Helicoidal surfaces of prescribed mean curvature in
$\mbox{${\mathbb{R}}$}^{3}$

Aires E. M. Barbieri

> Abstract Given a function $\mathcal{H}\in C^{1}(\mbox{${\mathbb{S}}$}^{2})$,
> an $\mathcal{H}$-surface $\Sigma$ is a surface in the Euclidean space
> $\mbox{${\mathbb{R}}$}^{3}$ whose mean curvature $H_{\Sigma}$ satisfies
> $H_{\Sigma}=\mathcal{H}\circ\eta$, where $\eta$ is the Gauss map of
> $\Sigma$. The purpose of this paper is to use a phase space analysis to give
> some classification results for helicoidal $\mathcal{H}$-surfaces, when
> $\mathcal{H}$ is rotationally symmetric, that is,
> $\mathcal{H}\circ\eta=\mathfrak{h}\circ\nu$, for some
> $\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$, where $\nu$ is the angle function
> of the surface. We prove a classification theorem for the case where
> $\mbox{$\mathfrak{h}$}(t)$ is even and increasing for $t\in[0,1]$. Finally,
> we provide examples of helicoidal $\mathcal{H}$-surfaces in cases where
> $\mathfrak{h}$ vanishes at some point.
>
>
> Mathematics Subject Classification: 53A10, 53C42, 34C05, 34C40.
> Keywords: helicoidal surfaces, prescribed mean curvature.

## 1\. Introduction

In this article, we study the existence and classification of helicoidal
surfaces $\Sigma$ of $\mbox{${\mathbb{R}}$}^{3}$ whose mean curvature
$H_{\Sigma}$ is given as a prescribed function of its Gauss map
$\eta:\Sigma\to\mbox{${\mathbb{S}}$}^{2}$. Specifically, given $\mathcal{H}\in
C^{1}(\mbox{${\mathbb{S}}$}^{2})$, we are interested in finding surfaces
$\Sigma$ that satisfy

$H_{\Sigma}=\mathcal{H}\circ\eta.$

Any such $\Sigma$ will be called a surface of prescribed mean curvature
$\mathcal{H}$, or simply, an $\mathcal{H}$-surface.

The study of $\mathcal{H}$-hypersurfaces in $\mbox{${\mathbb{R}}$}^{n+1}$ is
motivated by the classical contributions of Alexandrov and Pogorelov in the
1950s (see [1, 7]). They started to investigate the existence and uniqueness
of ovaloids in $\mbox{${\mathbb{R}}$}^{n+1}$ determined by a prescribed
curvature function through its Gauss map.

When specific functions $\mathcal{H}$ are chosen as prescribed, some geometric
theories receive deeply research. The following examples are emphasized:

  1. (1)

Surfaces of constant mean curvature, or simply, CMC surfaces. In this case,
$\mathcal{H}\equiv H_{0}\in\mbox{${\mathbb{R}}$}$ is a constant. When
$H_{0}=0$, they are minimal surfaces;

  2. (2)

$\lambda$-translators, which corresponds to the case $\mathcal{H}(x)=\langle
x,v\rangle+\lambda$, where $v\in\mbox{${\mathbb{S}}$}^{n}$ and
$\lambda\in\mbox{${\mathbb{R}}$}$. The vector $v$ is called the density
vector. When $\lambda=0$, we have translating solitons of the mean curvature
flow.

The theory of complete, non-compact $\mathcal{H}$-surfaces for general choice
of $\mathcal{H}$ has been recently developed by A. Bueno, J. Gálvez and P.
Mira [2, 3], taking as starting point the theories of CMC surfaces and
translating solitons. In the first article, the authors started the
development of the global theory of complete $\mathcal{H}$-hypersurfaces in
$\mbox{${\mathbb{R}}$}^{n+1}$, demonstrating that, under mild symmetry and
regularity assumptions on the function $\mathcal{H}$, the theory of
$\mathcal{H}$-hypersurfaces sometimes admits a uniform treatment that
resembles the constant mean curvature case.

In [3], focusing on the study of rotational $\mathcal{H}$-hypersurfaces in
$\mbox{${\mathbb{R}}$}^{n+1}$, in case that the function $\mathcal{H}\in
C^{1}(\mbox{${\mathbb{S}}$}^{n})$ is rotationally symmetric, i.e.
$\mathcal{H}(x)=\mbox{$\mathfrak{h}$}(\langle x,e_{n+1}\rangle)$ for some
$\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$, the same authors treated the
resulting ordinary differential equation as a nonlinear autonomous system and
conducted a qualitative study of its solutions through a phase space analysis.
The aim was to provide classification results for these hypersurfaces.

Motivated by these ideas, our objective in this article is to use a similar
phase space analysis to study, classify and exhibit some helicoidal surfaces
of prescribed mean curvature in $\mbox{${\mathbb{R}}$}^{3}$.

First, in Section 2, we define a helicoidal surface $\Sigma$ in
$\mbox{${\mathbb{R}}$}^{3}$ (Def. 2.1) and explicitly calculate in (2) the
mean curvature of $\Sigma$ in terms of its profile curve. Since the mean
curvature of a helicoidal surface does not depend on its rotational parameter,
we focus on the case where $\mathcal{H}\in C^{1}(\mbox{${\mathbb{S}}$}^{2})$
is rotationally symmetric, that is, up to Euclidian change of coordinates,
there is a function $\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$ such that
$\mathcal{H}(x)=\mbox{$\mathfrak{h}$}(\langle x,e_{3}\rangle)$. Then, the mean
curvature of a helicoidal $\mathcal{H}$-surface $\Sigma$ is given by

$H_{\Sigma}=\mathcal{H}\circ\eta=\mbox{$\mathfrak{h}$}\circ\nu.$

In terms of the profile curve of $\Sigma$, we treat next the resulting ODE as
the nonlinear autonomous system (2.12), carrying out a qualitative study of
its solutions through a phase space analysis. We also prove in Proposition 2.4
that, up to rotations around the $e_{3}$-axis, there exists at most a unique
helicoidal $\mathcal{H}$-surface in $\mbox{${\mathbb{R}}$}^{3}$ that
intersects its rotation axis.

Building on the ideas of Theorem 4.1 of [3], we study in Section 3 the
helicoidal $\mathcal{H}$-surfaces in cases where $\mathcal{H}\in
C^{1}(\mbox{${\mathbb{S}}$}^{2})$ is positive, rotationally invariant and
even, i.e. $\mathcal{H}(-x)=\mathcal{H}(x)>0$ for all
$x\in\mbox{${\mathbb{S}}$}^{2}$. Different from the rotational case, we also
need to suppose that the function $\mbox{$\mathfrak{h}$}(t)$ related to
$\mathcal{H}$ is increasing for $t\in[0,1]$. Then, we arrive at the
classification Theorem 3.3.

Finally, in Section 4, we provide examples of helicoidal
$\mathcal{H}$-surfaces in cases where $\mathcal{H}$ vanishes at some point.
The biggest problem to extend the orbits behavior of the examples to general
choices of $\mathcal{H}$ is not knowing exactly how the phase space looks
like. It is important to observe that, even though these examples are
specific, similar functions $\mathcal{H}$ can lead to similar behaviors of the
resulting helicoidal $\mathcal{H}$-surfaces. Thus, attempting to generalize
these models to various choices of $\mathcal{H}$ could inspire future research
in this area.

## 2\. Phase space analysis of helicoidal $\mathcal{H}$-surfaces

Let $\Sigma$ be an immersed, oriented surface in $\mbox{${\mathbb{R}}$}^{3}$
and $\eta:\Sigma\to\mbox{${\mathbb{S}}$}^{2}$ its Gauss map. Given a function
$\mathcal{H}\in C^{1}(\mbox{${\mathbb{S}}$}^{2})$, we will say that $\Sigma$
is a surface of prescribed mean curvature $\mathcal{H}$, or simply,
$\mathcal{H}$-surface, if its mean curvature $H_{\Sigma}$ is given by

(2.1) $H_{\Sigma}=\mathcal{H}\circ\eta.$

When $\Sigma$ is given as a graph $x_{3}=u(x_{1},x_{2})$, then (2.1) is
written as the elliptic, second order quasilinear PDE

(2.2)
$\mathrm{div}\left(\frac{\mathrm{D}u}{\sqrt{1+|\mathrm{D}u|^{2}}}\right)=2\mathcal{H}(\mathrm{Z}_{u}),\quad\mathrm{Z}_{u}:=\frac{(-\mathrm{D}u,1)}{\sqrt{1+|\mathrm{D}u|^{2}}},$

where $\mathrm{div}$ and $\mathrm{D}$ denote respectively the divergence and
gradient operators on $\mbox{${\mathbb{R}}$}^{2}$ and $\mathrm{Z}_{u}$ is the
unit normal of the graph.

######  Definition 2.1.

A helicoidal surface $\Sigma$ in $\mbox{${\mathbb{R}}$}^{3}$ is an immersed,
oriented surface obtained as the orbit of a regular planar curve parametrized
by arc-length

$\alpha(s)=(x(s),0,z(s)):I\subset\mbox{${\mathbb{R}}$}\to\mbox{${\mathbb{R}}$}^{3}$

under a helicoidal motion, that is, up to isometries of
$\mbox{${\mathbb{R}}$}^{3}$, there exists a constant $c_{0}\neq 0$ such that

(2.3)
$\Psi(s,\theta)=\left(x(s)\cos{\theta},x(s)\sin{\theta},z(s)+c_{0}\theta\right):I\times\mbox{${\mathbb{R}}$}\to\mbox{${\mathbb{R}}$}^{3}$

is a parametrization of $\Sigma$.

Up to a change of orientation, the Gauss map of a helicoidal surface $\Sigma$
is given by

(2.4)
$\eta(s,\theta)=\frac{\left(c_{0}\sin{\theta}x^{\prime}(s)-\cos{\theta}z^{\prime}(s)x(s),-\sin{\theta}x(s)z^{\prime}(s)-c_{0}\cos{\theta}x^{\prime}(s),x(s)x^{\prime}(s)\right)}{\sqrt{c_{0}^{2}x^{\prime}(s)^{2}+x(s)^{2}}}.$

Therefore, the angle function $\nu:=\langle\eta,e_{3}\rangle$ of $\Sigma$ is
given by

(2.5)
$\nu(s)=\nu(s,\theta)=\frac{x(s)x^{\prime}(s)}{\sqrt{c_{0}^{2}x^{\prime}(s)^{2}+x(s)^{2}}}.$

Note that $\|\Psi_{s}(s,\theta)\wedge\Psi_{\theta}(s,\theta)\|=0$ if, and only
if, $x(s)=x^{\prime}(s)=0$. So, we assume that $x^{\prime}(s)\neq 0$ for $s\in
I$ such that $x(s)=0$ to $\Sigma$ be a regular surface.

######  Remark 2.2.

In a helicoidal surface $\Sigma$, its angle function satisfies $|\nu|<1$.
Indeed, for $s\in I$ such that $x^{\prime}(s)=0$, we directly obtain
$\nu(s)=0<1$. For $s\in I$ such that $x^{\prime}(s)\neq 0$, since
$|x^{\prime}(s)|\leq 1$ and $c_{0}\neq 0$, we have

$|\nu(s)|=\frac{|x(s)||x^{\prime}(s)|}{\sqrt{c_{0}^{2}x^{\prime}(s)^{2}+x(s)^{2}}}\leq\frac{|x(s)|}{\sqrt{c_{0}^{2}x^{\prime}(s)^{2}+x(s)^{2}}}<1.$

∎

The coefficients of the first fundamental form of $\Sigma$ are

$\displaystyle E$
$\displaystyle=\langle\Psi_{s},\Psi_{s}\rangle=\cos^{2}(\theta)x^{\prime}(s)^{2}+\sin^{2}(\theta)x^{\prime}(s)^{2}+z^{\prime}(s)^{2}=1,$
$\displaystyle F$
$\displaystyle=\langle\Psi_{s},\Psi_{\theta}\rangle=-\sin\theta\cos\theta
x(s)x^{\prime}(s)+\sin\theta\cos\theta
x(s)x^{\prime}(s)+c_{0}z^{\prime}(s)=c_{0}z^{\prime}(s),$ $\displaystyle G$
$\displaystyle=\langle\Psi_{\theta},\Psi_{\theta}\rangle=\sin^{2}(\theta)x^{2}(s)+\cos^{2}(\theta)x^{2}(s)+c_{0}^{2}=x(s)^{2}+c_{0}^{2},$

and its coefficients of the second fundamental form are

$\displaystyle e$
$\displaystyle=\langle\Psi_{ss},\eta\rangle=\frac{x(s)\left(z^{\prime\prime}(s)x^{\prime}(s)-z^{\prime}(s)x^{\prime\prime}(s)\right)}{\sqrt{c_{0}^{2}x^{\prime}(s)^{2}+x(s)^{2}}},$
$\displaystyle f$
$\displaystyle=\langle\Psi_{s\theta},\eta\rangle=-\frac{c_{0}x^{\prime}(s)^{2}}{\sqrt{c_{0}^{2}x^{\prime}(s)^{2}+x(s)^{2}}},$
$\displaystyle g$
$\displaystyle=\langle\Psi_{\theta\theta},\eta\rangle=\frac{x(s)^{2}z^{\prime}(s)}{\sqrt{c_{0}^{2}x^{\prime}(s)^{2}+x(s)^{2}}}.$

Therefore, the mean curvature of a helicoidal surface $\Sigma$ is given by

$\displaystyle H_{\Sigma}$
$\displaystyle=\frac{eG-2fF+gE}{2(EG-F^{2})}=\frac{x(z^{\prime\prime}x^{\prime}-z^{\prime}x^{\prime\prime})(x^{2}+c_{0}^{2})+2c_{0}^{2}x^{\prime
2}z^{\prime}+x^{2}z^{\prime}}{2\sqrt{c_{0}^{2}x^{\prime
2}+x^{2}}\left(x^{2}+c_{0}^{2}-c_{0}^{2}z^{\prime 2}\right)}$ (2.6)
$\displaystyle=\frac{(x^{\prime}z^{\prime\prime}-z^{\prime}x^{\prime\prime})(x^{3}+c_{0}^{2}x)+z^{\prime}(2c_{0}^{2}x^{\prime
2}+x^{2})}{2(c_{0}^{2}x^{\prime 2}+x^{2})^{\frac{3}{2}}}.$

Since the mean curvature of a helicoidal surface does not depend on the
rotational parameter $\theta$, let $\mathcal{H}\in
C^{1}(\mbox{${\mathbb{S}}$}^{n})$ be a rotationally symmetric function, that
is, $\mathcal{H}(x)=\mbox{$\mathfrak{h}$}(\langle x,v\rangle)$, for some
$v\in\mbox{${\mathbb{S}}$}^{2}$ and $\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$.
Up to an Euclidean change of coordinates, we can assume that $v=e_{3}$, and
then $\mathcal{H}(x)=\mbox{$\mathfrak{h}$}(\langle x,e_{3}\rangle)$.

Let now $\Sigma$ be a helicoidal $\mathcal{H}$-surface in
$\mbox{${\mathbb{R}}$}^{3}$. From (2.1), the mean curvature of $\Sigma$ is
given by

(2.7) $H_{\Sigma}=\mathcal{H}\circ\eta=\mbox{$\mathfrak{h}$}\circ\nu.$

Therefore, from (2), the profile curve $\alpha(s)$ of $\Sigma$ satisfies

(2.8) $2\mbox{$\mathfrak{h}$}\left(\frac{xx^{\prime}}{\sqrt{c_{0}^{2}x^{\prime
2}+x^{2}}}\right)=\frac{(x^{\prime}z^{\prime\prime}-z^{\prime}x^{\prime\prime})(x^{3}+c_{0}^{2}x)+z^{\prime}(2c_{0}^{2}x^{\prime
2}+x^{2})}{(c_{0}^{2}x^{\prime 2}+x^{2})^{\frac{3}{2}}}.$

Taking into account that $x^{\prime 2}+z^{\prime 2}=1$, we can denote
$z^{\prime}=\varepsilon\sqrt{1-x^{\prime 2}}$, where
$\varepsilon=\operatorname{sign}(z^{\prime})$. Thus,

(2.9)
$z^{\prime\prime}=-\varepsilon\frac{x^{\prime}x^{\prime\prime}}{\sqrt{1-x^{\prime
2}}},$

on every subinterval $J\subset I$ where $z^{\prime}(s)\neq 0$, for all $s\in
J$. So, from (2.8), we obtain that $x(s)$ is a solution to the autonomous
second order ordinary differential equation

$\displaystyle 2(c_{0}^{2}x^{\prime
2}+x^{2})^{\frac{3}{2}}\mbox{$\mathfrak{h}$}\left(\frac{xx^{\prime}}{\sqrt{c_{0}^{2}x^{\prime
2}+x^{2}}}\right)$ $\displaystyle=-\varepsilon
x^{\prime\prime}\left(\frac{x^{\prime 2}}{\sqrt{1-x^{\prime
2}}}+\sqrt{1-x^{\prime 2}}\right)(x^{3}+c_{0}^{2}x)$
$\displaystyle\phantom{==}+\varepsilon\sqrt{1-x^{\prime
2}}(2c_{0}^{2}x^{\prime 2}+x^{2})$
$\displaystyle=-\varepsilon\left(\frac{x^{\prime\prime}}{\sqrt{1-x^{\prime
2}}}(x^{3}+c_{0}^{2}x)-\sqrt{1-x^{\prime 2}}(2c_{0}^{2}x^{\prime
2}+x^{2})\right)$ (2.10)
$\displaystyle=-\varepsilon\left(\frac{x^{\prime\prime}(x^{3}+c_{0}^{2}x)-(1-x^{\prime
2})(2c_{0}^{2}x^{\prime 2}+x^{2})}{\sqrt{1-x^{\prime 2}}}\right),$

that is, when $x\neq 0$,

(2.11) $x^{\prime\prime}=\frac{(1-x^{\prime 2})(x^{2}+2c_{0}^{2}x^{\prime
2})-2\varepsilon\mbox{$\mathfrak{h}$}\left(\frac{xx^{\prime}}{\sqrt{c_{0}^{2}x^{\prime
2}+x^{2}}}\right)\sqrt{1-x^{\prime 2}}(c_{0}^{2}x^{\prime
2}+x^{2})^{\frac{3}{2}}}{x^{3}+c_{0}^{2}x}.$

Denoting $y=x^{\prime}$, the ODE (2.11) transforms into the first order
autonomous system

(2.12) $\left(\begin{array}[]{c}x\\\
y\end{array}\right)^{\prime}=\left(\begin{array}[]{c}y\\\
\frac{(1-y^{2})(x^{2}+2c_{0}^{2}y^{2})-2\varepsilon\mbox{$\mathfrak{h}$}\left(\frac{xy}{\sqrt{c_{0}^{2}y^{2}+x^{2}}}\right)\sqrt{1-y^{2}}(c_{0}^{2}y^{2}+x^{2})^{\frac{3}{2}}}{x^{3}+c_{0}^{2}x}\end{array}\right)=:F(x,y).$

The phase space of the system is

$\Theta_{\varepsilon}:=\mbox{${\mathbb{R}}$}\smallsetminus\\{0\\}\times(-1,1)$

with coordinates $(x,y)$ denoting, respectively, the distance to the rotation
axis and its variation.

If the profile curve $\alpha(s)=(x(s),0,z(s))$ intersects the $e_{3}$-axis,
that is, $x(s)=0$ for some $s\in I$, we obtain from (2.10) that

$\varepsilon\mbox{$\mathfrak{h}$}(0)=\frac{2(1-x^{\prime}(s)^{2})c_{0}^{2}x^{\prime}(s)^{2}}{2\sqrt{1-x^{\prime}(s)^{2}}|c_{0}|^{3}|x^{\prime}(s)|^{3}}=\frac{\sqrt{1-x^{\prime}(s)^{2}}}{|c_{0}||x^{\prime}(s)|}\quad\Longrightarrow\quad\mbox{$\mathfrak{h}$}(0)^{2}=\frac{1-x^{\prime}(s)^{2}}{c_{0}^{2}x^{\prime}(s)^{2}},$

that is, $\varepsilon\mbox{$\mathfrak{h}$}(0)\geq 0$ and $\alpha(s)$ satisfies

(2.13)
$x^{\prime}(s)=\pm\frac{1}{\sqrt{1+c_{0}^{2}\mbox{$\mathfrak{h}$}(0)^{2}}}.$

In particular, its associated orbit in $\Theta_{\varepsilon}$, such that
$\varepsilon\mbox{$\mathfrak{h}$}(0)\geq 0$, must intersect the $x=0$ axis at
the points

$p_{\varepsilon}:=\left(0,\frac{1}{\sqrt{1+c_{0}^{2}\mathfrak{h}(0)^{2}}}\right)\quad\text{or}\quad-
p_{\varepsilon}=\left(0,-\frac{1}{\sqrt{1+c_{0}^{2}\mathfrak{h}(0)^{2}}}\right).$

Moreover, observe that no orbit in $\Theta_{\varepsilon}$ can have a limit
point of the form $(0,y)$, with
$y\neq\pm\frac{1}{\sqrt{1+c_{0}^{2}\mathfrak{h}(0)^{2}}}$. Indeed, since
$\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$ and the equation (2.10) is continuous
around $\Theta_{\varepsilon}$, we can take $x(s)\to 0$ and obtain
$y(s)\to\pm\frac{1}{\sqrt{1+c_{0}^{2}\mathfrak{h}(0)^{2}}}$.

######  Remark 2.3.

Let $\gamma_{1}(s)=(x(s),y(s))$ be any solution to (2.12). Observe that
$\gamma_{2}(s)=(-x(s),-y(s))$ is also a solution to the system. This means
that the phase space $\Theta_{\varepsilon}$ is symmetric with respect to the
origin. Thus, without loss of generality, it is enough to analyse the case
$x(s)>0$ and consider the phase space

$\Theta_{\varepsilon}=(0,\infty)\times(-1,1).$

If the profile curve intersects the $e_{3}$-axis, its associated orbit
$(x(s),y(s))$ in $\Theta_{\varepsilon}$ must pass through the point $\pm
p_{\varepsilon}$ at some $s_{0}\in I$ ($x(s_{0})=0$). Since
$x^{\prime}(s_{0})\neq 0$, let $\delta>0$ such that, for
$s\in(s_{0},s_{0}+\delta)$ or $s\in(s_{0}-\delta,s_{0})$, we would have
$x(s)<0$. Note that we can analyse the orbit behavior from $\mp
p_{\varepsilon}$, considering the symmetry, and obtain $x(s)>0$ for
$s\in(s_{0}-\delta,s_{0}+\delta)\smallsetminus\\{s_{0}\\}$.

Moreover, let $\alpha_{i}(s)=(x_{i}(s),0,z_{i}(s))$, $i=1,2$, be the profile
curve of the helicoidal $\mathcal{H}$-surface $\Sigma_{i}$ associated to
$\gamma_{i}(s)$ and let $\Psi_{i}(s,\theta)$ be the parametrization of
$\Sigma_{i}$, as in (2.3). Letting $z_{0}:=z_{1}(0)$ and considering, up to
vertical translations, that $z_{2}(0)=z_{0}+c_{0}\pi$, we obtain that

$\Psi_{2}(s,\theta)=\Psi_{1}(s,\theta+\pi),$

that is, up to a change of coordinates that corresponds to a rotation by $\pi$
around the $e_{3}$-axis, $\Sigma_{1}=\Sigma_{2}$.

Indeed, note that $x_{2}(s)=-x_{1}(s)=-x(s)$ and

$z_{2}(s)=\int_{0}^{s}\varepsilon\sqrt{1-(-y(t))^{2}}dt+z_{2}(0)=\int_{0}^{s}\varepsilon\sqrt{1-(y(t))^{2}}dt+z_{0}+c_{0}\pi=z_{1}(s)+c_{0}\pi.$

Therefore,

$\displaystyle\Psi_{2}(s,\theta)$
$\displaystyle=\left(x_{2}(s)\cos{\theta},\,x_{2}(s)\sin{\theta},\,z_{2}(s)+c_{0}\theta\right)$
$\displaystyle=\left(-x(s)\cos{\theta},\,-x(s)\sin{\theta},\,z_{1}(s)+c_{0}\pi+c_{0}\theta\right)$
$\displaystyle=\left(x(s)\cos{(\theta+\pi)},\,x(s)\sin{(\theta+\pi)},\,z_{1}(s)+c_{0}(\pi+\theta)\right)$
$\displaystyle=\Psi_{1}(s,\theta+\pi).$

∎

######  Proposition 2.4.

There exists at most a unique (up to rotations around the $e_{3}$-axis)
helicoidal $\mathcal{H}$-surface in $\mbox{${\mathbb{R}}$}^{3}$ that
intersects its rotation axis.

###### Proof.

Let $\Sigma_{1}$ and $\Sigma_{2}$ be helicoidal $\mathcal{H}$-surfaces that
meet the $e_{3}$-axis. Then, this axis is contained in both surfaces. Let
$\Psi_{1}(s,\theta)$ and $\Psi_{2}(s,\theta)$ be the parametrization of
$\Sigma_{1}$ and $\Sigma_{2}$, respectively, as in (2.3), where, up to
rotations around the $e_{3}$-axis and a change of coordinates,
$\alpha_{1}(s)=(x_{1}(s),0,z_{1}(s))$ and
$\alpha_{2}(s)=(x_{2}(s),0,z_{2}(s))$ are the profile curves of $\Sigma_{1}$
and $\Sigma_{2}$, respectively, in the plane $\Pi=\\{y=0\\}$, such that
$\Psi_{i}(s,0)=\alpha_{i}(s)$, $i=1,2$, and
$\alpha_{1}(0)=\alpha_{2}(0)=p_{0}$, for some
$p_{0}=(0,0,z_{0})\in\Sigma_{1}\cap\Sigma_{2}$.

Consider $\delta_{i}:=\operatorname{sign}(x_{i}^{\prime}(0))$, $i=1,2$. By
(2.4), we obtain that the Gauss map of $\Sigma_{i}$ in $p_{0}$ is given by

$\eta(0,0)=\frac{\left(c_{0}\sin(0)x_{i}^{\prime}(0),-c_{0}\cos(0)x_{i}^{\prime}(0),0\right)}{|c_{0}||x_{i}^{\prime}(0)|}=-\operatorname{sign}(c_{0})\left(0,\delta_{i},0\right).$

Let $\varepsilon\in\\{-1,1\\}$ such that
$\varepsilon\mbox{$\mathfrak{h}$}(0)\geq 0$, where $\mathfrak{h}$ is given by
(2.7) in terms of $\mathcal{H}$. By (2.13), we have

$|x_{1}^{\prime}(0)|=|x_{2}^{\prime}(0)|=\frac{1}{\sqrt{1+c_{0}^{2}\mathfrak{h}(0)^{2}}},$

that is, the orbit associated to the profile curve of $\Sigma_{i}$ passes
through the point $\delta_{i}p_{\varepsilon}\in\Theta_{\varepsilon}$. If
$\delta_{1}\neq\delta_{2}$, we can rotate $\Sigma_{2}$ by $\pi$ around the
$e_{3}$-axis and, then, by Remark 2.3, obtain that $\Sigma_{1}$ and
$\Sigma_{2}$ have the same unit normal in $p_{0}$.

On a sufficiently small ball $B_{\delta}(p_{0})\subset\Pi$, we can see
$\Sigma_{1}$ and $\Sigma_{2}$ as graphs of differential functions $u_{1}(x,z)$
and $u_{2}(x,z)$, respectively, with

$u_{1}(0,z)=u_{2}(0,z)=0,\quad\forall z\in(z_{0}-\delta,z_{0}+\delta),$

where the difference $v:=u_{1}-u_{2}$ is a pseudoanalytic function, since
$u_{1}$ and $u_{2}$ are solutions to the same elliptic PDE (2.2) ([4, p.
469]).

If there exists some $(x_{0},z_{0})\in B_{\delta}(p_{0})$, with $x_{0}\neq 0$,
such that $u_{1}(x_{0},z_{0})=u_{2}(x_{0},z_{0})$, consider
$s_{0},\theta_{0}\in\mbox{${\mathbb{R}}$}$ such that
$\Psi_{1}(s_{0},\theta_{0})=u_{1}(x_{0},z_{0})=\Psi_{2}(s_{0},\theta_{0})$.
This means that $\Psi_{1}(s_{0},\theta)=\Psi_{2}(s_{0},\theta)$, for all
$\theta\in\mbox{${\mathbb{R}}$}$, that is, $u_{1}=u_{2}$ along the curve
$\gamma_{s_{0}}(\theta):=\Psi_{1}(s_{0},\theta)$ in $\Sigma_{1}$ and
$\Sigma_{2}$. In case there exists a set of accumulated curves
$\gamma_{s_{0}}(\theta)$, we obtain that $u_{1}\equiv u_{2}$ by
pseudoanalyticity, i.e. $\Sigma_{1}=\Sigma_{2}$. Therefore, the curves
$\gamma_{s_{0}}$ are isolated and we can get a sufficiently small ball
$B_{\epsilon}(p_{0})$ such that $u_{1}\leq u_{2}$ in

$B_{\epsilon}(p_{0})\cap\\{x\leq 0\\}.$

By (2.2), the functions $u_{1}$ and $u_{2}$ satisfy the same absolutely
elliptic partial differential equation. It follows from [5, Corollary 4.6]
that $u_{1}\equiv u_{2}$ and, therefore, up to rotations around the
$e_{3}$-axis, $\Sigma_{1}=\Sigma_{2}$. ∎

A point $e_{0}=(x_{0},y_{0})\in\Theta_{\varepsilon}$ is a equilibrium of
(2.12) if $F(x_{0},y_{0})=0$, i.e. $y_{0}=0$ and

$x_{0}^{2}-2\varepsilon\mbox{$\mathfrak{h}$}\left(0\right)x_{0}^{3}=0\quad\Longrightarrow\quad
x_{0}=\frac{1}{2\varepsilon\mbox{$\mathfrak{h}$}(0)}.$

Thus, if $\varepsilon\mbox{$\mathfrak{h}$}(0)>0$, the equilibrium in
$\Theta_{\varepsilon}$ is the point

$e_{0}=\left(\frac{1}{2\varepsilon\mbox{$\mathfrak{h}$}(0)},0\right)$

and it corresponds to the case where $\Sigma$ is a right circular cylinder
$\mbox{${\mathbb{S}}$}^{1}(1/2\mbox{$\mathfrak{h}$}(0))\times\mbox{${\mathbb{R}}$}$
in $\mbox{${\mathbb{R}}$}^{3}$ of constant mean curvature
$\mbox{$\mathfrak{h}$}(0)$ and vertical rulings. Note that the cylinder does
not depend on the pitch $c_{0}$. If $\varepsilon\mbox{$\mathfrak{h}$}(0)\leq
0$, there is not a equilibrium in $\Theta_{\varepsilon}$.

The orbits $(x(s),y(s))$ provide a foliation by regular proper $C^{1}$ curves
of $\Theta_{\varepsilon}$, or of
$\Theta_{\varepsilon}\smallsetminus\\{e_{0}\\}$, in case $e_{0}$ exists. Since
$\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$, the uniqueness of the initial value
problem for (2.12) implies that if an orbit $(x(s),y(s))$ converges to
$e_{0}$, the value of the parameter $s$ goes to $\pm\infty$.

The nullcline which corresponds to $x^{\prime}=0$ is given by the axis $y=0$
in $\Theta_{\varepsilon}$. Therefore, $x^{\prime}(s)>0$ if $y(s)>0$ and
$x^{\prime}(s)<0$ if $y(s)<0$. Denote by $\Gamma_{\varepsilon}$ the nullcline
in $\Theta_{\varepsilon}$ which corresponds to $y^{\prime}=0$, that is, the
set of points $(x,y)\in\Theta_{\varepsilon}$ such that

(2.14)
$\frac{\sqrt{1-y^{2}}(x^{2}+2c_{0}^{2}y^{2})}{2(x^{2}+c_{0}^{2}y^{2})^{\frac{3}{2}}}=\varepsilon\mbox{$\mathfrak{h}$}\left(\frac{xy}{\sqrt{c_{0}^{2}y^{2}+x^{2}}}\right).$

Alternatively, we can describe $\Gamma_{\varepsilon}$ as the implicit equation
$F_{\varepsilon}(x,y)=0$, where $(x,y)\in\Theta_{\varepsilon}$ and
$F_{\varepsilon}$ is the differential function given by

(2.15)
$F_{\varepsilon}(x,y):=2\varepsilon\mbox{$\mathfrak{h}$}\left(\frac{xy}{\sqrt{c_{0}^{2}y^{2}+x^{2}}}\right)(x^{2}+c_{0}^{2}y^{2})^{\frac{3}{2}}-(x^{2}+2c_{0}^{2}y^{2})\sqrt{1-y^{2}}.$

Note that $\Gamma_{\varepsilon}$ might be empty, for example, in the case
$\mbox{$\mathfrak{h}$}\leq 0$ and $\varepsilon=1$. Furthermore, observe that,
if $\Gamma_{\varepsilon}$ exists, it must intersect the $x=0$ axis at the
points $\pm p_{\varepsilon}$ and intersect the $y=0$ axis only at the
equilibrium $e_{0}$.

The values $s\in J$ where the profile curve $\alpha(s)=(x(s),0,z(s))$ of
$\Sigma$ has zero geodesic curvature $\kappa_{\alpha}$ are those where its
associated orbit $(x(s),y(s))$ belongs to $\Gamma_{\varepsilon}$. Indeed,
using (2.9), we obtain that for each $s\in J$,

$\displaystyle\kappa_{\alpha}(s)$
$\displaystyle=x^{\prime}(s)z^{\prime\prime}(s)-x^{\prime\prime}(s)z^{\prime}(s)$
$\displaystyle=-\varepsilon\frac{x^{\prime}(s)^{2}x^{\prime\prime}(s)}{\sqrt{1-x^{\prime}(s)^{2}}}-\varepsilon
x^{\prime\prime}(s)\sqrt{1-x^{\prime}(s)^{2}}$ $\displaystyle=-\varepsilon
y^{\prime}(s)\left(\frac{y(s)^{2}}{\sqrt{1-y(s)^{2}}}+\sqrt{1-y(s)^{2}}\right)$
$\displaystyle=-\frac{\varepsilon y^{\prime}(s)}{\sqrt{1-y(s)^{2}}},$

that is, $\kappa_{\alpha}(s)=0$ if, and only if, $y^{\prime}(s)=0$.

######  Remark 2.5.

Given $\gamma(s)=(x(s),y(s))\in\Theta_{\varepsilon}$ an orbit, if
$\gamma(s)\mapsto(x_{0},\pm 1)$, for some $x_{0}\geq 0$, then there exists
$s_{0}\in\mbox{${\mathbb{R}}$}$ such that $\gamma(s_{0})=(x_{0},\pm 1)$.
Indeed, if $\gamma(s)\mapsto(x_{0},\pm 1)$ only when $s\to\infty$, we can get
a sequence {$x(s_{n}):n\in\mathbb{N}\\}$, monotonous for $n\geq n_{0}$
sufficiently large, such that $x(s_{n})\to x_{0}$. It follows that
$x^{\prime}(s_{n})=x(s_{n+1})-x(s_{n})\to 0$, which is a contradiction,
because $x^{\prime}(s)=y(s)\to\pm 1$. ∎

## 3\. Positive mean curvature helicoidal surfaces

First, observe that if $\mathcal{H}\equiv H_{0}>0$, that is, the resulting
$\mathcal{H}$-surface has constant mean curvature $H_{0}$, we get that the
curve $\Gamma_{1}$ is given by the points $(x,y)\in\Theta_{1}$ such that

$x^{2}+2c_{0}^{2}y^{2}=\frac{2H_{0}(x^{2}+c_{0}^{2}y^{2})^{\frac{3}{2}}}{\sqrt{1-y^{2}}}\quad\Longrightarrow\quad
H_{0}=\frac{(x^{2}+2c_{0}^{2}y^{2})\sqrt{1-y^{2}}}{2(x^{2}+c_{0}^{2}y^{2})^{\frac{3}{2}}}.$

So, $\Gamma_{1}$ corresponds to the level curve $f_{1}\equiv H_{0}$, where

(3.1)
$f_{\varepsilon}(x,y):=\frac{\varepsilon(x^{2}+2c_{0}^{2}y^{2})\sqrt{1-y^{2}}}{2(x^{2}+c_{0}^{2}y^{2})^{\frac{3}{2}}}.\text{
(See Fig. \ref{curvnivel}).}$

Figure 3.1. Level curves of $f_{\varepsilon}$, for different values of
$H_{0}$.

######  Remark 3.1.

The intersection points of the level curve $f_{\varepsilon}\equiv H_{0}$ with
the $x=0$ axis are $p_{\varepsilon}$ and $-p_{\varepsilon}$. Indeed, note that
for $x=0$ we have

$H_{0}=f_{\varepsilon}(0,y)=\frac{\varepsilon\sqrt{1-y^{2}}}{|c_{0}||y|}\quad\Longrightarrow\quad
y=\pm\frac{1}{\sqrt{1+c_{0}^{2}H_{0}^{2}}},$

and $\varepsilon H_{0}>0$. ∎

We would like to get similar results from Theorem 4.1 of [3] for helicoidal
$\mathcal{H}$-surfaces in case that $\mathcal{H}\in
C^{1}(\mbox{${\mathbb{S}}$}^{2})$ is positive, rotationally invariant and
even, i.e. $\mathcal{H}(-x)=\mathcal{H}(x)>0$ for all
$x\in\mbox{${\mathbb{S}}$}^{2}$. In terms of the function $\mathfrak{h}$
associated to $\mathcal{H}$, this means that
$\mbox{$\mathfrak{h}$}(-y)=\mbox{$\mathfrak{h}$}(y)>0$ for all $y\in[-1,1]$.
We need to add one more hypothesis, different from the rotational case, to
guarantee the behavior of $\Gamma_{1}$: we suppose that the function
$\mbox{$\mathfrak{h}$}(t)$ is increasing for $t\in[0,1]$ and, so, decreasing
for $t\in[-1,0]$. It is possible to get similar results for
$\mbox{$\mathfrak{h}$}(t)$ decreasing for $t\in[0,1]$ and increasing for
$t\in[-1,0]$.

In this case, the curve $\Gamma_{-1}$ does not exist and $\Gamma_{1}$ is given
by the implicit equation $F_{1}(x,y)=0$, where $F_{1}$ is defined in (2.15).

######  Proposition 3.2.

If $\mbox{$\mathfrak{h}$}(t)\in C^{1}([-1,1])$ is positive, even and
increasing for $t\in[0,1]$, the curve $\Gamma_{1}$ is regular, compact and
connected.

###### Proof.

First, note that we can describe $\Gamma_{1}$ as the set

$\left\\{(x,y)\in\Theta_{1}:f_{1}(x,y)=\mbox{$\mathfrak{h}$}\left(\frac{xy}{\sqrt{c_{0}^{2}y^{2}+x^{2}}}\right)\right\\},$

where $f_{1}$ is the smooth function defined in (3.1). Moreover, observe that
$\Gamma_{1}$ is symmetric with respect to the $y=0$ axis, by reason of
$\mathfrak{h}$ is even.

Since $[-1,1]$ is compact and $\mathfrak{h}$ is continuous, there are minimum
$H_{m}$ and maximum $H_{M}$ values of the $\mathfrak{h}$ function, that is,
$0<H_{m}\leq\mbox{$\mathfrak{h}$}\leq H_{M}$. Thus, considering that $\pm
p_{1}\in\Gamma_{1}$, we get that $\Gamma_{1}$ is compact because it is a
closed set contained in the compact region

$D:=\\{H_{m}\leq f_{1}(x,y)\leq H_{M}\\}\cap\\{x\geq 0\\}.\text{ (See Fig.
\ref{regionfunction}).}$

Figure 3.2. A sketch of the $D$ region.

Consider the new variables

$u=x^{2}\quad\text{and}\quad v=\frac{xy}{\sqrt{x^{2}+c_{0}^{2}y^{2}}}.$

Note that the application
$G:\mbox{${\mathbb{R}}$}^{2}\to\mbox{${\mathbb{R}}$}^{2}$ given by
$G(x,y)=\left(x^{2},\frac{xy}{\sqrt{x^{2}+c_{0}^{2}y^{2}}}\right)$ is a
diffeomorphism from $\Theta_{1}$ to $G(\Theta_{1})$ that preserves
orientation, because the Jacobian determinant

$\mbox{$\mathrm{J}$}G(x,y)=\left|\begin{array}[]{cc}2x&0\\\
\frac{c_{0}^{2}y^{3}}{(x^{2}+c_{0}^{2}y^{2})^{\frac{3}{2}}}&\frac{x^{3}}{(x^{2}+c_{0}^{2}y^{2})^{\frac{3}{2}}}\end{array}\right|=\frac{2x^{4}}{(x^{2}+c_{0}^{2}y^{2})^{\frac{3}{2}}}>0$

is positive, for all $(x,y)\in\Theta_{1}$. Furthermore, we obtain

$G(\Theta_{1})=\left\\{(u,v)\in\mbox{${\mathbb{R}}$}^{2}:u>0,|v|<\sqrt{\frac{u}{u+c_{0}^{2}}}\right\\}.$

Considering the changing of variables, we obtain

$v^{2}=\frac{x^{2}y^{2}}{x^{2}+c_{0}^{2}y^{2}}=\frac{uy^{2}}{u+c_{0}^{2}y^{2}}\quad\Longrightarrow\quad
y^{2}=\frac{v^{2}u}{u-c_{0}^{2}v^{2}}.$

It is important to observe that $u-c_{0}^{2}v^{2}>0$ in $G(\Theta_{1})$,
because

$u-c_{0}^{2}v^{2}=x^{2}-\frac{c_{0}^{2}x^{2}y^{2}}{x^{2}+c_{0}^{2}y^{2}}=\frac{x^{4}}{x^{2}+c_{0}^{2}y^{2}}>0,$

for all $(x,y)\in\Theta_{1}$. Therefore, by (2.15), we get that $\Gamma_{1}$
is characterized by the implicit equation $\overline{F}_{1}(u,v)=0$, $(u,v)\in
G(\Theta_{1})$, where

$\displaystyle\overline{F}_{1}(u,v)$
$\displaystyle=2\mbox{$\mathfrak{h}$}(v)\left(u+\frac{c_{0}^{2}v^{2}u}{u-c_{0}^{2}v^{2}}\right)^{\frac{3}{2}}-\left(u+\frac{2c_{0}^{2}v^{2}u}{u-c_{0}^{2}v^{2}}\right)\sqrt{1-\frac{v^{2}u}{u-c_{0}^{2}v^{2}}}$
$\displaystyle=\frac{1}{\left(u-c_{0}^{2}v^{2}\right)^{\frac{3}{2}}}\left(2\mbox{$\mathfrak{h}$}(v)u^{3}-(u^{2}+c_{0}^{2}v^{2}u)\sqrt{u(1-v^{2})-c_{0}^{2}v^{2}}\right)$
$\displaystyle=\frac{u}{\left(u-c_{0}^{2}v^{2}\right)^{\frac{3}{2}}}\left(2\mbox{$\mathfrak{h}$}(v)u^{2}-(u+c_{0}^{2}v^{2})\sqrt{u(1-v^{2})-c_{0}^{2}v^{2}}\right).$

To simplify, we can also describe $\Gamma_{1}$ as the implicit equation
$F_{2}(u,v)=0$, $(u,v)\in G(\Theta_{1})$, where $F_{2}$ is the $C^{1}$
function given by

$F_{2}(u,v):=2\mbox{$\mathfrak{h}$}(v)u^{2}-(u+c_{0}^{2}v^{2})\sqrt{u(1-v^{2})-c_{0}^{2}v^{2}}.$

Let us prove that $0$ is a regular value of $F_{2}$. Note that

$\displaystyle\frac{\partial F_{2}}{\partial u}(u,v)$
$\displaystyle=\frac{c_{0}^{2}v^{2}(v^{2}+1)+3u(v^{2}-1)+8\mbox{$\mathfrak{h}$}(v)u\sqrt{u(1-v^{2})-c_{0}^{2}v^{2}}}{2\sqrt{u(1-v^{2})-c_{0}^{2}v^{2}}},$
(3.2) $\displaystyle\frac{\partial F_{2}}{\partial v}(u,v)$
$\displaystyle=\frac{v\left(u^{2}+3c_{0}^{4}v^{2}+c_{0}^{2}u(3v^{2}-1)\right)}{\sqrt{u(1-v^{2})-c_{0}^{2}v^{2}}}+2u^{2}\mbox{$\mathfrak{h}$}^{\prime}(v).$

Let $(u,v)\in G(\Theta_{1})$ be a point of $\Gamma_{1}$ such that
$\frac{\partial F_{2}}{\partial u}(u,v)=0$. Then, a computation shows that

(3.3)
$u=\frac{c_{0}^{2}v^{2}\left(3v^{2}-1+\sqrt{\left(3v^{2}-\frac{11}{3}\right)^{2}+\frac{32}{9}}\right)}{2(1-v^{2})}.$

Observe that the equilibrium $\left(\frac{1}{4\mathfrak{h}(0)^{2}},0\right)$,
the only point of the $v=0$ axis contained in $\Gamma_{1}$, does not satisfy
the above equation, so we can assume that $v\neq 0$.

By replacing (3.3) in (3.2), a straightforward computation shows that
$\frac{\partial F_{2}}{\partial v}(u,v)>0$, if $v>0$, and $\frac{\partial
F_{2}}{\partial v}(u,v)<0$, if $v<0$, since
$\mbox{$\mathfrak{h}$}^{\prime}(v)\geq 0$, for $v>0$, and
$\mbox{$\mathfrak{h}$}^{\prime}(v)\leq 0$, for $v<0$, by hypothesis. That
means that there does not exist a critical point of $F_{2}$ contained in
$\Gamma_{1}$, which proves that $0$ is a regular value of $F_{2}$, that is,
the curve $\Gamma_{1}=F_{2}^{-1}(0)$ is regular.

Now, let us prove that $\Gamma_{1}$ is connected. Let $\Gamma_{1}^{1}$ be the
connected component of $\Gamma_{1}$ that passes through the equilibrium
$e_{0}$. Since $\Gamma_{1}$ is contained in the compact $D$ region and $e_{0}$
is the only point in the $y=0$ axis that belongs to $\Gamma_{1}$, we get that
$\Gamma_{1}^{1}$ has $\pm p_{1}$ as endpoints. This means that $\Gamma_{1}$
divides the phase space $\Theta_{1}$ into two regions $\Lambda_{0}$ and
$\Lambda_{\infty}$, where the second one is unbounded.

Suppose that there exists a different connected component $\Gamma_{1}^{2}$ of
$\Gamma_{1}$, such that $\Gamma^{2}_{1}$ is not contained in the inner domain
bounded by some other possible connected component. Then, observe that
$\Gamma_{1}^{2}$ must be a closed curve contained in $\Lambda_{0}\cap\\{y\neq
0\\}$ or $\Lambda_{\infty}\cap\\{y\neq 0\\}$. Since $\Gamma_{1}$ is symmetric
with respect to the $y=0$ axis, it is sufficient to analyse the case $y>0$.
First, consider $\Gamma_{1}^{2}\subset\Lambda_{0}\cap\\{y>0\\}$.

Observe that, for points $(x,0)\in\Theta_{1}$ in the $y=0$ axis and
$\Lambda_{0}$ region, that is, points such that
$x<\frac{1}{2\mathfrak{h}(0)}$, it follows from (2.15) that

$F_{1}(x,0)=2\mbox{$\mathfrak{h}$}(0)x^{3}-x^{2}=x^{2}\left(2\mbox{$\mathfrak{h}$}(0)x-1\right)<0.$

This means, by regularity of $\Gamma_{1}=F_{1}^{-1}(0)$, that we can take a
point $p_{0}=(x_{0},y_{0})\in\Theta_{1}$, $y_{0}>0$, contained in the interior
region determined by $\Gamma_{1}^{2}$ such that $F_{1}(x_{0},y_{0})>0$, that
is,

(3.4)
$\mbox{$\mathfrak{h}$}\left(\frac{x_{0}y_{0}}{\sqrt{x_{0}^{2}+c_{0}^{2}y_{0}^{2}}}\right)>\frac{(x_{0}^{2}+2c_{0}^{2}y_{0}^{2})\sqrt{1-y_{0}^{2}}}{2(x_{0}^{2}+c_{0}^{2}y_{0}^{2})^{\frac{3}{2}}}.$

Note that we can also take a point $(x_{1},y_{0})\in\Theta_{1}$ with same
height of $(x_{0},y_{0})$ contained in the $\Lambda_{0}$ region, with
$x_{1}>x_{0}$, such that $F_{1}(x_{1},y_{0})<0$. See Figure 3.3.

Figure 3.3. A sketch of the connected components of $\Gamma_{1}$.

Define $\mbox{$\mathfrak{h}$}_{0}$ and $g_{0}$ by

$\mbox{$\mathfrak{h}$}_{0}(x):=\mbox{$\mathfrak{h}$}\left(\frac{y_{0}x}{\sqrt{x^{2}+c_{0}^{2}y_{0}^{2}}}\right)\quad\text{and}\quad
g_{0}(x):=\frac{(x^{2}+2c_{0}^{2}y_{0}^{2})\sqrt{1-y_{0}^{2}}}{2(x^{2}+c_{0}^{2}y_{0}^{2})^{\frac{3}{2}}},\quad
x>0.$

Since $\frac{y_{0}x}{\sqrt{x^{2}+c_{0}^{2}y_{0}^{2}}}$ is increasing for $x>0$
and, by hypothesis, the function $\mbox{$\mathfrak{h}$}(t)$ is increasing for
$t>0$, we get that $\mbox{$\mathfrak{h}$}_{0}$ is an increasing function.
Moreover, a computation shows that $g_{0}$ is strictly decreasing. Hence,

$\mbox{$\mathfrak{h}$}_{0}(x_{0})\leq\mbox{$\mathfrak{h}$}_{0}(x_{1})\quad\text{and}\quad
g_{0}(x_{0})>g_{0}(x_{1}).$

From (3.4) and using that $F_{1}(x_{1},y_{0})<0$, we get that

$\mbox{$\mathfrak{h}$}_{0}(x_{0})>g_{0}(x_{0})\quad\text{and}\quad\mbox{$\mathfrak{h}$}_{0}(x_{1})<g_{0}(x_{1}).$

Consequently,

$\mbox{$\mathfrak{h}$}_{0}(x_{0})\leq\mbox{$\mathfrak{h}$}_{0}(x_{1})<g_{0}(x_{1})<g_{0}(x_{0})<\mbox{$\mathfrak{h}$}_{0}(x_{0}),$

which is contradiction. If
$\Gamma_{1}^{2}\subset\Lambda_{\infty}\cap\\{y>0\\}$, it is possible to get a
similar contradiction, taking a point $(x_{1},y_{0})\in\Lambda_{\infty}$ such
that $x_{1}<x_{0}$.

Therefore, $\Gamma_{1}^{1}$ is the only connected component of $\Gamma_{1}$,
that is, $\Gamma_{1}$ is connected. ∎

######  Theorem 3.3.

Let $\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$ be given by (2.7) in terms of
$\mathcal{H}$, and suppose that $\mathfrak{h}$ is positive, even
($\mbox{$\mathfrak{h}$}(t)=\mbox{$\mathfrak{h}$}(-t)>0$, for all $t\in[-1,1]$)
and increasing for $t\in[0,1]$. Then, the following list exhausts, up to
vertical translations, all existing helicoidal $\mathcal{H}$-surfaces in
$\mbox{${\mathbb{R}}$}^{3}$ around the $e_{3}$-axis:

  1. (1)

The right circular cylinder
$C_{\mathcal{H}}:=\mbox{${\mathbb{S}}$}^{1}\left(\frac{1}{2\mathfrak{h}(0)}\right)\times\mbox{${\mathbb{R}}$}$;

  2. (2)

The orbit under a helicoidal motion of a curve $\alpha(s)=(x(s),0,z(s))$, with
$x(s)$ periodic and $x(s_{0}+kT)=0$, for some
$s_{0},T\in\mbox{${\mathbb{R}}$}$, and for all $k\in\mathbb{N}$, such that
$z^{\prime}(s)>0$, for all $s\in\mbox{${\mathbb{R}}$}$;

  3. (3)

The orbit under a helicoidal motion of a closed curve
$\alpha(s)=(x(s),0,z(s))$, diffeomorphic to the circle
$\mbox{${\mathbb{S}}$}^{1}$ , such that $x(s)>0$, for all
$s\in\mbox{${\mathbb{R}}$}$;

  4. (4)

A one-parameter family of helicoidal surfaces with a nodoid structure, that
is, the orbit under a helicoidal motion of a curve $\alpha(s)=(x(s),0,z(s))$,
such that $x(s)>0$ is periodic and the curve $\alpha$ is not embedded;

  5. (5)

A one-parameter family of helicoidal surfaces with a unduloid structure, that
is, the orbit under a helicoidal motion of a curve $\alpha(s)=(x(s),0,z(s))$,
such that $x(s)>0$ is periodic and $z^{\prime}(s)>0$, for all
$s\in\mbox{${\mathbb{R}}$}$.

###### Proof.

Let $(x(s),y(s))$ be any solution to (2.12). Taking into account that
$(x(-s),-y(-s))$ is also a solution to the system, we obtain that any orbit of
the phase space $\Theta_{\varepsilon}$ is symmetric with respect to the $y=0$
axis. By Proposition 3.2, the curve $\Gamma_{1}$ in $\Theta_{1}$ together with
$y=0$ divides the phase space $\Theta_{1}$ into four monotonicity regions
$\Lambda_{1},\dots,\Lambda_{4}$, all of them meeting at the equilibrium
$e_{0}$. Moreover, $\Theta_{-1}$ has only two monotonicity regions
$\Lambda^{+}=\Theta_{-1}\cap\\{y>0\\}$ and
$\Lambda^{-}=\Theta_{-1}\cap\\{y<0\\}$, because there does not exist
$\Gamma_{-1}$ in $\Theta_{-1}$. See Figure 3.4.

Figure 3.4. The phase spaces $\Theta_{1}$ and $\Theta_{-1}$ for
$\mbox{$\mathfrak{h}$}(t)$ positive, even and increasing for $t\in[0,1]$.

By monotonicity properties, any orbit in $\Theta_{-1}$ is given by a
horizontal $C^{1}$ graph $x=g(y)$, with $g(y)=g(-y)>0$ for all
$y\in(-\delta,\delta)$, for some $\delta\in(0,1]$, and such that $g$
restricted to $[0,\delta)$ is strictly increasing.

In case that $g(y)\to\infty$ as $y\to\delta$ for some orbit, let
$\alpha(s)=(x(s),0,z(s))$ be its associated profile curve and $r_{0}=g(0)$.
Consider $\Psi(s,\theta)$ a parametrization for the corresponding helicoidal
$\mathcal{H}$-surface $\Sigma$, as in (2.3). If we restrict
$\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$, the resulting surface $\Sigma$
would be a bi-graph over the set

$\Omega=\mbox{${\mathbb{R}}$}^{2}\smallsetminus B_{r_{0}}(0)\cap\\{x>0\\}.$

This is impossible, since $\mathcal{H}\geq H_{0}>0$ and it is well know, by
the maximum principle comparing with the sphere
$\mathbb{S}^{2}(\frac{1}{2H_{0}})$, that there do not exist graphs in
$\mbox{${\mathbb{R}}$}^{3}$ over $\Omega$ with mean curvature bounded from
below by a positive constant. Therefore, $\delta=1$ and any orbit in
$\Theta_{-1}$ has two limit endpoints of the form $(x_{0},\pm 1)$ for some
$x_{0}>0$. The resulting $\mathcal{H}$-surface $\Sigma_{-1}$ in
$\mbox{${\mathbb{R}}$}^{3}$ associated to any such orbit is then a surface
with boundary parametrized by the curves

$\gamma_{1}(t)=(x_{0}\cos t,x_{0}\sin
t,a+c_{0}t)\quad\text{and}\quad\gamma_{2}(t)=(x_{0}\cos t,x_{0}\sin
t,b+c_{0}t),\quad t\in\mbox{${\mathbb{R}}$},$

for some $a<b$. The $z(s)$-coordinate of the profile curve $\alpha(s)$ of
$\Sigma_{-1}$ is strictly decreasing and the unit normal to $\Sigma_{-1}$
along $\gamma_{1}$ and $\gamma_{2}$ are, respectively,

$\eta_{1}(t)=\frac{(c_{0}\sin t,-c_{0}\cos
t,x_{0})}{\sqrt{c_{0}^{2}+x_{0}^{2}}}\quad\text{and}\quad\eta_{2}(t)=\frac{(c_{0}\sin
t,-c_{0}\cos t,-x_{0})}{\sqrt{c_{0}^{2}+x_{0}^{2}}}.$

Figure 3.5. Phase spaces for the choice $c_{0}=1$ and
$\mbox{$\mathfrak{h}$}(y)=y^{2}+1$, $\Theta_{1}$ on the left and $\Theta_{-1}$
on the right. The dotted curve corresponds to $\Gamma_{1}$ and the red curve
to the orbit $\gamma_{0}$. The blue curve is the orbit corresponding to the
$\mathcal{H}$-surface with unduloid structure. The purple curves correspond to
the $\mathcal{H}$-surfaces with nodoid structure.

Now, we analyse the orbits in $\Theta_{1}$ and their associated
$\mathcal{H}$-surfaces. First, observe that the equilibrium
$e_{0}=\left(\frac{1}{2\mathfrak{h}(0)},0\right)\in\Theta_{1}$ corresponds to
the cylinder
$C_{\mathcal{H}}=\mathbb{S}^{1}\left(\frac{1}{2\mathfrak{h}(0)}\right)\times\mbox{${\mathbb{R}}$}$.

The linearized system at $e_{0}$ associated to the nonlinear system (2.12) for
$\varepsilon=1$ is

(3.5) $\left(\begin{array}[]{c}u\\\
v\end{array}\right)^{\prime}=\left(\begin{array}[]{cc}0&1\\\
-\frac{4\mathfrak{h}(0)^{2}}{1+4c_{0}^{2}\mathfrak{h}(0)^{2}}&0\end{array}\right)\left(\begin{array}[]{c}u\\\
v\end{array}\right),$

whose orbits are ellipses around the origin. By classical theory of nonlinear
autonomous systems, since the eigenvalues of the Jacobian matrix in (3.5) are
pure complex numbers
$\pm\frac{2\mathfrak{h}(0)}{\sqrt{1+4\mathfrak{h}(0)^{2}}}i$, we get that the
equilibrium $e_{0}$ has a center structure. This means that all orbits around
$e_{0}$ are closed curves, since $\Theta_{1}$ is symmetric with respect to the
$y=0$ axis. In particular, we deduce that all orbits of $\Theta_{1}$ stay at a
positive distance from $e_{0}$.

By Proposition 2.4, there exists an unique orbit $\gamma_{0}$ in the phase
space $\Theta_{1}$ that passes through
$p_{1}=\left(0,\frac{1}{\sqrt{1+c_{0}^{2}\mathfrak{h}(0)^{2}}}\right)$. By
monotonicity properties, $\gamma_{0}$ lies near $p_{1}$ inside the
monotonicity region $\Lambda_{2}$ and it cannot stay forever in that region,
by a same previous argument using that $\mathcal{H}\geq H_{0}>0$. Moreover,
$\gamma_{0}$ stays at a positive distance from $e_{0}$ and it does not
intersect $\Gamma_{1}$. Thus, $\gamma_{0}$ must intersect the $y=0$ axis at
some point $(x_{0},0)$ with $x_{0}>\frac{1}{2\mathfrak{h}(0)}$.

By symmetry of the phase space, we deduce that $\gamma_{0}$ can be extended to
be an orbit in $\Theta_{1}$ that joins the limit points $p_{1}$ and $-p_{1}$,
and that lies in the region $\Lambda_{2}\cup\Lambda_{3}\cup\\{y=0\\}$. Taking
into account that we can extend this orbit to $x(s)<0$, using that the phase
space is symmetric by Remark 2.3, the helicoidal $\mathcal{H}$-surface
associated to the orbit $\gamma_{0}$ is given by the orbit of a curve
$\alpha(s)=(x(s),0,z(s))$ under a helicoidal motion, with $x(s)$ periodic and
$x(s_{0}+kT)=0$, for some $s_{0},T\in\mbox{${\mathbb{R}}$}$ and for all
$k\in\mathbb{N}$, such that $z^{\prime}(s)>0$, for all
$s\in\mbox{${\mathbb{R}}$}$. See Figure 3.6.

Figure 3.6. The orbit $\gamma_{0}$ in $\Theta_{1}$, its associated profile
curve and its resulting helicoidal $\mathcal{H}$-surface.

The orbit $\gamma_{0}$ divides $\Theta_{1}$ into two connected components. The
first one containing the equilibrium $e_{0}$, which we will denote by
$\mathcal{W}_{0}$ and the second one, where $x>0$ is unbounded, which we will
denote by $\mathcal{W}_{\infty}$. By uniqueness of a system solution, any
orbit of $\Theta_{1}$ other than $\gamma_{0}$ lies entirely in
$\mathcal{W}_{0}$ or $\mathcal{W}_{\infty}$.

We can observe, by symmetry and monotonicity, that any orbit in
$\mathcal{W}_{\infty}$ is a symmetric horizontal graph $x=g(y)$ with
$g(1)=g(-1)=x_{0}$ for some $x_{0}>0$, and such that $g$ is strictly
increasing in $(-1,0)$ and strictly decreasing in $(0,1)$. The resulting
$\mathcal{H}$-surface $\Sigma_{1}$ in $\mbox{${\mathbb{R}}$}^{3}$ associated
to any such orbit is then a surface with boundary parametrized by the curves

$\gamma_{1}(t)=(x_{0}\cos t,x_{0}\sin
t,c+c_{0}t)\quad\text{and}\quad\gamma_{2}(t)=(x_{0}\cos t,x_{0}\sin
t,d+c_{0}t),\quad t\in\mbox{${\mathbb{R}}$},$

for some $c<d$. The $z(s)$-coordinate of the profile curve $\alpha(s)$ of
$\Sigma_{1}$ is strictly increasing and the unit normal to $\Sigma_{1}$ along
$\gamma_{1}$ and $\gamma_{2}$ are, respectively,

$\eta_{1}(t)=\frac{(c_{0}\sin t,-c_{0}\cos
t,x_{0})}{\sqrt{c_{0}^{2}+x_{0}^{2}}}\quad\text{and}\quad\eta_{2}(t)=\frac{(c_{0}\sin
t,-c_{0}\cos t,-x_{0})}{\sqrt{c_{0}^{2}+x_{0}^{2}}}.$

Consequently, by uniqueness of the solution to the Cauchy problem for
helicoidal $\mathcal{H}$-surfaces, we can deduce that, given $x_{0}>0$, the
$\mathcal{H}$-surfaces $\Sigma_{-1}$ and $\Sigma_{1}$ constructed associated
to $x_{0}$ can be smoothly glued together along any of boundary components
where their unit normal coincide, to form a larger $\mathcal{H}$-surface. For
this, we can assume without loss of generality in the previous construction
that $a=c$ or $b=d$, and hence $\Sigma_{1}$ and $\Sigma_{-1}$ have the same
Cauchy data.

If we have simultaneously $a=c$ and $b=d$, the profile curve of the
$\mathcal{H}$-surface obtained is diffeomorphic to the circle
$\mathbb{S}^{1}$. Thus, the resulting $\mathcal{H}$-surface $\Sigma$ will be a
tube diffeomorphic to the cylinder
$\mbox{${\mathbb{S}}$}^{1}\times\mbox{${\mathbb{R}}$}$. Note that $\Sigma$
could be properly embedded and it is important to observe that, if
$\mathcal{H}\equiv H_{0}>0$, this is impossible by Theorem 2.10 of [6].

If we have $a=c$ and $b\neq d$, or $a\neq c$ and $b=d$, by iterating the
previous process we obtain a non-embedded helicoidal $\mathcal{H}$-surface,
with a nodoid structure, invariant by a vertical translation. See Figure 3.7.

Figure 3.7. On the left, the red curve corresponds to an orbit in the
$\Theta_{-1}$ phase space and the purple curve to an orbit in $\Theta_{1}$
that lies in $\mathcal{W_{\infty}}$. On the middle, their associated profile
curves. On the right, the red surfaces correspond to the resulting helicoidal
$\mathcal{H}$-surfaces $\Sigma_{-1}$ and the purple surfaces to $\Sigma_{1}$,
composing a helicoidal $\mathcal{H}$-surface with nodoid structure.

Now, we consider an orbit $\gamma$ of $\Theta_{1}$ that is contained in the
region $\mathcal{W}_{0}$. Since $\gamma$ stays at a positive distance from
$e_{0}$ and cannot approach a point of the form $(0,y)$, with
$y\neq\pm\frac{1}{\sqrt{1+c_{0}^{2}\mathfrak{h}(0)^{2}}}$, we deduce, taking
into account the monotonicity properties and the symmetry with respect to the
$y=0$ axis, that $\gamma$ is a closed curve containing $e_{0}$ in its inner
region. That means that the profile curve $\alpha(s)=(x(s),0,z(s))$ of the
helicoidal $\mathcal{H}$-surface associated to any such orbit has an unduloid
structure, that is, satisfies that $s$ is defined for all real values, such
that $z^{\prime}(s)>0$ for all $s\in\mbox{${\mathbb{R}}$}$ and $x(s)$ is
periodic. See Figure 3.8.

Figure 3.8. An orbit that lies in $\mathcal{W}_{0}$ in the $\Theta_{1}$ phase
space, its associated profile curve and its resulting helicoidal
$\mathcal{H}$-surface, which has an unduloid structure.

∎

######  Remark 3.4.

It is important to observe that the fact that $\mathfrak{h}$ is increasing in
$[0,1]$ is only used to know that $\Gamma_{1}$ is regular and connected. If it
were established that this behavior of $\Gamma_{1}$ persists even when the
function $\mathfrak{h}$ does not exhibit monotonicity, the above theorem would
remain valid.

For instance, the Figure 3.9 illustrates the behavior of $\Gamma_{1}$ for the
specific choice $\mbox{$\mathfrak{h}$}(t)=\cos(40t)+2$, an even and positive
$C^{1}$ function that is not increasing in $[0,1]$. Observe that $\Gamma_{1}$
is not connected in this case.

Figure 3.9. The curve $\Gamma_{1}$ for the choice
$\mbox{$\mathfrak{h}$}(t)=\cos(40t)+2$.

∎

## 4\. Another examples of helicoidal $\mathcal{H}$-surfaces

In this section, we will exhibit some examples of helicoidal
$\mathcal{H}$-surfaces which there exists some $t_{0}\in(-1,1)$ ($t_{0}=\pm 1$
is not included by Remark 2.2) such that $\mbox{$\mathfrak{h}$}(t_{0})=0$ and
$\mbox{$\mathfrak{h}$}(t)\neq 0$, for all $t\neq t_{0}$, where
$\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$ is given by (2.7) in terms of
$\mathcal{H}$.

By (2.5), the angle function of $\Sigma$ is constant along the curves

$\beta_{0}:=\left\\{(x,y)\in\Theta_{\varepsilon}:\frac{xy}{\sqrt{x^{2}+c_{0}^{2}y^{2}}}=t_{0}\right\\}$

in each $\Theta_{\varepsilon}$ phase space. It is important to observe that,
in the $x=0$ axis, the angle function vanishes, in particular, at the points
$\pm p_{\varepsilon}$, when they exist. Since $x>0$ in $\Theta_{\varepsilon}$,
note that we can describe $\beta_{0}$ as a graph $y=f(x)$, where

$f(x)=\frac{t_{0}x}{\sqrt{x^{2}-c_{0}^{2}t_{0}^{2}}}.$

By letting $x\to\infty$ in $f(x)$, we get that $\beta_{0}$ converges to the
straight line $y=t_{0}$. Indeed,

$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}\frac{t_{0}x}{\sqrt{x^{2}-c_{0}^{2}t_{0}^{2}}}=\lim_{x\to\infty}\frac{t_{0}}{\sqrt{1-\frac{t_{0}^{2}c_{0}^{2}}{x^{2}}}}=t_{0}.$

Along $\beta_{0}$, letting $y\to\pm 1$, we would have
$\frac{x}{\sqrt{c_{0}^{2}+x(s)^{2}}}\to\pm t_{0}$, that is,
$x\to\frac{\pm|c_{0}|t_{0}}{\sqrt{1-t_{0}^{2}}}$. Set

$p_{0}^{+}:=\left(\frac{|c_{0}|t_{0}}{\sqrt{1-t_{0}^{2}}},1\right),\text{ if
}t_{0}\geq 0,\text{ and
}p_{0}^{-}:=\left(\frac{-|c_{0}|t_{0}}{\sqrt{1-t_{0}^{2}}},-1\right),\text{ if
}t_{0}\leq 0.\text{ (See Fig. \ref{beta0fig}).}$

Figure 4.1. A sketch for the curve $\beta_{0}$ for each $t_{0}>0$, $t_{0}=0$
and $t_{0}<0$.

It is important to observe that, if $\Gamma_{\varepsilon}$ exists, it must not
intersect the curve $\beta_{0}$. Indeed, if $(x,y)\in\beta_{0}$, we would get
that
$\mbox{$\mathfrak{h}$}\left(\frac{xy}{\sqrt{x^{2}+c_{0}^{2}y^{2}}}\right)=0$
in (2.14), which is impossible. Moreover, letting $y\to\pm 1$ in (2.14),
observe that $\Gamma_{\varepsilon}$ converges to $p_{0}^{+}$ or $p_{0}^{-}$.

### 4.1. Case $\mbox{$\mathfrak{h}$}(0)=0$

Suppose $t_{0}=0$. In this case, we get that $\pm p_{\varepsilon}=(0,\pm 1)$
and there is not a equilibrium in both $\Theta_{\varepsilon}$ phase spaces.
Moreover, if $\Gamma_{\varepsilon}$ exists, it must intersect $(0,\pm 1)$.
Letting $y\to 0$ in (2.14), we obtain that $\Gamma_{\varepsilon}$ converges at
infinity to the $y=0$ axis.

First, consider $\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$ such that
$\mbox{$\mathfrak{h}$}(0)=0$ and $\mbox{$\mathfrak{h}$}(t)>0$, for all $t\neq
0$. This means that $\Gamma_{-1}$ does not exist and $\Gamma_{1}$ have two
connected components, each one contained in the regions $\\{y>0\\}$ and
$\\{y<0\\}$. Consider the example $\mbox{$\mathfrak{h}$}(t)=t^{2}$.

The curve $\Gamma_{1}$ in $\Theta_{1}$ together with $y=0$ divides the phase
space $\Theta_{1}$ into four monotonicity regions
$\Lambda_{1},\dots,\Lambda_{4}$, where $\Lambda_{1}$ and $\Lambda_{2}$ are
contained in the $\\{y>0\\}$ region and $\Lambda_{3}$ and $\Lambda_{4}$ are
contained in $\\{y<0\\}$, such that $\Lambda_{1}$ and $\Lambda_{4}$ meet at
the $y=0$ axis. The curve $\Gamma_{-1}$ in $\Theta_{-1}$ does not exist, so
$\Theta_{-1}$ has only two monotonicity regions
$\Lambda_{+}:=\Theta_{-1}\cap\\{y>0\\}$ and
$\Lambda_{-}:=\Theta_{-1}\cap\\{y<0\\}$. See Figure 4.2.

Figure 4.2. The phase spaces $\Theta_{1}$ and $\Theta_{-1}$ for the choice
$\mbox{$\mathfrak{h}$}(t)=t^{2}$.

By Proposition 2.4, let $\gamma_{0}(s)=(x_{0}(s),y_{0}(s))$ be the orbit that
corresponds to the profile curve of the helicoidal $\mathcal{H}$-surface that
meets its rotation axis. Up to a change of orientation, assume that
$\gamma_{0}$ meets the $x=0$ axis at the point $p_{1}=(0,1)$. By Remark 2.3,
we can analyse its behavior for $x_{0}(s)<0$ in the regions $\Lambda_{j}^{*}$,
$j=1,\dots,4$, where $\Lambda_{j}^{*}$ is the reflection of $\Lambda_{j}$ with
respect to the origin.

For $x(s)>0$, we obtain, by monotonicity properties, that $\gamma_{0}$ lies
entirely in the region $\Lambda_{2}$ and converges to the axis $y=0$ as
$s\to\infty$. For $x(s)<0$, we get the same behavior, noting that $\gamma_{0}$
lies in the region $\Lambda_{3}^{*}$. See Figure 4.3.

Figure 4.3. The orbit $\gamma_{0}(s)$ in $\Theta_{1}$, where the blue
component correspond to $x_{0}(s)>0$ and the red component to $x_{0}(s)<0$,
its associated profile curve and its resulting helicoidal
$\mathcal{H}$-surface, for the choice $\mbox{$\mathfrak{h}$}(t)=t^{2}.$

Now, for each $\varepsilon=\pm 1$, let
$\gamma_{\varepsilon}(s)=(x_{\varepsilon}(s),y_{\varepsilon}(s))$ be the orbit
that passes through $(r_{0},0)$, for some $r_{0}>0$, and belongs to the phase
space $\Theta_{\varepsilon}$ around that point. By monotonicity properties,
note that, for $y_{1}>0$, the orbit $\gamma_{1}$ must intersect $\Gamma_{1}$,
moving for the $\Lambda_{2}$ region and, then, converges at infinity to the
$y=0$ axis, getting the same behavior for $y_{1}<0$. See Figure 4.4.

Figure 4.4. The orbit $\gamma_{1}$ in $\Theta_{1}$, for the choices
$\mbox{$\mathfrak{h}$}(t)=t^{2}$ and $r_{0}=1$, its associated profile curve
and its resulting helicoidal $\mathcal{H}$-surface.

Observe that the orbit $\gamma_{-1}$ in $\Theta_{-1}$ is given by a horizontal
graph $x=g(y)$, where $g\in C^{1}((a,b))$, for some $-1\leq a<0<b\leq 1$, such
that $g(0)=r_{0}$ and $g$ restricted to $[0,b]$ is strictly increasing (resp.
$g$ restricted to $[a,0]$ is strictly decreasing). Suppose that
$g(y)\to\infty$ as $y\to\\{a,b\\}$. Consider the parametrization
$\Psi(s,\theta)$ of the resulting helicoidal $\mathcal{H}$-surface $\Sigma$,
as in (2.3). If we restrict $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$ and
exclude the helix $\gamma(t)=(r_{0}\cos t,r_{0}\sin t,z_{0}+c_{0}t)$, the
surface $\Sigma$ would be a bi-graph over the set

$\Omega=\mbox{${\mathbb{R}}$}^{2}\smallsetminus
B_{r_{0}+\delta}(0)\cap\\{x>0\\},$

for some $\delta>0$, such that $\mathcal{H}\geq H_{0}>0$ in $\Omega$. This is
impossible, by previous discussions using the maximum principle.

Therefore, $a=-1$, $b=1$ and any orbit in $\Theta_{-1}$ has two limit
endpoints of the form $(x_{1},1)$ and $(x_{2},-1)$, for some $x_{1},x_{2}>0$.
In these points, the coordinate $z(s)$ attains, respectively, local maximum
and local minimum values, so we can analyse orbits $\gamma_{-1}^{(1)}$ and
$\gamma_{-1}^{(2)}$ in $\Theta_{1}$ that have, respectively, $(x_{1},1)$ and
$(x_{2},-1)$ as limit endpoints and, then, smoothly glue together the
resulting $\mathcal{H}$-surfaces. By monotonicity properties of $\Theta_{1}$,
the orbit $\gamma_{-1}^{(1)}$ stays in the region $\Lambda_{2}$ and converges
at infinity to the $y=0$ axis as well as $\gamma_{-1}^{(2)}$ stays in the
region $\Lambda_{3}$ and converges at infinity to the same axis. See Figure
4.5.

Figure 4.5. On the left, the red curve corresponds to the orbit $\gamma_{-1}$
in $\Theta_{-1}$, for the choices $\mbox{$\mathfrak{h}$}(t)=t^{2}$ and
$r_{0}=1$. The blue curves correspond to the orbits $\gamma_{-1}^{(1)}$ and
$\gamma_{-1}^{(2)}$ in $\Theta_{1}$. On the middle, their associated profile
curve and, on the right, their resulting helicoidal $\mathcal{H}$-surface.

Now, consider $\mbox{$\mathfrak{h}$}\in C^{1}([-1,1])$ such that
$\mbox{$\mathfrak{h}$}(0)=0$, $\mbox{$\mathfrak{h}$}(t)>0$, for all $t>0$ and
$\mbox{$\mathfrak{h}$}(t)<0$, for all $t<0$. This means that $\Gamma_{1}$ is
contained in the region $\\{y>0\\}$ and $\Gamma_{-1}$ is contained in the
region $\\{y<0\\}$. Consider the example $\mbox{$\mathfrak{h}$}(t)=t$.

The curve $\Gamma_{1}$ in $\Theta_{1}$ together with $y=0$ divides the phase
space $\Theta_{1}$ into three monotonicity regions $\Lambda_{1}^{+}$,
$\Lambda_{2}^{+}$ and $\Lambda_{3}^{+}$, where
$\Lambda_{1}^{+}:=\Theta_{1}\cap\\{y<0\\}$ and $\Lambda_{2}^{+}$ and
$\Lambda_{3}^{+}$ are contained in the $\\{y>0\\}$ region, such that
$\Lambda_{2}^{+}$ and $\Lambda_{1}^{+}$ meet at the $y=0$ axis. In the phase
space $\Theta_{-1}$, it happens an analogous partition, the curve
$\Gamma_{-1}$ together with $y=0$ divides the phase space $\Theta_{-1}$ into
three monotonicity regions $\Lambda_{1}^{-}$, $\Lambda_{2}^{-}$ and
$\Lambda_{3}^{-}$, where $\Lambda_{3}^{-}:=\Theta_{-1}\cap\\{y>0\\}$ and
$\Lambda_{1}^{-}$ and $\Lambda_{2}^{-}$ are contained in the $\\{y<0\\}$
region, such that $\Lambda_{2}^{-}$ and $\Lambda_{3}^{-}$ meet at the $y=0$
axis. See Figure 4.6.

Figure 4.6. The phase spaces $\Theta_{1}$ and $\Theta_{-1}$ for the choice
$\mbox{$\mathfrak{h}$}(t)=t$.

By Proposition 2.4, let $\gamma_{0}(s)=(x_{0}(s),y_{0}(s))$ be the orbit that
corresponds to the profile curve of the helicoidal $\mathcal{H}$-surface that
meets its rotation axis. Up to a change of orientation, assume that
$\gamma_{0}$ meets the $x=0$ axis at the point $p_{1}=(0,1)$. We can analyse
the orbit behavior for $x_{0}(s)<0$ in the reflected monotonicity regions with
respect to the origin, by previous discussion.

Observe that, by monotonicity properties, the orbit $\gamma_{0}$ cannot stay
in the phase space $\Theta_{1}$ for $x_{0}(s)<0$. This means that the
coordinate $z(s)$ of its associated profile curve attains a local minimum in
the rotation axis. So, we can analyse the orbit $\gamma_{0}^{-}$ in
$\Theta_{-1}$ that has $(0,1)$ as an endpoint and smoothly glue together the
resulting $\mathcal{H}$-surfaces. Note that $\gamma_{0}$ lies entirely in the
region $\Lambda_{3}^{+}$ and converges at infinity to the $y=0$ axis as well
as $\gamma_{0}^{-}$ lies entirely in the reflected region of $\Lambda_{1}^{-}$
with respect to the origin and converges to the $y=0$ axis as $s\to-\infty$.
See Figure 4.7.

Figure 4.7. On the left, the blue curve corresponds to the orbit $\gamma_{0}$
in $\Theta_{1}$ and the red curve to the orbit $\gamma_{0}^{-}$ in
$\Theta_{-1}$, for the choice $\mbox{$\mathfrak{h}$}(t)=t$. On the middle,
their associated profile curve and, on the right, their resulting helicoidal
$\mathcal{H}$-surface.

Now, for each $\varepsilon=\pm 1$, let
$\gamma_{\varepsilon}(s)=(x_{\varepsilon}(s),y_{\varepsilon}(s))$ be the orbit
that passes through $(r_{0},0)$, for some $r_{0}>0$, and belongs to the phase
space $\Theta_{\varepsilon}$ around this point. By monotonicity properties,
note that, for $y_{1}>0$, the orbit $\gamma_{1}$ must intersect $\Gamma_{1}$,
moving to the region $\Lambda_{3}^{+}$ and, then, converges at infinity to the
$y=0$ axis. For $y_{1}<0$, the orbit $\gamma_{1}$ has a limit endpoint of the
form $(x_{0},-1)$, for some $x_{0}>r_{0}$. Therefore, we analyse an orbit
$\gamma_{1}^{-}$ in $\Theta_{-1}$ that has $(x_{1},-1)$ as a limit endpoint
and smoothly glue the resulting $\mathcal{H}$-surfaces. By monotonicity
properties, $\gamma_{1}^{-}$ stays in the region $\Lambda_{1}^{-}$ and
converges to the $y=0$ axis as $s\to-\infty$. See Figure 4.8. The behavior of
$\gamma_{-1}$ is pretty similar to $\gamma_{1}$.

Figure 4.8. On the left, the blue curve corresponds to the orbit $\gamma_{1}$
in $\Theta_{1}$ and the red curve to the orbit $\gamma_{1}^{-}$ in
$\Theta_{-1}$, for the choice $\mbox{$\mathfrak{h}$}(t)=t$. On the middle,
their associated profile curve and, on the right, their resulting helicoidal
$\mathcal{H}$-surface.

### 4.2. Case $\mbox{$\mathfrak{h}$}(t_{0})=0$, for some $t_{0}\neq 0$

In this case, suppose that there exists one $t_{0}\in(0,1)$ such that
$\mbox{$\mathfrak{h}$}(t_{0})=0$ and $\mbox{$\mathfrak{h}$}(t)\neq 0$, for all
$t\neq t_{0}$. Thus, $\mathfrak{h}$ vanishes in the curve $\beta_{0}$ in both
phase spaces. First, we will show a example of $\mathfrak{h}$ such that
$\mbox{$\mathfrak{h}$}(t)>0$, for all $t\neq t_{0}$. Finally, we will show a
example of $\mathfrak{h}$ such that $\mbox{$\mathfrak{h}$}(t)>0$, for all
$t>t_{0}$, and $\mbox{$\mathfrak{h}$}(t)<0$, for all $t<t_{0}$. The biggest
problem to extend the orbits behavior of these examples to the general case is
not knowing exactly how $\Gamma_{\varepsilon}$ looks like.

First, consider the example $\mbox{$\mathfrak{h}$}(t)=(t-0.6)^{2}$, that is,
$t_{0}=0.6$. Thus,

$p_{0}^{+}=\left(\frac{|c_{0}|0.6}{\sqrt{1-0.6^{2}}},1\right)=\left(\frac{3|c_{0}|}{4},1\right)\quad\text{and}\quad
p_{1}=\left(0,\frac{1}{\sqrt{1+0.36^{2}c_{0}^{2}}}\right).$

Since $\mbox{$\mathfrak{h}$}\geq 0$, the curve $\Gamma_{-1}$ does not exist.
Therefore, $\Theta_{-1}$ has two monotonicity regions:
$\Lambda^{+}:=\Theta_{-1}\cap\\{y>0\\}$ and
$\Lambda^{-}:=\Theta_{-1}\cap\\{y<0\\}$. The curve $\Gamma_{1}$ has three
connected components, the first one connecting $p_{1}$ and $p_{0}^{+}$, the
second one above the curve $\beta_{0}$ and the last one containing the
equilibrium $e_{0}\in\Theta_{1}$. Then, the curve $\Gamma_{1}$, together with
the $y=0$ axis, divides the phase space $\Theta_{1}$ into six monotonicity
regions
$\Lambda_{1}^{+},\dots,\Lambda_{4}^{+},\Lambda_{1}^{-},\Lambda_{2}^{-}$, four
of them above the $y=0$ axis and two of them below $y=0$. See Figure 4.9.

Figure 4.9. The phase spaces $\Theta_{1}$ and $\Theta_{-1}$ for the choice
$\mbox{$\mathfrak{h}$}(t)=(t-0.6)^{2}$.

By Proposition 2.4, let $\gamma_{0}(s)=(x_{0}(s),y_{0}(s))$ be the orbit that
corresponds to the $\mathcal{H}$-surface that meets its rotation axis. Up to a
change of orientation, suppose that $\gamma_{0}$ intersects the point
$p_{1}\in\Theta_{1}$. By monotonicity properties, for $x_{0}(s)>0$, we get
that $\gamma_{0}$ lies in the $\Lambda_{2}^{+}$ region, intersect the curve
$\Gamma_{1}$ and, then, stays in the region $\Lambda_{3}^{+}$ converging at
infinity to the straight line $y=t_{0}$. For $x_{0}(s)<0$, we get that
$\gamma_{0}$ converges to the equilibrium $-e_{0}$. See Figure 4.10.

Figure 4.10. The orbit $\gamma_{0}(s)$ in $\Theta_{1}$, where the blue
component correspond to $x_{0}(s)>0$ and the red component to $x_{0}(s)<0$,
its associated profile curve and its resulting helicoidal
$\mathcal{H}$-surface, for the choice $\mbox{$\mathfrak{h}$}(t)=(t-0.6)^{2}.$

Now, let $\gamma_{1}(s)$ be an orbit in $\Theta_{1}$ that passes through a
point $(x_{1},0)\in\Theta_{1}$, for some $0<x_{1}<\frac{1}{2\mathfrak{h}(0)}$,
at some moment $s=s_{0}$. For $s>s_{0}$, we get by monotonicity properties
that $\gamma_{1}$ lies in the $\Lambda_{2}^{+}$ region, intersect the curve
$\Gamma_{1}$ and, then, stays in the region $\Lambda_{3}^{+}$ converging at
infinity to the straight line $y=t_{0}$. For $s<s_{0}$, we get that
$\gamma_{1}$ converges to the equilibrium $e_{0}$ as $s\to-\infty$. See Figure
4.11.

Figure 4.11. The orbit $\gamma_{1}$ in $\Theta_{1}$, its associated profile
curve and its resulting helicoidal $\mathcal{H}$-surface, for the choice
$\mbox{$\mathfrak{h}$}(t)=(t-0.6)^{2}$.

Let now $\gamma_{-1}(s)$ be an orbit in $\Theta_{-1}$. By previous discussion,
note that there exist some moments $s_{1},s_{2}\in\mbox{${\mathbb{R}}$}$, with
$s_{1}<s_{2}$, such that $\gamma_{-1}(s_{1})=(x_{1},-1)$ and
$\gamma_{-1}(s_{2})=(x_{2},1)$, for some $x_{1},x_{2}>0$. We can analyse
orbits in $\Theta_{1}$ that have these points as limit endpoints and smoothly
glue the resulting $\mathcal{H}$-surfaces. Let $\gamma_{-1}^{1}$ be the orbit
in $\Theta_{1}$ that has $(x_{1},-1)$ as endpoint. By monotonicity properties,
we obtain that $\gamma_{-1}^{1}$ converges to the equilibrium
$e_{0}\in\Theta_{1}$ as $s\to-\infty$.

Next, let $\gamma_{-1}^{2}$ be the orbit in $\Theta_{2}$ that has $(x_{2},1)$
as a limit endpoint. There are two possibilities for the behavior of
$\gamma_{-1}^{2}$. If $x_{2}\geq\frac{3|c_{0}|}{4}$, that is, $(x_{2},1)$ is
$p_{0}^{+}$ or is to the right of $p_{0}^{+}$, then $\gamma_{-1}^{2}$ stays in
the region $\Lambda_{3}^{+}$ and converges at infinity to the straight line
$y=t_{0}$. See Figure 4.12.

Figure 4.12. On the left, the red curve corresponds to the orbit $\gamma_{-1}$
in $\Theta_{-1}$, the blue curve to the orbit $\gamma_{-1}^{1}$ in
$\Theta_{1}$ and the green curve to the orbit $\gamma_{-1}^{2}$ in
$\Theta_{1}$, with $x_{2}\geq\frac{3|c_{0}|}{4}$, for the choice
$\mbox{$\mathfrak{h}$}(t)=(t-0.6)^{2}$. On the middle, their associated
profile curve and, on the right, their resulting helicoidal
$\mathcal{H}$-surface.

Now, if $x_{2}<\frac{3|c_{0}|}{4}$, that is, $(x_{2},1)$ is to the left of
$p_{0}^{+}$, then, by monotonicity properties, the orbit $\gamma_{-1}^{2}$
lies in the region $\Lambda_{1}^{+}$ around $(x_{2},1)$, intersects the curve
$\Gamma_{1}$ moving to the $\Lambda_{2}^{+}$ region, intersects again the
curve $\Gamma_{1}$ and stays in the region $\Lambda_{3}^{+}$, converging at
infinity to the straight line $y=t_{0}$. See Figure 4.13.

Figure 4.13. On the left, the red curve corresponds to the orbit $\gamma_{-1}$
in $\Theta_{-1}$, the blue curve to the orbit $\gamma_{-1}^{1}$ in
$\Theta_{1}$ and the green curve to the orbit $\gamma_{-1}^{2}$ in
$\Theta_{1}$, with $x_{2}<\frac{3|c_{0}|}{4}$, for the choice
$\mbox{$\mathfrak{h}$}(t)=(t-0.6)^{2}$. On the middle, their associated
profile curve and, on the right, their resulting helicoidal
$\mathcal{H}$-surface.

Finally, consider the example $\mbox{$\mathfrak{h}$}(t)=(t-0.5)(t+2)$, that
is, $t_{0}=0.5$. Thus,

$p_{0}^{+}=\left(\frac{|c_{0}|0.5}{\sqrt{1-0.5^{2}}},1\right)=\left(\frac{|c_{0}|\sqrt{3}}{3},1\right)\quad\text{and}\quad
p_{-1}=\left(0,\frac{1}{\sqrt{1+c_{0}^{2}}}\right).$

Note that, in this case, we have that $\mbox{$\mathfrak{h}$}>0$ above the
curve $\beta_{0}$ and $\mbox{$\mathfrak{h}$}<0$ below it. This means that the
curve $\Gamma_{1}$ is connected and stays above $\beta_{0}$, starting from the
point $p_{0}^{+}$ and converging at infinity to the straight line $y=t_{0}$.
Therefore, the phase space $\Theta_{1}$ has three monotonicity regions
$\Lambda_{1}^{+},\Lambda_{2}^{+},\Lambda_{3}^{+}$, where
$\Lambda_{1}^{+}:=\Theta_{1}\cap\\{y<0\\}$ and $\Lambda_{2}^{+}$ and
$\Lambda_{3}^{+}$ are contained in the region $\\{y>0\\}$, with
$\Lambda_{3}^{+}$ above $\Gamma_{1}$.

Moreover, the curve $\Gamma_{-1}$ has two connected components, where the
first one connects the points $p_{-1}$ and $p_{0}^{+}$ and the second one
contains the equilibrium $e_{0}$. Therefore, the phase space $\Theta_{-1}$ has
five monotonicity regions $\Lambda_{1}^{-},\dots,\Lambda_{5}^{-}$, three of
them above the $y=0$ axis and two of them below it. See Figure 4.14.

Figure 4.14. The phase spaces $\Theta_{1}$ and $\Theta_{-1}$ for the choice
$\mbox{$\mathfrak{h}$}(t)=(t-0.5)(t+2)$.

By Proposition 2.4, let $\gamma_{0}(s)=(x_{0}(s),y_{0}(s))$ be the orbit that
corresponds to the $\mathcal{H}$-surface that meets its rotation axis. Up to a
change of orientation, suppose that $\gamma_{0}$ intersects the point
$p_{-1}\in\Theta_{-1}$. By monotonicity properties, for $x_{0}(s)<0$, we get
that $\gamma_{0}$ converges to the equilibrium $-e_{0}$ as $s\to-\infty$. For
$x_{0}(s)>0$, we get that $\gamma_{0}$ lies in the $\Lambda_{2}^{-}$ region
and, at some moment $s_{0}\in\mbox{${\mathbb{R}}$}$, goes to a limit endpoint
$(x_{1},1)$, with $x_{1}\geq\frac{|c_{0}|\sqrt{3}}{3}$. Considering
$\gamma_{0}^{+}$ the orbit in $\Theta_{1}$ that also has $(x_{1},1)$ as an
endpoint, we can smoothly glue the resulting $\mathcal{H}$-surfaces. It
follows from monotonicity properties that $\gamma_{0}^{+}$ stays in the
$\Lambda_{3}^{+}$ region and converges at infinity to the straight line
$y=t_{0}$. See Figure 4.15.

Figure 4.15. On the left, the red curve corresponds to the orbit $\gamma_{0}$
in $\Theta_{-1}$ and the blue curve to the orbit $\gamma_{0}^{+}$ in
$\Theta_{1}$, for the choice $\mbox{$\mathfrak{h}$}(t)=(t-0.5)(t+2)$. On the
middle, their associated profile curve and, on the right, their resulting
helicoidal $\mathcal{H}$-surface.

Let $\gamma_{1}(s)$ be an orbit in $\Theta_{1}$ that passes through a point
$(x_{0},0)\in\Theta_{1}$, for some $x_{0}>0$, at some moment $s=s_{0}$. By
previous discussion and Remark 2.5, note that there exists
$s_{1}\in\mbox{${\mathbb{R}}$}$, with $s_{1}<s_{0}$, such that
$\gamma_{1}(s_{1})=(x_{1},-1)$, for some $x_{1}>x_{0}$. Consider
$\gamma_{1}^{-}$ the orbit in $\Theta_{-1}$ that has $(x_{1},-1)$ as an
endpoint, which we can smoothly glue together the resulting
$\mathcal{H}$-surfaces. By monotonicity properties, $\gamma_{1}^{-}$ converges
to the equilibrium $e_{0}$ as $s\to-\infty$.

For $s>s_{0}$, there are two possibilities for the orbit $\gamma_{1}$
behavior. The first one is that $\gamma_{1}$ intersects the curve $\Gamma_{1}$
and, then, stays in the $\Lambda_{3}^{+}$ converging at infinity to the
straight line $y=t_{0}$. See Figure 4.16.

Figure 4.16. On the left, the blue curve corresponds to the orbit $\gamma_{1}$
in $\Theta_{1}$ and the red curve to the orbit $\gamma_{1}^{-}$ in
$\Theta_{-1}$, for the choice $\mbox{$\mathfrak{h}$}(t)=(t-0.5)(t+2)$. On the
middle, their associated profile curve and, on the right, their resulting
helicoidal $\mathcal{H}$-surface.

The second possibility is that the orbit $\gamma_{1}$ has an endpoint
$(x_{2},1)$, with $x_{2}<\frac{|c_{0}|\sqrt{3}}{3}$, that is, $(x_{2},1)$ is
to the left of $p_{0}^{+}$. Consider $\gamma_{1}^{2}$ the orbit in
$\Theta_{-1}$ that has $(x_{2},1)$ as an endpoint, which we can smoothly glue
together the resulting $\mathcal{H}$-surfaces. Thus, we get, by monotonicity
properties, that $\gamma_{1}^{2}$ lies in the $\Lambda_{1}^{-}$ region for
points near $(x_{2},1)$, intersects the curve $\Gamma_{-1}$ at some moment
moving to the $\Lambda_{2}^{-}$ region and, then, goes to another endpoint
$(x_{3},1)$, with $x_{3}>\frac{|c_{0}|\sqrt{3}}{3}$. Let $\gamma_{1}^{3}$ be
the orbit in $\Theta_{1}$ that has $(x_{3},1)$ as an endpoint. Observe that
$\gamma_{1}^{3}$ stays in the $\Lambda_{3}^{+}$ region and converges at
infinity to the straight line $y=t_{0}$. See Figure 4.17.

Figure 4.17. On the left, the blue curves correspond to the orbits
$\gamma_{1}$ and $\gamma_{1}^{3}$ in $\Theta_{1}$ and the red curves to the
orbits $\gamma_{1}^{-}$ and $\gamma_{1}^{2}$ in $\Theta_{-1}$, for the choice
$\mbox{$\mathfrak{h}$}(t)=(t-0.5)(t+2)$. On the middle, their associated
profile curve and, on the right, their resulting helicoidal
$\mathcal{H}$-surface.

Lastly, let $\gamma_{-1}(s)$ be an orbit in $\Theta_{-1}$ that passes at some
moment $s=s_{1}$ through a point $(x_{-1},0)$, with
$0<x_{-1}<\frac{1}{2\mathfrak{h}(0)}$, that is, $(x_{-1},0)$ is to the left of
the equilibrium $e_{0}\in\Theta_{-1}$. By monotonicity properties, the orbit
$\gamma_{-1}(s)$ converges to the equilibrium $e_{0}$ as $s\to-\infty$.
Moreover, for $s>s_{1}$, the orbit $\gamma_{-1}$ stays in the
$\Lambda_{2}^{-}$ region and has an endpoint $(x_{4},1)$, with
$x_{4}>\frac{|c_{0}|\sqrt{3}}{3}$. Considering $\gamma_{-1}^{+}$ the orbit in
$\Theta_{1}$ that has $(x_{4},1)$ as an endpoint, which we can smoothly glue
together the resulting $\mathcal{H}$-surfaces, we get that $\gamma_{-1}^{+}$
stays in the $\Lambda_{3}^{+}$ region and converges at infinity to the
straight line $y=t_{0}$. See Figure 4.18.

Figure 4.18. On the left, the red curve corresponds to the orbit $\gamma_{-1}$
in $\Theta_{-1}$ and the blue curve to the orbit $\gamma_{-1}^{+}$ in
$\Theta_{1}$, for the choice $\mbox{$\mathfrak{h}$}(t)=(t-0.5)(t+2)$. On the
middle, their associated profile curve and, on the right, their resulting
helicoidal $\mathcal{H}$-surface.

## Acknowledgements

This work was done in Instituto de Matemáticas of University of Granada and
the author is supported by BEPE - FAPESP (São Paulo Research Foundation) Grant
number 2022/14381-5. The author is grateful to Prof. José A. Gálvez for his
supervisory on the project and all his collaboration.

## References

  * [1] A. D. Aleksandrov.  Uniqueness theorems for surfaces in the large. I.  Vestnik Leningrad. Univ., 11(19):5–17, 1956.
  * [2] Antonio Bueno, José A. Gálvez, and Pablo Mira.  The global geometry of surfaces with prescribed mean curvature in $\mbox{${\mathbb{R}}$}^{3}$.  Trans. Amer. Math. Soc., 373(6):4437–4467, 2020.
  * [3] Antonio Bueno, José A. Gálvez, and Pablo Mira.  Rotational hypersurfaces of prescribed mean curvature.  J. Differential Equations, 268(5):2394–2413, 2020.
  * [4] Philip Hartman and Aurel Wintner.  On the local behavior of solutions of non-parabolic partial differential equations.  Amer. J. Math., 75:449–476, 1953.
  * [5] Heinz Hopf.  Differential geometry in the large, volume 1000 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, second edition, 1989.
  * [6] Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon.  The structure of complete embedded surfaces with constant mean curvature.  J. Differential Geom., 30(2):465–503, 1989.
  * [7] A. V. Pogorelov.  Extension of a general uniqueness theorem of A. D. Aleksandrov to the case of nonanalytic surfaces.  Doklady Akad. Nauk SSSR (N.S.), 62:297–299, 1948.

Aires E. M. Barbieri

Instituto de Ciências Matemáticas e de Computação,
Universidade de São Paulo, São Carlos (Brazil).

e-mail<EMAIL_ADDRESS>


# Are EEG-to-Text Models Working?

###### Abstract

This work critically analyzes existing models for open-vocabulary EEG-to-Text
translation. We identify a crucial limitation: previous studies often employed
implicit teacher-forcing during evaluation, artificially inflating performance
metrics. Additionally, they lacked a critical benchmark – comparing model
performance on pure noise inputs. We propose a methodology to differentiate
between models that truly learn from EEG signals and those that simply
memorize training data. Our analysis reveals that model performance on noise
data can be comparable to that on EEG data. These findings highlight the need
for stricter evaluation practices in EEG-to-Text research, emphasizing
transparent reporting and rigorous benchmarking with noise inputs. This
approach will lead to more reliable assessments of model capabilities and pave
the way for robust EEG-to-Text communication systems.

##  1 Introduction

The field of brain-computer interfaces (BCIs) has witnessed significant
progress, particularly in translating electroencephalography (EEG) signals
into written text. This technology offers immense potential for individuals
with communication disabilities by enabling direct conversion of brain
activity into language. The rapid progress in this area has stimulated
extensive research aimed at refining the accuracy, speed, and overall
usability of EEG-to-Text models, with the ultimate aim of facilitating smooth
and effective communication for a wider population.††*Equal contribution
†Corresponding author
Corresponding author email<EMAIL_ADDRESS><EMAIL_ADDRESS>

However, despite advancements, existing studies on EEG-to-Text translation
exhibit methodological shortcomings that limit the validity and reliability of
their findings. One major concern is the use of implicit teacher-forcing
during model evaluation [1]. This technique can artificially inflate
performance metrics, resulting in an inaccurate assessment of a model’s true
capability. Additionally, previous studies [1, 2, 3] have not compared EEG
inputs with pure noise, a crucial benchmark for evaluating the effectiveness
of EEG-based communication systems. While prior research has reported high-
performance metrics using EEG data (e.g., BLEU scores [4]), they have not
compared these results to models fed with pure noise of the same length. This
omission is critical because similar performance on noise suggests the model
is not effectively learning from EEG inputs and might be memorizing text
instead.

Addressing these limitations, Yang et al. [5] compared their model with noise
and magnetoencephalography (MEG) inputs, but their analysis did not encompass
mainstream EEG-to-Text models [1]. This paper addresses this gap by conducting
a comprehensive examination of existing EEG-to-Text models, particularly
focusing on their performance in open-vocabulary settings. Our analysis
reveals a significant deficiency in previous evaluation methodologies,
specifically the lack of comparison between EEG inputs and pure noise. This
comparison is essential to accurately assess a model’s ability to extract
meaningful information from EEG signals, as it differentiates genuine learning
from simple memorization.

Through our comparative analysis, we demonstrate that current EEG-to-Text
models achieve similar or even better performance on pure noise compared to
actual EEG inputs. This finding raises concerns about the true learning
capabilities of these models and underscores the need for more transparent and
rigorous evaluation methodologies within the field.

The main contributions of this paper are summarized as follows:

  1. 1.

We introduce a novel methodology to assess whether EEG-to-Text models are
genuinely learning from EEG inputs or simply memorizing patterns.

  2. 2.

By shedding light on the learning behavior of existing models, our findings
offer valuable insights for future research and development efforts in EEG-
based communication systems.

  3. 3.

We emphasize the importance of improved transparency and rigorous benchmarking
practices in EEG-to-Text research, paving the way for more reliable and
trustworthy research in this rapidly evolving field.

Figure 1: Schematic illustration of the pipeline for a comprehensive assessment of EEG-to-Text models across four distinct training and evaluation setups [1]. These setups explore various combinations of training with either EEG data or random noise as input, followed by evaluation on the same type of data. This approach reveals how models perform under different input conditions. Each setup is further divided to show the influence of teacher-forcing on text generation. Table 1: Number of training, development, and testing samples for each task within the ZuCo datasets. SR: Normal Reading (movie reviews), NR: Normal Reading (Wikipedia), TSR: Task Specific Reading (Wikipedia). Reading task |  | #Training
---
samples
| #Development
---
samples
| #Testing
---
samples
| #Total
---
samples
SR v1.0 | 3609 | 467 | 456 | 4533
NR v1.0 | 2645 | 343 | 350 | 3343
NR v2.0 | 3697 | 458 | 392 | 4547
TSR v1.0 | 4456 | 522 | 601 | 5579

##  2 Materials and Methods

###  2.1 Task Definition

In this replication study, we decoded sentences $\mathcal{S}$ using a sequence
of word-level EEG features $\mathcal{E}$ within a sequence-to-sequence
(seq2seq) framework. Specifically, we employed EEG-text pairs
<$\mathcal{E},\mathcal{S}$> collected during natural reading tasks, as
detailed in Section 2.2.

###  2.2 ZuCo Datasets

We followed the approach of Wang and Ji [1] and utilized two publicly
available datasets: ZuCo 1.0 [6] and ZuCo 2.0 [7]. These datasets consist of
EEG signals and eye-tracking data collected while participants read movie
reviews and Wikipedia articles naturally. Essentially, ZuCo provides a
collection of EEG recordings and corresponding eye movement data captured
during natural reading tasks. The reading materials encompass movie reviews
and Wikipedia articles. From the EEG signals, we extracted 840 features. These
features were segmented based on eye fixations for each word encountered
during reading. Each feature represents the average amplitude obtained after
applying a Hilbert transform across eight predefined frequency bands: theta1
(4-6 Hz), theta2 (6.5-8 Hz), alpha1 (8.5-10 Hz), alpha2 (10.5-13 Hz), beta1
(13.5-18 Hz), beta2 (18.5-30 Hz), gamma1 (30.5-40 Hz), and gamma2 (40-49.5
Hz). Sentences containing NaN values were excluded from the analysis.
Subsequently, we partitioned the data for each reading task into three sets:
80$\%$ for training, 10$\%$ for development, and 10$\%$ for testing.
Importantly, these partitions were based on unique sentences to ensure no
overlap between training, development, and testing data. Table 1 provides a
detailed breakdown of the final sample sizes within the dataset. We leveraged
the top-performing task combinations identified by Wang and Ji [1] for both
training and evaluating our models. These combinations included SR v1.0 + NR
v1.0 + TSR v1.0 and SR v1.0 + NR v1.0 + NR v2.0 (refer to the original paper
by Wang and Ji [1] for detailed descriptions of these task combinations).

###  2.3 EEG-to-Text Decoding Evaluation

#### EEG-to-Text Decoding

Our analysis builds upon the EEG-to-Text model proposed by Wang and Ji [1].
This model is a combination of two key components: 1) An encoder module
consists of a multi-layered transformer encoder with six layers and eight
attention heads. It processes word-level EEG features as input and generates
embeddings through a feed-forward network; 2) A pretrained BART model takes
the embeddings from the encoder and utilizes its decoder to generate text
sequences by predicting tokens from its vocabulary. The model is trained to
minimize text reconstruction cross-entropy loss, essentially aiming to
accurately reproduce the original text based on the provided EEG features.

The original approach by Wang and Ji [1] employed teacher-forcing during
training, raising questions about its impact on generalizability and true
model capabilities††https://github.com/MikeWangWZHL/EEG-to-Text. To ensure
reliable testing and explore the true learning capabilities of the models, we
extended our experiments beyond BART. We included two additional pretrained
seq2seq models based on the transformer architecture, including PEGASUS [8],
and T5 [9]. All models were used in their LARGE variant only. To assess the
model’s ability to learn from EEG data and address concerns about teacher-
forcing, we designed four distinct training and testing scenarios:

  1. 1.

EEG (training and testing): In this scenario, the model is trained and tested
entirely on word-level EEG features. This setup directly assesses the model’s
capability to genuinely generate text based on EEG data. It reveals whether
the model can extract meaningful information from the EEG signals and
translate it into coherent language.

  2. 2.

Random (training and testing): The model is trained and tested using random
noise instead of actual EEG features. This establishes a crucial baseline
performance level. By comparing the model’s performance on random noise to its
performance on EEG data (in scenario 1), we can gain insights into whether the
model is learning from the EEG information or simply memorizing patterns. If
the model performs equally well on both EEG and random noise, it suggests it
might not be effectively utilizing the EEG data.

  3. 3.

EEG (training) + Random (testing): The model is initially trained using EEG
features. However, during evaluation, it is tested on random noise data. This
setup helps us understand how well the model generalizes its learned patterns
from EEG data to unseen data, even if that data is not actual EEG.

  4. 4.

Random (training) + EEG (testing): The model is trained on random noise,
essentially providing it with no meaningful information. However, during
evaluation, it is tested on actual EEG features. This scenario serves as a
control to assess the model’s ability to generate any meaningful text output
in the complete absence of relevant training data.

By analyzing the model’s performance across these four scenarios, we can gain
valuable insights into how effectively the models learn from EEG data and
address the concerns raised about teacher-forcing techniques. This
comprehensive evaluation approach allows us to differentiate between genuine
learning and potential memorization of training patterns.

#### Training

All models were trained on Nvidia RTX 4090 GPUs using a batch size 32 and a
learning rate of $2e^{-5}$. Training was conducted for 30 epochs for each
model. The model with the lowest loss on the development dataset was chosen as
the best-performing model. Stochastic gradient descent was employed as the
optimizer for all training processes.

#### Evaluation

This study investigated the influence of teacher-forcing on text generation
from EEG features during the evaluation phase. teacher-forcing is a training
technique commonly used in recurrent neural networks (RNNs) [10]. It involves
using the ground truth (correct output) from the training data as the input
for the next time step, rather than relying solely on the model’s own
predictions. This approach can accelerate learning and improve training
performance because it ensures the model receives accurate information during
the training process. However, in the inference phase (when the model
generates new text), teacher-forcing is not typically used. Instead, the model
generates predictions based on its own outputs from previous steps. This might
not reflect the model’s capabilities in real-world scenarios. To assess the
impact of teacher-forcing, we evaluated the model’s performance under two
conditions: 1) With teacher-forcing: here, the model inferences with the
actual output from the previous time step during evaluation, replicating the
approach used in the original study [1]; 2) Without teacher-forcing: This
condition tested the model’s ability to generate text independently. The model
relied solely on its predictions from previous steps, without any input of the
actual ground truth. To mitigate repetition in the generated text without
teacher-forcing, we employed beam search with a beam size of 5, along with a
repetition penalty of 5 and a restriction on the size of repeated n-grams
(sequences of words) to 2. Following prior research [1, 2, 11, 12], we
evaluated the model’s translation performance on the ZuCo dataset using
established natural language processing (NLP) metrics, including BLEU [4],
ROUGE [13], and WER [14]. These metrics provide quantitative measures of how
well the generated text aligns with the reference text from the dataset.

##  3 Results and Discussion

Our analysis yielded two significant findings regarding EEG-to-Text models and
the impact of teacher-forcing: The use of teacher-forcing during evaluation
resulted in a substantial performance boost (at least a three-fold increase)
compared to models evaluated without teacher-forcing (Tables 2 and 3). This is
evident in Table 4, where models consistently generated text closely
resembling the ground truth when teacher-forcing was used, regardless of
training or evaluation data type. In contrast, models without teacher-forcing
frequently began their outputs with "He was…" irrespective of the actual
content. This highlights the significant influence of teacher-forcing on
reported performance metrics, suggesting previous studies may have
overestimated model capabilities due to this technique. We observed consistent
evaluation results across models trained on either EEG or random input data,
regardless of the evaluation data type (EEG or random). This consistency
suggests that the pretrained seq2seq models primarily rely on the label tokens
provided during training, not the actual content of the input data (EEG or
random). Furthermore, the T5 model consistently achieved BLEU-1 scores
approximately 3$\%$ higher than other models, suggesting its text generation
capabilities are heavily influenced by its pretrained architecture. The
consistent performance across EEG and random inputs raises concerns about
whether the models are genuinely learning text-related information from EEG
data. It suggests that they might be simply memorizing patterns from the
training labels rather than extracting meaningful content from the EEG
signals. To definitively determine this, it’s crucial to compare model
performance using EEG data against a baseline established with random noise
input. These findings underscore the need for more rigorous evaluation
methodologies in EEG-to-Text research that avoid relying on teacher-forcing
and ensure models are assessed based on their ability to learn from EEG data
itself.

##  4 Conclusion

This study underscores the importance of meticulous evaluation practices in
EEG-to-Text research. Our analysis revealed a critical shortcoming in previous
research methodologies – the lack of a robust baseline for assessing model
performance. By demonstrating that models perform similarly on pure noise
compared to actual EEG data, we highlight the potential for inflated
performance metrics when using teacher-forcing and previous evaluation
methods. Our proposed methodology offers a more reliable way to assess a
model’s ability to extract meaningful information from EEG signals. It helps
differentiate between models that truly learn from EEG data and those that
might simply be memorizing patterns from training labels.

Looking ahead, we advocate for stricter evaluation practices within the EEG-
to-Text research community. Researchers should ensure transparent reporting of
methodologies and results, fostering open scientific discourse. Regularly
incorporating noise baselines during evaluation is crucial for establishing a
reliable performance benchmark. By employing more rigorous methodologies,
researchers can shift the focus towards demonstrably learning text
representations from EEG data, paving the way for advancements in this field.
The adoption of these practices will lead to more reliable research findings
and ultimately accelerate the development of robust and effective EEG-to-Text
communication systems.

Table 2: EEG-to-Text model evaluation on the ZuCo datasets, incorporating
reading tasks from SR v1.0, NR v1.0, and TSR v1.0. "w/tf" denotes results
obtained using teacher-forcing during evaluation as utilized in the original
study [1]. In the training and evaluation phases, "EEG" denotes the use of
word-level EEG features, while "Random" refers to the employment of random
numbers generated from a normal distribution.

|  |  | BLEU-N (%) | ROUGE-1 (%) | WER (%)
---|---|---|---|---|---
Pretrained model | Training | Evaluation | N=1 | N=2 | N=3 | N=4 | P | R | F |
BART | EEG | EEG | 13.69 | 2.97 | 0.82 | 0.32 | 11.98 | 13.43 | 11.87 | 108.43
| EEG | Random | 13.87 | 3.09 | 0.77 | 0.25 | 12.23 | 13.60 | 12.14 | 108.31
| Random | EEG | 14.05 | 3.12 | 1.00 | 0.41 | 11.46 | 12.37 | 11.14 | 110.96
| Random | Random | 14.22 | 3.06 | 0.93 | 0.39 | 11.62 | 12.29 | 11.19 | 110.98
BART w/tf [1] | EEG | EEG | 39.31 | 22.09 | 12.49 | 7.27 | 26.41 | 31.40 | 28.58 | 78.08
| EEG | Random | 39.34 | 22.13 | 12.52 | 7.29 | 26.44 | 31.43 | 28.61 | 78.07
| Random | EEG | 39.67 | 22.15 | 12.49 | 7.12 | 26.29 | 31.00 | 28.34 | 78.09
| Random | Random | 39.69 | 22.17 | 12.50 | 7.12 | 26.32 | 31.03 | 28.37 | 78.09
Pegasus | EEG | EEG | 8.47 | 2.48 | 0.81 | 0.25 | 0.00 | 0.00 | 0.00 | 99.69
| EEG | Random | 8.58 | 2.48 | 0.78 | 0.00 | 0.00 | 0.00 | 0.00 | 99.89
| Random | EEG | 9.12 | 2.70 | 0.91 | 0.23 | 0.00 | 0.00 | 0.00 | 98.73
| Random | Random | 9.06 | 2.60 | 0.84 | 0.00 | 0.00 | 0.00 | 0.00 | 99.24
Pegasus w/tf | EEG | EEG | 38.18 | 21.04 | 11.50 | 6.09 | 26.72 | 30.51 | 28.38 | 78.57
| EEG | Random | 38.30 | 21.09 | 11.57 | 6.12 | 26.84 | 30.65 | 28.51 | 78.56
| Random | EEG | 39.10 | 21.74 | 11.97 | 6.17 | 27.43 | 31.26 | 29.11 | 78.09
| Random | Random | 39.17 | 21.70 | 11.96 | 6.18 | 27.41 | 31.34 | 29.14 | 78.10
T5 | EEG | EEG | 16.64 | 5.80 | 1.96 | 0.81 | 12.28 | 12.88 | 11.85 | 111.13
| EEG | Random | 15.42 | 4.78 | 1.57 | 0.65 | 10.57 | 11.45 | 10.35 | 112.00
| Random | EEG | 15.95 | 5.71 | 2.01 | 0.91 | 11.90 | 12.61 | 11.47 | 111.37
| Random | Random | 15.54 | 5.22 | 1.70 | 0.67 | 11.48 | 12.23 | 11.10 | 111.74
T5 w/tf | EEG | EEG | 43.50 | 25.50 | 15.18 | 8.69 | 22.92 | 28.23 | 25.11 | 81.39
| EEG | Random | 43.53 | 25.56 | 15.15 | 8.68 | 22.76 | 27.82 | 24.87 | 81.43
| Random | EEG | 43.47 | 25.34 | 15.03 | 8.67 | 23.02 | 27.96 | 25.06 | 81.64
| Random | Random | 43.63 | 25.57 | 15.23 | 8.78 | 23.36 | 28.40 | 25.45 | 81.46

Table 3: EEG-to-Text model evaluation on the ZuCo datasets, incorporating
reading tasks from SR v1.0, NR v1.0, and NR v2.0. "w/tf" denotes results
obtained using teacher-forcing during evaluation as utilized in the original
study [1]. In the training and evaluation phases, "EEG" denotes the use of
word-level EEG features, while "Random" refers to the employment of random
numbers generated from a normal distribution.

|  |  | BLEU-N (%) | ROUGE-1 (%) | WER (%)
---|---|---|---|---|---
Pretrained model | Training | Evaluation | N=1 | N=2 | N=3 | N=4 | P | R | F |
BART | EEG | EEG | 11.58 | 3.40 | 1.33 | 0.54 | 11.33 | 15.44 | 12.40 | 99.68
| EEG | Random | 11.26 | 3.19 | 1.28 | 0.57 | 11.07 | 15.08 | 12.06 | 100.03
| Random | EEG | 11.84 | 3.36 | 1.41 | 0.68 | 11.39 | 15.17 | 12.24 | 100.88
| Random | Random | 11.78 | 3.18 | 1.23 | 0.52 | 11.22 | 15.05 | 12.07 | 101.12
BART w/tf [1] | EEG | EEG | 41.22 | 24.18 | 13.87 | 7.77 | 29.52 | 36.31 | 32.42 | 75.33
| EEG | Random | 41.22 | 24.18 | 13.87 | 7.77 | 29.52 | 36.32 | 32.42 | 75.31
| Random | EEG | 41.63 | 24.70 | 14.44 | 8.45 | 29.56 | 35.83 | 32.30 | 74.76
| Random | Random | 41.61 | 24.69 | 14.40 | 8.40 | 29.55 | 35.77 | 32.27 | 74.76
Pegasus | EEG | EEG | 10.57 | 2.82 | 0.86 | 0.27 | 0.00 | 0.00 | 0.00 | 99.21
| EEG | Random | 11.14 | 2.75 | 0.88 | 0.30 | 0.00 | 0.00 | 0.00 | 99.70
| Random | EEG | 9.06 | 2.57 | 1.00 | 0.36 | 0.00 | 0.00 | 0.00 | 98.37
| Random | Random | 9.08 | 2.54 | 0.92 | 0.22 | 0.00 | 0.00 | 0.00 | 98.40
Pegasus w/tf | EEG | EEG | 41.18 | 23.42 | 13.12 | 7.22 | 30.32 | 34.83 | 32.31 | 75.30
| EEG | Random | 41.06 | 23.17 | 12.82 | 6.86 | 30.08 | 34.59 | 32.07 | 75.51
| Random | EEG | 41.89 | 23.90 | 13.27 | 7.09 | 30.88 | 35.52 | 32.93 | 74.67
| Random | Random | 41.73 | 23.80 | 13.29 | 7.22 | 30.69 | 35.31 | 32.73 | 74.88
T5 | EEG | EEG | 16.76 | 6.15 | 2.56 | 1.26 | 13.69 | 15.27 | 13.71 | 104.50
| EEG | Random | 15.70 | 5.44 | 2.04 | 0.93 | 12.25 | 13.84 | 12.35 | 106.01
| Random | EEG | 15.86 | 5.47 | 2.17 | 1.02 | 12.88 | 14.75 | 13.03 | 104.30
| Random | Random | 16.26 | 5.76 | 2.23 | 1.03 | 13.58 | 15.44 | 13.66 | 104.49
T5 w/tf | EEG | EEG | 46.03 | 28.23 | 17.35 | 10.55 | 28.14 | 33.84 | 30.53 | 77.06
| EEG | Random | 45.55 | 27.71 | 16.85 | 10.12 | 27.59 | 32.99 | 29.87 | 77.63
| Random | EEG | 45.62 | 27.78 | 16.79 | 10.15 | 27.26 | 32.93 | 29.65 | 77.44
| Random | Random | 45.75 | 27.96 | 16.99 | 10.21 | 27.39 | 33.05 | 29.77 | 77.20

Table 4: Decoding examples of EEG-to-Text models [1]. "EEG" and "Random" represent that the model is trained and tested on word-level EEG features and random numbers, respectively. BART is used as the pretrained seq2seq model for all models. "w/tf" denotes results obtained using teacher-forcing during evaluation, as utilized in the original study [1]. Bold indicates words exactly matching the ground truth. Underline denotes words consistently generated by models without teacher-forcing. Ground truth |  It’s not a particularly good film, but neither is it a monsterous one.
---|---
EEG w/tf [1] |  ’s a a bad good movie, but it is it bad bad. one.
Random w/tf |  ’s a a bad good movie, but it is it bad bad. one.
EEG |  He was elected to the United States House of Representatives in 1946.
Random |  He was educated at the University of Virginia, where he earned a Bachelor of Arts degree in political science.
Ground truth |  Everything its title implies, a standard-issue crime drama spat out from the Tinseltown assembly line.
EEG w/tf [1] |  about predecessor implies is and movie,issue, drama. between of the depthsseltown set line.
Random w/tf |  about predecessor implies is and movie,issue, drama. between of the sameseltown set..
EEG |  He was a member of the Democratic National Committee (DNC) from 1952 until his death in 1968.
Random |  He was educated at Trinity College, Cambridge and the University of Oxford.
Ground truth |  Joseph H. Ball (November 3, 1905 - December 18, 1993) was an American politician.
EEG w/tf [1] |  was. Smith (born 23, 18 - April 23, 1993) was an American actor,
Random w/tf |  wasux W (born 23, 18 - April 23, 1977) was an American actor.
EEG |  He was elected to the United States House of Representatives in 1920.
Random |  He was educated at the University of Wisconsin-Madison, and received a Bachelor of Arts degree in English literature from Stony Brook University.

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