# Horo-shrinkers in the hyperbolic space

Antonio Bueno, Rafael López  Departamento de Ciencias
Centro Universitario de la Defensa de San Javier. 30729 Santiago de la Ribera,
Spain<EMAIL_ADDRESS>Departamento de Geometría y Topología
Universidad de Granada. 18071 Granada, Spain<EMAIL_ADDRESS>
###### Abstract.

A surface $\Sigma$ in the hyperbolic space $\mathbb{H}^{3}$ is called a horo-
shrinker if its mean curvature $H$ satisfies $H=\langle
N,\partial_{z}\rangle$, where $(x,y,z)$ are the coordinates of
$\mathbb{H}^{3}$ in the upper half-space model and $N$ is the unit normal of
$\Sigma$. In this paper we study horo-shrinkers invariant by one-parameter
groups of isometries of $\mathbb{H}^{3}$ depending if these isometries are
hyperbolic, parabolic or spherical. We characterize totally geodesic planes as
the only horo-shrinkers invariant by a one-parameter group of hyperbolic
translations. The grim reapers are defined as the horo-shrinkers invariant by
a one-parameter group of parabolic translations. We describe the geometry of
the grim reapers proving that they are periodic surfaces. In the last part of
the paper, we give a complete classification of horo-shrinkers invariant by
spherical rotations, distinguishing if the surfaces intersect or not the
rotation axis.

###### Key words and phrases:

hyperbolic space, mean curvature flow, conformal vector field, grim reaper,
spherical rotation

###### 1991 Mathematics Subject Classification:

Primary 53E10; Secondary 53C44, 53A10, 53C21, 53C42.

##  1\. Introduction

The theory of the mean curvature flow (MCF for short) is an area of great
activity in geometric analysis in the last decades: see, for example, the
surveys [9, 11, 19] and references therein. In Euclidean space
$\mathbb{R}^{3}$, let be $\Sigma$ an oriented smooth surface and
$\Psi\colon\Sigma\to\mathbb{R}^{3}$ an isometric immersion. A MCF for $\Psi$
is a smooth family of immersions
$\\{\Psi_{t}:\Sigma\to\mathbb{R}^{3}:t\in[0,T)\\}$ satisfying

$\left\\{\begin{array}[]{ll}\displaystyle\frac{\partial\Psi_{t}}{\partial
t}&=H(\Psi_{t})N(\Phi_{t}),\\\ \Psi_{0}&=\Psi,\end{array}\right.$

where $H(\Phi_{t})$ and $N(\Phi_{t})$ are the mean curvature and the unit
normal of $\Phi_{t}$ respectively. Solutions of the MCF develop singularities
at finite time, which may cause a change in the topology of the surface. There
are two types of singularities. Translators of the MCF (also called
translating solitons) appear as the equation of the limit flow by a blow-up
procedure near type II singularities, according to Huisken and Sinestrari
[14]. In $\mathbb{R}^{3}$, a translator is a surface $\Sigma$ characterized by
the equation $H=\langle N,\vec{v}\rangle$, where $H$ and $N$ are the mean
curvature and the unit normal of $\Sigma$, respectively, and $\vec{v}$ is a
direction of the ambient space. This direction $\vec{v}$ indicates that the
shape of $\Sigma$ does not change during the evolution because $\Sigma$ is
translated by the MCF at constant velocity.

Huisken initiated the study of the MCF in general Riemannian manifolds [13].
When the ambient space is the hyperbolic space, pioneering research on the MCF
is [6, 7]; see also [3, 4, 8, 12]. Singularities of MCF have been less studied
[16, 22]. Nevertheless, the same notion of translator can be defined by
replacing $\vec{v}\in\mathbb{R}^{3}$ by a Killing vector field
$X\in\mathfrak{X}(\mathbb{H}^{3})$ whose flow of isometries consists of
translations of $\mathbb{H}^{3}$. A surface $\Sigma\subset\mathbb{H}^{3}$ is a
translator with respect to $X$ if $H=\langle N,X\rangle$. In the hyperbolic
space $\mathbb{H}^{3}$ there are two types of translations [10]. Parabolic
translations are isometries of $\mathbb{H}^{3}$ that fix one point of the
ideal boundary $\mathbb{H}^{3}_{\infty}$. Hyperbolic translations are
isometries of $\mathbb{H}^{3}$ that fix two points of
$\mathbb{H}^{3}_{\infty}$. The corresponding translators have been recently
studied in [5] and [17], respectively.

Besides Killing vector fields, another vector fields of $\mathbb{H}^{3}$ of
special relevance are the conformal vector fields. In order to give explicit
example of such vector fields, we will use the upper half-space model of
$\mathbb{H}^{3}$, that is, $(\mathbb{R}^{3}_{+},\langle\cdot,\cdot\rangle)$,
where $\mathbb{R}_{+}^{3}=\\{(x,y,z)\in\mathbb{R}^{3}:z>0\\}$,
$\langle\cdot,\cdot\rangle$ is the hyperbolic metric

$\langle\cdot,\cdot\rangle=\frac{1}{z^{2}}\langle\cdot,\cdot\rangle_{e},$

and $\langle\cdot,\cdot\rangle_{e}=dx^{2}+dy^{2}+dz^{2}$ is the Euclidean
metric of $\mathbb{R}^{3}$. In this model, the vector field
$\partial_{z}\in\mathfrak{X}(\mathbb{H}^{3})$ is a conformal vector field
because its Lie derivative is
$\mathcal{L}_{\partial_{z}}\langle,\rangle=-\frac{2}{z}\langle,\rangle$.
Motivated by the notion of translator in $\mathbb{H}^{3}$, we give the
following definition.

######  Definition 1.1.

A horo-shrinker in $\mathbb{H}^{3}$ is an isometric immersion
$\Psi\colon\Sigma\to\mathbb{H}^{3}$ of an oriented smooth surface $\Sigma$
whose mean curvature $H$ satisfies

(1) $H=\langle N,\partial_{z}\rangle,$

where $N$ is the unit normal of $\Sigma$.

It was in [1] where the authors proposed the study of self-similar solutions
of the mean curvature flow in the presence of a conformal vector field. More
precisely, the conformal vector field $\partial_{z}$ corresponds with the
vector field $e^{t}\partial_{t}$ in Example 3.3 of [1] when the hyperbolic
space $\mathbb{H}^{3}$ is viewed as the warped product
$\mathbb{R}\times_{e^{t}}\mathbb{R}^{2}$. This was motivated by the examples
of self-shrinkers of the mean curvature flow in $\mathbb{R}^{3}$, where the
vector field is the position vector field. In fact, horo-shrinkers are the
analogues of the self-shrinkers of the MCF theory in Euclidean space
$\mathbb{R}^{3}$ [9, 11, 19]. Instead the vector field $\partial_{z}$, in [20]
the authors considered the conformal vector field $-\partial_{z}$ and the
corresponding solutions of (1) are called horo-expanders. In $\mathbb{R}^{3}$,
self-shrinkers and self-expanders are surfaces which move by homotheties
(contractions or expansions, respectively) when they evolve by the MCF. In
contrast to the translators of $\mathbb{H}^{3}$, the shape of horo-shrinkers
and horo-expanders is not preserved along the MCF. Another reason to consider
surfaces satisfying (1) is because of its formal similarity with Ricci
solitons [18, 19].

A first observation is that horo-shrinkers and horo-expanders are minimal
surfaces in the sense of Ilmanen [15]. Specifically, if we define the function
$\phi(x,y,z)=-2/z$, then a minimal surface for the conformal metric
$e^{\phi}\langle,\rangle$ is characterized by Eq. (1), that is, the surface is
a horo-shrinker. In case of horo-expanders the function is $\phi(x,y,z)=2/z$.
Recall that being minimal in a conformal space is a property that also fulfill
self-shrinkers and self-expanders of $\mathbb{R}^{3}$. However, and in
contrast to the theory of the MCF in Euclidean space, it is unknown if the
translators of $\mathbb{H}^{3}$ determined by parabolic and hyperbolic
translations and described in [5, 17] are minimal surfaces in the sense of
Ilmanen.

The purpose of this paper is to investigate the horo-shrinkers. To be precise,
we are interested in those horo-shrinkers that are invariant by a one-
parameter group of isometries of $\mathbb{H}^{3}$. We detail the organization
of the paper and highlight some of the main results. In Section 2 we show the
first examples of horo-shrinkers, such as vertical planes (totally geodesic
planes) and the horosphere $\mathsf{H}_{1}$ of equation $z=1$. Taking these
examples as comparison surfaces and by the tangency principle, we prove that
there are no closed horo-shrinkers. We also classify in Thm. 2.3 all horo-
shrinkers invariant by hyperbolic translations. In Section 3 we define the
grim reapers as those horo-shrinkers invariant by a one-parameter group of
parabolic translations. The classification of the grim reapers is given in
Thm. 3.5, being these surfaces vertical planes, the horosphere
$\mathsf{H}_{1}$, and a one-parameter family of periodic surfaces along a
horizontal direction orthogonal to the parabolic translations. As a
consequence of the properties of the grim reapers, we will prove in Thm. 3.7
that there are no solutions of the Dirichlet problem at infinity associated to
the non-parametric equation for (1).

Section 4 is devoted to rotational horo-shrinkers of spherical type. We
distinguish if the surfaces intersect or not the rotation axis. In the first
case, the existence of these surfaces is not a direct consequence of standard
theory, since the ODE fulfilled is degenerated when the surface intersects the
rotation axis. In Thm. 4.1 we prove such existence using Banach’s fixed point
theorem. In Thm. 4.3 we prove that they are parametrized by one parameter,
namely, the initial height at which they intersect the rotation axis. We also
prove that these surfaces oscillate around $\mathsf{H}_{1}$ converging to it.
Finally, in Thm. 4.4 we describe the spherical rotational horo-shrinkers that
do not intersect the rotation axis. These surfaces are parametrized by two
parameters and they oscillate around $\mathsf{H}_{1}$.

##  2\. Preliminaries

In this paper we use the upper half-space model of $\mathbb{H}^{3}$. We will
employ the terminology parallel in the Euclidean sense and by vertical and
horizontal we mean to be parallel to the $z$-axis or parallel to the
$xy$-plane, respectively. The ideal boundary $\mathbb{H}_{\infty}^{3}$ of
$\mathbb{H}^{3}$ is the one-point compactification of the plane $z=0$. We show
some explicit examples of horo-shrinkers.

  1. (1)

Vertical totally geodesic planes. These surfaces are minimal ($H=0$) and the
unit normal $N$ is orthogonal to $\partial_{z}$.

  2. (2)

The horosphere of equation $z=1$. This horosphere will be denoted by
$\mathsf{H}_{1}$. In general, horospheres in the upper half-space model can be
viewed as horizontal planes of equation $z=c$, $c>0$. The mean curvature is
$H=1$ with the orientation $N=c\partial_{z}$. Then $\langle
N,\partial_{z}\rangle=1/c$ and consequently, the only horosphere $z=c$
satisfying (1) is when $c=1$.

Next, we express the condition of being a horo-shrinker in a non-parametric
way. For this, we will use a relation between the hyperbolic mean curvature
$H$ of a surface $\Sigma$ in $\mathbb{H}^{3}$ and its Euclidean mean curvature
$H_{e}$ when $\Sigma$ is regarded as a surface in
$(\mathbb{R}^{3}_{+},\langle,\rangle_{e})$. This relation is given by

(2) $H(x,y,z)=zH_{e}(x,y,z)+(N^{e})_{3}(x,y,z),\quad(x,y,z)\in\Sigma,$

where $N^{e}$ is the Euclidean unit normal of $\Sigma$ and the subindex
$(\cdot)_{3}$ denotes the third coordinate of the vector. From the viewpoint
of PDE theory, equation (1) is of second order and elliptic. Indeed, if
$\Sigma$ is locally expressed as $z=u(x,y)$, in virtue of (2), equation (1)
writes as

(3)
$\mbox{div}\left(\frac{Du}{\sqrt{1+|Du|^{2}}}\right)=\frac{2}{\sqrt{1+|Du|^{2}}}\frac{1-u}{u^{2}}.$

This elliptic equation is of quasilinear type. As stated in the introduction,
horo-shrinkers are minimal surfaces in the conformal space
$(\mathbb{H}^{3},e^{-2/z}\langle,\rangle)$. This allows to use the tangency
principle similarly as for minimal surfaces of $\mathbb{R}^{3}$. This implies
that if two horo-shrinkers have a common tangent point and one horo-shrinker
locally lies at one side of the other around that point, then both horo-
shrinkers agree in an open set. A first consequence of the tangency principle
is a certain control of the height of the points of a horo-shrinker that are
critical points of height function.

######  Proposition 2.1.

Let $\Sigma$ be a horo-shrinker, $\Sigma\not=\mathsf{H}_{1}$. If $p\in\Sigma$
is a local maximum (resp. minimum) of the function
$z\colon\Sigma\to\mathbb{R}$, then $z(p)>1$ (resp. $z(p)<1$).

###### Proof.

Let $p=(x_{0},y_{,}z_{0})$ be a local maximum of the function $z$. Since
$Du(x_{0},y_{0})=(0,0)$, Eq. (3) becomes simply

$\Delta u(x_{0},y_{0})=2\frac{1-z_{0}}{z_{0}^{2}},$

where $\Delta$ is the Euclidean Laplacian of $\mathbb{R}^{2}$. Thus $\Delta
u(x_{0},y_{0})\leq 0$ implies $z_{0}\geq 1$. We prove that, in fact,
$z_{0}>1$. On the contrary, if $z_{0}=1$, then the horo-shrinker $\Sigma$ lies
in one side of $\mathsf{H}_{1}$ in an open set of $p$. Then the tangency
principle would imply $\Sigma\subset\mathsf{H}_{1}$, which it is a
contradiction. In case that $p$ is a local minimum, the arguments are
analogous. ∎

A second consequence of the tangency principle is the following result.

######  Proposition 2.2.

There are no closed (compact without boundary) horo-shrinkers.

###### Proof.

Arguing by contradiction, assume that $\Sigma$ is a closed horo-shrinker. Take
$\Pi$ a vertical plane of $\mathbb{R}^{3}_{+}$ that does not intersect
$\Sigma$. We move $\Pi$ towards $\Sigma$ until we arrive to a first contact
point between both surfaces. Since $\Pi$ is a horo-shrinker, the tangency
principle asserts that $\Sigma$ and $\Pi$ agree in the largest neighborhood of
both surfaces containing the tangency point. This implies that
$\Sigma\subset\Pi$, which is a contradiction. ∎

We finish this section giving the classification of horo-shrinkers invariant
by a one-parameter group $\mathcal{H}$ of hyperbolic translations. A
hyperbolic translation of $\mathbb{H}^{3}$ is an isometry that leaves fixed
two points of the ideal boundary $\mathbb{H}_{\infty}^{3}$. In the upper-
halfspace model of $\mathbb{H}^{3}$, and after an isometry, we can assume that
these two points are the origin $O$ of $\mathbb{R}^{3}$ and the infinity. Then
a hyperbolic translation is an Euclidean homothety from $O$ and the
corresponding group is $\mathcal{H}=\\{(x,y,z)\mapsto t(x,y,z):t>0\\}$. In
particular, a surface invariant by $\mathcal{H}$ can be viewed as a radial
graph on the hemisphere
$\mathbb{S}^{2}_{+}=\\{(x,y,z)\in\mathbb{R}^{3}_{+}:x^{2}+y^{2}+z^{2}=1\\}$.

######  Theorem 2.3.

Let $\mathcal{H}$ be a one-parameter group of hyperbolic translations fixing
two points $p,q\in\mathbb{H}^{3}_{\infty}$. Then, totally geodesic planes
containing both points are the only horo-shrinkers invariant by the group
$\mathcal{H}$.

###### Proof.

Without loss of generality, we can assume that $\mathcal{H}$ is the group
Euclidean homotheties from the origin $O\in\mathbb{R}^{3}$. Let $\Sigma$ be a
horo-shrinker invariant by $\mathcal{H}$. Since $\Sigma$ is a radial graph on
some domain of $\mathbb{S}^{2}_{+}$, a parametrization of $\Sigma$ is

$\Psi(s,t)=t\alpha(s),\quad s\in I\subset\mathbb{R},\ t\in\mathbb{R},$

where $\alpha\colon I\to\mathbb{S}^{2}_{+}$ is a curve parametrized by the
Euclidean arc-length. Then $|\alpha(s)|_{e}=|\alpha^{\prime}(s)|_{e}=1$ for
all $s\in I$. The Euclidean mean curvature $H_{e}$ and the Euclidean unit
vector $N^{e}$ are

$H_{e}=\frac{1}{2}\frac{\langle\alpha^{\prime}\times\alpha,\alpha^{\prime\prime}\rangle_{e}}{t},\quad
N^{e}=\alpha^{\prime}\times\alpha.$

Then, (1) is

$2\alpha_{3}\langle\alpha^{\prime}\times\alpha,\alpha^{\prime\prime}\rangle_{e}+(\alpha^{\prime}\times\alpha)_{3}=\frac{\langle\alpha^{\prime}\times\alpha,\partial_{z}\rangle_{e}}{t\alpha_{3}}=\frac{(\alpha^{\prime}\times\alpha)_{3}}{t\alpha_{3}},$

where the last equality is because
$\langle\alpha^{\prime}\times\alpha,\partial_{z}\rangle_{e}=(\alpha^{\prime}\times\alpha)_{3}$.
If we write this equation as

$t\alpha_{3}\left(2\alpha_{3}\langle\alpha^{\prime}\times\alpha,\alpha^{\prime\prime}\rangle_{e}+(\alpha^{\prime}\times\alpha)_{3}\right)-(\alpha^{\prime}\times\alpha)_{3}=0,$

we have a polynomial equation on $t$. Thus, we deduce
$\langle\alpha^{\prime}\times\alpha,\alpha^{\prime\prime}\rangle_{e}=0$ and
$(\alpha^{\prime}\times\alpha)_{3}=0$ because $\alpha_{3}\not=0$. Since
$\langle\alpha^{\prime}\times\alpha,\alpha^{\prime\prime}\rangle_{e}=0$, and
$\alpha$ is a unit speed curve in $\mathbb{S}^{2}_{+}$, we have
$\alpha^{\prime\prime}=-\alpha$. Thus $\alpha$ is an Euclidean geodesic of
$\mathbb{S}^{2}_{+}$, that is, a great (hemi) circle of $\mathbb{S}^{2}_{+}$.
Using that $(\alpha^{\prime}\times\alpha)_{3}=0$, we deduce that $\alpha$ is
included in a vertical plane through $O$. In consequence, $\Sigma$ is a
vertical plane containing $O$. ∎

######  Remark 2.4.

Notice that the same proof is valid for horo-expanders, proving that totally
geodesic planes (vertical planes) are the only horo-expanders that are
invariant under hyperbolic translations. This completes the classification
given in [20] of all horo-expanders invariant by a one-parameter group of
isometries of $\mathbb{H}^{3}$.

##  3\. The grim reapers

In this section we classify the horo-shrinkers invariant by a one-parameter
group $\mathcal{P}$ of parabolic translations. A parabolic translation of
$\mathbb{H}^{3}$ is an isometry that leaves fixed one double point of the
ideal boundary $\mathbb{H}_{\infty}^{3}$. After an isometry of
$\mathbb{H}^{3}$, we can assume that this point is
$\infty\in\mathbb{H}^{3}_{\infty}$ and thus a parabolic translation is simply
a horizontal Euclidean translation. Then the group $\mathcal{P}$ is determined
by a horizontal direction $(a,b,0)\in\mathbb{R}^{3}$ being
$\mathcal{P}=\\{(x,y,z)\mapsto(x,y,z)+t(a,b,0):t\in\mathbb{R}\\}$. Hence, a
surface invariant by $\mathcal{P}$ is a ruled surface of $\mathbb{R}^{3}_{+}$
whose all rulings are horizontal straight-lines parallel to $(a,b,0)$. In
analogy with the Euclidean context, we give the following definition.

######  Definition 3.1.

A grim reaper is a horo-shrinker that is invariant by a one-parameter group of
parabolic translations.

Let $\Sigma$ be a grim reaper. Without loss of generality, we can assume that
the rulings of $\Sigma$ are parallel to the direction $(0,1,0)$. A
parametrization of $\Sigma$ is

(4) $\Psi(s,t)=(x(s),t,z(s)),\quad t\in\mathbb{R},s\in I\subset\mathbb{R},$

where $\alpha(s)=(x(s),0,z(s))$ is a planar curve contained in the $xz$\-
plane. Suppose that $s$ is the hyperbolic arc-length, that is
$\langle\alpha^{\prime}(s),\alpha^{\prime}(s)\rangle=1$. This reads as

$\frac{1}{z(s)^{2}}(x^{\prime}(s)^{2}+z^{\prime}(s)^{2})=1,$

hence there is a smooth function $\theta=\theta(s)$ such that

$x^{\prime}(s)=z(s)\cos\theta(s),\qquad z^{\prime}(s)=z(s)\sin\theta(s).$

The unit normal is $N=zN^{e}=(-z^{\prime},0,x^{\prime})$. The Euclidean mean
curvature $H_{e}$ of $\Sigma$ is $H_{e}=\kappa/2$, where $\kappa$ is the
Euclidean curvature of $\alpha$. Since $\kappa=\theta^{\prime}/z$, then
$H_{e}=\theta^{\prime}/(2z)$. Using (2), we have

$H=\frac{\theta^{\prime}}{2}+\frac{x^{\prime}}{z}.$

Since $\langle N,\partial_{z}\rangle=x^{\prime}/z^{2}$, then (1) is

(5) $\frac{\theta^{\prime}}{2}+\cos\theta=\frac{\cos\theta}{z}.$

Therefore, the coordinate functions $x(s)$, $z(s)$ and $\theta(s)$ satisfy

(6) $\left\\{\begin{split}x^{\prime}(s)&=z(s)\cos\theta(s),\\\
z^{\prime}(s)&=z(s)\sin\theta(s),\\\
\theta^{\prime}(s)&=2\cos\theta(s)\frac{1-z(s)}{z(s)}.\end{split}\right.$

Since the aim of this section is the geometric description of the grim
reapers, we study the shape of the solution curves of (6). First, we see that
each solution of (6) remains at a bounded distance to the plane $z=0$.

######  Proposition 3.2.

Let $(x(s),z(s),\theta(s))$ be a solution to (6). Then there is $\delta>0$
such that $z(s)\geq\delta$ for all $s\in I$.

###### Proof.

Multiplying the last equation of (6) by $\cos\theta\sin\theta$ and taking into
account that $z^{\prime}/z=\sin\theta$, we have

$\frac{\sin\theta(\sin\theta)^{\prime}}{1-\sin^{2}\theta}=2\sin\theta\frac{1-z}{z}=2\frac{1-z}{z^{2}}z^{\prime}.$

Hence we deduce that there exists $c\in\mathbb{R}$ such that

(7) $\cos\theta=cz^{2}e^{2/z}.$

Now, arguing by contradiction, assume that there is a sequence $s_{n}\to
s_{1}$ such that $z(s_{n})\rightarrow 0$ as $n\rightarrow\infty$, being
$s_{1}$ either finite or infinite. Substituting in (7) the right-hand side
diverges to $\infty$, which it is not possible because the left-hand side is
bounded. ∎

As a consequence, each solution of (6) is defined in $\mathbb{R}$ because the
functions on the right-hand side of (6) are bounded.

Recall that the function $x(s)$ does not explicitly appear in (6), but only
its derivative. Geometrically this implies that any solution of (6) remains a
solution after a parabolic translation $(x,0,z)\mapsto(x+t,0,z)$.
Consequently, in order to study the properties of the solutions of (6) it is
enough to consider the nonlinear autonomous system

(8) $\left(\begin{array}[]{c}z^{\prime}\\\
\theta^{\prime}\end{array}\right)=\left(\begin{array}[]{c}z\sin\theta\\\
\displaystyle{2\cos\theta\frac{1-z}{z}}\end{array}\right),$

defined in the domain $\\{(z,\theta):z>0,\theta\in\mathbb{R}\\}$. Indeed, if
we fix $(x_{0},z_{0},\theta_{0}),\ z_{0}>0$, let $(z,\theta)$ be the unique
solution to (6) for the initial data $z(0)=z_{0}>0$, $\theta(0)=\theta_{0}$,
and define $x$ as the solution to $x^{\prime}=z\cos\theta,\ x(0)=x_{0}$. Then,
$\alpha(s)=(x(s),0,z(s))$ is the generating curve of a surface parametrized by
(4) that is a solution to (1).

By periodicity of the trigonometric functions, we define the _orbits_ as the
solutions $\gamma(s)=(z(s),\theta(s))$ of (8), which are defined for $z>0$ and
$\theta\in(-\pi,\pi)$. By uniqueness of the Cauchy problem, two different
orbits cannot intersect, hence the $(z,\theta)$-domain
$(0,\infty)\times(-\pi,\pi)$ is foliated by all the orbits.

The following result exhibits that we can reduce the study of the orbits
essentially to $\theta\in(0,\pi/2)$. Its proof follows immediately, hence it
is omitted.

######  Proposition 3.3.

The following properties hold:

  1. (1)

If $\gamma(s)=(z(s),\theta(s))$ is an orbit, then
$\overline{\gamma}(s)=(z(-s),-\theta(-s))$ is also an orbit. Consequently,
every orbit $\gamma$ is symmetric with respect to the line $\theta=0$.

  2. (2)

If $\gamma(s)=(z(s),\theta(s))$ is an orbit for $\theta\in(-\pi/2,\pi/2)$,
then $\overline{\gamma}(s)=(z(s),-\theta(s)+\pi)$ is an orbit for
$\theta\in(\pi/2,3\pi/2)$.

We define the phase plane of (8) as the set

$\Theta=\\{(z,\theta)\colon z>0,\theta\in(-\pi/2,\pi/2)\\}.$

The coordinates $(z,\theta)$ are in one-to-one correspondence to the orbits of
(8). The motion of any orbit in $\Theta$ is uniquely determined by the sign of
the functions $z^{\prime}$ and $\theta^{\prime}$. From (8), the signs of
$z^{\prime}$ and $\theta^{\prime}$ are determined by the signs of $\cos\theta$
and $\sin\theta$ as well as of the function $z-1$. For example, any orbit
intersecting the line $z=1$ changes the monotonicity of its second coordinate,
attaining a local maximum or minimum. In the remaining of the phase plane, the
second coordinate of any orbit is strictly monotonous.

The explicit examples of horo-shrinkers given in Section 2 are now viewed as
trivial solutions of (6) in the following result.

######  Proposition 3.4.

Explicit examples of orbits of (8) are the following:

  1. (1)

The point $(1,0)$. This orbit corresponds to the horosphere $\mathsf{H}_{1}$.

  2. (2)

The lines $\theta=\pm\pi/2$. These orbits correspond to vertical planes
(totally geodesic planes) of equation $x=x_{0}$, where $x_{0}\in\mathbb{R}$.
The parameter $\theta=\pi/2$ implies that the vertical plane is parametrized
with increasing height, and for $\theta=-\pi/2$ the height is decreasing.

Figure 1. Left: the phase plane of (8) and different orbits portrayed. Right:
the generating curves of the corresponding orbits. The initial height are
$z_{0}=0.2$, $1.1$, $2$ and $5$.

As usual, for the description of the orbits of the autonomous system (8), we
analyze its equilibrium points. It is clear that the point $P_{0}=(1,0)$ is
the unique equilibrium. The linearized of the system around $P_{0}$ is

$\left(\begin{matrix}0&1\\\ -2&0\end{matrix}\right).$

Since the eigenvalues are imaginary numbers with zero real part, the
equilibrium $P_{0}$ has a center structure. Thus the orbits of the linearized
system are ellipses enclosing $P_{0}$ in their inner regions. Consequently,
the orbits that are close enough to $P_{0}$ either spiral around $P_{0}$ or
are closed curves enclosing $P_{0}$ in their inner regions. However, by Prop.
3.3 the orbits are symmetric about $\theta=0$, hence they cannot spiral around
$P_{0}$. In particular all the orbits stay at a positive distance from
$P_{0}$: see Fig. 1, left. Note: all figures in this paper have been plotted
using the software Mathematica.

We now derive the classification of the grim reapers.

######  Theorem 3.5.

The classification of the grim reapers is the following:

  1. (1)

Vertical planes (totally geodesic planes).

  2. (2)

The horosphere $\mathsf{H}_{1}$.

  3. (3)

A one-parameter family of entire graphs $\mathcal{G}(z_{0}),\ z_{0}\in(0,1)$
that are periodic along the $x$-direction. The value $z_{0}$ indicates the
Euclidean distance of $\mathcal{G}(z_{0})$ at $z=0$. Moreover:

    1. (a)

If $z_{0}\rightarrow 0$, then $\mathcal{G}(z_{0})$ converges to a double
covering of a vertical plane.

    2. (b)

If $z_{0}\rightarrow 1$ then $\mathcal{G}(z_{0})$ converges to the horosphere
$\mathsf{H}_{1}$.

    3. (c)

For each $z_{0}\in(0,1)$ there exists a unique $z_{0}^{*}\in(1,\infty)$ that
corresponds to the Euclidean height of $\mathcal{G}(z_{0})$ at $z=0$. Hence
$\mathcal{G}(z_{0})$ can be also parametrized in terms of $z_{0}^{*}$, being
equivalent. In fact, if $z_{0}\rightarrow 0$ (resp. $z_{0}\rightarrow 1$) then
$z_{0}^{*}\rightarrow\infty$ (resp. $z_{0}^{*}\rightarrow 1$).

###### Proof.

The first two types of surfaces were already depicted in Prop. 3.4. Now, fix
some $z_{0}\in(0,1)$. We call $\mathcal{G}(z_{0})$ the grim reaper which is
the graph of the solution of (6) for initial conditions
$x(0),z(0),\theta(0))=(0,z_{0},0)$. Let $\gamma_{z_{0}}$ be the orbit with
initial data $\gamma_{z_{0}}(0)=(z_{0},0)$ and let $\alpha_{z_{0}}$ be the
corresponding generating curve of the grim reaper. Then $\gamma_{z_{0}}$ is
vertical at $(z_{0},0)$, for $s>0$ and by monotonicity both coordinates
functions $z(s)$ and $\theta(s)$ of $\gamma_{z_{0}}$ strictly increase. Since
$\gamma_{z_{0}}$ cannot intersect the orbit $\theta=\pi/2$, necessarily
$\gamma_{z_{0}}$ intersects the vertical line $z=1$ where its
$\theta$-coordinate attains a local maximum. Then, $\theta$ decreases and
since $\gamma_{z_{0}}$ cannot converge to $P_{0}$ due to its center structure,
$\gamma_{z_{0}}$ intersects again the line $\theta=0$ at some $(z_{0}^{*},0)$,
where $z_{0}^{*}>1$. Finally, by symmetry of the phase plane with respect to
the line $\theta=0$, $\gamma_{z_{0}}$ closes again at the point $(z_{0},0)$.
See Fig. 1, left. Note that if $z_{0}\rightarrow 1$ then
$\gamma_{z_{0}}\rightarrow P_{0}$, while if $z_{0}\rightarrow 0$ then
$\gamma_{z_{0}}$ converges to both $\theta=\pm\pi/2$.

At this point, for each $z_{0}\in(0,1)$ the point $z_{0}^{*}\in(1,\infty)$
corresponds with a local maximum of the function $z=z(s)$ by Prop. 2.1. It
could happen that $(z_{0})_{n}\rightarrow 0$ and
$(z_{0}^{*})_{n}\rightarrow(z_{0}^{*})_{\infty}<\infty$. However, this
possibility cannot happen in virtue of Prop. 3.2. Indeed, assume by
contradiction that this behavior occurs, take some
$z^{*}>(z_{0}^{*})_{\infty}$ and let $\gamma_{z^{*}}$ be the orbit passing
through $(z^{*},0)$ at $s=0$. Then, when $s$ increases $\gamma_{z^{*}}$ cannot
intersect $\theta=-\pi/2$, hence $\gamma_{z^{*}}$ intersects $z=1$. Since it
cannot intersect again the line $\theta=0$ (because it would correspond to
some $z_{*}\in(0,1)$, a contradiction), the only possibility for
$\gamma_{z^{*}}$ is to converge to $z=0$. But this contradicts Prop. 3.2. As a
consequence, for each $z_{0}\in(0,1)$ there exists exactly one
$z_{0}^{*}\in(1,\infty)$, being both intervals in a one-to-one correspondence.

Therefore, the $z$-coordinate of the associated generating curve
$\alpha_{z_{0}}$ is periodic. Since $\theta\in(-\pi/2,\pi/2)$ we conclude that
$x^{\prime}>0$. This implies that $x$ is strictly increasing, hence the curve
$\alpha_{z_{0}}$ is periodic along the $x$-axis, in particular, invariant
under a discrete group of translations along the $x$-axis. The curve
$\alpha_{z_{0}}$ is a graph on the $x$-axis because $x^{\prime}>0$. The
Euclidean height of $\alpha_{z_{0}}$ at $z=0$ is $z_{0}^{*}$ and its distance
to $z=0$ is $z_{0}$. If $z_{0}\rightarrow 1$ then the curve $\alpha_{z_{0}}$
converges to the horizontal line $z=1$. If $z_{0}\rightarrow 0$ then
$\alpha_{z_{0}}$ converges to a double covering of a vertical line, since
$\gamma_{z_{0}}$ converges to both $\theta=\pm\pi/2$. See Fig. 1, right. This
concludes the proof. ∎

With the same notation as in the proof of Thm. 3.5, let us consider initial
conditions $(x(0),z(0),\theta(0))=(0,z_{1},0)$ with $z_{1}>1$ in system (6),
and let $\gamma_{z_{1}}$ be the corresponding orbit passing through
$(z_{1},0)$. Then, $\gamma_{z_{1}}$ passes through some $(z_{1}^{0},0)$, with
$z_{1}^{0}<1$ being the minimum value of the height function $z$.
Definitively, $(z_{1}^{0})^{*}=z_{1}$ and therefore up to a parabolic
translation orthogonal to the ruling direction $(0,1,0)$, the grim reaper
$\mathcal{G}(z_{1})$ agrees with $\mathcal{G}((z_{1}^{0})^{*})$.

######  Corollary 3.6.

Let $z_{0}\in(0,1)$. Then there is a unique $z_{1}\in(1,\infty)$ such that
$\mathcal{G}(z_{0})$ and $\mathcal{G}(z_{1})$ coincide up to a parabolic
translation orthogonal to $(0,1,0)$.

To finish this section, we address the Dirichlet problem at infinity for Eq.
(2). More precisely, let
$\Omega\subset\mathbb{R}^{2}=\\{z=0\\}\subset\mathbb{H}^{3}_{\infty}$ be a
bounded domain with smooth boundary. We are asking for functions $u\in
C^{\infty}(\Omega)\cap C^{0}(\overline{\Omega})$ such that $u$ satisfies (3)
in $\Omega$ with $u>0$ in $\Omega$ and $u=0$ along $\partial\Omega$. This is
motivated by the pioneering works of the theory of constant mean curvature
surfaces in hyperbolic space $\mathbb{H}^{3}$ due to Anderson ($H=0$) and to
Tonegawa ($0<H<1$) [2, 21]. The Dirichlet problem at infinity for horo-
expanders was considered in [20] proving existence under general hypothesis on
convexity of $\Omega$. However, for Eq. (3) we show that the Dirichlet problem
at infinity is not solvable. The key is that the grim reapers of Thm. 3.5
allow us to compare with the possible solutions of the Dirichlet problem at
infinity.

######  Theorem 3.7.

There are no solutions of the Dirichlet problem at infinity for Eq. (3).

###### Proof.

By contradiction, suppose that $u$ is a solution of (3), $u>0$ in $\Omega$ and
with initial condition $u=0$ along $\partial\Omega$. Let $\Sigma$ be the graph
of $u$ when $u$ is defined in $\Omega$ and let $u_{M}>0$ be the maximum of $u$
in $\Omega$ which exists because $\overline{\Omega}$ is compact and $u$ is
continuous in $\overline{\Omega}$. After a parabolic translation along the
$x$-direction, we can assume that $\Sigma$ is included in the half-space
$\\{(x,y,z)\in\mathbb{R}^{3}_{+}:x>0\\}$. By Prop. 2.1, let $u_{M}>1$ be the
maximum value of $u$ in $\Omega$. Using Thm. 3.5, let $z_{0}\in(0,1)$
sufficiently close to $0$ such that $z_{0}^{*}$, the maximum height of
$\mathcal{G}(z_{0})$, satisfies $z_{0}^{*}>u_{M}$. Take the piece of
$\mathcal{G}(z_{0})$ comprised between two consecutive maximum of
$\mathcal{G}(z_{0})$. To be precise, if we write the generating curve of
$\mathcal{G}(z_{0})$ as $z=z(r)$, let $r_{0}>0$ be such that $z(0)=z_{0}$,
$z(\pm r_{0})=z_{0}^{*}$ and $r=\pm r_{0}$ are the only maximum of $z(r)$ in
the interval $[-r_{0},r_{0}]$. Consider $\mathcal{G}(z_{0})^{F}$ the piece of
$\mathcal{G}(z_{0})$ determined by $z(r)$ in the interval
$r\in[-r_{0},r_{0}]$, that is,
$\mathcal{G}(z_{0})^{F}=\mathcal{G}(z_{0})\cap\\{(x,y,z)\in\mathbb{R}^{3}:-r_{0}\leq
x\leq r_{0}\\}$. Notice that the boundary of $\mathcal{G}(z_{0})^{F}$ are two
straight-lines parallel to the $y$-axis and both situated at height
$z_{0}^{*}$.

Let us move $\mathcal{G}(z_{0})^{F}$ by translations along the $x$-direction
with $x\searrow-\infty$ until $\mathcal{G}(z_{0})^{F}$ does not intersect
$\Sigma$. This is possible because $\Sigma$ is included in the half-space
$x>0$ and the rulings of $\mathcal{G}(z_{0})^{F}$ are parallel to the
$y$-axis. Next, we move $\mathcal{G}(z_{0})^{F}$ by translations along the
$x$-direction with $x\nearrow\infty$ until the first contact point $p$ with
$\Sigma$. Let $\widetilde{\mathcal{G}(z_{0})^{F}}$ denote the position of
$\mathcal{G}(z_{0})^{F}$ when it touches $\Sigma$. This point exists because
$\overline{\Omega}$ is compact. Since $z_{0}>0$, then $z(p)\geq z_{0}$, so it
is an interior point of $\Sigma$ (or equivalently, $z(p)\not=0$). On the other
hand, $z(p)\leq u_{M}<z_{0}^{*}$, so $p$ is an interior point of
$\widetilde{\mathcal{G}(z_{0})^{F}}$. Definitively, $p$ is a common interior
point of $\Sigma$ and $\widetilde{\mathcal{G}(z_{0})^{F}}$. The tangency
principle implies that $\Sigma$ is included in
$\widetilde{\mathcal{G}(z_{0})^{F}}$. This is a contradiction because
$\widetilde{\mathcal{G}(z_{0})^{F}}$ is contained in the half-space
$\\{(x,y,z)\in\mathbb{R}^{3}_{+}:z>z_{0}\\}$ where $z_{0}>0$. ∎

##  4\. Spherical rotational horo-shinkers

In this section we classify all horo-shrinkers invariant by one-parameter
group $\mathcal{S}$ of spherical rotations of $\mathbb{H}^{3}$. After an
isometry of $\mathbb{H}^{3}$, we can suppose that the common rotation axis of
the elements of $\mathcal{S}$ is the $z$-axis. Then the elements of
$\mathcal{S}$ are simply Euclidean rotations about the $z$-axis, being
$\mathcal{S}=\\{(x,y,z)\mapsto(x\cos t-y\sin t,x\sin t+y\cos
t,z):t\in\mathbb{R}\\}$. Thus, a parametrization of a spherical rotational
surface $\Sigma$ is

$\Psi(s,t)=(x(s)\cos t,x(s)\sin t,z(s)),\quad s\in
I\subset\mathbb{R},t\in\mathbb{R},$

where $\alpha(s)=(x(s),0,z(s))$ is the generating curve. Notice that
horospheres, viewed as horizontal planes of equation $z=c$, $c>0$, also are
spherical rotational surfaces. Consequently, the horosphere $\mathsf{H}_{1}$
is a spherical rotational horo-shrinker.

Assume that $\alpha$ is parametrized by the Euclidean arc-length. Then
$\alpha^{\prime}(s)=(\cos\theta(s),0,\sin\theta(s))$, for some smooth function
$\theta=\theta(s)$. The Euclidean mean curvature $H_{e}$ and the Euclidean
unit normal $N^{e}$ of $\Sigma$ are, respectively,

$H_{e}=\frac{1}{2}\left(\theta^{\prime}+\frac{\sin\theta}{x}\right),\quad
N^{e}=(-\sin\theta\cos t,-\sin\theta\sin t,\cos\theta).$

By (2), the equation (1) writes as

(9)
$\frac{z}{2}\left(\theta^{\prime}+\frac{\sin\theta}{x}\right)+\cos\theta=\frac{\cos\theta}{z}.$

Thus, Eq. (1) is equivalent to say that the coordinate functions of the curve
$\alpha$ satisfy

(10) $\left\\{\begin{split}x^{\prime}(s)&=\cos\theta(s),\\\
z^{\prime}(s)&=\sin\theta(s),\\\
\theta^{\prime}(s)&=-\frac{\sin\theta(s)}{x(s)}+2\cos\theta(s)\frac{1-z(s)}{z(s)^{2}}.\end{split}\right.$

The study of the spherical rotational horo-shrinkers, or equivalently, of the
solutions of (10), is separated in two cases depending if the surface, or
equivalently the generating curve, intersects or not the rotation axis.

First consider the case the surface intersects the rotational axis. We will
prove the existence of such surfaces and, in such a case, that this
intersection must be orthogonal. Since $\alpha$ intersects the $z$-axis, then
at the initial value, say $s=0$, for (10), $x(0)$ must be $0$ and
$z^{\prime}(0)=0$. However, the existence of solutions of (10) is not assured
by the standard ODE theory because (10) is degenerate at $x=0$. To such a
existence, we parametrize the curve $\alpha$ by $r\mapsto(r,0,z(r))$,
$z(r)>0$, then Eq. (9) writes as

(11) $\frac{z}{2}\left(\frac{z^{\prime\prime}}{(1+z^{\prime
2})^{3/2}}+\frac{z^{\prime}}{r(1+z^{\prime
2})^{1/2}}\right)+\frac{1}{(1+z^{\prime 2})^{1/2}}=\frac{1}{z(1+z^{\prime
2})^{1/2}}.$

Next we prove that there exist solutions of (11) defined at $r=0$ such that
$z^{\prime}(0)=0$.

######  Theorem 4.1.

If $z_{0}>0$, then there exist $R>0$ and a solution $z\in C^{2}([0,R])$ of
(11) with initial conditions

(12) $z(0)=z_{0}>0,\quad z^{\prime}(0)=0.$

###### Proof.

Multiplying (11) by $r$, we can write (11) as

(13)
$\left({\displaystyle\frac{rz^{\prime}(r)}{\sqrt{1+z^{\prime}(r)^{2}}}}\right)^{\prime}={\displaystyle
2r\frac{1-z(r)}{z(r)^{2}\sqrt{1+z^{\prime}(r)^{2}}}}.$

Define the functions

$g:\mathbb{R}^{+}\times\mathbb{R}\rightarrow\mathbb{R},\quad
g(x,y)=\frac{2(1-x)}{x^{2}\sqrt{1+y^{2}}},$
$\varphi:\mathbb{R}\rightarrow\mathbb{R},\quad\varphi(y)=\frac{y}{\sqrt{1+y^{2}}}.$

From (13), a function $z=z(r)\in C^{2}([0,R])$ satisfies (11)-(12) if and only
if $(r\varphi(z^{\prime}))^{\prime}=rg(z,z^{\prime})$ under initial conditions
(12). The inverse function of $\varphi$ is $\varphi^{-1}(x)=x/\sqrt{1-x^{2}}$,
which is defined in $(-1,1)$. Fix $R>0$ that will be determined later and
define the operator ${\mathsf{T}}:C^{1}([0,R])\to C^{1}([0,R])$ by

(14)
$({\mathsf{T}}z)(r)=z_{0}+\int_{0}^{r}\varphi^{-1}\left(\frac{1}{s}\int_{0}^{s}tg(z,z^{\prime})dt\right)ds.$

It is clear that a fixed point of ${\mathsf{T}}$ is a solution of the initial
value problem (12)-(13). First, we prove the existence of $\epsilon>0$ such
that ${\mathsf{T}}$ is well defined in a closed ball
$\overline{B(z_{0},\epsilon)}$ of $C^{1}([0,R])$. Here we understand that the
space $C^{1}([0,R])$ is endowed the usual sup-norm
$\|z\|=\|z\|_{\infty}+\|z^{\prime}\|_{\infty}$. For this, let $\epsilon>0$
such that $\epsilon<z_{0}$, and consider $g$ defined in
$[z_{0}-\epsilon,z_{0}+\epsilon]\times\mathbb{R}$. Let $M>0$ such that
$|\frac{2(1-z)}{z^{2}}|\leq M$ for all $|z-z_{0}|\leq\epsilon$. Let
$R\leq\min\\{\frac{1}{M},\frac{\sqrt{3}\epsilon}{2},\frac{\sqrt{3}\epsilon}{2M}\\}$.
We have

$\int_{0}^{s}\frac{t}{s}g(z,z^{\prime})\,dt\leq\int_{0}^{s}\frac{t}{s}M\,dt\leq\frac{RM}{2}\leq\frac{1}{2},$

because $R\leq 1/M$. This allows to apply $\varphi^{-1}$ in the parenthesis of
(14). In order to use the Banach fixed point theorem, we need the two
following steps.

(1) The map ${\mathsf{T}}$ satisfies
${\mathsf{T}}(\overline{B(z_{0},\epsilon)})\subset\overline{B(z_{0},\epsilon)}$.
To prove this inclusion, let $z\in\overline{B(z_{0},\epsilon)}$. By using that
$\varphi^{-1}$ is increasing, we have

$\begin{split}|({\mathsf{T}}z)(r)-z_{0}|&\leq\int_{0}^{r}\varphi^{-1}\left(\int_{0}^{s}\frac{t}{s(z_{0}-\epsilon)}\,dt\right)\,ds<\varphi^{-1}\left(\frac{1}{2}\right)R=\frac{R}{\sqrt{3}}\leq\frac{\epsilon}{2},\\\
|({\mathsf{T}}z)^{\prime}(r)|&\leq\varphi^{-1}\left(\int_{0}^{s}\frac{t}{s}M\,dt\right)\leq\varphi^{-1}\left(\frac{R}{2}M\right)=M\frac{R}{\sqrt{4-RM^{2}}}\leq\frac{RM}{\sqrt{3}}\leq\frac{\epsilon}{2}\end{split}$

because $R\leq\sqrt{3}\epsilon/2$ and $R\leq\sqrt{2}\epsilon/(2M)$,
respectively. As a conclusion, $\|{\mathsf{T}}z\|\leq\epsilon$.

(2) The map ${\mathsf{T}}$ is a contraction. The functions $g$ and
$\varphi^{-1}$ are Lipschitz continuous in
$[z_{0}-\epsilon,z_{0}+\epsilon]\times[-\epsilon,\epsilon]$ and
$[-\epsilon,\epsilon]$, respectively provided $0<\epsilon<\min\\{z_{0},1\\}$.
Let $L=\min\\{L_{g},L_{\varphi^{-1}}\\}$, where $L_{g}$ and $L_{\varphi^{-1}}$
stand for the Lipschitz constants of $g$ and $\varphi^{-1}$, respectively.
Given $z,\tilde{z}\in\overline{B(z_{0},\epsilon)}$, for all $r\in[0,R]$ we
have

$\begin{split}|({\mathsf{T}}z)(r)-({\mathsf{T}}\tilde{z})(r)|&\leq
L\int_{0}^{r}\frac{1}{s}\int_{0}^{s}t\left|g(z,z^{\prime})-g(\tilde{z},\tilde{z}^{\prime})\right|\,dt\,ds\\\
&\leq
L^{2}\int_{0}^{r}\frac{1}{s}\int_{0}^{s}t\left(\|z-\tilde{z}\|_{\infty}+\|z^{\prime}-\tilde{z}^{\prime}\|_{\infty}\right)\,dt\,ds\\\
&=L^{2}\|z-\tilde{z}\|\int_{0}^{r}\frac{s}{2}\,ds=\frac{r^{2}L^{2}}{4}\|z-\tilde{z}\|.\end{split}$

Analogously,

$|({\mathsf{T}}z)^{\prime}(r)-({\mathsf{T}}\tilde{z})^{\prime}(r)|\leq\frac{rL^{2}}{2}\|z-\tilde{z}\|.$

Therefore

$\|{\mathsf{T}}z-{\mathsf{T}}\tilde{z}\|\leq\min\\{\frac{R^{2}L^{2}}{4},\frac{RL^{2}}{2}\\}\|z-\tilde{z}\|.$

Since $L$ is fixed, by choosing $R>0$ small enough, we conclude that
${\mathsf{T}}$ is a contraction.

The solution $z=z(r)$ obtained by the Banach fixed point theorem lies in
$C^{1}([0,R])\cap C^{2}((0,R])$. We prove that $z(r)$ can be extended up to
$C^{2}$-regularity at $r=0$. From (11), the L’Hôpital rule gives

(15) $\lim_{r\rightarrow 0}z^{\prime\prime}(r)=\frac{1-z_{0}}{z_{0}^{2}}.$

This completes the proof of the theorem. ∎

Once we have proved the existence of spherical rotational horo-shrinkers
intersecting orthogonally the rotation axis, our next goal is to achieve a
full classification of such surfaces. First, we prove the following result
which is valid for any solution of (10).

######  Proposition 4.2.

Let $\alpha(s)=(x(s),0,z(s))$ be a solution of (10). If $\alpha$ is not a
graph, then the $x$-coordinate has exactly one critical point which is a
minimum. In consequence, if $\alpha$ intersects the rotation axis then
$\alpha$ is a graph.

###### Proof.

If $\alpha$ is not a graph, then there is a critical point $s_{0}$ of $x$,
$x^{\prime}(s_{0})=0$ with $x(s_{0})>0$. From (10) we have
$\theta^{\prime}(s_{0})=-\sin\theta(s_{0})/x(s_{0})=\pm\frac{1}{x(s_{0})}$ and
thus

$x^{\prime\prime}(s_{0})=\frac{1}{x(s_{0})}>0,$

which yields that $s_{0}$ is a local minimum of $x(s)$. This proves that
$s_{0}$ must be a local minimum and in such a case, no more critical points of
$x(s)$ exist.

If $\alpha$ intersects the rotation axis at $s=0$, then $x(0)=0$ and
$x^{\prime}(0)=1$. If $s_{0}>0$ is the first critical point of $x(s)$, then
$s_{0}$ would be a local minimum, a contradiction. ∎

As a consequence of Prop. 4.2, the generating curve $\alpha$ of a spherical
rotational horo-shrinker that intersects the rotation axis can be globally
parametrized by $z=z(r)$. By Eq. (11), $z(r)$ is a solution of the initial
value problem

(16) $\left\\{\begin{split}&\frac{z^{\prime\prime}}{1+z^{\prime
2}}+\frac{z^{\prime}}{r}=2\frac{1-z}{z^{2}}\\\ &z(0)=z_{0}>0,\quad
z^{\prime}(0)=0.\end{split}\right.$

Let $J=[0,r_{max})$ stand for the maximal domain of the solutions of (10),
where $r_{\max}\in\mathbb{R}\cup\\{\infty\\}$. We denote by
$\mathcal{B}(z_{0})$ the spherical rotational horo-shrinker generated by
$z(r)$, whose intersection with the rotation axis occurs at $z=z_{0}$. The
following result exhibits the properties of $\mathcal{B}(z_{0})$ and its
classification. Numerical examples are depicted in Fig. 2.

######  Theorem 4.3.

The spherical rotational horo-shinkers interesecting the rotation axis are the
surfaces $\mathcal{B}(z_{0})$, where the parameter $z_{0}>0$ indicates the
height of the intersection point of the surface with the rotation axis. Each
$\mathcal{B}(z_{0})$ is an entire graph that oscillates around the horosphere
$\mathsf{H}_{1}$. Furthermore,

  1. (1)

If $z_{0}=1$, then $\mathcal{B}(1)=\mathsf{H}_{1}$.

  2. (2)

If $z_{0}\in(0,1)$, then $\mathcal{B}(z_{0})$ is strictly convex at $r=0$.

  3. (3)

If $z_{0}\in(1,\infty)$, then $\mathcal{B}(z_{0})$ is strictly concave at
$r=0$.

###### Proof.

The case $z_{0}=1$ follows immediately by just checking that the constant
function $z(r)=1$ fulfills (16). This proves the assertion (1). Suppose now
$z_{0}\neq 1$. Substituting at (16), we have

$z^{\prime\prime}(0)=\frac{1-z_{0}}{z_{0}}^{2}.$

If $z_{0}\in(0,1)$ (resp. $z_{0}\in(1,\infty)$) then $z^{\prime\prime}(0)>0$,
the function $z(r)$ has a local minimum (resp. local maximum) at $r=0$ and
$z(r)$ is strictly convex (resp. concave) for $r>0$ small enough. This proves
(2) and (3).

We now prove that the solutions $z(r)$ of (16) are entire graphs (that is,
$r_{max}=\infty$) that oscillate around the horizontal line $z=1$ in the
$xz$-plane. We assume $z_{0}\in(0,1)$, as the arguments when
$z_{0}\in(1,\infty)$ are analogous. The behavior of $z(r)$ will be deduced by
proving a series of claims.

  1. (1)

The function $z(r)$ cannot be a convex graph for $r>0$. On the contrary,
because $z^{\prime},z^{\prime\prime}$ are positive, we have that
$z\rightarrow\infty$ as $r\to r_{max}$. The left-hand side of (16) remains
always positive, but its right-hand side is eventually negative, a
contradiction.

  2. (2)

The function $z(r)$ cannot fail to be a graph at finite time $r_{0}>0$.
Arguing by contradiction, as $r\rightarrow r_{0}$ it happens $z(r)\rightarrow
z(r_{0})$, $z^{\prime}(r)\rightarrow\infty$, $z^{\prime\prime}(r)>0$ for $r$
close to $r_{0}$. Recall that $\lim_{r\rightarrow r_{0}}z^{\prime\prime}(r)$
can be either finite or infinite, but in any case it is positive. Taking
limits in (16) as $r\rightarrow r_{0}$ we see that the left-hand side of (16)
goes to $\infty$, while its right-hand side is a finite value, a
contradiction.

  3. (3)

As a consequence, $z$ must change its convexity, which implies
$z^{\prime\prime}(r_{1})=0$ and $z^{\prime\prime}(r_{1})<0$ for $r>r_{1}$
close enough to $r_{1}$. In particular, $z^{\prime}(r_{1})>0$ and from (16),
we deduce that $z(r_{1})=z_{1}<1$.

Figure 2. Generating curves of spherical rotational horo-shrinkers
intersecting the rotational axis, and the horosphere $z=1$ between them. Here
$z_{0}=0,5$, $1$ and $2$.

  4. (4)

There are $0<z_{m}<z_{M}<\infty$ such that $z_{m}\leq z(r)\leq z_{M}$ for all
$r\in J$. Moreover, $z_{m}=z_{0}$ (if $z_{0}>1$, then $z_{M}=z_{0}$). In order
to prove the claim, let us multiply (16) by $z^{\prime}$ and integrate from
$0$ to $r$. Then we obtain

(17) $\frac{1}{2}\log(1+z^{\prime
2})+\int_{0}^{r}\frac{z^{\prime}(t)^{2}}{t}dt=-2\left(\frac{1}{z}+\log
z\right)+2\left(\frac{1}{z_{0}}+\log z_{0}\right).$

If there exists a sequence $r_{n}\to r_{max}$ such that $z(r_{n})\to\infty$,
then the right hand-side of (17) goes $-\infty$, a contradiction because the
left hand-side is positive. This proves that $z(r)$ is bounded from above. A
similar argument shows that $z(r)$ is bounded from below, by taking a sequence
$r_{n}\rightarrow r_{max}$ such that $z(r_{n})\to 0$.

We now prove that $z_{m}=z_{0}$. On the contrary, let $r_{*}>0$ be such that
$z(r_{*})=z_{*}<z_{0}$. Letting $r=r_{*}$ in (17), the right hand-side must be
positive. Consider the function $f(t)=-2(\frac{1}{t}+\log t)$, which is
negative and increasing in $(0,1)$. The right hand-side of (17) writes as
$f(z_{*})-f(z_{0})>0$. Hence $z_{*}>z_{0}$, a contradiction.

  5. (5)

We have $r_{max}=\infty$. We write the first equation of (16) as

(18) $\left(\begin{array}[]{l}z\\\
z^{\prime}\end{array}\right)^{\prime}=\left(\begin{array}[]{c}z^{\prime}\\\
\displaystyle(1+z^{\prime
2})\left(2\frac{1-z}{z^{2}}-\frac{z^{\prime}}{r}\right)\end{array}\right).$

Then $r_{max}=\infty$ if we show that the right hand-side of (18) is bounded.
In fact, by the above claim, it is enough to prove that the function
$z^{\prime}(r)$ is bounded. If there is a sequence $(r_{n})\to r_{max}$ such
that $|z^{\prime}(r_{n})|\to\infty$, then evaluating (17) at $r=r_{n}$ and
letting $n\to\infty$, we have that the left hand-side of (17) diverges.
However, the right hand-side is bounded by Claim 4. This contradiction proves
the claim.

  6. (6)

The function $z(r)$ attains a local maximum at some $r_{2}>r_{1}$. By
contradiction, assume that such maximum does not exist, which implies that
$z^{\prime\prime}(r)\leq 0$ and $z^{\prime}(r)>0$ for every $r>r_{1}$. By the
previous claim, since $z(r)$ is strictly increasing and bounded from above,
then $z(r)$ has a limit which, without loss of generality, we can suppose that
it is $z_{M}$. Moreover, $z^{\prime}(r)\to 0$ as $r\rightarrow\infty$.

Next, letting $r\to\infty$ in (16), we deduce that $z^{\prime\prime}(r)$ has a
limit. Since $z^{\prime\prime}(r)\leq 0$, we conclude that this limit is $0$.
Thus $\lim_{r\to\infty}z^{\prime\prime}(r)=2\frac{1-z_{M}}{z_{M}^{2}}=0$. This
yields $z_{M}=1$. In particular, the left hand-side of (16) is positive.
Therefore, by dividing in (16) by $z^{\prime}$ and integrating from $r_{1}$ to
$r$ for $r$ big enough yields

$\frac{1}{2}\log\frac{z^{\prime 2}}{1+z^{\prime 2}}+\log r+c_{1}>0,.$

for some integration constant $c_{1}$. After some manipulations we arrive to

$z^{\prime}>\frac{c_{2}}{\sqrt{r^{2}-c_{2}^{2}}},\quad c_{2}=e^{-c_{1}}>0.$

Finally, integrating from $r_{1}$ to $r$ we obtain

$z(r)>c_{2}\,\mathrm{arctanh}\frac{r}{\sqrt{r^{2}-c_{2}^{2}}}+c_{3},\quad\
c_{3}\in\mathbb{R}.$

Letting $r\to\infty$, the right hand-side in this inequality diverges and thus
$z(r)\to\infty$ as $r\to\infty$. This it is not possible by Claim 4.

After these arguments we ensure the existence of $r_{2}>r_{1}$ such that
$z^{\prime}(r_{2})=0$. Note that $z(r_{2})\neq 1$ since otherwise,
$\mathcal{B}(z_{0})=\mathsf{H}_{1}$ by uniqueness in (16): this is not
possible because $z_{0}<1$. Since $z^{\prime\prime}(r)\leq 0$ for $r<r_{2}$
close enough to $r_{2}$ we conclude $z_{2}>1$. Then
$z^{\prime\prime}(r_{2})<0$, which implies that $z(r)$ attains a local maximum
at $r_{2}$.

  7. (7)

From the above claims, we deduce that the function $z(r)$ cannot end being a
graph at some finite $r_{3}>r_{2}$ because the left-hand side of (16) would be
$-\infty$ but the right-hand side is finite. Consequently, $z$ keeps being a
graph and for $r>r_{2}$ small enough we have $z^{\prime\prime}(r),\
z^{\prime}(r)<0$. We show that $z$ cannot keep this behavior. Otherwise, for
$r\rightarrow r_{3}>r_{2}$ we would have $\lim_{r\rightarrow r_{3}}z(r)=0$
with $z^{\prime},z^{\prime\prime}<0$. But this contradicts the Claim 4. Thus
$z(r)$ has to change its curvature, i.e. there exists $r_{3}>r_{2}$ such that
$z^{\prime\prime}(r_{3})=0$, for which $z(r_{3})>1$ since
$z^{\prime}(r_{3})<0$. At this point, the only possibilities for $z$ are the
following:

    1. (a)

$z(r)\rightarrow z_{\infty}>0$ as $r\rightarrow\infty$;

    2. (b)

$z^{\prime}(r_{4})=0,\ z^{\prime\prime}(r_{4})>0$ for some $r_{4}>r_{3}$.

We prove that the latter is the one that holds. By contradiction, if the
former holds, the following would occur

$\lim_{r\rightarrow\infty}z^{\prime}(r)=\lim_{r\rightarrow\infty}z^{\prime\prime}(r)=0,$

which yields a contradiction after substituting in (16). We conclude that
necessarily $z^{\prime}(r_{4})=0$ at some $r_{4}>r_{3}$, where by a similar
argument as in the case of the maximum we get that
$z^{\prime\prime}(r_{4})>0$, i.e. $z$ attains a local minimum at $r_{4}$, and
$z(r_{4})<1$ by Prop. 2.1.

At this point, we have a similar structure as when $z(r)$ started at the
rotation axis with an orthogonal intersection at a local minimum of $z(r)$.
Therefore, this process is repeated and we see that $z=z(r)$ is an entire
graph that oscillates around the horosphere $\mathsf{H}_{1}$.

∎

The last result of this section is devoted to show the geometric properties of
the spherical rotational horo-shrinkers that do not intersect the rotation
axis. See Fig. 3.

######  Theorem 4.4.

Let $\Sigma$ be a spherical rotational horo-shrinker about the $z$-axis such
that $\Sigma$ does not intersect the rotation axis. Then $\Sigma$ belongs to a
two-parameter family of spherical rotational horo-shrinkers,
$\mathcal{W}(x_{0},z_{0})$, where the parameter $x_{0}\in(0,\infty)$ indicates
the Euclidean distance of $\mathcal{W}(x_{0},z_{0})$ to the $z$-axis and
$z_{0}$ is the Euclidean distance to $z=0$. Moreover:

  1. (1)

The surfaces $\mathcal{W}(x_{0},z_{0})$ are bi-graphs on the $xy$-plane.

  2. (2)

Each $\mathcal{W}(x_{0},z_{0})$ is contained in the closure of the non-bounded
domain determined by the Euclidean cylinder about the $z$-axis and of radius
$x_{0}$.

  3. (3)

Each $\mathcal{W}(x_{0},z_{0})$ has the topology of an annulus, and its ends
oscillate around the horosphere $\mathsf{H}_{1}$.

###### Proof.

Let us write Eq. (9) considering that the generating curve $\alpha$ is a graph
$x=x(r)$ on the $z$-axis, $z>0$. Then $x(r)$ satisfies

(19) $x^{\prime\prime}=\frac{(r^{2}+2(r-1)xx^{\prime})(1+x^{\prime
2})}{r^{2}x^{2}}.$

For $x_{0},z_{0}>0$, let $x=x(r)$ be the solution of (19) with initial
conditions $x(z_{0})=x_{0}$, $x^{\prime}(z_{0})=0$. Since
$x^{\prime\prime}(z_{0})=1/x_{0}$, the function $x(r)$ is strictly convex
locally around $r_{0}$. This curve generates a spherical rotational horo-
shrinker which will be denoted by $\mathcal{W}(x_{0},z_{0})$. Then
$\mathcal{W}(x_{0},z_{0})$ starts as a bi-graph over the $xy$-plane around
$(x_{0},z_{0})$. Let $\mathcal{W}_{+}(x_{0},z_{0})$ denote the upper graphical
component, which is the graph of a function $z=z_{+}(r)$. Similarly, its lower
graphical component is denoted by $\mathcal{W}_{-}(x_{0},z_{0})$ and it is the
graph of a function $z=z_{-}(r)$.

Figure 3. Generating curves of spherical rotational horo-shrinkers which do no
intersect the rotational axis. In orange, the initial condition is
$(x_{0},z_{0})=(1,1)$. In blue, the initial condition is
$(x_{0},z_{0})=(1,2)$.

Let us analyze the behavior of $\mathcal{W}_{+}(x_{0},z_{0})$, that is, of the
function $z_{+}(r)$: for $z_{-}(r)$ the arguments are analogous. The function
$z_{+}$ satisfies $z_{+}(x_{0})=z_{0}$, $z_{+}^{\prime}(x_{0})=\infty$ and for
$r>x_{0}$ close enough to $x_{0}$ we have $z_{+}^{\prime}(r)>0$ and
$z_{+}^{\prime\prime}(r)<0$. At this point, we follow similar ideas that the
ones developed in the proof of Thm. 4.3, so the graph $z_{+}$ must attain a
local maximum, decrease, change its curvature and then attain a local minimum.
This process is repeated proving that $z_{+}(r)$ oscillates around $z=1$ as
$r\to\infty$. ∎

######  Remark 4.5.

The surfaces $\mathcal{B}(z_{0})$ and $\mathcal{W}(x_{0},z_{0})$ of Thms. 4.3
and 4.4 can be thought as the analogous to the bowl soliton and the wing-like
examples of translators in the theory of the mean curvature flow in
$\mathbb{R}^{3}$. Up to a translation of $\mathbb{R}^{3}$, the bowl soliton is
the unique translator in $\mathbb{R}^{3}$ intersecting orthogonally the
rotation axis, while the wing-like examples form a one-parameter family of
annuli, parametrized in terms of the distance to the rotation axis. In
contrast, the situation for horo-shrinkers is a bit different. If the rotation
axis is the $z$-axis (such as it has been considered in this section), the
hyperbolic translations of $\mathbb{H}^{3}$ from the origin $O$ (Euclidean
homotheties) do not preserve equation (1). Therefore, two horo-shrinkers
$\mathcal{B}(z_{0})$ and $\mathcal{B}(z_{1})$, $z_{0}\not=z_{1}$, do not
coincide by a hyperbolic translation of $\mathbb{H}^{3}$. For this reason, the
family of surfaces $\mathcal{B}(z_{0})$ is one-parametric. Similarly, the
family $\mathcal{W}(x_{0},z_{0})$ is two-parametric.

We end this paper with the following observation. Using Mathematica, it is
possible to observe that the surfaces $\mathcal{B}(z_{0})$ and
$\mathcal{W}(x_{0},z_{0})$ not only oscillate around $\mathsf{H}_{1}$ but they
converge to it at infinity. However, the authors have not been able to prove
this convergence. The difficulty is that if we project the system (10) on the
$(z,\theta)$-plane, the $2$-dimensional system is not autonomous by the
presence of $x$. Or equivalently, the system (18) is non-autonomous. Anyway,
it is important to point out that $(z,\theta)=(1,0)$ (resp.
$(z,z^{\prime})=(1,0)$) is an equilibrium point of (10) (resp. (18))
regardless of the value of $x$ (resp. of $r$). This equilibrium point
corresponds to the horosphere $\mathsf{H}_{1}$.

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