# Vertically invariant minimal surfaces in unimodular semidirect products

David Moya 111This work is supported in part by the IMAG–Maria de Maeztu grant
CEX2020-001105-M / AEI /10.13039/501100011033, MICINN grant
PID2020-117868GB-I00 and Junta de Andalucía grant A-FQM-139-UGR18.

###### Abstract

A surface in a three-dimensional metric Lie group $G$ is said invariant if it
is invariant with respect to a one-dimensional subgroup $\Gamma$ of the
isometry group of $G$. Is this work we focus on unimodular metric Lie groups
$G$ that can be written as a semidirect product of the form
$\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ for certain matrix
$A\in\mathcal{M}_{2}(\mathbb{R})$ and study the minimal surfaces which are
invariant under the group $\Gamma$ generated by left translations by elements
in the vertical axis $\\{0\\}\rtimes\mathbb{R}$. We will call these surfaces
vertically invariant. In particular, we describe new examples of minimal
surfaces in $\widetilde{E}(2)$ which are vertically invariant.

##  1 Introduction

Constant mean curvature (CMC) surfaces (including the minimal case) in
homogeneous spaces is a field of interest in recent years, specially after the
generalization of the holomorphic Hopf differential by Abresch and Rosenberg
[1] for CMC surfaces in homogeneous spaces with isometry group of dimension 4,
the so-called $\mathbb{E}(\kappa,\tau)$ spaces. A non-holomorphic Hopf-type
quadratic differential for CMC surfaces in three-dimensional metric Lie groups
has also been developed in [10].

Among the family of CMC surfaces in a three-dimensional Riemannian manifold,
it is natural to study and describe the surfaces that satisfy a further
geometric extrinsic condition. For instance, if the ambient space $G$ is
globally endowed with coordinates $(x,y,z)\in G$, we can look for surfaces
given by a graph $z(x,y)=f(x)+g(y)$, where $f$, $g$ are smooth functions on
some interval of $\mathbb{R}$ (they are called translation surfaces), as it is
done in the works [5] and [6]; we can also impose the condition of being
rotationally symmetric, as for example in [13]; or, as we do in this study, we
can assume that the ambient space is a metric Lie group and our surface is
invariant under a one-parameter group of left translations of this metric Lie
group, see [7],[8],[12]. It is also natural to impose these geometric
conditions together with constant Gaussian curvature, see [14]. Imposing these
conditions usually leads to specific examples of surfaces with the desired
properties.

The background spaces of this work are metric Lie groups. We will call metric
Lie group to a simply connected 3-dimensional Lie group endowed with a left
invariant metric. On a Lie group $G$ there is a measure invariant under left
translations which is called the Haar measure. When the Haar measure is also
invariant under right translations, the Lie group $G$ is called unimodular.
See [10] for a complete description of unimodular and non-unimodular metric
Lie groups. A particularly interesting algebraic condition on metric Lie
groups is the property of being written as a semidirect product. Semidirect
products comprise all the non-unimodular metric Lie groups and among the
unimodular family there are only two cases that can not be written as
semidirect products: the special unitary group $SU(2)$ and the universal cover
$\widetilde{SL}(2,\mathbb{R})$ of the special linear group.

We organise this work as follows. In Section 2 we give some preliminaries and
introduce notation involving metric Lie groups, focusing on the properties we
need to develop the remaining sections. In Section 3 we get an ODE for the
horizontal curve which generates, through vertical translations, an invariant
CMC surface. In Section 4 we particularize this equation for CMC surfaces when
the metric Lie group is unimodular and can be written as a semidirect product.
Finally, in Sections 5 and 6 we describe the vertically invariant subclass of
minimal surfaces in the Heisenberg group ${\rm Nil}_{3}$ and in the universal
cover $\widetilde{E}(2)$ of the group of orientation-preserving rigid motions
of the Euclidean plane. When the ambient space is
$\mathbb{R}^{3}=\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ with
$A=0\in\mathcal{M}_{2}(\mathbb{R})$, it is well-known that vertical planes are
the only vertically invariant minimal surfaces. When $G={\rm Sol}_{3}$, the
vertically invariant minimal surfaces were studied in [7] and [8]. From
another point of view, the vertically invariant CMC surfaces of the Heisenberg
group $\text{Nil}_{3}$ were classified in [3]. In Section 7, we also describe
vertically invariant surfaces with zero Gaussian curvature in these ambient
spaces.

##  2 Preliminaries on metric Lie groups

The results in this preliminary section are detailed in [10]. For a matrix
$A\in\mathcal{M}_{2}(\mathbb{R})$ of the form

$A=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right),$

where $a,b,c,d\in\mathbb{R}$, we can consider the semidirect product
$G=\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$.

Consider $(p_{1},z_{1}),(p_{2},z_{2})\in\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$,
the operation $*$ of the semidirect product $G$ is given by:

$(p_{1},z_{1})*(p_{2},z_{2})=(p_{1}+e^{z_{1}A}p_{2},z_{1}+z_{2}).$ (1)

We choose global coordinates $(x,y)\in\mathbb{R}^{2}$, $z\in\mathbb{R}$ so
that $\partial_{x}=\frac{\partial}{\partial
x},\partial_{y}=\frac{\partial}{\partial
y},\partial_{z}=\frac{\partial}{\partial z}$ generates the space of
differentiable vector fields on $G$, $\mathfrak{X}(G)$, and denote

$e^{zA}=\left(\begin{array}[]{cc}a_{11}(z)&a_{12}(z)\\\
a_{21}(z)&a_{22}(z)\end{array}\right).$

With this notation, the set $\\{E_{1},E_{2},E_{3}\\}$ given by:

$E_{1}(x,y,z)=a_{11}(z)\partial_{x}+a_{21}(z)\partial_{y},\quad
E_{2}(x,y,z)=a_{12}(z)\partial_{x}+a_{22}(z)\partial_{y},\quad
E_{3}=\partial_{z}$ (2)

is a basis of the Lie algebra $\mathfrak{g}$ of $G$. We consider the canonical
left invariant metric $\langle,\rangle$ on
$\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$, which is the one that extends via left
translations the usual inner product on $T_{e}G$, $e=(0,0,0)$ or equivalently
for which the left invariant basis $\\{E_{1},E_{2},E_{3}\\}$ is orthonormal.
$\langle,\rangle$ takes the following form when we consider the basis
$\\{\partial_{x},\partial_{y},\partial_{z}\\}$:

$\displaystyle\langle,\rangle$ $\displaystyle=$
$\displaystyle[a_{11}(-z)^{2}+a_{21}(-z)^{2}]\,dx^{2}+[a_{12}(-z)^{2}+a_{22}(-z)^{2}]\,dy^{2}+dz^{2}$
(3) $\displaystyle+[a_{11}(-z)a_{12}(-z)+a_{21}(-z)a_{22}(-z)](dx\otimes
dy+dy\otimes dx).$

The Levi-Civita connection $\nabla$ for the canonical metric of $G$ is
determined by:

$\begin{array}[]{l|l|l}\displaystyle\nabla_{E_{1}}E_{1}=aE_{3}&\displaystyle\nabla_{E_{1}}E_{2}=\frac{b+c}{2}E_{3}&\displaystyle\nabla_{E_{1}}E_{3}=-aE_{1}-\frac{b+c}{2}E_{2}\\\
\displaystyle\nabla_{E_{2}}E_{1}=\frac{b+c}{2}E_{3}&\displaystyle\nabla_{E_{2}}E_{2}=dE_{3}&\displaystyle\nabla_{E_{2}}E_{3}=-\frac{b+c}{2}E_{1}-dE_{2}\\\
\displaystyle\nabla_{E_{3}}E_{1}=\frac{c-b}{2}E_{2}&\displaystyle\nabla_{E_{3}}E_{2}=\frac{b-c}{2}E_{1}&\displaystyle\nabla_{E_{3}}E_{3}=0.\end{array}$
(4)

Moreover, we also need the following right invariant basis of vector fields of
$G$ (in particular, these are killing vector fields):

$F_{1}=\partial_{x},\quad F_{2}=\partial_{y},\quad
F_{3}(x,y,z)=(ax+by)\partial_{x}+(cx+dy)\partial_{y}+\partial_{z}.$ (5)

In this context, we are focused on studying CMC surfaces which are invariant
under certain 1-parameter subgroups of left translations. In our case we
choose vertical translations. We begin by considering a surface $\Sigma$ which
is generated by a curve $\gamma\subset\mathbb{R}^{2}\rtimes_{A}\\{0\\}$,
$\gamma(t)=(x(t),y(t),0)$ via left translations by elements of the form
$(0,0,z)$, $z\in\mathbb{R}$. A parametrization of $\Sigma$ can be given by
$\Phi(t,s)=\phi_{s}(\gamma(t))$ where

$\phi_{s}(x,y,z)=(0,0,s)*(x,y,z)=(e^{sA}(x,y)^{T},s+z),$ (6)

where $(x,y)^{T}$ is the transpose of the row $(x,y)$.

######  Definition 1.

If ${\gamma}$ is a curve of the form $\gamma(t)=(x(t),y(t),0)$, we will call
vertically invariant surface to the image through the parametrization
$\Phi(t,s)=\phi_{s}(\gamma(t))$.

##  3 The mean curvature of an invariant surface through vertical translations

Let $\Sigma$ be a vertically invariant surface. We next compute the
coefficients of the first and second fundamental forms of $\Sigma$, and relate
them to the geometry of $\gamma$.

$\Phi_{t}(t,0)=\gamma^{\prime}(t)=\left(\begin{array}[]{c}x^{\prime}(t)\\\
y^{\prime}(t)\\\
0\end{array}\right)^{T}=\left[\begin{array}[]{c}x^{\prime}(t)\\\
y^{\prime}(t)\\\ 0\end{array}\right]^{T}$ (7)

where we are using parenthesis for coordinates with respect to
$\\{\partial_{x},\partial_{y},\partial_{z}\\}$ and brackets for coordinates
with respect to $\\{E_{1},E_{2},E_{3}\\}$. We impose that $\gamma$ is arc
length parameterized (i.e. $(x^{\prime})^{2}+(y^{\prime})^{2}=1$, observe that
$\langle,\rangle$ coincides with the usual inner product of $\mathbb{R}^{3}$
along $\mathbb{R}^{2}\rtimes_{A}\\{0\\}$), and let $\theta=\theta(t)$ be a
differentiable function such that

$x^{\prime}(t)=\cos\theta(t),\quad y^{\prime}(t)=\sin\theta(t).$ (8)

In order to compute $\Phi_{s}$, we use that
$\Phi_{s}(t,s)=(F_{3})_{\Phi(t,s)}$:

$F_{3}(x,y,z)=\left(\begin{array}[]{c}\delta\\\ \varepsilon\\\
1\end{array}\right)^{T}=\left[\begin{array}[]{c}\delta a_{11}(-z)+\varepsilon
a_{12}(-z)\\\ \delta a_{21}(-z)+\varepsilon a_{22}(-z)\\\
1\end{array}\right]^{T},$ (9)

(see equation 5.9 in [9]) where $\delta(x,y)=ax+by$, $\varepsilon(x,y)=cx+dy$.
Thus,

$\Phi_{s}(t,0)=(F_{3})_{\gamma(t)}=\left[\begin{array}[]{c}\delta(t)\\\
\varepsilon(t)\\\ 1\end{array}\right]^{T},$ (10)

where $\delta(t)=\delta(\gamma(t))$ and
$\varepsilon(t)=\varepsilon(\gamma(t))$. With $\Phi_{t}$ and $\Phi_{s}$ we can
compute the coefficients of the first fundamental form:

$E=(x^{\prime})^{2}+(y^{\prime})^{2}=1,\quad
F=\delta(t)\cos\theta(t)+\varepsilon(t)\sin\theta(t),\quad
G=1+\delta(t)^{2}+\varepsilon(t)^{2}.$ (11)

Using the expressions for $\nabla_{E_{i}}E_{j}$, $i,j=1,2,3$ given in (4) we
compute:

$\displaystyle\begin{split}\cdot&\nabla_{\Phi_{t}}\Phi_{t}(t,0)=\frac{D}{dt}\left(\cos\theta(t){E_{1}}_{\gamma(t)}+\sin\theta(t){E_{2}}_{\gamma(t)}\right)\\\
&=-\theta^{\prime}(t)\sin\theta(t)E_{1}+\theta^{\prime}(t)\cos\theta(t)E_{2}+\cos\theta(t)\nabla_{\gamma^{\prime}(t)}E_{1}+\sin\theta(t)\nabla_{\gamma^{\prime}(t)}E_{2}\\\
&=\left[\begin{array}[]{c}-\theta^{\prime}(t)\sin\theta(t)\\\
\theta^{\prime}(t)\cos\theta(t)\\\
0\end{array}\right]^{T}+\cos^{2}\theta(t)aE_{3}+\cos\theta(t)\sin\theta(t)\frac{b+c}{2}E_{3}\\\
&+\sin\theta(t)\cos\theta(t)\frac{b+c}{2}E_{3}+\sin^{2}\theta(t)dE_{3}\\\
&=\left[\begin{array}[]{c}-\theta^{\prime}(t)\sin\theta(t)\\\
\theta^{\prime}(t)\cos\theta(t)\\\
a\cos^{2}\theta(t)+d\sin^{2}\theta(t)+\cos\theta(t)\sin\theta(t)(b+c)\end{array}\right]^{T}\end{split}$
(12)
$\displaystyle\begin{split}\cdot&\nabla_{\Phi_{t}}\Phi_{s}(t,0)=\frac{D}{dt}(\delta(t){E_{1}}_{\gamma(t)}+\varepsilon(t){E_{2}}_{\gamma(t)}+{E_{3}}_{\gamma(t)})\\\
&=\left[\begin{array}[]{c}\delta^{\prime}(t)\\\ \varepsilon^{\prime}(t)\\\
0\end{array}\right]^{T}+\delta(t)\nabla_{\gamma^{\prime}(t)}E_{1}+\varepsilon(t)\nabla_{\gamma^{\prime}(t)}E_{2}+\nabla_{\gamma^{\prime}(t)}E_{3}\\\
&=\left[\begin{array}[]{c}\delta^{\prime}(t)\\\ \varepsilon^{\prime}(t)\\\
0\end{array}\right]^{T}+\delta(t)\cos\theta(t)aE_{3}+\delta(t)\sin\theta(t)\frac{b+c}{2}E_{3}+\varepsilon(t)\cos\theta(t)\frac{b+c}{2}E_{3}\\\
&+\varepsilon(t)\sin\theta(t)dE_{3}+\cos\theta(t)\left(-aE_{1}-\frac{b+c}{2}E_{2}\right)+\sin\theta(t)\left(-\frac{b+c}{2}E_{1}-dE_{2}\right)\\\
&=\left[\begin{array}[]{c}\sin\theta(t)\frac{b-c}{2}\vspace{0.3cm}\\\
\cos\theta(t)\frac{c-b}{2}\vspace{0.3cm}\\\
\delta(t)\left(\cos\theta(t)a+\sin\theta(t)\frac{b+c}{2}\right)+\varepsilon(t)\left(\cos\theta(t)\frac{b+c}{2}+\sin\theta(t)d\right)\end{array}\right]^{T}\end{split}$
(13)
$\displaystyle\begin{split}\cdot&\nabla_{\Phi_{s}}\Phi_{s}(t,0)=\left(\nabla_{\delta(t)E_{1}+\varepsilon(t)E_{2}+E_{3}}\delta(t)E_{1}+\varepsilon(t)E_{2}+E_{3}\right)(t,0)\\\
&=\delta(t)^{2}aE_{3}+\delta(t)\varepsilon(t)\frac{b+c}{2}E_{3}+\delta(t)\left(-aE_{1}-\frac{b+c}{2}E_{2}\right)+\varepsilon(t)\delta(t)\frac{b+c}{2}E_{3}+\varepsilon(t)^{2}dE_{3}\\\
&+\varepsilon(t)\left(-\frac{b+c}{2}E_{1}-dE_{2}\right)+\delta(t)\frac{c-b}{2}E_{2}+\varepsilon(t)\frac{b-c}{2}E_{1}\\\
&=\left[\begin{array}[]{c}-c\varepsilon(t)-a\delta(t)\\\
-b\delta(t)-d\varepsilon(t)\\\
\delta(t)\varepsilon(t)(b+c)+a\delta(t)^{2}+d\varepsilon(t)^{2}\end{array}\right]^{T}\end{split}$
(14)

A unitary normal field to $\Sigma$ is

$N=\frac{\Phi_{t}\times\Phi_{s}}{|\Phi_{t}\times\Phi_{s}|}=\frac{\sin\theta
E_{1}-\cos\theta
E_{2}+(\varepsilon\cos\theta-\delta\sin\theta)E_{3}}{\sqrt{1+(\varepsilon\cos\theta-\delta\sin\theta)^{2}}}.$
(15)

With these computations the coefficients of the second fundamental form of
$\Sigma$ at $(t,0)$ become:

$e=\langle
N,\nabla_{\Phi_{t}}\Phi_{t}\rangle=\frac{(\delta\sin\theta-\varepsilon\cos\theta)[-(a-d)\cos(2\theta)-a-(b+c)\sin(2\theta)-d]-2\theta^{\prime}}{2\sqrt{1+(\sin\theta\delta-\cos\theta\varepsilon)^{2}}}$
(16) $f=\langle
N,\nabla_{\Phi_{t}}\Phi_{s}\rangle=\frac{(\varepsilon\cos\theta-\delta\sin\theta)[\cos\theta(2a\delta+(b+c)\varepsilon)+\sin\theta((b+c)\delta+2d\varepsilon)]+b-c}{2\sqrt{1+(\delta\sin\theta-\varepsilon\cos\theta)^{2}}}$
(17)
$g=\frac{\cos\theta\left[a\delta^{2}\varepsilon+\delta\left((b+c)\varepsilon^{2}+b\right)+d\varepsilon\left(\varepsilon^{2}+1\right)\right]-\sin\theta\left[\delta\left(a\delta^{2}+a+(b+c)\delta\varepsilon+d\varepsilon^{2}\right)+c\varepsilon\right]}{\sqrt{1+(\delta\sin\theta-\varepsilon\cos\theta)^{2}}}$
(18)

The mean curvature of $\Sigma$ is given by

$H=\frac{Eg-2Ff+eG}{2(EG-F^{2})},$

so imposing constant mean curvature $H\in\mathbb{R}$ for $\Sigma$ is
equivalent to the following system of ODEs:

$\left\\{\begin{array}[]{l}x^{\prime}=\cos\theta\\\ y^{\prime}=\sin\theta\\\
\eqref{21}\end{array},\right.$ (19)

where (20) is the ODE

$\begin{split}&8\left[((cx+dy)\cos\theta-(ax+by)\sin\theta)^{2}+1\right]^{3/2}H=-4\theta^{\prime}\left[1+(ax+by)^{2}+(cx+dy)^{2}\right]\\\
&-\sin\theta[(ax+by)\left[3(a+d)\left((ax+by)^{2}+(cx+dy)^{2}\right)+5a+3d\right]+(3b-c)(cx+dy)]\\\
&+\sin(3\theta)[(ax+by)\left[(a+d)\left((ax+by)^{2}-3(cx+dy)^{2}\right)-a+d\right]+(b+c)(cx+dy)]\\\
&+\cos\theta[(cx+dy)\left[3(a+d)(ax+by)^{2}+3a+5d\right]-(b-3c)(ax+by)+3(a+d)(cx+dy)^{3}]\\\
&+\cos(3\theta)[-(cx+dy)\left[3(a+d)(ax+by)^{2}-a+d\right]+(b+c)(ax+by)+(a+d)(cx+dy)^{3}].\end{split}$
(20)

##  4 Unimodular semidirec products

Our goal is to study the solutions of the system (19) depending of the ambient
space (or equivalently depending of the matrix
$A\in\mathcal{M}_{2}(\mathbb{R})$). We are going to focus on unimodular metric
Lie groups. Using Theorem 2.15 of [10] we know that, after scaling the metric,
the matrix $A$ can be chosen as:

$A=0,\quad A=\left(\begin{array}[]{cc}0&\pm c\\\
\frac{1}{c}&0\end{array}\right),\quad c\in[1,\infty),\quad\text{or}\quad
A=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right),$

in the sense that a unimodular metric Lie group $G$ is isomorphic and
isometric to $\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ with its canonical metric.
Depending on the chosen matrix $A$, we find one of the following metric Lie
groups: $\mathbb{R}^{3}$, the solvable group ${\rm Sol}_{3}$ of orientation-
preserving rigid motions of the Lorentz-Minkowski plane, the universal cover
$\widetilde{E}(2)$ of the group of orientation-preserving rigid motions of the
Euclidean plane or the Heisenberg group ${\rm Nil}_{3}$. Next, we write the
equation (20) in these ambient spaces (we do not include here $\mathbb{R}^{3}$
since the only vertically invariant surfaces in $\mathbb{R}^{3}$ are vertical
planes):

  * •

If we consider $\text{Sol}_{3}$ with the metric that makes it isometric and
isomorphic to $\mathbb{R}^{2}\rtimes_{A(c)}\mathbb{R}$ with
$A(c)=\left(\begin{array}[]{cc}0&c\\\ 1/c&0\end{array}\right)$ with
$c\in[1,\infty)$, then (20) writes as

$H=\frac{((1+c^{2})\cos(2\theta)+1-c^{2})(c^{2}y\cos\theta+x\sin\theta)-2\theta^{\prime}(c^{2}+x^{2}+c^{4}y^{2})}{4c^{2}\left(1+\left(\frac{x\cos{\theta}}{c}-cy\sin\theta\right)^{2}\right)^{\frac{3}{2}}}.$
(21)

  * •

If $G=\text{Sol}_{3}$ with its standard metric, then $G$ is isometric and
isomorphic to $\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ with
$A=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$ (this is equivalent
to choose $A$ as in the previous item with $c=1$). Equation (20) writes as:

$H=\frac{\cos(2\theta)(x\sin\theta+y\cos\theta)-(1+x^{2}+y^{2})\theta^{\prime}}{2(1+(x\cos\theta-y\sin\theta)^{2})^{\frac{3}{2}}}.$
(22)

We are not going to deal with vertically invariant minimal surfaces in
$\text{Sol}_{3}=\mathbb{R}^{2}\rtimes_{A(1)}\mathbb{R}$ since they have
already been studied in [7].

  * •

If $G=\text{Nil}_{3}$ with its standard metric, which is isometric to
$\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ with $A=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)$, then equation (20) takes the form

$H=-\frac{y\cos\theta\sin^{2}\theta+(1+y^{2})\theta^{\prime}}{2(1+y^{2}\sin^{2}\theta)^{\frac{3}{2}}}.$
(23)

  * •

If we consider $\widetilde{E}(2)$ with the metric that makes it isometric and
isomorphic to $\mathbb{R}^{2}\rtimes_{A(c)}\mathbb{R}$ with
$A(c)=\left(\begin{array}[]{cc}0&-c\\\ 1/c&0\end{array}\right)$ with
$c\in[1,\infty)$, then (20) writes as

$H=\frac{\left[\left(c^{2}-1\right)\cos(2\theta)-c^{2}-1\right]\left(c^{2}y\cos\theta-x\sin\theta\right)-2\theta^{\prime}\left(c^{2}+x^{2}+c^{4}y^{2}\right)}{4\left[\left(x\cos\theta+c^{2}y\sin\theta\right)^{2}+c^{2}\right]\sqrt{1+\left(\frac{x\cos\theta}{c}+cy\sin\theta\right)^{2}}}.$
(24)

##  5 Vertically invariant minimal surfaces in $\text{Nil}_{3}$

In order to keep computations simple, we will focus on minimal surfaces, which
means that we will impose $H=0$. If $\Sigma$ is minimal then system (19) can
be written as:

$\left\\{\begin{array}[]{l}x^{\prime}=\cos\theta\\\ y^{\prime}=\sin\theta\\\
\theta^{\prime}=\frac{-y\cos\theta\sin^{2}\theta}{1+y^{2}}\end{array}.\right.$
(25)

The next proposition is clear by killing the previous numerator.

######  Proposition 2.

The only solutions of the system (25) that are obtained by choosing $\theta$
as a constant are given by

$(x(t),y(t),\theta)=(t,0,0)+(x(0),y(0),0),$ (26)
$(x(t),y(t),\theta)=(0,t,\pi/2)+(x(0),y(0),0).$ (27)

######  Proposition 3.

If $\left\\{x,y,\theta\right\\}$ is a solution of (25) with a nonconstant
$\theta$, then $\theta$ never attains the values
$\left\\{\frac{k\pi}{2},\,k\in\mathbb{Z}\right\\}$.

###### Proof.

Suppose that $\left\\{x,y,\theta\right\\}$ is a solution of (25) with a
nonconstant $\theta$ which satisfies that
$\theta\in\left\\{\frac{k\pi}{2},\,k\in\mathbb{Z}\right\\}$ at $t_{0}$. We can
consider the following solution of (25):

$\left\\{x(t_{0})+\cos\left(\frac{k\pi}{2}\right)(t-t_{0}),y(t_{0})+\sin\left(\frac{k\pi}{2}\right)(t-t_{0}),\frac{k\pi}{2}\right\\}.$

Due to the uniqueness of the initial value problem associated to (25), we have
$\theta(t)=\frac{k\pi}{2}$ which is a contradiction. ∎

######  Remark 4.

From Proposition 3 we have that the generating curve $\gamma$ of a solution of
(25) that is not in Proposition 2 is a graph over the axes $x$ and $y$.
Moreover, since (25) implies that the derivatives of the functions $x$, $y$
and $\theta$ are bounded, then an application of the Picard-Lindelöf Theorem
gives that the maximal interval of definition of $\gamma$ is $\mathbb{R}$.

In order to describe the solutions of (25) with a nonconstant $\theta$, we can
assume that $\theta(t)\in(0,\pi/2)$ for all $t\in\mathbb{R}$. We give a first
integral for the system (25):

$J(t):=(1+y(t)^{2})\tan^{2}\theta(t),$

which means that $J(t)$ is constant along any solution of (25). We rewrite $J$
as

$J=(1+y^{2})\frac{y^{\prime 2}}{1-y^{\prime 2}},$

which makes solving (25) equivalent to solve

$\left\\{\begin{array}[]{l}x^{\prime}=\cos\theta\\\ y^{\prime}=\sin\theta\\\
(1+y^{2})\frac{y^{\prime 2}}{1-y^{\prime
2}}=a\in[0,\infty)\end{array}.\right.$ (28)

We now define the following diffeomorphism from $\mathbb{R}$ to $\mathbb{R}$:

$\begin{array}[]{cccl}f\colon&\mathbb{R}&\rightarrow&\mathbb{R}\\\
&t&\mapsto&\frac{1}{2}(1+a)\ln(t+\sqrt{1+a+t^{2}})+\frac{1}{2}t\sqrt{1+a+t^{2}},\end{array}$

where $a\geq 0$. The last equation of (28) can be solved using direct
integration:

$y(t)=f^{-1}(\sqrt{a}t+c_{1}),\quad c_{1}\in\mathbb{R}.$ (29)

If $a=0$, from (28) we deduce that the solution of this system is given by
(27). Otherwise we can explicitly integrate the function $x$ in (28):

$x(t)=c_{2}+\frac{1}{2\sqrt{a}}\left(\text{arcsinh}(f^{-1}(\sqrt{a}t+c_{1}))+f^{-1}(\sqrt{a}t+c_{1})\sqrt{1+(f^{-1}(\sqrt{a}t+c_{1}))^{2}}\right),$
(30)

with $c_{1},c_{2}\in\mathbb{R}$. These expressions describe all vertically
invariant minimal surfaces in $\text{Nil}_{3}$.

######  Remark 5.

  1. Figure 1: The solution ${\gamma}$ of (28) with the initial conditions ${\gamma}(0)=(0,0)$, $a=1$.
  2. 1.

These surfaces are well-known; in fact CMC surfaces in ${\rm Nil}_{3}$
invariant under any one-dimensional subgroup of the isometry group of ${\rm
Nil}_{3}$ were classified by Figueroa-Mercuri-Pedrosa in [3]. To be more
precise, the surfaces given through (29) and (30) are described in Theorem 6
of [3], as we explain next.

In [3] Heisenberg space ${\rm Nil}_{3}$ is given as $\mathbb{R}^{3}$ with the
following inner product:

$(x_{1},y_{1},z_{1})(x_{2},y_{2},z_{2})=(x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}+\frac{1}{2}(x_{1}y_{2}-x_{2}y_{1})).$

Considering the canonical basis $\left\\{e_{1},e_{2},e_{3}\right\\}$ of
$\mathbb{R}^{3}$ then

$E_{1}=\frac{\partial}{\partial x}-\frac{y}{2}\frac{\partial}{\partial
z},\quad E_{2}=\frac{\partial}{\partial y}+\frac{x}{2}\frac{\partial}{\partial
z},\quad E_{3}=\frac{\partial}{\partial z}$

are the only left invariant vector fields verifying $(E_{k})_{0}=e_{k}$,
$k\in\left\\{1,2,3\right\\}$. The canonical metric of ${\rm Nil}_{3}$ is
defined as the only left invariant metric for which
$\left\\{e_{1},e_{2},e_{3}\right\\}$ is an orthonormal basis and, for this
metric, $\left\\{E_{1},E_{2},E_{3}\right\\}$ has to be an orthonormal basis of
vector fields of $\mathbb{R}^{3}$.

We can give an isomorphism (which is also a isometry) between this model of
$\text{Nil}_{3}$ and the semidirect product $\mathbb{R}\rtimes_{A}\mathbb{R}$
with $A=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right)$ (and therefore,
it identifies ${\rm Nil}_{3}$ and $\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ as
metric Lie groups) as follows:

$\begin{array}[]{cccl}\Xi:&{\rm
Nil_{3}}&\rightarrow&\mathbb{R}^{2}\rtimes_{A}\mathbb{R}\\\
&(x,y,z)&\mapsto&\left(z+\frac{xy}{2},y,x\right).\end{array}$

The minimal surfaces described in Theorem 6 of [3] are either vertical planes
or they are given through the equation

$z(x,y)=\frac{xy}{2}-c\left[\frac{y\sqrt{1+y^{2}}}{2}+\frac{1}{2}\ln\left(y+\sqrt{1+y^{2}}\right)\right].$

Using $\Xi$, the parametrization $\left(s,u,z(s,u)\right)$, $s,u\in\mathbb{R}$
is mapped into

$\left(su-c\left[\frac{u\sqrt{1+u^{2}}}{2}+\frac{1}{2}\ln\left(u+\sqrt{1+u^{2}}\right)\right],u,s\right)\in\mathbb{R}^{2}\rtimes_{A}\mathbb{R}.$

The generating curve contained in the plane $z=0$ of this surface is

$\left(-c\left[\frac{u\sqrt{1+u^{2}}}{2}+\frac{1}{2}\ln\left(u+\sqrt{1+u^{2}}\right)\right],u,0\right)\in\mathbb{R}^{2}\rtimes_{A}\mathbb{R},$

which generates (27) if $c=0$ and (29) and (30) when $c\not=0$ after using the
change of variables $u=y(t)$ and the equality
$\text{arcsinh}(x)=\ln\left(x+\sqrt{1+x^{2}}\right)$ for all $x\in\mathbb{R}$.
Vertical planes in $\text{Nil}_{3}$ are mapped by $\Xi$ into the vertical
planes generated by (26).

Figure 2: Vertically invariant minimal surface in $\text{Nil}_{3}$ generated
through $\Phi(t,s)=\phi_{s}(\gamma(t))$, with $\gamma(t)=(x(t),y(t),0)$, where
$x,y$ are solutions of the system (28) with initial conditions $x(0)=y(0)=0$
and $a=1$.

  3. 2.

The solution (26) in Proposition 2 gives through the parametrization
$\Phi(t,s)=\phi_{s}(\gamma(t))$, described in (6), a vertical plane. All
solutions of (25) that are not in Proposition 2 look similar to the one in
Figure 2.

  4. 3.

When $a\to+\infty$, from the equation $(1+y^{2})\cot\theta=a$ we have that
$\theta\to 0$ uniformly. Therefore the family of solutions described by (29)
and (30) converge to the solution (26) in Proposition 2 when $a\to+\infty$.

##  6 Vertically invariant minimal surfaces in $\widetilde{E}(2)$

If $G=\widetilde{E}(2)$ with its standard metric (isometric to the flat
$\mathbb{R}^{3}$), which is isometric and isomorphic to
$\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ with $A=\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)$, equation (20) can be written as

$H=\frac{x\sin\theta-y\cos\theta-(1+x^{2}+y^{2})\theta^{\prime}}{2(1+(x\cos\theta+y\sin\theta)^{2})^{\frac{3}{2}}}$
(31)

When $G=\widetilde{E}(2)$, the orbit of a point $(x,y,0)\in G$ is

$\phi_{s}(x,y,0)=(x\cos s-y\sin s,x\sin s+y\cos s,s),$

$s\in\mathbb{R}$, which describes a helicoid. Recall that $\widetilde{E}(2)$
is endowed with its standard flat metric, therefore, vertically invariant CMC
surfaces in $\widetilde{E}(2)$ are a subset of the classical CMC surfaces in
$\mathbb{R}^{3}$ invariant through a screw motion. In particular the surfaces
of this section are minimal surfaces of $\mathbb{R}^{3}$. Again by imposing
$H=0$, equation (31) is written as:

$\theta^{\prime}=\frac{x\sin\theta-y\cos\theta}{1+x^{2}+y^{2}}$ (32)

and the system (19) becomes

$\left(\begin{array}[]{c}x\\\ y\\\
{\theta}\end{array}\right)^{\prime}=\left(\begin{array}[]{c}\cos\theta\\\
\sin\theta\\\
\frac{x\sin\theta-y\cos\theta}{1+x^{2}+y^{2}}\end{array}\right).$ (33)

(33) is a system of first order ODEs in the three-dimensional manifold
$\mathbb{R}^{2}\times\mathbb{S}^{1}=\\{(x,y,{\theta})\ |\
x,y\in\mathbb{R},{\theta}\in\mathbb{S}^{1}\\}$ that also can be viewed as a
second order system in $\mathbb{R}^{2}$:

$x^{\prime\prime}=(\cos{\theta})^{\prime}=-{\theta}^{\prime}\sin{\theta}=-\frac{x\sin\theta-y\cos\theta}{1+x^{2}+y^{2}}y^{\prime},\qquad
y^{\prime\prime}=\frac{x\sin\theta-y\cos\theta}{1+x^{2}+y^{2}}x^{\prime}.$

###  6.1 Type I solutions

Given a fixed $\theta\in\mathbb{S}^{1}$, the straight line
$(x,y)(t)=t(\cos{\theta},\sin{\theta})$, $t\in\mathbb{R}$, is a solution of
(33). By moving ${\theta}\in\mathbb{S}^{1}$ we find all solutions of (33) with
${\gamma}(0)=(0,0)$. If we fix
$(x_{0},y_{0})\in\mathbb{R}^{2}\setminus\\{(0,0)\\}$ then we can not give
explicitly the solutions of (33) with ${\gamma}(0)=(x_{0},y_{0})$, unless the
condition ${\gamma}^{\prime}(0)=\frac{{\gamma}(0)}{\|{\gamma}(0)\|}$ is
verified (in this situation ${\gamma}$ is a reparametrization of the straight
line joining $(0,0)$ and $(x_{0},y_{0})$). A solution of (33) will be called
of type I if it parameterizes an affine line passing through the origin. The
solutions of (33) that are not of type I will be called type II solutions.

###  6.2 Solutions of type II

The right-hand side of (32) can be written as

$\frac{x\sin\theta-y\cos\theta}{1+x^{2}+y^{2}}=\frac{\langle
J{\gamma},{\gamma}^{\prime}\rangle}{1+|{\gamma}|^{2}},$

where $J(x,y)=(-y,x)$. This leads us to write (33) as

$\left(\begin{array}[]{c}{\gamma}\\\
{\gamma}^{\prime}\end{array}\right)^{\prime}=\left(\begin{array}[]{c}x^{\prime}\\\
y^{\prime}\\\ \frac{\langle
J{\gamma},{\gamma}^{\prime}\rangle}{1+|{\gamma}|^{2}}J{\gamma}^{\prime}\end{array}\right).$
(34)

This means that, if we call $\kappa$ to the curvature of ${\gamma}$ with
respect to the flat metric of the plane $\left\\{z=0\right\\}$, our ODE
becomes $\kappa J{\gamma}^{\prime}={\gamma}^{\prime\prime}=\frac{\langle
J{\gamma},{\gamma}^{\prime}\rangle}{1+|{\gamma}|^{2}}J{\gamma}^{\prime}$, or
equivalently

$\kappa=\frac{\langle
J{\gamma},{\gamma}^{\prime}\rangle}{1+|{\gamma}|^{2}}=-\frac{\xi}{1+|{\gamma}|^{2}},$
(35)

where $\xi=\langle{\gamma},J{\gamma}^{\prime}\rangle$ is the support function
of ${\gamma}$. The next theorem gives a complete description of solutions of
(35).

######  Theorem 6.

If ${\gamma}$ verifies (35), then

  1. 1.

$|\kappa(t)|\leq 1/2$, $\forall t\in\mathbb{R}$.

  2. 2.

If ${\gamma}$ intersects the origin $(0,0)$, then ${\gamma}$ is type I.
Otherwise, ${\gamma}$ is a strictly convex curve, i.e. $\kappa$ is a nowhere
vanishing function. In particular $\kappa$ has constant sign.

  3. 3.

The maximal interval of definition of ${\gamma}$ is $\mathbb{R}$, and
$\kappa(t)$ goes to 0 when $t\to\pm\infty$.

  4. 4.

The curvature $\kappa$ verifies the ODE
$\langle{\gamma},{\gamma}^{\prime}\rangle\kappa+(1+|{\gamma}|^{2})\kappa^{\prime}=0$.

  5. 5.

Given $A\in SO(2)$, the curve $A{\gamma}$ is also a solution of (35).

  6. 6.

If ${\gamma}$ does not pass through the origin, then there exists a unique
$t_{0}\in\mathbb{R}$ such that $\kappa^{\prime}(t_{0})=0$. Moreover:

     * •

The distance from ${\gamma}$ to the origin reaches its unique critical point
at ${\gamma}(t_{0})$, which is a global minimum.

     * •

The image of ${\gamma}$ is symmetric with respect to the reflection through
the normal line to ${\gamma}$ at $t_{0}$.

     * •

The support function $\xi$ of ${\gamma}$ has a unique critical point at
$t_{0}$.

     * •

All the self-intersection points of ${\gamma}$ occur for opposite values of
its parameter (i.e. ${\gamma}(t_{1})={\gamma}(t_{2})$ $\Rightarrow$ $t_{1}=\pm
t_{2}$) and they are all in the normal line $r$ to ${\gamma}$ at $t_{0}$.
Moreover the angle of intersection between $r$ and ${\gamma}$ at $t$ is
$\pi/2$ if and only if $t=t_{0}$.

###### Proof.

From (35) and the Schwarz inequality we have that

$|\kappa|\leq\frac{|\xi|}{1+|{\gamma}|^{2}}\leq\frac{|{\gamma}|}{1+|{\gamma}|^{2}},$
(36)

which proves item 1.

If $\kappa(t_{0})=0$ for some $t_{0}\in\mathbb{R}$, then using (35) we have
$\xi(t_{0})=0$ . Therefore, either ${\gamma}(t_{0})=(0,0)$ (in this case
${\gamma}$ is of type I) or ${\gamma}(t_{0})\neq(0,0)$ is orthogonal to
$J{\gamma}^{\prime}(t_{0})$, and therefore ${\gamma}(t_{0})$ is collinear with
${\gamma}^{\prime}(t_{0})$. In this case, the uniqueness of the initial value
problem associated to (34) implies that ${\gamma}$ is of type I. This proves
item 2.

We can view (34) as an ODE of the following form: $X^{\prime}=F(t,X)$ with
$X=({\gamma},{\gamma}^{\prime})\in\mathbb{R}^{4}$. The fact that $F$ is
bounded in our equation (because ${\gamma}$ is arc length parameterized and
because of item 1) and Picard-Lindelöf Theorem ensure that the maximal
interval of definition of $X$ is $\mathbb{R}$.

${\gamma}$ cannot be bounded, otherwise the surface
$\Sigma\subset\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ generated by ${\gamma}$
from (6) would be inside a vertical straight cylinder
$\mathbb{D}(R)\times\mathbb{R}\subset\mathbb{R}^{3}$, where $\mathbb{D}(R)$ is
the open disc of radius $R>0$ in $\mathbb{R}^{2}$. Since
$\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ is isometric to $\mathbb{R}^{3}$ with
its standard metric, we would reach a contradiction by applying the maximum
principle at infinity (see [11]) to $\Sigma$ and a vertical plane that keeps
$\mathbb{D}(R)\times\mathbb{R}$ on one side. Since $|{\gamma}(t)|\to\infty$
when $|t|\to\infty$, from (36) we conclude that $\kappa(t)\to 0$ when
$t\to\pm\infty$.

By cross multiplying in (35) and taking derivatives we get

$\begin{split}0&=[(1+|{\gamma}|^{2})\kappa+\xi]^{\prime}=2\langle{\gamma},{\gamma}^{\prime}\rangle\kappa+(1+|{\gamma}|^{2})\kappa^{\prime}+(\langle{\gamma},J{\gamma}^{\prime}\rangle)^{\prime}\\\
&=2\langle{\gamma},{\gamma}^{\prime}\rangle\kappa+(1+|{\gamma}|^{2})\kappa^{\prime}+\langle{\gamma},-\kappa{\gamma}^{\prime}\rangle=\langle{\gamma},{\gamma}^{\prime}\rangle\kappa+(1+|{\gamma}|^{2})\kappa^{\prime},\end{split}$
(37)

which proves item 4.

To prove item 5, let $A\in SO(2)$, and $\widetilde{{\gamma}}=A{\gamma}$. The
curvature $\widetilde{\kappa}$ of $\widetilde{{\gamma}}$ is
$\widetilde{\kappa}=\kappa$, and $\langle
J\widetilde{{\gamma}},\widetilde{{\gamma}}^{\prime}\rangle=\langle
JA{\gamma},A{\gamma}^{\prime}\rangle=\langle
AJ{\gamma},A{\gamma}^{\prime}\rangle=\langle
J{\gamma},{\gamma}^{\prime}\rangle$ therefore $\widetilde{{\gamma}}$ satisfies
(34).

To prove item 6: due to item 2 we can suppose (up to orientation) that the
curvature of ${\gamma}$ is positive. Using item 3 we know that
$\kappa\colon\mathbb{R}\to(0,\infty)$ reaches a maximum for some
$t_{0}\in\mathbb{R}$. Since $\kappa^{\prime}(t_{0})=0$ and $\kappa(t_{0})\neq
0$, item 4 implies that the position vector ${\gamma}(t_{0})$ is orthogonal to
${\gamma}^{\prime}(t_{0})$ (both vectors are nonzero). Item 5 implies that up
to a rotation around the origin, we can assume that
${\gamma}(t_{0})=(x(t_{0}),0)$ and therefore, $x^{\prime}(t_{0})=0$,
$y^{\prime}(t_{0})=\pm 1$. Also, without loss of generality, using a
translation of the parameter $t$, we can suppose that $t_{0}=0$.

We now prove that ${\gamma}$ is symmetric with respect to the real axis. Let
$t\in\mathbb{R}\mapsto\widetilde{{\gamma}}(t)=(x(t),-y(t))$ be the reflection
of ${\gamma}$ through the real axis. The curvature $\widetilde{\kappa}$ of
$\widetilde{{\gamma}}$ is $\widetilde{\kappa}=-\kappa$, and

$\langle J\widetilde{{\gamma}},\widetilde{{\gamma}}^{\prime}\rangle=\langle
J(x,-y),(x^{\prime},-y^{\prime})\rangle=\langle(y,x),(x^{\prime},-y^{\prime})\rangle=x^{\prime}y-xy^{\prime}=-\langle
J{\gamma},{\gamma}^{\prime}\rangle,$

therefore $\widetilde{{\gamma}}$ satisfies (34). Since
$\widetilde{{\gamma}}(t_{0})={\gamma}(t_{0})$ and
$\widetilde{{\gamma}}^{\prime}(t_{0})=-{\gamma}^{\prime}(t_{0})$, we conclude
that ${\gamma}$ is symmetric through the reflection in the real axis. In
particular, ${\gamma}$ intersects orthogonally the real axis at ${\gamma}(0)$.

In order to prove that $\kappa^{\prime}$ only vanishes at $t_{0}$ we are going
to suppose that $\kappa^{\prime}(t_{1})=0$ for some $t_{1}\neq 0$. With the
same argument as before, ${\gamma}$ is symmetric with respect to the
reflection of the normal line $r(t_{1})$ to ${\gamma}$ at $t_{1}$. In this
situation we have three options:

  * •

$r(t_{1})$ is the real axis. In this situation, ${\gamma}$ is orthogonal to
the real axis at $t_{1}$, therefore, using the symmetry of ${\gamma}$ with
respect to the real axis, ${\gamma}$ is a closed curve. This contradicts item
3.

  * •

$r(t_{1})$ is parallel to (and different from) the real axis. This implies
that ${\gamma}$ is symmetric with respect to reflections through two distinct
parallel lines, therefore ${\gamma}$ is invariant by a non trivial vertical
translation obtained by applying both reflections. This implies that $\kappa$
is periodic, which contradicts item 3.

  * •

$r(t_{1})$ is not parallel to the real axis. Therefore, ${\gamma}$ is
symmetric with respect to both the real axis and $r(t_{1})$, which implies
that ${\gamma}$ is symmetric with respect to the composition of both
reflections which is a rotation of angle $\alpha\neq 0,\pi$. This implies that
$\kappa$ is periodic, which contradicts item 3.

Thus, $\kappa^{\prime}$ only vanishes at $t_{0}=0$. Since we are supposing
$\kappa>0$, we have that $\kappa$ reaches its global maximum at $t=0$.

Since, because of item 4,
$(|{\gamma}|^{2})^{\prime}=2\langle{\gamma},{\gamma}^{\prime}\rangle$ only
vanishes at the zeros of $\kappa^{\prime}$, this implies that the distance to
the origin restricted to ${\gamma}$ has a unique critical point at $t=0$. This
critical point is the global minimum of $|{\gamma}|$, because ${\gamma}$ is
not bounded.

Now we study the behaviour of each symmetric half
${\gamma}^{+}={\gamma}(t_{0},\infty)$, ${\gamma}^{-}=S({\gamma}^{+})$ of
${\gamma}$ that starts at the point ${\gamma}(0)$. In this way, ${\gamma}^{+}$
is a branch of ${\gamma}$ which is orthogonal to the real axis at its initial
point $t_{0}=0$. Up to a rotation of angle $\pi$ around ${\gamma}(0)$, we can
suppose that $y(t)>0$ for $t\in(0,\varepsilon)$ and for a small enough
$\varepsilon>0$, which means that ${\gamma}^{+}$ starts lying in the upper
half plane; which is equivalent to write $y^{\prime}(0)=1$. Since $\kappa>0$,
${\gamma}^{+}$ starts being at the left-hand side of the vertical line
$\\{x=x(0)\\}$.

In the case we suppose that the height function
$\langle{\gamma}(t),{\gamma}^{\prime}(0)\rangle=y(t)$ over the real axis has
no critical points then ${\gamma}^{+}$ is a convex global graph over its
projection $(-\infty,x(0))$ over the real axis (this graph may be unbounded).
Therefore, the unit normal $J{\gamma}^{\prime}=n=(n_{1},n_{2})$ along
${\gamma}^{+}$ is in one of the semicircumferences defined by
$\mathbb{S}^{1}\setminus\\{n_{2}=0\\}$. Notice that ${\gamma}$ is a global
graph with respect to the straight line which is tangent to ${\gamma}$ at
${\gamma}(0)$ (because $y(t)$ has no critical points). In particular,
${\gamma}$ is embedded. This implies that the surface
$\Sigma\subset\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ generated by ${\gamma}$
through (6) is a complete embedded simply-connected minimal surface in
$\mathbb{R}^{3}$ with its standard metric and, therefore, $\Sigma$ is a plane
or an helicoid, which is a contradiction.

In conclusion, the height function $y(t)$ over the real axis has at least one
critical point at $t_{1}\in(0,\infty)$. This implies that the tangent line to
${\gamma}$ at $t_{1}$ is horizontal, and since $\kappa(t_{1})>0$, $y(t)$ has a
local maximum at $t_{1}$. Using that $\kappa$ can not change sign, we deduce
that ${\gamma}^{+}$ has to intersect the real axis at a new point
${\gamma}(t_{2})$ with $t_{2}>t_{1}$. In this situation,
${\gamma}^{\prime}(t_{2})$ can not be orthogonal to the real axis (if it were,
we would conclude that ${\gamma}$ is a closed curve which is a contradiction
with item 3). Therefore, ${\gamma}(t_{2})$ is a point of self-intersection at
which ${\gamma}^{+}$ and ${\gamma}^{-}$ intersect transversely as shown in
Figure 3.

Figure 3: The numerical solution ${\gamma}$ of (35) with the initial
conditions ${\gamma}(0)=(1,0)$, ${\gamma}^{\prime}(0)=(0,1)$. It shows
${\gamma}$ for the values $t\in[-250,250]$. For this interval we found two
self-intersecting points.

Since ${\gamma}$ is symmetric with respect to the real axis and $t_{0}=0$, we
have that $x(t)$ is an even function (respectively $y(t)$ is an odd function).
Therefore, $x^{\prime}(t)$ is odd (respectively $y^{\prime}(t)$ is even), and
we deduce that the support function
$\xi=\langle{\gamma},J{\gamma}^{\prime}\rangle$ is even. Since $\kappa$ is
positive, from (35) we have that $\xi$ is negative. Moreover,

$\xi^{\prime}=(\langle{\gamma},J{\gamma}^{\prime}\rangle)^{\prime}=\langle{\gamma},(J{\gamma}^{\prime})^{\prime}\rangle=-\langle{\gamma},{\gamma}^{\prime}\rangle\kappa=(1+|{\gamma}|^{2})\kappa^{\prime},$

where, for the last equality, item 4 is needed. Since $\kappa$ is strictly
increasing on $(-\infty,0)$ (respectively is strictly decreasing on
$(0,\infty)$), we have that $\kappa^{\prime}>0$ on $(-\infty,0)$ (respectively
$\kappa^{\prime}<0$ on $(0,\infty)$), therefore $\xi^{\prime}>0$ on
$(-\infty,0)$ (respectively $\xi^{\prime}<0$ on $(0,\infty)$), which means
that $\xi$ is strictly increasing on $(-\infty,0)$ and strictly decreasing on
$(0,\infty)$, and $\xi$ has a unique critical point which is a negative global
maximum at $t=0$.

Next, we prove that the map $t\mapsto|{\gamma}(t)|$ is inyective on
$(0,\infty)$ (the same argument works on $(-\infty,0)$). If there exists
$t_{1},t_{2}$ such that $0<t_{1}<t_{2}$ and
$|{\gamma}(t_{1})|=|{\gamma}(t_{2})|$, this implies the existence of
$t_{3}\in(t_{1},t_{2})$ such that $\xi(t_{3})=|{\gamma}(t_{3})|$. Since
$\xi=\left\langle{\gamma},J{\gamma}\right\rangle\leq|{\gamma}|$, then
$\left\langle{\gamma}(t_{3}),{\gamma}(t_{3})\right\rangle=0$. Using equation
(37) we get that $\kappa^{\prime}(t_{3})=0$ which contradicts the fact that
$\kappa^{\prime}$ only vanishes at 0.

Figure 4: Vertically invariant minimal surface in $\widetilde{E}(2)$ (of type
II) generated through $\Phi(t,s)=\phi_{s}(\gamma(t))$, with
$\gamma(t)=(x(t),y(t),0)$, where $x,y$ are solutions of the system (33) with
initial conditions $x(0)=1$, $y(0)=0$, $\theta(0)=1$.

We still have to prove that the self-intersection points of ${\gamma}$ are all
in the real axis and they occur for opposite values of the parameter of
${\gamma}$. If ${\gamma}(t_{1})={\gamma}(t_{2})$ but $t_{1}\neq t_{2}$, then
$t_{1},t_{2}$ can not have the same sign (because the map
$t\mapsto|{\gamma}(t)|$ is inyective on $(0,\infty)$ and on $(-\infty,0)$),
therefore we can suppose $t_{1}<0<t_{2}$. Since
$|{\gamma}(t_{1})|=|{\gamma}(t_{2})|=|{\gamma}(-t_{2})|$ (because $|{\gamma}|$
is even) then $t_{1}=-t_{2}$, and therefore
$y(t_{2})=y(t_{1})=y(-t_{2})=-y(t_{2})$ which implies $y(t_{2})=0$). ∎

###  6.3 Vertically invariant minimal surfaces in $\widetilde{E}(2)$ with a
non-flat metric

We consider $\widetilde{E}(2)$ with the metric that makes it isometric and
isomorphic to $\mathbb{R}^{2}\rtimes_{A(c)}\mathbb{R}$ with

$A(c)=\left(\begin{array}[]{cc}0&-c\\\ 1/c&0\end{array}\right)$

with $c\in(1,\infty)$. If we impose $H=0$ in equation (24) we get the
following system:

$\left\\{\begin{array}[]{l}x^{\prime}=\cos\theta\\\ y^{\prime}=\sin\theta\\\
{\theta}^{\prime}=\frac{\left[1+c^{2}+\left(1-c^{2}\right)\cos(2\theta)\right]\left(x\sin\theta-c^{2}y\cos\theta\right)}{2\left(c^{2}+x^{2}+c^{4}y^{2}\right)}\end{array}.\right.$
(38)

######  Proposition 7.

If ${\gamma}$ verifies (38) then the maximal interval of definition of
${\gamma}$ is $\mathbb{R}$.

###### Proof.

The function $a\in[-1,1]\mapsto\frac{1}{2}[1+c^{2}+(1-c^{2})a]$ is
nonincreasing because $c\geq 1$, therefore its maximum is
$\frac{1}{2}[1+c^{2}-(1-c^{2})]=c^{2}$. Moreover, using Schwarz inequality,

$\left|\frac{x\sin{\theta}-c^{2}y\cos{\theta}}{c^{2}+x^{2}+c^{4}y^{2}}\right|\leq\frac{\sqrt{x^{2}+c^{4}y^{2}}}{c^{2}+x^{2}+c^{4}y^{2}}\leq\frac{1}{2c}.$
(39)

This proves that the derivative of ${\theta}$ is bounded. Again, (38) can be
seen as an ODE of the form $X^{\prime}=F(t,X)$, where $F$ is bounded and the
result follows from the Picard-Lindelöf Theorem. ∎

Viewing (38) as an ODE of the form $X^{\prime}=F(t,X,c)$ for all
$c\in[1,\infty)$, since for each $c\in[1,\infty)$ the maximal interval of
definition of the solution $\gamma_{c}$ of (38) (once we have fixed the
initial conditions) is $\mathbb{R}$, and the map
$F:\mathbb{R}\times\mathbb{R}^{3}\times[1,\infty)\rightarrow\mathbb{R}^{3}$,
$(t,X,c)\mapsto F(t,X,c)$ is locally Lipschitz (in fact is differentiable), we
can use the continuous dependence on the parameter to conclude that, once we
fix the initial conditions, for every $\varepsilon>0$ and for every interval
$I\subset\mathbb{R}$ there exists $\delta>0$ such that if $1\leq c<1+\delta$
then $|\gamma(t)-\gamma_{c}(t)|<\varepsilon$ for every $t\in I$, where
$\gamma$ is the solution of (38) with $c=1$ which have been described in
Subsections 6.1 and 6.2.

Next result describes further properties of the generating curve $\gamma$ in
$\widetilde{E}(2)$ with a non-flat metric.

######  Proposition 8.

  1. 1.

If $c>1$ then the x-axis and the y-axis are the only solution of (38) with
constant $\theta$.

  2. 2.

The curvature $\kappa=\theta^{\prime}$ goes to $0$ when $t\to\pm\infty$ and
$|\kappa(t)|\leq c/2$, $\forall t\in\mathbb{R}$.

###### Proof.

To prove item 1, notice that solving the equation $1+c^{2}+(1-c^{2})C=0$ gives
$C=\frac{c^{2}+1}{c^{2}-1}$ which is strictly greater than 1. Therefore the
factor $1+c^{2}+(1-c^{2})\cos(2\theta)$ can not be 0. Thus, a solution of (38)
with constant $\theta=\theta_{0}\in\mathbb{R}$ must verify

$x\sin\theta_{0}-c^{2}y\cos\theta_{0}\equiv 0,$

which means that the generating curve $\gamma$ is contained in a straight line
passing through the origin. Write $\gamma$ as $\gamma(t)=tv$, where
$v=(v_{1},v_{2})\in\mathbb{S}^{1}$, for all $t\in\mathbb{R}$ and therefore we
have $tv_{1}\sin\theta_{0}-c^{2}tv_{2}\cos\theta_{0}=t(-1+c^{2})v_{1}v_{2}=0$
for all $t\in\mathbb{R}$. As we are assuming $c>1$, then $v_{1}v_{2}=0$.

To prove item 2, we know that $|\kappa(t)|\leq c/2$, $\forall t\in\mathbb{R}$
because of the proof of Proposition 7. To see that $\kappa=\theta^{\prime}$
goes to $0$ when $t\to\pm\infty$, the same proof of Theorem 7 does not work
here. However, vertical planes are still minimal surfaces in
$\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$, for any
$A\in\mathcal{M}_{2}(\mathbb{R})$ (see Remark 2.10 in [10]). Suppose that
$\gamma$ is bounded, then the surface $\Sigma\subset\widetilde{E}(2)$
generated by $\gamma$ from (6) would be inside a vertical cylinder
$\mathbb{D}(R)\times\mathbb{R}\subset\mathbb{R}^{3}$, $R>0$. We can take a
vertical plane disjoint from $\Sigma$ and left translate it until we find a
first contact point. If the first contact point is at infinity (see Remark 9)
then there is also a first finite contact point. This comes from the fact that
the vertical translation of a point $(x,y,z)\in\widetilde{E}(2)$ describes a
helix $s\mapsto\left(x\cos s-cy\sin s,\frac{x\sin s}{c}+y\cos s,z+s\right)$.
Therefore the surface $\Sigma$ is periodic along the vertical axis, which
implies that there are infinitely many finite contact points when we find a
first contact point. Thus, we can apply the classical maximum principle to
reach a contradiction and therefore $|\gamma(t)|\to\infty$ when
$t\to\pm\infty$. Using (39) we have

$\kappa\leq c^{2}F\left(\sqrt{x^{2}+c^{4}y^{2}}\right),$

where $F(b)=\frac{b}{c^{2}+b^{2}}$. Since $\sqrt{x^{2}+c^{4}y^{2}}\to\infty$
if $|\gamma|\to\infty$ and $F(b)$ goes to $0$ when $b\to\infty$ we have that
$k(t)\to 0$ if $|t|\to\infty$. ∎

######  Remark 9.

It is still not proved a maximum principle at infinity result or a half-space
theorem for vertical planes in $\widetilde{E}(2)$. However a half-space
theorem for horizontal planes was proved in [15].

##  7 Surfaces with zero Gaussian curvature

Since in Section 3 we have computed the coefficients of the first and second
fundamental forms, it is natural to compute the Gaussian curvature of a
surface generated by (6) and study the vertically invariant surfaces on
$\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$ with constant or zero Gaussian
curvature.

Again, we assume that our metric Lie group
$G=\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$, $A\in\mathcal{M}_{2}(\mathbb{R})$ is
unimodular. In the literature, invariant surfaces of zero Gaussian curvature
in metric Lie groups that can be written as a semidirect product has been
studied in the following ambient spaces:

  1. 1.

In $\mathbb{R}^{3}$, Hartman and Nirenberg proved that (Theorem III in [4])
all the complete, zero Gaussian curvature surfaces are given by right
cylinders over possibly non compact curves in $\mathbb{R}^{2}\times\\{0\\}$,
which describes all the possible examples.

  2. 2.

The invariant surfaces with constant extrinsic Gaussian curvature in
$\text{Nil}_{3}$ were studied by Belarbi in [2], where he considered all
possible 1-parameter subgroups of left translations.

  3. 3.

Invariant surfaces with respect to any 1-parameter subgroup of constant
intrinsic or extrinsic Gaussian curvature in $\text{Sol}_{3}$ with its
standard metric, $A=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$,
were studied in López [7] and López-Munteanu [8].

When we consider $\widetilde{E}(2)$ with its standard (flat) metric, we know,
by Theorem III in [4] that except for cylinders over a circumference centered
at the origin, there are no complete zero Gaussian curvature invariant
surfaces because vertical translations in $\widetilde{E}(2)$ do not produce
right cylinders except for these vertical cylinders over circumferences. Even
though, we can still give a description of the vertically invariant surfaces
in $\widetilde{E}(2)$ with zero Gaussian curvature (since the metric is flat
we are dealing with both, intrinsic and extrinsic, cases). None of these
surfaces can be complete except for the vertical cylinders over circumferences
centered at the origin.

Using the expressions for the first and second fundamental forms of Section 3,
and the same notation as in previous sections, since the Gaussian curvature
$K$ can be computed as

$K=\frac{eg-f^{2}}{EG-F^{2}},$

we get the following system of ODEs for a curve $\gamma=(x,y,0)$ that
generates, through vertical translations, a constant Gaussian curvature
surface in the standard $\widetilde{E}(2)$:

$\left\\{\begin{array}[]{l}x^{\prime}=\cos\theta\\\ y^{\prime}=\sin\theta\\\
K=-\frac{1+(y\cos\theta-x\sin\theta)\theta^{\prime}}{(1+(x\cos\theta+y\sin\theta))^{2}}\end{array}.\right.$
(40)

If we impose $K=0$ this system becomes:

$\left\\{\begin{array}[]{l}x^{\prime}=\cos\theta\\\ y^{\prime}=\sin\theta\\\
(y\cos\theta-x\sin\theta)\theta^{\prime}=-1\end{array}.\right.$ (41)

As in Section 5, we can give a first integral for the system (41):

$J(t)=x(t)\cos\theta(t)+y(t)\sin\theta(t)=:J\in\mathbb{R}.$

$J$ can be written as

$J(t)=\frac{1}{2}\frac{d}{dt}\left(x(t)^{2}+y(t)^{2}\right).$

Integrating and assuming $x(0)=x_{0}$, $y(0)=y_{0}$, $\theta(0)=\theta_{0}$,
$x_{0},y_{0},\theta_{0}\in\mathbb{R}$ gives that for any $t$ in the maximal
interval of definition of $\gamma$ we have

$|\gamma(t)|^{2}=x(t)^{2}+y(t)^{2}=at+x_{0}^{2}+y_{0}^{2},$ (42)

where $a:=2J=2(x_{0}\cos\theta_{0}+y_{0}\sin\theta_{0})$. We know that if
$a\not=0$ there are no solutions of (41) defined for all $t\in\mathbb{R}$.
(42) implies that the maximal interval of definition of $\gamma$ is contained
in $\left[-(x_{0}^{2}+y_{0}^{2})/a,\infty\right)$ if $a>0$ or in
$\left(-\infty,-(x_{0}^{2}+y_{0}^{2})/a\right]$ if $a<0$. Now write $\gamma$
in polar coordinates:

$x(t)=r(t)\cos\alpha(t),\quad y(t)=r(t)\sin\alpha(t).$

Due to (42) we know that

$r(t)=\sqrt{at+x_{0}^{2}+y_{0}^{2}}.$ (43)

Since

$\displaystyle x^{\prime}$
$\displaystyle=r^{\prime}\cos\alpha-r\alpha^{\prime}\sin\alpha,$
$\displaystyle y^{\prime}$
$\displaystyle=r^{\prime}\sin\alpha+r\alpha^{\prime}\cos\alpha,$

equation $1=(x^{\prime})^{2}+(y^{\prime})^{2}$ becomes

$1=(r^{\prime})^{2}+r^{2}(\alpha^{\prime})^{2},$

which implies

$\alpha^{\prime}(t)=\frac{1}{2}\frac{\sqrt{-a^{2}+4at+4\left(x_{0}^{2}+y_{0}^{2}\right)}}{\left|at+x_{0}^{2}+y_{0}^{2}\right|}.$

The maximal interval of definition of $\alpha^{\prime}$ (and therefore the
maximal interval of definition of $\alpha$) is
$I_{1}:=\left[a/4-(x_{0}^{2}+y_{0}^{2})/a,\infty\right)$ if $a>0$,
$I_{2}:=\left(-\infty,a/4-(x_{0}^{2}+y_{0}^{2})/a\right]$ if $a<0$ and
$\mathbb{R}$ if $a=0$. Since
$I_{1}\subset\left[-(x_{0}^{2}+y_{0}^{2})/a,\infty\right)$ when $a>0$ (resp.
$I_{2}\subset\left(-\infty,-(x_{0}^{2}+y_{0}^{2})/a\right]$) then the maximal
interval of definition of a solution of (41) is given by $I_{1}$ if $a>0$,
$I_{2}$ if $a<0$, or $\mathbb{R}$ if $a=0$.

Figure 5: Left: solution $\gamma$ of (41) with initial conditions $x_{0}=2$,
$y_{0}=1$, $\theta_{0}=\frac{\pi}{4}$ ($a>0$). Right: zero Gaussian curvature
surface generated through (6) with this $\gamma$.

Set

$g(t)=\sqrt{-a^{2}+4at+4(x_{0}^{2}+y_{0}^{2})},$

then, integrating, we finally describe the function $\alpha$ and, therefore,
all the invariant surfaces in $\widetilde{E}(2)$ with zero Gaussian curvature:

$\begin{split}\alpha(t)=&\arctan\left(\frac{y_{0}}{x_{0}}\right)+\frac{1}{2}\int_{0}^{t}\frac{\sqrt{-a^{2}+4a\tau+4\left(x_{0}^{2}+y_{0}^{2}\right)}}{\left|a\tau+x_{0}^{2}+y_{0}^{2}\right|}d\tau\\\
=&\arctan\left(\frac{y_{0}}{x_{0}}\right)+\left.\frac{\text{sgn}\left(a\tau+x_{0}^{2}+y_{0}^{2}\right)}{2a}\left(g(\tau)-a\;\text{arccot}\left(\frac{a}{g(\tau)}\right)\right)\right|_{\tau=0}^{\tau=t}.\end{split}$
(44)

In summary, we have proved the following result:

######  Proposition 10.

Let $\Sigma\subset\widetilde{E}(2)$ be a vertically invariant surface with
zero Gaussian curvature, then either $\Sigma$ is a vertical cylinder over a
circumference in the plane $\left\\{z=0\right\\}$ centered at the origin (and
this is the only complete example) or the generating curve $\gamma$ of
$\Sigma$ is given in polar coordinates by (43) and (44).

## References

  * [1] Uwe Abresch and Harold Rosenberg.  A Hopf differential for constant mean curvature surfaces in $\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}$.  Acta Mathematica, 193(2):141–174, 2004.
  * [2] Lakehal Belarbi.  Surfaces with constant extrinsically Gaussian curvature in the Heisenberg group.  In Annales Mathematicae et Informaticae, volume 50, pages 5–17. Eszterházy Károly Egyetem Líceum Kiadó, 2019.
  * [3] Christiam B Figueroa, Francesco Mercuri, and Renato HL Pedrosa.  Invariant surfaces of the Heisenberg groups.  Annali di Matematica pura ed applicata, 177(1):173–194, 1999.
  * [4] Philip Hartman and Louis Nirenberg.  On spherical image maps whose Jacobians do not change sign.  American Journal of Mathematics, 81(4):901–920, 1959.
  * [5] Jun-ichi Inoguchi, Rafael López, and Marian-Ioan Munteanu.  Minimal translation surfaces in the Heisenberg group $\text{{N}il}_{3}$.  Geometriae Dedicata, 161:221–231, 2012.
  * [6] Huili Liu.  Translation surfaces with constant mean curvature in 3-dimensional spaces.  Journal of Geometry, 64:141–149, 1999.
  * [7] Rafael López.  Invariant surfaces in $\text{{S}ol}_{3}$ with constant mean curvature and their computer graphics.  Advances in Geometry, 14(1):31–48, 2014.
  * [8] Rafael López and Marian Ioan Munteanu.  Invariant surfaces in the homogeneous space $\text{{S}ol}_{3}$ with constant curvature.  Mathematische Nachrichten, 287(8-9):1013–1024, 2014.
  * [9] William H Meeks III, Pablo Mira, and Joaquín Pérez.  The geometry of stable minimal surfaces in metric Lie groups.  Transactions of the American Mathematical Society, 372(2):1023–1056, 2019.
  * [10] William H Meeks III and Joaquín Pérez.  Constant mean curvature surfaces in metric Lie groups.  Geometric Analysis, 570:25–110, 2012.
  * [11] William H Meeks III and Harold Rosenberg.  The maximum principle at infinity for minimal surfaces in flat three manifolds.  Commentarii Mathematici Helvetici, 65(1):255–270, 1990.
  * [12] Stefano Montaldo and Irene I Onnis.  Invariant CMC surfaces in $\mathbb{H}^{2}\times\mathbb{R}$.  Glasgow Mathematical Journal, 46(2):311, 2004.
  * [13] Francisco Torralbo.  Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds.  Differential Geometry and its Applications, 28(5):593–607, 2010\.
  * [14] Francisco Torralbo and Joeri Van der Veken.  Rotationally invariant constant Gauss curvature surfaces in Berger spheres.  Journal of Mathematical Analysis and Applications, 489(2):124183, 2020.
  * [15] Yiming Zang.  Constructions of helicoidal minimal surfaces and minimal annuli in $\widetilde{E(2)}$.  Annals of Global Analysis and Geometry, 62(3):693–719, 2022.

David Moya, Department of Geometry and Topology & IMAG, University of Granada,
18001 Granada, Spain

E-mail address<EMAIL_ADDRESS>