# Solitons of the mean curvature flow in $\mathbb{S}^{2}\times\mathbb{R}$

Rafael López, Marian Ioan Munteanu  Departamento de Geometría y Topología
Universidad de Granada. 18071 Granada, Spain<EMAIL_ADDRESS>University Al. I.
Cuza’ of Iasi, Faculty of Mathematics, Bd. Carol I, no. 11, 700506 Iasi,
Romania<EMAIL_ADDRESS>
###### Abstract.

A soliton of the mean curvature flow in the product space
$\mathbb{S}^{2}\times\mathbb{R}$ is a surface whose mean curvature $H$
satisfies the equation $H=\langle N,X\rangle$, where $N$ is the unit normal of
the surface and $X$ is a Killing vector field. In this paper we consider the
vector field tangent to the fibers and the vector field associated to
rotations about an axis of $\mathbb{S}^{2}$, respectively. We give a
classification of the solitons with respect to these vector fields assuming
that the surface is invariant under a one-parameter group of vertical
translations or rotations of $\mathbb{S}^{2}$.

###### Key words and phrases:

solitons, mean curvature flow, $\mathbb{S}^{2}\times\mathbb{R}$, one-parameter
group

###### 1991 Mathematics Subject Classification:

53A10, 53C42, 53C44.

##  1\. Introduction

Let $\psi\colon\Sigma\to\mathbb{R}^{3}$ be an immersion of a surface $\Sigma$
in Euclidean space $\mathbb{R}^{3}$. A variation
$\\{\psi_{t}\colon\Sigma\to\mathbb{R}^{3}:t\in[0,T)\\}$, $T>0$,
$\psi_{0}=\psi$, evolves by the mean curvature flow (MCF in short) if
$\displaystyle\frac{\partial\psi_{t}}{\partial t}=H(\psi_{t})N(\psi_{t})$,
where $H(\psi_{t})$ is the mean curvature of $\psi_{t}$ and $N(\psi_{t})$ is
its unit normal. The surface $\Sigma$ is called a soliton of MCF if the
evolution of $\Sigma$ under a one-parameter family of dilations or isometries
remains constant. An important type of solitons are the translators whose
shape is invariant by translations along a direction
$\vec{v}\in\mathbb{R}^{3}$. Translators are characterized by the equation
$H=\langle N,\vec{v}\rangle$, where $H$ and $N$ are the mean curvature and
unit normal of $\Sigma$ respectively. Translators play a special role in the
theory of MCF because they are, after rescaling, a type of singularities of
the MCF according to Huisken and Sinestrari [6]. In the meantime, the
development of the theory of solitons of the MCF in other ambient space has
been developed. Without to be complete, we refer: a general product space
$M^{2}\times\mathbb{R}$ [9]; hyperbolic space [3, 4, 8, 11]; the product
$\mathbb{H}^{2}\times\mathbb{R}$ [1, 2, 5, 7]; the Sol space [14]; the
Heisenberg group [15]; the special linear group [10].

In this paper, we focus on solitons in the product space
$\mathbb{S}^{2}\times\mathbb{R}$, where $\mathbb{S}^{2}$ is the unit sphere of
$\mathbb{R}^{3}$. Looking in the equation $H=\langle N,\vec{v}\rangle$, it is
natural to replace $\vec{v}$ by a Killing vector of the space, which motivates
the following definition.

######  Definition 1.1.

Let $X\in\mathfrak{X}(\mathbb{S}^{2}\times\mathbb{R})$ be a Killing vector
field. A surface $\Sigma$ in $\mathbb{S}^{2}\times\mathbb{R}$ is said to be a
$X$-soliton if its mean curvature $H$ and unit normal vector $N$ satisfy

$H=\langle N,X\rangle.$ (1)

In this paper, $H$ is the sum of the principal curvatures of the surface. The
dimension of Killing vector fields in the space
$\mathbb{S}^{2}\times\mathbb{R}$ is $4$. Taking coordinates $(x,y,z,t)$ in
$\mathbb{S}^{2}\times\mathbb{R}$, a relevant Killing vector field is
$V=\partial_{t}$. Here $V$ is tangent to the fibers of the natural submersion
$\mathbb{S}^{2}\times\mathbb{R}\to\mathbb{S}^{2}$. Other Killing vector fields
come from the rotations of $\mathbb{S}^{2}$. After renaming coordinates,
consider the vector field $R=-y\partial_{x}+x\partial_{y}$ about the $z$-axis
of $\mathbb{S}^{2}$. Examples of solitons are the following.

  1. (1)

Cylinders over geodesic of $\mathbb{S}^{2}$ are $V$-solitons. Indeed, let
$\Sigma=C\times\mathbb{R}$ be a surface constructed as a cylinder over a curve
$C$ of $\mathbb{S}^{2}$. Then the mean curvature of $\Sigma$ is $H=\kappa$,
where $\kappa$ the curvature of $C$. Since the unit normal vector $N$ of
$\Sigma$ is orthogonal to $\partial_{t}$, then $\langle N,V\rangle=0$. Thus
$\Sigma$ is a $V$-soliton if and only if $\kappa=0$. Thus $C$ is a geodesic of
$\mathbb{S}^{2}$.

  2. (2)

Slices $\mathbb{S}^{2}\times\\{t_{0}\\}$, $t_{0}\in\mathbb{R}$, are
$R$-solitons. Notice that $H=0$ because a slice is totally geodesic. Since
$N=\partial_{t}$, then $\langle N,R\rangle=0$, proving that $H=\langle
N,R\rangle$.

In this paper, we are interested in examples of $V$-solitons and $R$-solitons
that are invariant by a one-parameter group of isometries of
$\mathbb{S}^{2}\times\mathbb{R}$. Here we consider two types of such surfaces.
First, vertical surfaces which are invariant by vertical translations in the
$t$-coordinate. Second, rotational surfaces, which are invariant by a group of
rotations about an axis of $\mathbb{S}^{2}$. Under these geometric conditions
on the surfaces, we give a full classifications of $V$-solitons (Sect. 3) and
$R$-solitons (Sect. 4).

##  2\. Preliminaries

In this section, we compute each one of the terms of Eq. (1) for vertical and
rotational surfaces. The isometry group of $\mathbb{S}^{2}\times\mathbb{R}$
isomorphic to $\mbox{Isom}(\mathbb{S}^{2})\times\mbox{Isom}(\mathbb{R})$. The
group $\mbox{Isom}(\mathbb{S}^{2})$ is generated by the identity, the
antipodal map, rotations and reflections. The group $\mbox{Isom}(\mathbb{R})$
contains the identity, translations, and reflections. Therefore there are two
important one-parameter groups of isometries in
$\mathbb{S}^{2}\times\mathbb{R}$: vertical translations in the factor
$\mathbb{R}$ and rotations in the factor $\mathbb{S}^{2}$. This leads two
types of invariant surfaces.

  1. (1)

Vertical surfaces. A vertical translation is a map of type
$T_{\lambda}\colon\mathbb{S}^{2}\times\mathbb{R}\to\mathbb{S}^{2}\times\mathbb{R}$
defined by $T_{\lambda}(p,t)=(p,t+\lambda)$, where $\lambda$ is fixed. This
defines a one-parameter group of vertical translations
$\mathcal{T}=\\{T_{\lambda}:\lambda\in\mathbb{R}\\}$. A vertical surface is a
surface $\Sigma$ invariant by the group $\mathcal{T}$, that is,
$T_{\lambda}(\Sigma)\subset\Sigma$ for all $\lambda\in\mathbb{R}$. The
generating curve of $\Sigma$ is a curve $\alpha\colon
I\subset\mathbb{R}\to\mathbb{S}^{2}$ in the unit sphere $\mathbb{S}^{2}$. Let
us write this curve as

$\alpha(s)=(\cos u(s)\cos v(s),\cos u(s)\sin v(s),\sin u(s)),$ (2)

for some smooth functions $u=u(s)$ and $v=v(s)$. Then a parametrization of
$\Sigma$ is

$\Psi(s,t)=(\cos u(s)\cos v(s),\cos u(s)\sin v(s),\sin u(s),t),\quad s\in
I,t\in\mathbb{R}.$ (3)

In what follows, we parametrize the curve $\beta(s)=(u(s),v(s))$ to have

$u^{\prime}(s)=\cos u(s)\cos\theta(s),\quad v^{\prime}(s)=\sin\theta(s).$

  2. (2)

Rotational surfaces. These surfaces are invariant by rotations of
$\mathbb{S}^{2}$. To be precise, and after a choice of coordinates on
$\mathbb{S}^{2}$, a rotation in $\mathbb{S}^{2}\times\mathbb{R}$ about the
$z$-axis is a map
$\mathcal{R}_{\varphi}\colon\mathbb{S}^{2}\times\mathbb{R}\to\mathbb{S}^{2}\times\mathbb{R}$,
given by

$R_{\varphi}=\left(\begin{array}[]{llll}\cos\varphi&-\sin\varphi&0&0\\\
\sin\varphi&\cos\varphi&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right).$

The set $\mathcal{R}=\\{\mathcal{R}_{\varphi}:\varphi\in\mathbb{R}\\}$ of all
$\mathcal{R}_{\varphi}$, is a one-parameter group of rotations, that is
${\mathrm{SO}}(2)$. A rotational surface is a surface $\Sigma$ invariant by
the group $\mathcal{R}$, that is, $\mathcal{R}_{\varphi}(\Sigma)\subset\Sigma$
for all $\varphi\in\mathbb{R}$. The generating curve of $\Sigma$ is a curve
$\alpha$ contained in the $xzt$-hyperplane which we suppose parametrized by

$\alpha(s)=(\cos u(s),0,\sin u(s),v(s)),\quad s\in I\subset\mathbb{R},$ (4)

where $u=u(s)$ and $v=v8s)$ are smooth functions. Then a parametrization of
$\Sigma$ is

$\Psi(s,\varphi)=(\cos u(s)\cos\varphi,\cos u(s)\sin\varphi,\sin
u(s),v(s)),\quad s\in I,\varphi\in\mathbb{R}.$ (5)

From now, suppose that the the curve $\beta(s)=(u(s),v(s))$ obtained from
$\alpha$ in (4) is parametrized by the Euclidean arc-length, that is,

$u^{\prime}(s)=\cos\theta(s),\quad v^{\prime}(s)=\sin\theta(s),$

for some smooth function $\theta=\theta(s)$. Notice that $\theta^{\prime}$ is
the curvature of $\beta$ as planar curve of $\mathbb{R}^{2}$.

We now compute the mean curvature $H$ and the unit normal vector $N$ of
vertical surfaces and rotational surfaces.

######  Proposition 2.1.

Suppose that $\Sigma$ is a vertical surface parametrized by (3). Then the unit
normal vector $N$ is

$N=(\cos\theta\sin v-\sin\theta\sin u\cos v,-\cos\theta\cos v-\sin\theta\sin
u\sin v,\sin\theta\cos u,0),$ (6)

and the mean curvature $H$ is

$H=\tan u\sin\theta-\frac{\theta^{\prime}}{\cos u}.$ (7)

###### Proof.

Suppose that $\Sigma$ is parametrized by (3). Then the tangent plane each
point of $\Sigma$ is spanned by $\\{\Psi_{s},\Psi_{t}\\}$, where

$\begin{split}\Psi_{s}&=(-\cos u(\cos\theta\sin u\cos v+\sin\theta\sin v),\cos
u(\sin\theta\cos v-\cos\theta\sin u\sin v),\cos\theta\cos^{2}u,0),\\\
\Psi_{t}&=(0,0,0,1).\end{split}$ (8)

A straightforward computation yields that the unit normal vector is (6).

As usually, denote by $g_{ij}$ the coefficients of the first fundamental form
of $\Psi$, where

$g_{11}=\langle\Psi_{s},\Psi_{s}\rangle,\quad
g_{12}=\langle\Psi_{s},\Psi_{t}\rangle,\quad
g_{22}=\langle\Psi_{t},\Psi_{t}\rangle.$

The formula of $H$ is

$H=\frac{g_{22}b_{11}-2g_{12}b_{12}+g_{11}b_{22}}{g_{11}g_{22}-{g_{12}}^{2}},$

where $b_{ij}$ are the coefficients of the second fundamental form. Here

$b_{11}=-\langle N_{s},\Psi_{s}\rangle,\quad b_{12}=-\langle
N_{s},\Psi_{t}\rangle,\quad b_{22}=-\langle N_{t},\Psi_{t}\rangle.$

A computation of $g_{ij}$ gives $g_{11}=g_{22}=(\cos u)^{2}$ and $g_{12}=0$.
In particular, $\cos u(s)\not=0$ for all $s\in I$. Then
$g_{11}g_{22}-g_{12}^{2}=(\cos u)^{4}$. For the coefficients of the second
fundamental, we have $b_{12}=b_{22}=0$ and

$b_{11}=\cos u(\sin\theta\sin-\theta^{\prime}).$ (9)

Then the mean curvature $H$ is (7). ∎

######  Proposition 2.2.

Suppose that $\Sigma$ is a rotational surface parametrized by (5). Then the
unit normal vector $N$ is

$N(s,\varphi)=(\sin\theta\sin u\cos\varphi,\sin\theta\sin
u\sin\varphi,-\sin\theta\cos u,\cos\theta),$ (10)

and the mean curvature $H$ is

$H=\theta^{\prime}-\sin\theta\tan u.$ (11)

###### Proof.

From (3), the basis $\\{\Psi_{s},\Psi_{t}\\}$ at each tangent plane of
$\Sigma$ is

$\begin{split}\Psi_{s}(s,\varphi)&=(-u^{\prime}\sin
u\cos\varphi,-u^{\prime}\sin u\sin\varphi,u^{\prime}\cos u,v^{\prime}),\\\
\Psi_{\varphi}(s,\varphi)&=(-\sin\varphi\cos u,\cos\varphi\cos
u,0,0).\end{split}$ (12)

Thus $g_{11}=1$, $g_{12}=0$ and $g_{22}=\cos^{2}u$, in particular, $\cos
u\not=0$. As a consequence, the unit normal vector $N$ is (10). The
computation of the coefficients of the second fundamental form gives

$\begin{split}b_{11}&=\theta^{\prime}\\\ b_{12}&=0\\\ b_{22}&=-\sin\theta\sin
u\cos u.\end{split}$ (13)

Hence we deduce that the expression of $H$ is (11).

∎

##  3\. The class of $V$-solitons

Let the vector field

$V=\partial_{t}.$ (14)

The fact that $V$ is tangent to the fibers of the submersion
$\mathbb{S}^{2}\times\mathbb{R}\to\mathbb{S}^{2}$ makes that $V$ has special
properties. For example, $V$-solitons of $\mathbb{S}^{2}\times\mathbb{R}$ can
be viewed as weighted minimal surfaces in a space with density. So, let
$e^{t}dA$ and $e^{t}dV$ the area and volume of
$\mathbb{S}^{2}\times\mathbb{R}$ with a weight $e^{t}$, where $t$ is the last
coordinate of the space. Considering the energy functional $\Omega\mapsto
E(\Omega)=\int_{\Omega}e^{t}dA$ defined for compact subdomains
$\Omega\subset\Sigma$, a critical point of this functional, also called a
weighted minimal surface, is a surface characterized by the equation
$H-\langle N,\nabla t\rangle=0$, where $\nabla$ is the gradient in
$\mathbb{S}^{2}\times\mathbb{R}$. Since $\nabla t=\partial_{t}=V$, we have
proved that a weighted minimal surface in
$(\mathbb{S}^{2}\times\mathbb{R},e^{t}\langle,\rangle)$ is a $V$-soliton. A
property of weighted minimal surfaces is they satisfy a principle of tangency
as a consequence of the Hopf maximum principle for elliptic equations of
divergence type. In our context, the tangency principle asserts that if two
$V$-solitons $\Sigma_{1}$ and $\Sigma_{2}$ touch at some interior point
$p\in\Sigma_{1}\cap\Sigma_{2}$ and one surface is in one side of the other
around $p$, then $\Sigma_{1}$ and $\Sigma_{2}$ coincide in a neighborhood of
$p$. The following result proves that slices are the only closed $V$-solitons.

######  Theorem 3.1.

Slices $\mathbb{S}^{2}\times\\{t_{0}\\}$, $t_{0}\in\mathbb{R}$, are the only
closed (compact without boundary) $V$-solitons in
$\mathbb{S}^{2}\times\mathbb{R}$.

###### Proof.

Let $\psi\colon\Sigma\to\mathbb{S}^{2}\times\mathbb{R}$ be a closed
$V$-soliton. Define on $\Sigma$ the height function
$h\colon\Sigma\to\mathbb{R}$ by $h(q)=\langle\psi(q),\partial_{t}\rangle$. It
is known that for any surface of $\mathbb{S}^{2}\times\mathbb{R}$, the
Laplacian of $h$ is $\Delta h=H\langle N,\partial_{t}\rangle$ [16].

Using that $\Sigma$ is a $V$-soliton, then $\Delta h=\langle
N,\partial_{t}\rangle^{2}=\langle N,V\rangle^{2}$. Integrating in $\Sigma$,
the divergence theorem yields $\int_{\Sigma}\langle N,V\rangle^{2}=0$. Thus
$\langle N,V\rangle=0$ on $\Sigma$ and $H=0$. In particular, $\Delta h=0$. By
the maximum principle, $h$ is a constant function, namely $h(q)=t_{0}$, for
some $t_{0}\in\mathbb{R}$. This proves that
$\Sigma\subset\mathbb{S}^{2}\times\\{t_{0}\\}$ and thus, both surfaces
coincide. ∎

We begin with the study of $V$-solitons invariant by the group $\mathcal{T}$.
We prove that any vertical $V$-soliton is trivial in the sense that it is a
minimal surface, even more, we prove that it is a cylinder of type
$\mathbb{S}^{1}\times\mathbb{R}\subset\mathbb{S}^{2}\times\mathbb{R}$.

######  Theorem 3.2.

Suppose that $\Sigma$ is a vertical surface. Then $\Sigma$ is a $V$-soliton if
and only its generating curve is a geodesic of $\mathbb{S}^{2}$ and $\Sigma$
is a vertical surface on a geodesic of $\mathbb{S}^{2}$.

###### Proof.

Let $\Sigma$ be a vertical surface. Since the vertical lines are fibers of the
submersion $\mathbb{S}^{2}\times\mathbb{R}\to\mathbb{S}^{2}$, the mean
curvature $H$ of $\Sigma$ is $H=\kappa$, where $\kappa$ is the curvature of
$\alpha$. Moreover, the unit normal vector is horizontal, hence $\langle
N,V\rangle=0$. This proves the result. ∎

We now study $V$-solitons of rotational type. As we have indicated in the
previous section, we can assume that the rotation axis is the $z$-axis. Thus a
rotational surface $\Sigma$ can be parametrized by (5).

An immediate example of rotational $V$-soliton is the cylinder
$\mathcal{C}=(\mathbb{S}^{1}\times\\{0\\})\times\mathbb{R}$. This surface
corresponds with the curve $(u(s),v(s))=(0,s)$, $s\in\mathbb{R}$, in (4). Thus
$\alpha(s)=(1,0,0,s)$ is the vertical line through the point
$(1,0,0)\in\mathbb{S}^{2}$. The unit normal $N$ is orthogonal to $V$. Since
the generating curve is a geodesic of $\mathbb{S}^{2}$, the surface is
minimal, proving that $\mathcal{C}$ is a $V$-soliton. This surface is also a
vertical $R$-soliton (Thm. 3.2). We now characterize rotational $V$-solitons
in terms of its generating curve $\alpha$.

######  Proposition 3.3.

Let $\Sigma$ be a rotational surface in $\mathbb{S}^{2}\times\mathbb{R}$. If
$\Sigma$ is parametrized by (5), then $\Sigma$ is a $V$-soliton if and only if
the generating curve $\alpha$ satisfies

$\left\\{\begin{split}u^{\prime}&=\cos\theta\\\ v^{\prime}&=\sin\theta\\\
\theta^{\prime}&=\sin\theta\tan u+\cos\theta\end{split}\right.$ (15)

###### Proof.

This is an immediate consequence of Prop. 2.2. Indeed, from the expression of
$N$ in (10), we have

$\langle N,V\rangle=\cos\theta.$

Using (11), then Eq. (1) is $\theta^{\prime}=\sin\theta\tan u+\cos\theta$.

∎

We now study the solutions of (15), describing their main geometric
properties. Recall that $\cos u\not=0$ by regularity of the surface (Prop.
2.2). Since the last equation of (15) does not depend on $v$, we can study the
solutions $\alpha$ of (15) projecting in the $(u,\theta)$-plane, which in turn
leads to the planar autonomous ordinary system

$\left\\{\begin{split}u^{\prime}&=\cos\theta\\\
\theta^{\prime}&=\sin\theta\tan u+\cos\theta.\end{split}\right.$ (16)

The phase plane of (16) is depicted in Fig. 1. By regularity of the surface,
$u(s)\in(-\pi/2,\pi/2)$. Thus the phase plane of (16) is the set

$A=\\{(u,\theta)\colon
u\in(\frac{\pi}{2},\frac{\pi}{2}),\theta\in\mathbb{R}\\}.$

The trajectories of (16) are the solutions $\gamma(s)=(u(s),\theta(s))$ of
(16) when regarded in $A$ and once initial conditions $(u_{0},\theta_{0})\in
A$ have been fixed. These trajectories foliate $A$ as a consequence of the
existence and uniqueness of the Cauchy problem of (16).

Figure 1. The $(u,\theta)$-phase plane of (16). The red points are the
equilibrium points $(0,\pm\frac{\pi}{2})$, where the surface is the cylinder
$\mathcal{C}$.

The equilibrium points of (16) are $(u,\theta)=(0,\frac{\pi}{2})$ and
$(u,\theta)=(0,-\frac{\pi}{2})$. The rest of equilibrium points can be
obtained by translations by multiples of $\pi$ in the $\theta$-coordinate. If
$(u,\theta)=(0,\frac{\pi}{2})$, then $u(s)=0$, $v(s)=s$. Thus the generating
curve $\alpha$ is the vertical fiber at $(1,0,0)\in\mathbb{S}^{2}$
parametrized with increasing variable $s$, $v(s)=s$. For this curve, the
corresponding surface is the vertical right cylinder $\mathcal{C}$ and this
solution is already known. If $(u,\theta)=(0,-\frac{\pi}{2})$, then $v(s)=-s$
and the generating curve is again the above vertical line but parametrized by
decreasing variable $s$. The surface is the cylinder $\mathcal{C}$ again.

The qualitative behaviour of the trajectories near the equilibrium points are
analyzed, as usually, by the linearized system (see [13] as a general
reference). At the point $(u,\theta)=(0,\frac{\pi}{2})$, we find

$\left(\begin{array}[]{ll}0&-1\\\ 1&-1\end{array}\right)$

as the matrix of the linearized system. The eigenvalues of this matrix are the
two conjugate complex numbers $\frac{1}{2}(-1\pm i\sqrt{3})$. Since the real
parts are negative, then the point $(0,\frac{\pi}{2})$ is a stable spiral.
Thus all the trajectories will move in towards the equilibrium point as $s$
increases. Similarly, for the point $(u,\theta)=(0,-\frac{\pi}{2})$, the
matrix of the corresponding linearized system is

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

Then the eigenvalues of this matrix are $\frac{1}{2}(1\pm i\sqrt{3})$ and the
point $(0,-\frac{\pi}{2})$ is an unstable spiral. Since there are no more
equilibrium points, every trajectory starts in the unstable spiral
$(0,-\frac{\pi}{2})$ and ends in the stable spiral $(0,\frac{\pi}{2})$.

In order to give initial conditions at $s=0$, notice that if we do a vertical
translation in $\mathbb{R}^{4}$ of the generating curve $\alpha$, the surface
is a translated from the original. This vertical translation is simply adding
a constant to the last coordinate function $v=v(s)$. Thus, at the initial time
$s=0$, we can assume $v(0)=0$. On the other hand, the fact that the
trajectories go from $(0,-\frac{\pi}{2})$ towards $(0,\frac{\pi}{2})$ implies
that the function $\theta$ attains the value $0$. As a first initial step, we
can consider initial condition $\theta(0)=0$ when the curve
$\beta(s)=(u(s),v(s))$ starts at the $v$-axis. So, let

$u(0)=v(0)=\theta(0)=0.$ (17)

It is immediate from (15) that
$(\bar{u}(s)=-u(-s),\bar{v}(s)=v(-s),\bar{\theta}(s)=-\theta(-s))$ is also a
solution of (15) with the same initial conditions (17). Thus the graphic of
$\beta$ is symmetric about the $v$-axis.

Given initial conditions (17), we know that
$(u(s),\theta(s))\to(0,\frac{\pi}{2})$. Then the right hand-sides of (16)
(also in (15)) are bounded functions, proving that the domain of solutions is
$\mathbb{R}$. Since $|v^{\prime}(s)|\to 1$, then
$\lim_{s\to\pm\infty}v(s)=\infty$ by symmetry of $\beta$. Thus
$\lim_{s\to\pm\infty}\beta(s)=(0,\infty)$, that is $\beta$ is asymptotic to
the $v$-axis at infinity. The projection of $\alpha$ in the factor
$\mathbb{S}^{2}$ converges to $(1,0,0)$. This implies that $\Sigma$ is
asymptotic to the cylinder $\mathcal{C}$.

Because $(0,\pi/2)$ is a stable spiral, the function $\theta(s)$ converges to
$\pi/2$ oscillating around this value, and the same occurs for the function
$u(s)$ around $u=0$. In particular, the graphic of $\beta$ intersects
infinitely many times the $v$-axis. By the symmetry of $\beta$ with respect to
the $v$-axis, we deduce that $\beta$ has (infinitely many) self-intersections.

We claim that the coordinate function $v(s)$ of $\beta$ has no critical points
except $s=0$. We know

$v^{\prime\prime}(s)=\theta^{\prime}(s)\cos\theta(s)=\sin\theta(s)\cos\theta(s)\tan
u(s)+\cos^{2}\theta(s).$

If $v^{\prime}(s)=0$ at $s=s_{0}$, then $\sin\theta(s_{0})=0$, hence
$v^{\prime\prime}(s_{0})=1$. Thus all critical points are local minimum
deducing that $s=0$ is the only minimum. Once we have prove that
$v^{\prime}\not=0$ for all $s\not=0$, then each branch of $\beta$, that is,
$\beta(0,\infty)$ and $\beta(-\infty,0)$, are graphs on the $v$-axis. This
proves that $\beta$ is a bi-graph on the $v$-axis. See Fig. 2, left.

It remains to study the case that the curve $\beta$ does not start in the
$v$-axis, that is, $u(0)=u_{0}$ with $u_{0}\not=0$. Since the equilibrium
points are of spiral type, the solutions of (16) under this initial condition
converge (or diverges) to the equilibrium points. Thus, the curve $\beta$
meets the $v$-axis being asymptotic to this axis.

We summarize the above arguments.

Figure 2. Generating curves of rotational $V$-solitons. Left: the curve
$\beta$. Middle and right: projection of the generating curve $\alpha$ on the
$xzt$-space (middle) and showing it as subset of the cylinder
$\mathbb{S}^{1}\times\mathbb{R}$ (right).

Figure 3.  A rotational $V$-soliton after the stereographic projection
$p_{r}$. The surface after rotating $\beta$ in the interval $[0,\infty)$
(left) and in the interval $(-\infty,0]$ (middle). Right: the full surface.

######  Theorem 3.4.

Let $\Sigma$ be a rotational $V$-soliton. Then $\Sigma$ is the cylinder
$\mathcal{C}$ or $\Sigma$ is parametrized by (5) with the following
properties:

  1. (1)

The curve $\beta(s)=(u(s),v(s))$ has self-intersections and it is asymptotic
to the $v$-axis at infinity. In case that $\beta$ satisfies the initial
conditions (17), then $\beta$ is a symmetric bi-graph on the $v$-axis.

  2. (2)

The surface $\Sigma$ is not embedded with infinitely many intersection points
with the $z$-axis.

  3. (3)

The surface $\Sigma$ is asymptotic to the cylinder $\mathcal{C}$.

In Fig. 3 we show the surface $\Sigma$ after a stereographic projection
$p_{r}$ of the first factor $\mathbb{S}^{2}$ into $\mathbb{R}^{2}$,
$p_{r}\colon\mathbb{S}^{2}\times\mathbb{R}\to\mathbb{R}^{2}\times\mathbb{R}$,
$p_{r}(x,y,z,t)=(\frac{x}{1-z},\frac{y}{1-z},t)$.

##  4\. The class of $R$-solitons

In this section we study $R$-solitons, where the vector field $R$ is

$R=-y\partial_{x}+x\partial_{y}.$ (18)

Following our scheme, we will classify $R$-solitons that are vertical surfaces
and next, rotational surfaces. First, suppose that $\Sigma$ is a vertical
$R$-soliton. We know that $\Sigma$ is parametrized by (3) and that the
generating curve $\alpha$ is contained in $\mathbb{S}^{2}$, see (2). A first
example of vertical $R$-soliton is a vertical cylinder over the geodesic
$\mathbb{S}^{1}\times\\{0\\}$ of $\mathbb{S}^{2}$. To be precise, let
$(u(s),v(s))=(0,s)$. Then $\alpha(s)=(\cos s,\sin s,0)$ in (2) and the surface
is the vertical cylinder over $\alpha$ which we have denoted by $\mathcal{C}$
in the previous section. This surface is minimal and it is immediate that $N$
is orthogonal to $R$. Thus $\mathcal{C}$ is a $R$-soliton. Recall that
$\mathcal{C}$ is also a rotational $V$-soliton.

######  Proposition 4.1.

Suppose that $\Sigma$ is a vertical surface parametrized by (3). Then $\Sigma$
is a $R$-soliton if and only if the generating curve $\alpha$ satisfies

$\left\\{\begin{split}u^{\prime}&=\cos u\cos\theta\\\
v^{\prime}&=\sin\theta\\\ \theta^{\prime}&=\sin\theta\sin u+(\cos
u)^{2}\cos\theta.\end{split}\right.$ (19)

###### Proof.

The expression of the unit normal $N$ is given in (3). Thus

$\langle N,R\rangle=-\cos u\cos\theta.$

Since the expression of $H$ is given in (7), then Eq. (1) is
$\theta^{\prime}=\sin\theta\sin u+(\cos u)^{2}\cos\theta$, proving the result.
∎

As in the previous section, we project the solutions of (19) on the
$(u,\theta)$-plane, obtaining the autonomous system

$\left\\{\begin{split}u^{\prime}&=\cos u\cos\theta\\\
\theta^{\prime}&=\sin\theta\sin u+(\cos u)^{2}\cos\theta.\end{split}\right.$
(20)

Equilibrium points are $(u,\theta)=(0,\pm\pi/2)$ again, together the points
$(u,\theta)=(\pm\frac{\pi}{2},0)$ and translations of length $\pi$ of these
points in the $\theta$ variable. For the points $(u,\theta)=(0,\pm\pi/2)$, we
have $u(s)=0$ and $v(s)=\pm s$ are solutions. In this case, we know that
$\Sigma$ is the vertical cylinder $\mathcal{C}$. The points
$(u,\theta)=(\pm\frac{\pi}{2},0)$ do not correspond with surfaces because
regularity is lost. In fact, coming back to the parametrization (3), the map
$\Psi$ is the parametrization of the vertical fiber at
$(0,0,1)\in\mathbb{S}^{2}$.

Figure 4. The $(u,\theta)$-phase plane of (20). The red points are the
equilibrium points $(0,\pm\frac{\pi}{2})$.

The phase plane is the set $A=(-\pi,\pi)\times(-\pi,\pi)$ in the
$(u,\theta)$-plane by the periodicity of $\theta$. If we now compute the
linearized system at the points $(u,\theta)=(0,\pm\frac{\pi}{2})$, we find
that they have the same character that the ones of the system (15). Thus we
have that $(u,\theta)=(0,\frac{\pi}{2})$ is a stable spiral and
$(u,\theta)=(0,-\frac{\pi}{2})$ is an unstable spiral.

For the points $(\frac{\pi}{2},0)$ and $(-\frac{\pi}{2},0)$, the linearized
system is

$\left(\begin{array}[]{ll}-1&0\\\
0&1\end{array}\right),\quad\left(\begin{array}[]{ll}1&0\\\
0&-1\end{array}\right),$

respectively. Since the eigenvalues are two real numbers with opposite signs,
then both equilibrium points are saddle points. See Fig. 4. However, this does
not affect to the reasoning since the arguments now are similar as in the
proof of Thm. 3.4. We omit the details. Figure 5 shows generating curves of
vertical $R$-solitons.

######  Theorem 4.2.

Let $\Sigma$ be a vertical $R$-soliton. Then $\Sigma$ is the cylinder
$\mathcal{C}$ or $\Sigma$ is parametrized by (3) with the following
properties:

  1. (1)

The curve $\beta(s)=(u(s),v(s))$ has self-intersections and it is asymptotic
to the $v$-axis at infinity. In case that $\beta$ satisfies the initial
conditions (17), then $\beta$ is a symmetric bi-graph on the $v$-axis.

  2. (2)

The surface $\Sigma$ is not embedded with infinitely many intersection points
with the $z$-axis.

  3. (3)

The surface $\Sigma$ is asymptotic to the cylinder $\mathcal{C}$.

Figure 5. Generating curves of vertical $R$-solitons. Left: solution curve
$\beta(s)=(u(s),v(s))$. Middle: the generating curve $\alpha$. Right: the
generating curve $\alpha$ contained in the unit sphere $\mathbb{S}^{2}$.

The second type of $R$-solitons are those surfaces that are invariant by a
one-parameter group of rotations of the first factor. Since we have defined in
(18) the vector field $R$ as the rotation about the direction
$(0,0,1)\in\mathbb{S}^{2}$, we cannot a priori fix the rotational axis of the
surface.

######  Theorem 4.3.

The only rotational $R$-solitons are:

  1. (1)

Slices $\mathbb{S}^{2}\times\\{t_{0}\\}$, $t_{0}\in\mathbb{R}$, viewed as
rotational surfaces with respect to any axis of $\mathbb{S}^{2}$ and;

  2. (2)

Rotational minimal surfaces about the $z$-axis.

###### Proof.

Let $\Sigma$ be a rotational $R$-soliton. In order to have manageable
computations of the mean curvature $H$ and the unit normal $N$ of $\Sigma$, we
will assume in this proof that the rotation axis of $\Sigma$ is the $z$-axis.
Thus we are assuming that the surface is parametrized by (5). Thus the vector
field $R$ is now arbitrary and with no a priori relation with the $z$-axis.
The vector field $R$ is determined by an orthonormal basis
$B=\\{E_{1},E_{2},E_{3}\\}$ of $\mathbb{R}^{3}$. With respect to $B$, the
vector field $R$ can be expressed by

$R(x_{1},x_{2},x_{3},t)=-x_{2}E_{1}+x_{1}E_{2},$

where $(x_{1},x_{2},x_{3})$ are coordinates of $\mathbb{S}^{2}$ with respect
to $B$.

We now write $E_{1}$ and $E_{2}$ with respect to the canonical basis of
$\mathbb{R}^{3}$,

$E_{i}=(\cos m_{i}\sin n_{i},\cos m_{i}\sin n_{i},\sin m_{i}),\quad i=1,2,$

where $m_{i},n_{i}\in\mathbb{R}$. The unit normal $N$ and the mean curvature
$H$ of $\Sigma$ were computed in (10) and (11), respectively. Then

$\begin{split}\langle N,R\rangle&=-\langle\Psi,E_{2}\rangle\langle
N,E_{1}\rangle+\langle\Psi,E_{1}\rangle\langle N,E_{2}\rangle\\\ =&(\sin
m_{1}\cos m_{2}\sin n_{2}-\cos m_{1}\sin n_{1}\sin
m_{2})\sin\theta\sin\varphi\\\ &+(\sin m_{1}\cos m_{2}\cos n_{2}-\cos
m_{1}\cos n_{1}\sin m_{2})\sin\theta\cos\varphi.\end{split}$ (21)

Looking now Eq. (1), we have that in the expression of right hand-side of (1),
that is, $\langle N,R\rangle$, formula (21), the variable $\varphi$ do appear.
However in the left hand-side of (1), the mean curvature $H$, formula (11),
does not depend on $\varphi$. This implies that the coefficients of
$\sin\varphi$ and $\cos\varphi$ in (21) must vanish. Both coefficients contain
the factor $\sin\theta$. This gives the following discussion of cases.

  1. (1)

Case $\sin\theta(s)=0$ for all $s$. Then $u(s)=s$ and $v(s)$ is a constant
function, $v(s)=t_{0}$, $t_{0}\in\mathbb{R}$. This proves that $\Sigma$ is a
slice $\mathbb{S}^{2}\times\\{t_{0}\\}$.

  2. (2)

Case $\sin\theta(s_{0})\not=0$ at some $s_{0}$. Then in an interval around
$s=s_{0}$, we deduce

$\sin m_{1}\cos m_{2}\sin n_{2}-\cos m_{1}\sin n_{1}\sin m_{2}=0,$ $\sin
m_{1}\cos m_{2}\cos n_{2}-\cos m_{1}\cos n_{1}\sin m_{2}=0.$

Both identities imply $E_{1}\times E_{2}=(0,0,1)$. Thus $R$ coincides with the
vector field defined in (18) and the rotation axis is the $z$-axis. Moreover,
the right hand-side of (1) is $0$, proving that the surface is minimal. This
proves the result.

∎

Minimal surfaces in $\mathbb{S}^{2}\times\mathbb{R}$ of rotational type with
respect to an axis in the first factor were classified by Pedrosa and Ritoré
[12].

## Acknowledgements

Rafael López is a member of the IMAG and of the Research Group “Problemas
variacionales en geometría”, Junta de Andalucía (FQM 325). This research has
been partially supported by MINECO/MICINN/FEDER grant no.
PID2020-117868GB-I00, and by the “María de Maeztu” Excellence Unit IMAG,
reference CEX2020-001105- M, funded by MCINN/AEI/10.13039/501100011033/
CEX2020-001105-M. Marian Ioan Munteanu is thankful to Romanian Ministry of
Research, Innovation and Digitization, within Program 1 – Development of the
national RD system, Subprogram 1.2 – Institutional Performance – RDI
excellence funding projects, Contract no.11PFE/30.12.2021, for financial
support.

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