{equation*}
then \(k=0\) and \(bc=0\). Then (12) gives \(\gamma ^{\prime
\prime }=0\) and \(\alpha ^{\prime \prime }=0\), a contradiction.
- If
\begin{equation*}
\left\{
\begin{array}{l}
\lambda k+bcB^{2}=0 \\
c=0 \\
\lambda =1,
\end{array}
\right.
\end{equation*}
we obtain \(k=0\). Then (12) gives \(\gamma ^{\prime \prime
}=0\) and \(\alpha ^{\prime \prime }=0\), a contradiction.
- If
\begin{equation*}
\left\{
\begin{array}{l}
(\lambda -2)k+bcB^{2}=0 \\
b=0 \\
\lambda =1,
\end{array}
\right.
\end{equation*}
we obtain \(k=0\). Then (12) gives \(\gamma ^{\prime \prime
}=0\) and \(\alpha ^{\prime \prime }=0\), a contradiction.
- If
\begin{equation*}
\left\{
\begin{array}{c}
\lambda -1-b\alpha ^{2}=0 \\
\lambda -1+c\gamma ^{2}=0
\end{array}
\right.
\end{equation*}
we deduce that \(\alpha \), \(\gamma \) are both constant functions, and so, \(
\gamma ^{\prime \prime }=0\) and \(\alpha ^{\prime \prime }=0\), a
contradiction.
- If \(b=0,\) \(c=0\) and \(\lambda =1,\) Equation (11)
writes as
\begin{equation}
\left\{
\begin{array}{c}
\gamma ^{\prime 2}=k\gamma ^{2} \\
\alpha ^{\prime 2}=k\alpha ^{2}.
\end{array}
\right. \label{surface minimal6}
\end{equation}
(13)
The equations (13) have the following solutions
\begin{equation*}
\alpha (v)=k_{1}e^{\sqrt{k}v},\gamma (u)=k_{2}e^{\sqrt{k}u},\text{
}k>0,
\end{equation*}
where \(k_{1}\), \(k_{2}\in\mathbb{R}\) are integration constants.
Hence
\begin{equation*}
g(v)=\lambda _{1}e^{\sqrt{k}v}-\frac{A}{B}, f(u)=\lambda _{2}e^{
\sqrt{k}u}-\frac{A}{B},\text{ }k>0.
\end{equation*}
Therefore, we have the following:
Theorem 6.
Let \(M^{2}\) be a TH-surface in \(\mathbb{E}_{1}^{3}.\) If \(M^{2}\) is minimal
surface, then \(M^{2}\) can be parameterized as
\begin{equation*}
r(u,v)=(u,\text{ }v,\text{ }A(f(u)+g(v))+Bf(u)g(v)),
\end{equation*}
where
\(1)\) either \(f(u)=-\frac{A}{B}\) and \(g(v)\) is a smooth function in \(v.\)
\(2)\) \(g(v)=-\frac{A}{B}\) and \(f(u)\) is a smooth function in \(u.\)
\(3)\) \(f(u)=\lambda _{1}u+\lambda _{2}\) and \(g(v)=\lambda _{3}\coth (\lambda
_{4}v+\lambda _{5})-\lambda _{6}\), \(\lambda _{i}\in\mathbb{R}.\)
\(4)\) \(f(u)=\frac{1}{\lambda }\tan (\lambda _{1}u+\lambda _{2})-\frac{A}{B},\)
\(\lambda _{1},\) \(\lambda _{2}\in\mathbb{R}\) and \(g(v)=\delta _{5}v+\delta _{6}\), \(\delta _{i}\in\mathbb{R}.\)
\(5)\) \(f(u)=\lambda _{2}e^{\sqrt{k}u}-\frac{A}{B}\) and \(g(v)=\lambda _{1}e^{
\sqrt{k}v}-\frac{A}{B}.\)
Let \(M^{2}\) be a TH-surface in \(\mathbb{E}_{1}^{3}\) parameterized by a patch
\begin{equation*}
r(u,\text{ }v)=(A(f(u)+g(v))+Bf(u)g(v),\text{ }u,\text{ }v),
\end{equation*}
where \(A\) and \(B\) are non-zero real numbers.
So
\begin{equation*}
r_{u}=(f^{\prime }\alpha ,\text{ }1,\text{ }0),\;r_{v}=(g^{\prime
}\gamma ,\text{ }0,\text{ }1),
\end{equation*}
where \(\alpha =A+Bg\) and \(\gamma =A+Bf.\)
We have
\begin{equation*}
E=\frac{-\gamma ^{\prime 2}\alpha ^{2}+B^{2}}{B^{2}},\; F=-\frac{
\alpha \gamma \alpha ^{\prime }\gamma ^{\prime }}{B^{2}},\text{ }G=\frac{
-\gamma ^{2}\alpha ^{\prime 2}+B^{2}}{B^{2}}.
\end{equation*}
The coefficients of the second fundamental form on \(M^{2}\) are obtained by
\begin{equation*}
L=\frac{\alpha \gamma ^{\prime \prime }}{BW}, M=\frac{\alpha
^{\prime }\gamma ^{\prime }}{BW},N=\frac{\gamma \alpha ^{\prime
\prime }}{BW}.
\end{equation*}
Then \(M^{2}\) is a minimal surface if and only if
\begin{equation}
\alpha \gamma ^{\prime \prime }(B^{2}-\alpha ^{\prime 2}\gamma ^{2})+2\alpha
\gamma \alpha ^{\prime 2}\gamma ^{\prime 2}-\gamma \alpha ^{\prime \prime
}(\gamma ^{\prime 2}\alpha ^{2}-B^{2})=0, \label{surface minimal B}
\end{equation}
(14)
where \(\alpha =A+Bg\) and \(\gamma =A+Bf.\)
Using the same algebraic techniques as in the case of surfaces (1
), we get:
Theorem 7.
Let \(M^{2}\) be a TH-surface in \(\mathbb{E}_{1}^{3}.\) If \(M^{2}\) is minimal
surface, then \(M^{2}\) can be parameterized as
\begin{equation*}
r(u,\text{ }v)=(A(f(u)+g(v))+Bf(u)g(v),\text{ }u,\text{ }v),
\end{equation*}
where
\(1)\) either \(f(u)=\frac{\zeta }{B}u+\alpha \) and \(g(v)=-\frac{1}{\zeta }
\coth (\lambda _{3}v+\lambda _{4})-\frac{A}{B}.\)
\(2)\) \(f(u)=-\frac{A}{B}\) and \(g(v)\) is a smooth function in \(v.\)
\(3)\) \(g(v)=-\frac{A}{B}\) and \(f(u)\) is a smooth function in \(u.\)
\(4)\) or \(g(v)=\frac{\delta }{B}v+\mu \) and \(f(u)=-\frac{1}{\delta }\coth
(\lambda _{1}u+\lambda _{2})-\frac{A}{B}.\)
4. TH-surfaces with zero Gaussian curvature in \(\mathbb{E}_{1}^{3}\)
A non-degenerate surface in \(\mathbb{E}_{1}^{3}\) is called flat, if its
Gaussian curvature vanishes identically.
A surface in \(\mathbb{E}_{1}^{3}\) parameterized by (1), after
eliminating \(f,\) \(g\) and their derivatives, has Gaussian curvature
\begin{equation*}
K=g_{L}(N,N)\frac{\alpha \gamma \alpha ^{\prime
\prime }\gamma ^{\prime \prime }-\gamma ^{\prime 2}\alpha ^{\prime 2}}{
B^{2}W^{4}}.
\end{equation*}
Suppose that \(M^{2}\) has zero Gaussian curvature. Then we have
\begin{equation}
\alpha \gamma \alpha ^{\prime \prime }\gamma ^{\prime \prime }-\gamma
^{\prime 2}\alpha ^{\prime 2}=0. \label{Gauss curvature}
\end{equation}
(15)
Case 1. Let \(\gamma ^{\prime }=0\). In this case \(\gamma \) is a
constant function \(\gamma (u)=u_{0}\) and the parametrization of (1) writes as
\begin{equation*}
r(u,v)=(u,\text{ }v,\text{ }\delta _{1}g(v)+\delta _{2});\text{ }\delta
_{1},\delta _{2}\in\mathbb{R}.
\end{equation*}
This means that \(M^{2}\) is a cylindrical surface with base curve a plane
curve in the \(vz-\) plane.
Case 2. Let \(\alpha ^{\prime }=0\). In this case \(\alpha \) is a
constant function \(\alpha (v)=v_{0}\) and the parametrization of (1) writes as
\begin{equation*}
r(u,v)=(u,\text{ }v,\text{ }\delta _{3}f(u)+\delta _{4});\text{ }\delta
_{3},\delta _{4}\in\mathbb{R}.
\end{equation*}
This means that \(M^{2}\) is a cylindrical surface with base curve a plane
curve in the \(uz-\) plane.
Case 3. Let \(\gamma ^{\prime \prime }=0\) and \(\gamma ^{\prime }\neq
0\). Then \(\gamma (u)=\lambda _{1}u+\lambda _{2},\) \((\lambda _{1},\) \(\lambda
_{2})\in\mathbb{R}\backslash \left\{ 0\right\} \times\mathbb{R}.\) Moreover, (15) gives \(\alpha ^{\prime }=0\) and \(\alpha
(v)=\) \(v_{0}\) is a constant function. Now \(M^{2}\) is the plane of equation \(
z(u,v)=\lambda _{3}u+\lambda _{4};\) \(\lambda _{3},\) \(\lambda _{4}\in\mathbb{R}\)
Case 4. Let \(\alpha ^{\prime \prime }=0\) and \(\alpha ^{\prime }\neq
0\). Then \(\alpha (v)=\lambda v+\delta _{1},\) \((\lambda ,\) \(\delta _{1})\in\mathbb{R}\setminus \left\{ 0\right\} \times\mathbb{R}.\)
Moreover, (15) gives \(\gamma ^{\prime }=0\) and \(\gamma
(u)=\) \(u_{0}\) is a constant function. Now \(M^{2}\) is the plane of equation \(
z(u,v)=\lambda _{5}u+\lambda _{6};\) \(\lambda _{5},\) \(\lambda _{6}\in\mathbb{R}.\)
Case 5. Let \(\gamma ^{\prime \prime }\neq 0\) and \(\alpha ^{\prime
\prime }\neq 0\).
Equation (15) writes as
\begin{equation*}
\frac{\gamma \gamma ^{\prime \prime }}{\gamma ^{\prime 2}}=\frac{\alpha
^{\prime 2}}{\alpha \alpha ^{\prime \prime }}.
\end{equation*}
Therefore, there exists a real number \(\lambda \in\mathbb{R}\setminus \left\{ 0\right\} \) uch that
\begin{equation*}
\frac{\gamma \gamma ^{\prime \prime }}{\gamma ^{\prime 2}}=\lambda =\frac{
\alpha ^{\prime 2}}{\alpha \alpha ^{\prime \prime }}.
\end{equation*}
Integrate these equations
\begin{equation}
\left\{
\begin{array}{l}
\gamma ^{\prime }=k_{1}\gamma ^{\lambda } \\
\alpha ^{\prime }=k_{2}\alpha ^{\frac{1}{\lambda }},
\end{array}
\right. \label{Equation1}
\end{equation}
(16)
where \(k_{1}\) and \(k_{2}\) are constants of integration.
- If \(\lambda =1\), the general solution of (16) is given by
\begin{equation*}
\left\{
\begin{array}{c}
\gamma (u)=\lambda _{1}e^{k_{1}u} \\
\alpha (v)=\lambda _{2}e^{k_{2}v},
\end{array}
\right.
\end{equation*}
where \(\lambda _{1}\) and \(\lambda _{2}\) are constants of integration.
Hence
\begin{equation*}
\left\{
\begin{array}{l}
f(u)=\lambda _{3}e^{k_{1}u}+\lambda _{4} \\
g(v)=\lambda _{5}e^{k_{2}v}+\lambda _{6},
\end{array}
\right.
\end{equation*}
where \(\lambda _{3}\), \(\lambda _{4},\) \(\lambda _{5}\), \(\lambda _{6}\in\mathbb{R}\).
- If \(\lambda \neq 1\), the general solution of (16) is given
by
\begin{equation*}
\left\{
\begin{array}{c}
\gamma (u)=((1-\lambda )k_{1}u+c_{1})^{\frac{1}{1-\lambda }} \\
\alpha (v)=((\frac{\lambda -1}{\lambda })k_{2}v+c_{2})^{\frac{\lambda }{
\lambda -1}},
\end{array}
\right.
\end{equation*}
where \(c_{1}\) and \(c_{2}\) are constants of integration.
Hence
\begin{equation*}
\left\{
\begin{array}{l}
f(u)=c_{3}((1-\lambda )k_{1}u+c_{1})^{\frac{1}{1-\lambda }}+c_{4} \\
g(v)=c_{5}((\frac{\lambda -1}{\lambda })k_{2}v+c_{2})^{\frac{\lambda }{
\lambda -1}}+c_{6},
\end{array}
\right.
\end{equation*}
where \(c_{3}\), \(c_{4},\) \(c_{5}\), \(c_{6}\in\mathbb{R}\).
Theorem 8.
Let \(M^{2}\) be a TH-surface in \(\mathbb{E}_{1}^{3}\) with constant Gauss
curvature \(K\). If \(M^{2}\)\ has zero Gaussian curvature, then \(M^{2}\) can be
parameterized as
\begin{equation*}
r(u,v)=(u,\text{ }v,\text{ }z(u,v)=A(f(u)+g(v))+Bf(u)g(v)),
\end{equation*}
where
\(1)\) either \(f(u)=\lambda _{1}e^{k_{1}u}+\lambda _{2}\) and \(g(v)=\lambda
_{3}e^{k_{2}v}+\lambda _{4},\)
\(2)\) or \(f(u)=\mu _{1}u+\mu _{2}\) and \(g(v)=\mu _{3},\)
\(3)\) or \(g(v)=\nu _{1}v+\nu _{2}\) and \(f(u)=\nu _{3},\)
\(4)\) or \(f(u)=\zeta _{1}((1-\lambda )k_{1}u+\zeta _{2})^{\frac{1}{1-\lambda }
}+\zeta _{3}\) and \(g(v)=\zeta _{4}((\frac{\lambda -1}{\lambda })k_{2}v+\zeta
_{5})^{\frac{\lambda }{\lambda -1}}+\zeta _{6}\).
5. Minimal TH-surfaces in \(\mathbb{E}^{3}\)
Let \(M^{2}\) be a TH-surface in the Euclidean 3-space \(\mathbb{E}^{3}.\) Then,
\(M^{2}\) is parameterized by
\begin{equation*}
r(u,\text{ }v)=(u,\text{ }v,\text{ }A(f(u)+g(v))+Bf(u)g(v)),
\end{equation*}
where \(A\) and \(B\) are non-zero real numbers.
We have the natural frame \(\left\{ r_{u},\text{ }r_{v}\right\} \) given by
\begin{equation*}
r_{u}=(1,\text{ }0,\text{ }f^{\prime }\alpha ),\text{ \ }r_{v}=(0,\text{ }1,
\text{ }g^{\prime }\gamma ),
\end{equation*}
where \(\alpha =A+Bg\) and \(\gamma =A+Bf.\)
From this, the unit normal vector field \(N\) of \(M^{2}\) is given
by
\begin{equation*}
N=\frac{1}{W}(-\alpha f^{\prime },\text{ }-\gamma g^{\prime },
\text{ }1),
\end{equation*}
where \(W=\sqrt{1+f^{\prime 2}\alpha ^{2}+g^{\prime 2}\gamma ^{2}}.\)
The coefficients of the first fundamental form of \(M^{2}\) are given by
\begin{equation*}
E=1+f^{\prime 2}\alpha ^{2},\text{ }G=1+g^{\prime 2}\gamma ^{2},\text{ }
F=f^{\prime }g^{\prime }\alpha \gamma .
\end{equation*}
The coefficients of the second fundamental form of the surface are
\begin{equation*}
L=\frac{\alpha f^{\prime \prime }}{W},\text{ \ }M=\frac{Bf^{\prime
}g^{\prime }}{W},\text{ }N=\frac{\gamma g^{\prime \prime }}{W}.
\end{equation*}
Hence, the mean curvature \(H\) and the Gaussian curvature \(K\) are given by,
respectively
\begin{equation}
H=\frac{\alpha f^{\prime \prime }(1+g^{\prime 2}\gamma ^{2})-2B\alpha \gamma
f^{\prime 2}g^{\prime 2}+\gamma g^{\prime \prime }(1+f^{\prime 2}\alpha ^{2})
}{2W^{3}}, \label{MEAN CURVA 1}
\end{equation}
(17)
\begin{equation}
K=\frac{\alpha \gamma f^{\prime \prime }g^{\prime \prime }-B^{2}f^{\prime
2}g^{\prime 2}}{EG-F^{2}}. \label{GAUSS CURVA}
\end{equation}
(18)
If the surface is minimal, that is, \(H=0\) on \(M^{2}\), we have from (17)
\begin{equation*}
\alpha f^{\prime \prime }(1+g^{\prime 2}\gamma ^{2})-2B\alpha \gamma
f^{\prime 2}g^{\prime 2}+\gamma g^{\prime \prime }(1+f^{\prime 2}\alpha
^{2})=0.
\end{equation*}
The previous equation may be rewritten as
\begin{equation}
\alpha \gamma ^{\prime \prime }(B^{2}+\alpha ^{\prime 2}\gamma ^{2})-2\alpha
\gamma \alpha ^{\prime 2}\gamma ^{\prime 2}+\gamma \alpha ^{\prime \prime
}(B^{2}+\gamma ^{\prime 2}\alpha ^{2})=0. \label{surface minimal v}
\end{equation}
(19)
Since the roles of \(\alpha \) and \(\gamma \) in (19) are
symmetric, we only discuss the cases according to the function \(\gamma \). We
distinguish cases.
Case 1. Let \(\gamma ^{\prime }=0\). In this case (19) gives \(B^{2}\gamma \alpha ^{\prime \prime }=0.\)
- If \(\gamma =0\), then \(f(u)=-\frac{A}{B}\), \(M^{2}\) is the horizontal
plane of equation \(z=-\frac{A^{2}}{B}\).
- If \(\alpha ^{\prime \prime }=0\), then \(g(v)=av+b,\) \(a,\) \(b\in\mathbb{R}\), and \(f(u)=c,\) \(c\in\mathbb{R}, M^{2}\) is the plane of equation \(z=c_{1}v+c_{2},\) \(c_{1},\) \(c_{2}\in\mathbb{R}\).
Case 2. Let \(\gamma ^{\prime \prime }=0\) and \(\gamma ^{\prime }\neq
0\), and by symmetry, \(\alpha ^{\prime }\neq 0\). Then \(\gamma (u)=\lambda
u+\delta _{1},\) \((\lambda ,\) \(\delta )\in\mathbb{R}^{\ast }\times\mathbb{R}\) and \(\alpha \) is a solution of the following
ODE
\begin{equation}
-2\lambda ^{2}\alpha \alpha ^{\prime 2}+\alpha ^{\prime \prime
}(B^{2}+\lambda ^{2}\alpha ^{2})=0. \label{surface minimal22}
\end{equation}
(20)
Then the general solution of (20) is given by
\begin{equation*}
\alpha (v)=\frac{B}{\lambda }\tan (\lambda _{1}v+\lambda _{2}),\text{ }
\lambda _{1},\text{ }\lambda _{2}\in\mathbb{R}.
\end{equation*}
Hence
\begin{equation*}
g(v)=\frac{1}{\lambda }\tan (\lambda _{1}v+\lambda _{2})-\frac{A}{B},\text{ }
\lambda _{1},\text{ }\lambda _{2}\in\mathbb{R}.
\end{equation*}
So, the parametrization of \(M^{2}\) can be written in the form
\begin{equation*}
r(u,\text{ }v)=(u,\text{ }v,\text{ }\lambda _{3}u+\delta _{2}+\frac{A}{
\lambda }\tan (\lambda _{1}v+\lambda _{2})-\frac{A^{2}}{B}+B(\lambda
_{3}u+\delta _{2})(\frac{1}{\lambda }\tan (\lambda _{1}v+\lambda _{2})-\frac{
A}{B})),
\end{equation*}
where \((\lambda _{3},\) \(\delta _{2})\in\mathbb{R}^{\ast }\times\mathbb{R}.\)
Case 3. Let \(\gamma ^{\prime \prime }\neq 0.\) By symmetry in the
discussion of the case, we also suppose \(\alpha ^{\prime \prime }\neq 0\). If
we divide (19) by \(\alpha \gamma \alpha ^{\prime
2}\gamma ^{\prime 2},\) we obtain
\begin{equation*}
\frac{B^{2}\gamma ^{\prime \prime }}{\gamma \alpha ^{\prime 2}\gamma
^{\prime 2}}+\frac{\gamma \gamma ^{\prime \prime }}{\gamma ^{\prime 2}}+
\frac{B^{2}\alpha ^{\prime \prime }}{\alpha \alpha ^{\prime 2}\gamma
^{\prime 2}}+\frac{\alpha \alpha ^{\prime \prime }}{\alpha ^{\prime 2}}-2=0.
\end{equation*}
Thus, after a derivation with respect to \(u\), followed by a derivation with
respect to \(v,\) we obtain
\begin{equation*}
\left( \frac{\gamma ^{\prime \prime }}{\gamma \gamma ^{\prime 2}}\right)
_{,u}\left( \frac{1}{\alpha ^{\prime 2}}\right) _{,v}+\left( \frac{\alpha
^{\prime \prime }}{\alpha \alpha ^{\prime 2}}\right) _{,v}\left( \frac{1}{
\gamma ^{\prime 2}}\right) _{,u}=0.
\end{equation*}
Hence we deduce the existence of a real number \(k\in\mathbb{R}\) such that
\begin{equation}
\left\{
\begin{array}{c}
\left( \frac{\gamma ^{\prime \prime }}{\gamma \gamma ^{\prime 2}}\right)
_{,u}=k\left( \frac{1}{\gamma ^{\prime 2}}\right) _{,u} \\
\left( \frac{\alpha ^{\prime \prime }}{\alpha \alpha ^{\prime 2}}\right)
_{,v}=-k\left( \frac{1}{\alpha ^{\prime 2}}\right) _{,v}.
\end{array}
\right. \label{surface minimal333}
\end{equation}
(21)
The first equation of (21) can integrate obtaining
\begin{equation}
\gamma ^{\prime \prime }=\gamma (k+b_{1}\gamma ^{\prime 2}).
\label{Equi 111}
\end{equation}
(22)
From the second equation in (21), we obtain
\begin{equation}
\alpha ^{\prime \prime }=-\alpha (k+b_{2}\alpha ^{\prime 2}).
\label{Equi 222}
\end{equation}
(23)
Substituting the above in (19), we get
\begin{equation*}
\alpha \gamma ((k+b_{1}\gamma ^{\prime 2})(B^{2}+\alpha ^{\prime 2}\gamma
^{2})-2\alpha ^{\prime 2}\gamma ^{\prime 2}-(k+b_{2}\alpha ^{\prime
2})(B^{2}+\gamma ^{\prime 2}\alpha ^{2}))=0.
\end{equation*}
If we simplify by \(\alpha \gamma \) and then we divide by \(\alpha ^{\prime
2}\gamma ^{\prime 2}\), we get
\begin{equation*}
\frac{k\gamma ^{2}-b_{2}B^{2}}{\gamma ^{\prime 2}}-2+b_{1}\gamma ^{2}=\frac{
k\alpha ^{2}-b_{1}B^{2}}{\alpha ^{\prime 2}}+b_{2}\alpha ^{2}.
\end{equation*}
Hence, we deduce the existence of a real number \(\lambda \in\mathbb{R}\) such that
\begin{equation}
\left\{
\begin{array}{c}
\gamma ^{\prime 2}=\frac{k\gamma ^{2}-b_{2}B^{2}}{\lambda +2-b_{1}\gamma ^{2}
} \\
\alpha ^{\prime 2}=\frac{k\alpha ^{2}-b_{1}B^{2}}{\lambda -b_{2}\alpha ^{2}}.
\end{array}
\right. \label{surface minimal444}
\end{equation}
(24)
Differentiating with respect to \(u\) and \(v\), respectively, we have
\begin{equation}
\left\{
\begin{array}{l}
\gamma ^{\prime \prime }=\frac{\gamma (\lambda k+2k-b_{1}b_{2}B^{2})}{
(\lambda +2-b_{1}\gamma ^{2})^{2}} \\
\alpha ^{\prime \prime }=\frac{\alpha (\lambda k-b_{1}b_{2}B^{2})}{(\lambda
-b_{2}\alpha ^{2})^{2}}.
\end{array}
\right. \label{surface minimal555}
\end{equation}
(25)
Let us compare these expressions of \(\alpha ^{\prime \prime }\) and \(\gamma
^{\prime \prime }\) with those ones that appeared in (22) and (
23) and replace the value of \(\gamma ^{\prime 2}\) and \(\alpha
^{\prime 2}\) obtained in (24). We get
\begin{equation*}
(\lambda k+2k-b_{1}b_{2}B^{2})(1+\lambda -b_{1}\gamma ^{2})=0,
\end{equation*}
\begin{equation*}
(\lambda k-b_{1}b_{2}B^{2})(\lambda -1-b_{2}\alpha ^{2})=0.
\end{equation*}
We discuss all possibilities.
- If \(\lambda k+2k-b_{1}b_{2}B^{2}=0\) and \(\lambda k-b_{1}b_{2}B^{2}=0\),
then \(k=0\) and \(b_{1}b_{2}=0\). Then (25) gives \(\gamma
^{\prime \prime }=0\) and \(\alpha ^{\prime \prime }=0\), a contradiction.
- If \(\lambda k+2k-b_{1}b_{2}B^{2}=0\), \(\lambda =1\) and \(b_{2}=0,\) we
obtain \(k=0\). Then (25) gives \(\gamma ^{\prime \prime
}=0\) and \(\alpha ^{\prime \prime }=0\), a contradiction.
- If \(\lambda k-b_{1}b_{2}B^{2}=0\), \(\lambda =-1\) and \(b_{1}=0,\) we
obtain \(k=0\). Then (25) gives \(\gamma ^{\prime \prime
}=0\) and \(\alpha ^{\prime \prime }=0\), a contradiction.
- If \(1+\lambda -b_{1}\gamma ^{2}=0\) and \(\lambda -1-b_{2}\alpha ^{2}=0,\)
we deduce that \(\alpha \), \(\gamma \) are both constant functions, and so, \(
\gamma ^{\prime \prime }=0\) and \(\alpha ^{\prime \prime }=0\), a
contradiction.
Therefore, we have the following:
Theorem 9.
Let \(M^{2}\) be a TH-surface in \(\mathbb{E}^{3}.\) If \(M^{2}\) is minimal
surface, then \(M^{2}\) is plane or parameterized as
\begin{equation*}
r(u,v)=(u,\text{ }v,\text{ }A(f(u)+g(v))+Bf(u)g(v)),
\end{equation*}
where
- either \(f(u)=\frac{\lambda _{1}}{B}u+\frac{\lambda _{2}-A}{B}\) and \(
g(v)=\frac{1}{\lambda _{1}}\tan (\lambda _{3}v+\lambda _{4})-\frac{A}{B}\) or
- [ii)] \(f(u)=\frac{1}{\lambda _{1}}\tan (\lambda _{2}u+\lambda _{3})-\frac{A}{
B}\) and \(g(v)=\frac{\lambda _{1}}{B}v+\frac{\lambda _{4}-A}{B}\).
6. TH-surfaces with zero Gaussian curvature in \(\mathbb{E}^{3}\)
A surface in Euclidean 3-space parameterized by (1) has Gaussian
curvature
\begin{equation*}
K=\frac{\alpha \gamma f^{\prime \prime }g^{\prime \prime }-B^{2}f^{\prime
2}g^{\prime 2}}{EG-F^{2}}.
\end{equation*}
Hence that if \(K=0\), then
\begin{equation}
\alpha \gamma \alpha ^{\prime \prime }\gamma ^{\prime \prime }-\gamma
^{\prime 2}\alpha ^{\prime 2}=0. \label{Gauss curvature 1}
\end{equation}
(26)
Since the roles of the function \(\gamma \) and \(\alpha \) are symmetric in (
26), we discuss the different cases according the
function \(\gamma .\)
Case 1. Let \(\gamma ^{\prime }=0\). In this case \(\gamma \) is a
constant function \(\gamma (u)=u_{0}\) and the parametrization of (1) writes as
\begin{equation*}
r(u,v)=(u,\text{ }v,\text{ }\delta _{1}g(v)+\delta _{2}).
\end{equation*}
This means that \(M^{2}\) is a cylindrical surface with base curve a plane
curve in the \(vz-\) plane.
Case 2. Let \(\gamma ^{\prime \prime }=0\) and \(\gamma ^{\prime }\neq
0\). Then \(\gamma (u)=\lambda u+\delta _{1},\) \((\lambda ,\) \(\delta )\in\mathbb{R}^{\ast }\times\mathbb{R}.\) Moreover, (26) gives \(\alpha ^{\prime }=0\) and \(
\alpha (v)=\) \(v_{0}\) is a constant function. Now \(M^{2}\) is the plane of
equation \(z(u,v)=\lambda u+\delta _{1},\) \(\lambda ,\) \(\delta _{1}\in\mathbb{R}.\)
Case 3. Let \(\gamma ^{\prime \prime }\neq 0.\) By the symmetry on
the arguments, we also suppose \(\alpha ^{\prime \prime }\neq 0\).
Equation (26) writes as
\begin{equation*}
\frac{\gamma \gamma ^{\prime \prime }}{\gamma ^{\prime 2}}=\frac{\alpha
^{\prime 2}}{\alpha \alpha ^{\prime \prime }}.
\end{equation*}
Therefore, there exists a real number \(\lambda \in\mathbb{R}^{\ast }\) such that
\begin{equation*}
\frac{\gamma \gamma ^{\prime \prime }}{\gamma ^{\prime 2}}=\lambda =\frac{
\alpha ^{\prime 2}}{\alpha \alpha ^{\prime \prime }}.
\end{equation*}
Integrate these equations
\begin{equation}
\left\{
\begin{array}{l}
\gamma ^{\prime }=k_{1}\gamma ^{\lambda } \\
\alpha ^{\prime }=k_{2}\alpha ^{\frac{1}{\lambda }},
\end{array}
\right. \label{Equation1a}
\end{equation}
(27)
where \(k_{1}\) and \(k_{2}\) are constants of integration.
- If \(\lambda =1\), the general solution of (27) is given by
\begin{equation*}
\left\{
\begin{array}{c}
\gamma (u)=\lambda _{1}e^{k_{1}u} \\
\alpha (v)=\lambda _{2}e^{k_{2}v},
\end{array}
\right.
\end{equation*}
where \(\lambda _{1}\) and \(\lambda _{2}\) are constants of integration.
Hence
\begin{equation*}
\left\{
\begin{array}{l}
f(u)=\lambda _{3}e^{k_{1}u}+\lambda _{4} \\
g(v)=\lambda _{5}e^{k_{2}v}+\lambda _{6},
\end{array}
\right.
\end{equation*}
where \(\lambda _{3}\), \(\lambda _{4},\) \(\lambda _{5}\), \(\lambda _{6}\in\mathbb{R}\).
- If \(\lambda \neq 1\), the general solution of (27) is
given by
\begin{equation*}
\left\{
\begin{array}{c}
\gamma (u)=((1-\lambda )k_{1}u+c_{1})^{\frac{1}{1-\lambda }} \\
\alpha (v)=((\frac{\lambda -1}{\lambda })k_{2}v+c_{2})^{\frac{\lambda }{
\lambda -1}},
\end{array}
\right.
\end{equation*}
where \(c_{1}\) and \(c_{2}\) are constants of integration.
Hence
\begin{equation*}
\left\{
\begin{array}{l}
f(u)=c_{3}((1-\lambda )k_{1}u+c_{1})^{\frac{1}{1-\lambda }}+c_{4} \\
g(v)=c_{5}((\frac{\lambda -1}{\lambda })k_{2}v+c_{2})^{\frac{\lambda }{
\lambda -1}}+c_{6},
\end{array}
\right.
\end{equation*}
where \(c_{3}\), \(c_{4},\) \(c_{5}\), \(c_{6}\in\mathbb{R}\).
Theorem 10.
Let \(M^{2}\) be a TH-surface in Euclidean \(3-\) space \(\mathbb{E}^{3}\) with
constant Gauss curvature \(K\). Then \(K=0\). Furthermore, the surface is plane
or is a cylindrical surface over a plane curve or parameterized as
\begin{equation*}
r(u,v)=(u,\text{ }v,\text{ }A(f(u)+g(v))+Bf(u)g(v)),
\end{equation*}
where
- either \(f(u)=\lambda _{3}e^{k_{1}u}+\lambda _{4}\) and \(g(v)=\lambda
_{5}e^{k_{2}v}+\lambda _{6}\) or
- \(f(u)=c_{3}((1-\lambda )k_{1}u+c_{1})^{\frac{1}{1-\lambda }}+c_{4}\)
and \(g(v)=c_{5}((\frac{\lambda -1}{\lambda })k_{2}v+c_{2})^{\frac{\lambda }{
\lambda -1}}+c_{6}\).
Acknowledgments
The authors would like to express their thanks to the referee for his useful remarks.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing Interests
The author(s) do not have any competing interests in the manuscript.
