# Solitons to Mean Curvature Flow
in the hyperbolic $3$-space

R. F. de Lima and A. K. Ramos and J. P. dos Santos  Departamento de Matemática
- UFRN<EMAIL_ADDRESS>Departamento de Matemática Pura e Aplicada -
UFRGS<EMAIL_ADDRESS>Departamento de Matemática - UnB
<EMAIL_ADDRESS>
###### Abstract.

We consider translators (i.e., initial condition of translating solitons) to
mean curvature flow (MCF) in the hyperbolic $3$-space $\mathbb{H}^{3}$,
providing existence and classification results. More specifically, we show the
existence and uniqueness of two distinct one-parameter families of complete
rotational translators in $\mathbb{H}^{3},$ one containing catenoid-type
translators, and the other parabolic cylindrical ones. We establish a tangency
principle for translators in $\mathbb{H}^{3}$ and apply it to prove that
properly immersed translators to MCF in $\mathbb{H}^{3}$ are not cylindrically
bounded. As a further application of the tangency principle, we prove that any
horoconvex translator which is complete or transversal to the $x_{3}$-axis is
necessarily an open set of a horizontal horosphere. In addition, we classify
all translators in $\mathbb{H}^{3}$ which have constant mean curvature. We
also consider rotators (i.e., initial condition of rotating solitons) to MCF
in $\mathbb{H}^{3}$ and, after classifying the rotators of constant mean
curvature, we show that there exists a one-parameter family of complete
rotators which are all helicoidal, bringing to the hyperbolic context a
distinguished result by Halldorsson, set in $\mathbb{R}^{3}.$

2020 Mathematics Subject Classification: 53E10 (primary), 53E99 (secondary).

Key words and phrases: soliton – mean curvature flow – hyperbolic space –
invariant surfaces.

The second and third authors were partially supported by the National Council
for Scientific and Technological Development – CNPq

##  1\. Introduction

The last decades flourished with great regard to the theory of extrinsic
geometric flows in Riemannian manifolds, especially to mean curvature flow in
Euclidean spaces, giving rise to a vast literature on the subject (cf. [2] and
the references therein). Extrinsic geometric flows constitute evolution
equations that describe hypersurfaces of a Riemannian manifold evolving in the
normal direction with velocity given by the corresponding extrinsic curvature.
A special class of solutions is that of the solitons, also known as the self-
similar solutions, which are characterized for being generated by the Killing
field defined by a one-parameter subgroup of isometries of the ambient
manifold. When these isometries are translations along a geodesic, we call the
corresponding self-similar solutions translating solitons to the giving flow,
and the initial hypersurfaces are known as translators. A main feature of
translators in Euclidean spaces is that they naturally appear as type II
singularities of certain compact solutions to mean curvature flow (cf. [12,
Theorem 4.1]).

There exist many examples of translators to mean curvature flow (MCF, for
short) in Euclidean space $\mathbb{R}^{3}.$ Three of the best known are the
cylinder over the graph of the function $f(t)=-\log(\cos t),$
$t\in(-\pi/2,\pi/2),$ called the grim reaper, the rotational entire graph over
$\mathbb{R}^{2}$ obtained by Altschuler and Wu [1] known as the translating
paraboloid or bowl soliton, and the one-parameter family of rotational annuli
obtained by Clutterbuck, Schnürer and Schulze [6], the so called translating
catenoids. On the other hand, little is known about translators in hyperbolic
spaces.

In this paper, we consider solitons to MCF in hyperbolic space
$\mathbb{H}^{3}$, and first we focus on the case of translators which move by
hyperbolic translations along a fixed geodesic. We classify all such surfaces
with constant mean curvature (Theorems 3.2 and 3.4) and also obtain new
families of examples (see Figure 1), which, by some similarities with the
translators in $\mathbb{R}^{3}$ described above, will be called the
translating catenoid (Theorem 3.9) and the grim reaper (Theorem 3.13). These
translators are then proven to be unique with respect to their fundamental
properties (Theorems 3.10 and 3.16).

Figure 1. A translating catenoid and a grim reaper in the half-space model of
$\mathbb{H}^{3}$.

A main tool in establishing uniqueness results for translators in Euclidean
space is the tangency principle. It asserts that two such translators which
are tangent at a point, with one on one side of the other in a neighborhood of
this point, must coincide in this neighborhood. In fact, this result is a
direct consequence of the tangency principle for minimal surfaces, since
translators in Euclidean space become minimal surfaces when the ambient space
is endowed with a suitable metric (cf. Section 3.3). Apparently, for
translators in hyperbolic space, such a metric is not available. Nevertheless,
we prove that a tangency principle holds for translators in $\mathbb{H}^{3}$
as well (Theorem 3.17). Then, by using the family of translating catenoids as
barriers, we apply it to prove that properly immersed translators in
$\mathbb{H}^{3}$ are never cylindrically bounded and, in particular, never
closed. As a further application, we show that any horoconvex translator which
is complete or transversal to the axis of the translation is necessarily an
open set of a horosphere (Theorem 3.20).

We study, as well, rotators to MCF, that is, initial data of solitons whose
associated isometries are rotations around a geodesic. In [10], Halldorsson
considered rotators in $\mathbb{R}^{3},$ obtaining a one-parameter family of
complete helicoidal rotators in $\mathbb{R}^{3}$ which are also translators.
Inspired by Halldorsson’s work, we obtain here an analogous result (Theorem
4.4), in which we construct a one-parameter family of helicoidal surfaces in
$\mathbb{H}^{3}$ that, under mean curvature flow, rotates around its axis and
translates downwards with velocity that equals its pitch, see Figure 2. As in
the case of translators, we also classify all rotators of constant mean
curvature in $\mathbb{H}^{3}$ (Theorem 4.1).

Figure 2. A helicoidal rotational soliton.

The paper is organized as follows. In Section 2, we set some notation and
formulae. In Section 3, we introduce translators to MCF in $\mathbb{H}^{3}$
and establish the aforementioned results related to them. In Section 4 we deal
with rotators to MCF in $\mathbb{H}^{3}.$ Finally, we use Section 5 to present
the classification of minimal translators in $\mathbb{H}^{3}$.

######  Acknowledgments.

We heartily thank Eric Toubiana for pointing us to the existence result
necessary for the final argument in Section 5, and also Marcus Marrocos, for
the enlightening conversations that led to the proofs of Lemmas 3.7 and 3.12.

##  2\. Preliminaries

Throughout the paper, we shall consider the upper half-space model of
$\mathbb{H}^{3},$ that is, $\mathbb{H}^{3}:=(\mathbb{R}_{+}^{3},ds^{2}),$
where $\mathbb{R}^{3}_{+}=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\mid
x_{3}>0\\}$, $ds^{2}:={d\bar{s}^{2}}/{x_{3}^{2}}$ and $d\bar{s}^{2}$ is the
standard Euclidean metric of $\mathbb{R}_{+}^{3}.$ We will also denote
$ds^{2}$ by $\langle\,,\,\rangle.$

Let $\varSigma$ be an oriented surface in a Riemannian 3-manifold
$\overline{M}$. Set $\overline{\nabla}$ for the Levi-Civita connection of
$\overline{M},$ $\eta$ for the unit normal field of $\varSigma,$ and $A$ for
its shape operator with respect to $\eta,$ so that

$AX=-\overline{\nabla}_{X}\eta,\,\,X\in T\varSigma,$

where $T\varSigma$ stands for the tangent bundle of $\varSigma$. The principal
curvatures of $\varSigma,$ that is, the eigenvalues of $A,$ will be denoted by
$k_{1},k_{2}$ and the mean curvature $H$ of $\varSigma$ is expressed by

$H=\frac{k_{1}+k_{2}}{2}.$

The mean curvature vector of $\varSigma$ is

$\mathbf{H}=H\eta,$

which is invariant under the choice of orientation $\eta\to-\eta$ and
satisfies $\|\mathbf{H}\|=|H|$.

Given an oriented surface $\varSigma\subset\mathbb{R}_{+}^{3},$ let
$\bar{\eta}=(\bar{\eta}_{1},\bar{\eta}_{2},\bar{\eta}_{3})$ be the unit normal
of $\varSigma$ with respect to the induced Euclidean metric $d\bar{s}^{2}.$ It
is easily checked that

$\eta(p)=x_{3}\bar{\eta}(p),\,\,\,p=(x_{1},x_{2},x_{3})\in\varSigma$

defines a unit normal of $\varSigma$ with respect to the hyperbolic metric
$ds^{2}.$ With these orientations, if we denote by $\mkern
1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu$ (resp. $H$) the mean
curvature of $\varSigma$ with respect to the Euclidean metric (resp.
hyperbolic metric) of $\mathbb{R}_{+}^{3},$ we have that $\mkern
1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu$ and $H$ satisfy the
following relation (cf. [15, Lemma 10.1.1]):

(1) $H(p)=x_{3}\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern
1.5mu(p)+\bar{\eta}_{3}(p)\,\,\,\forall p=(x_{1},x_{2},x_{3})\in\varSigma.$

###  2.1. Mean curvature flow

We say that a family of oriented surfaces $\varSigma_{t}=X_{t}(M)$ of a
Riemannian $3$-manifold $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu$ _evolves under mean
curvature flow_ if the corresponding one-parameter family of immersions

$X_{t}\colon M\rightarrow\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu,\,\,\,t\in[0,\delta),\,\,\,0<\delta\leq+\infty,$

satisfies the following condition:

(2) $\frac{\partial X_{t}}{\partial
t}^{\perp}(p)=H_{t}(p)\eta_{t}(p)\,\,\,\forall p\in M,$

where $\eta_{t}$ is the unit normal to $X_{t},$ $H_{t}$ is the mean curvature
of $X_{t}$ with respect to $\eta_{t},$ and $\frac{\partial X_{t}}{\partial
t}^{\perp}$ denotes the normal component of $\frac{\partial X_{t}}{\partial
t},$ that is,

$\frac{\partial X_{t}}{\partial t}^{\perp}=\left\langle\frac{\partial
X_{t}}{\partial t},\eta_{t}\right\rangle\eta_{t}\,.$

In particular, the equality (2) is equivalent to

$\left\langle\frac{\partial X_{t}}{\partial t},\eta_{t}\right\rangle=H_{t}.$

We call such a family $X_{t}:M\rightarrow\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu,$ $t\in[0,\delta)$ a
_mean curvature flow_ (MCF, for short) in $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu$ with initial data
$X_{0}.$ In this setting, we say that $\varSigma_{t}=X_{t}(M)$ is a _soliton_
or a _self-similar solution_ to MCF if there exists a one-parameter subgroup
$\mathcal{G}:=\\{\Gamma_{t}\mid t\in\mathbb{R}\\}$ of the group of isometries
of $\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu,$ such that
$\Gamma_{0}$ is the identity map of $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu$ and

$\varSigma_{t}=\Gamma_{t}(\varSigma)\,\,\,\forall t\in\mathbb{R}$

is a MCF. More specifically, we shall call such a family $\varSigma_{t}$ a
$\mathcal{G}$-_soliton_.

Let $\xi$ be the Killing field determined by the subgroup $\mathcal{G},$ i.e.,
for any $p\in\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu,$

$\xi(p):=\frac{\partial}{\partial
t}\Gamma_{t}(p)\,\,\,\,\,\text{at}\,\,\,t=0.$

It can be proved (see, e.g., [13]) that the surface $\varSigma=X_{0}(M)$ with
unit normal $\eta$ is the initial condition of a $\mathcal{G}$-_soliton_
generated by $\xi$ in $\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu$ if and only if the equality

(3) ${H=\langle\xi,\eta\rangle}$

holds everywhere on $\varSigma.$ So, in the class of solitons, equation (2) is
in fact a prescribed mean curvature problem.

##  3\. Translators to MCF in $\mathbb{H}^{3}$

Consider in hyperbolic space $\mathbb{H}^{3}$ the group
$\mathcal{G}=\\{\Gamma_{t}\mid t\in\mathbb{R}\\}\subset{\rm
Iso}(\mathbb{H}^{3})$ of hyperbolic translations along the $x_{3}$-axis,
defined by

$\Gamma_{t}(p)=e^{t}p,\,\,\,p\in\mathbb{H}^{3}.$

In this setting, an initial condition of a $\mathcal{G}$-soliton will be
called a _translating soliton_ or simply a _translator_. Using the abuse of
notation

$p=(x_{1},x_{2},x_{3})\in\mathbb{H}^{3}\leftrightarrow
x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}}\in
T_{p}\mathbb{H}^{3},$

the Killing field associated to $\mathcal{G}$ is $\xi(p)=p,$
$p\in\mathbb{H}^{3}$. Thus, it follows from (3) that a surface
$\varSigma\subset\mathbb{H}^{3}$ is a translator to MCF if and only if

(4) $H(p)=\langle p,\eta(p)\rangle\,\,\,\forall p\in\varSigma.$

######  Example 3.1.

Let $\Pi$ be a totally geodesic vertical plane of $\mathbb{H}^{3}$ which
contains $(0,0,1)$. Since $H$ vanishes on $\Pi$, it is clear that (4) holds
for $\varSigma=\Pi.$ Thus, $\Pi$ is a stationary translator to MCF in
$\mathbb{H}^{3}.$

In fact, equation (3) implies that a minimal surface
$\varSigma\subset\mathbb{H}^{3}$ is a (stationary) translator to MCF if and
only if it is invariant under the group $\mathcal{G}$ of hyperbolic isometries
as above. A complete classification of such surfaces is given by the following
description.

######  Theorem 3.2.

There exists a one-parameter family $\varSigma_{\theta}$, $\theta\in(0,\pi],$
of properly embedded minimal surfaces in $\mathbb{H}^{3}$ with the following
properties:

  * i)

$\varSigma_{\theta}$ is invariant under the one-parameter group
$\\{\Gamma_{t}\\}_{t\in\mathbb{R}}$ of hyperbolic translations

${p\in\mathbb{H}^{3}\mapsto\Gamma_{t}(p):=e^{t}p\in\mathbb{H}^{3},}$

and so it is a stationary translator to MCF in $\mathbb{H}^{3}.$

  * ii)

$\partial_{\infty}\varSigma_{\theta}\cap\mathbb{R}^{2}$ is the union of two
half-lines making an angle $\theta.$

  * iii)

$\varSigma_{\pi}$ is a vertical plane.

Conversely, if $\varSigma$ is a properly embedded minimal surface of
$\mathbb{H}^{3}$ which is invariant under the group $\Gamma_{t},$ then
$\varSigma=\varSigma_{\theta}$ for some $\theta\in(0,\pi].$

The proof of Theorem 3.2, for convenience, will be presented separately in
Section 5. Concerning the case of translators with nonzero constant mean
curvature, we start with the next example.

######  Example 3.3.

Let $\mathscr{H}_{h}$ be the horosphere of
$\mathbb{H}^{3}=(\mathbb{R}^{3}_{+}\,,\,ds^{2})$ at height $h>0,$ i.e.,

$\mathscr{H}_{h}=\\{(x_{1},x_{2},h)\in\mathbb{H}^{3}\mid
x_{1},x_{2}\in\mathbb{R}\\}.$

At any point $p=(x_{1},x_{2},h)\in\mathscr{H}_{h},$ we have that $H(p)=1$ and
$\eta(p)=he_{3},$ so that

$\langle p,\eta(p)\rangle=\frac{1}{h^{2}}h^{2}=1=H(p)\,\,\ \forall
p\in\mathscr{H}_{h}.$

Hence, $\mathscr{H}_{h}$ is a translator to MCF in $\mathbb{H}^{3}.$

In our next result we show that horospheres are the only translators to MCF
which have nonzero constant mean curvature. In the proof, we shall use the
following evolution formula for the mean curvature $H_{t}$ (notation as in
Section 2) of a mean curvature flow $X_{t}:M\to\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu$:

(5) $\frac{\partial H_{t}}{\partial t}=\Delta H_{t}+H_{t}(\|A_{t}\|^{2}+\mkern
1.5mu\overline{\mkern-1.5mu\rm Ric\mkern-1.5mu}\mkern
1.5mu(\eta_{t},\eta_{t})),$

where $\mkern 1.5mu\overline{\mkern-1.5mu\rm Ric\mkern-1.5mu}\mkern 1.5mu$
denotes the Ricci tensor of $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu$ (see [11, Theorem
3.2-(v)]).

######  Theorem 3.4.

Let $\varSigma$ be a connected translator to MCF in $\mathbb{H}^{3}$ which has
nonzero constant mean curvature. Then, $\varSigma$ is an open subset of a
horosphere.

###### Proof.

After a change of orientation, we may assume without loss of generality that
the mean curvature $H$ of $\varSigma$ is positive. Let
$X_{t}:M\to\mathbb{H}^{3},$ $t>0,$ be the MCF such that $X_{0}(M)=\varSigma$
and

$X_{t}(p)=e^{t}X_{0}(p),\,\,\,p\in M.$

Since $X_{t}(M)$ differs from $X_{0}(M)$ by an ambient isometry, $H_{t}=H>0$
is constant in space and time, thus ${\partial H_{t}}/{\partial t}=\Delta
H_{t}=0.$ Also, in $\mathbb{H}^{3},$ $\mkern 1.5mu\overline{\mkern-1.5mu\rm
Ric\mkern-1.5mu}\mkern 1.5mu(\eta_{t},\eta_{t})=-2.$ Then, formula (5) yields
$\|A_{t}\|^{2}=2$ for all $t\geq 0.$ Taking $t=0,$ we conclude that the
principal curvatures $k_{1},k_{2}$ of $\varSigma$ satisfy:

${\left\\{\begin{array}[]{cccccl}k_{1}&+&k_{2}&=&2H,\\\\[4.30554pt]
k_{1}^{2}&+&k_{2}^{2}&=&2,\end{array}\right.}$

from where it follows that $H\in(0,1]$ and, after possibly reindexing,

${k_{1}=H+\sqrt{1-H^{2}},\quad k_{2}=H-\sqrt{1-H^{2}}.}$

Since $H$ is constant, both $k_{1}$ and $k_{2}$ are constant, so $\varSigma$
is isoparametric. The isoparametric surfaces of $\mathbb{H}^{3}$ are
classified (see [3, Theorem 3.14]) and the fact that $H\in(0,1]$ imply that
$\varSigma$ is either an open subset of a horosphere or of an equidistant
surface to a totally geodesic plane. However, $k_{1}^{2}+k_{2}^{2}=2$ only
holds when $\varSigma$ is contained in a horosphere, which finishes the proof
of the theorem. ∎

######  Remark 3.5.

Since (5) holds for any $\mathcal{G}$-soliton, the proof of Theorem 3.4
applies to show that any initial condition of a $\mathcal{G}$-soliton in
$\mathbb{H}^{3}$ with nonzero constant mean curvature is necessarily an open
subset of a horosphere.

###  3.1. Rotational translators.

In this section, we focus on translators to MCF in $\mathbb{H}^{3}$ which are
invariant under rotations about the $x_{3}$-axis. With this purpose, we first
consider vertical rotational graphs. More precisely, let $\phi$ be a positive
smooth function on an open interval $I\subset(0,+\infty),$ and assume its
graph $\varSigma$ in $\mathbb{H}^{3}$ is invariant under rotations about the
$x_{3}$-axis. Then, $\varSigma$ admits a parameterization of the form

$X(u,v)=(v\cos u,v\sin u,\phi(v)),\,\,\,\,(u,v)\in U:=\mathbb{R}\times
I\subset\mathbb{R}^{2}.$

We shall call $\varSigma=X(U)$ the _rotational vertical graph determined by_
$\phi.$

For a rotational graph $\varSigma$ as above, a direct computation gives that

$\bar{\eta}:=(\bar{\eta}_{1},\bar{\eta}_{2},\bar{\eta}_{3})=\varrho(-\phi^{\prime}\cos
u,-\phi^{\prime}\sin
u,1),\quad\varrho:=\frac{1}{\sqrt{1+(\phi^{\prime}))^{2}}},$

is a unit normal with respect to the induced Euclidean metric, and that the
corresponding Euclidean mean curvature is

$\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern
1.5mu=\frac{\varrho}{2}\left(\frac{\phi^{\prime\prime}}{1+(\phi^{\prime})^{2}}+\frac{\phi^{\prime}}{v}\right).$

Thus, from (1), the mean curvature $H$ of $\varSigma$ in $\mathbb{H}^{3}$ with
respect to $\eta:=\phi\bar{\eta}$ is

(6) $H=\phi\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern
1.5mu+\bar{\eta}_{3}=\varrho\left(\frac{\phi}{2}\left(\frac{\phi^{\prime\prime}}{1+(\phi^{\prime})^{2}}+\frac{\phi^{\prime}}{v}\right)+1\right).$

It is also straightforward to see that the equality

(7) $\langle X,\eta\rangle=\frac{\varrho}{\phi}(\phi-v\phi^{\prime})$

holds everywhere on $\varSigma.$

From (6) and (7), we conclude that equation (4) for the vertical graph
$\varSigma$ is equivalent to the second order ODE:

(8)
$\phi^{\prime\prime}=-\phi^{\prime}(1+(\phi^{\prime})^{2})\left(\frac{2v}{\phi^{2}}+\frac{1}{v}\right).$

Defining $\Omega:=(0,+\infty)\times(0,+\infty)\times\mathbb{R},$ and

$\Psi(x,y,z)=-z(1+z^{2})\left(\frac{2x}{y^{2}}+\frac{1}{x}\right),\,\,\,(x,y,z)\in\Omega,$

we get from (8) the following

######  Lemma 3.6.

A vertical rotational graph determined by a smooth function $\phi$ is a
translator to MCF in $\mathbb{H}^{3}$ if and only if $\phi$ is a solution to
the second order ODE:

(9) $y^{\prime\prime}=\Psi(x,y,y^{\prime}).$

Next, we establish some properties of the solutions to (9).

######  Lemma 3.7.

For any $x_{0},y_{0}>0$ and any $\lambda\in\mathbb{R},$ the initial value
problem

(10)
$\left\\{\begin{array}[]{l}y^{\prime\prime}=\Psi(x,y,y^{\prime})\\\\[4.30554pt]
y(x_{0})=y_{0}\\\\[4.30554pt] y^{\prime}(x_{0})=\lambda\end{array}\right.$

has a unique smooth solution $\phi$ on $[x_{0},+\infty)$ which has the
following properties:

  * (i)

$\phi$ is constant if $\lambda=0.$

  * (ii)

$\phi$ is increasing, concave and bounded above by a positive constant if
$\lambda>0.$

  * (iii)

$\phi$ is decreasing, convex and bounded below by a positive constant if
$\lambda<0.$

###### Proof.

Since $\Psi$ is $C^{\infty}$ in $\Omega,$ the standard results on solutions
for ODE’s ensure the existence and uniqueness of a $C^{\infty}$ solution
$\phi$ defined in a maximal interval $I_{\rm max}:=[x_{0},x_{\rm
max}),\,x_{\rm max}\leq+\infty,$ in the sense that the equality

(11)
$\phi^{\prime\prime}=\Psi(x,\phi,\phi^{\prime})=-\phi^{\prime}(1+(\phi^{\prime})^{2})\left(\frac{2x}{\phi^{2}}+\frac{1}{x}\right)$

holds in $I_{\rm max}.$

If $\lambda=0,$ it is clear from (11) that the solution $\phi$ is constant, in
which case $x_{\rm max}=+\infty.$ This proves (i).

Assume now that $\lambda>0.$ Then, $\phi$ is increasing near $x_{0}.$ Also,
from property (i) and the uniqueness of solutions, $\phi$ has no critical
points. Hence, $\phi$ is increasing in $I_{\rm max}.$ In addition, equality
(11) gives that $\phi$ is concave in $I_{\rm max},$ which yields $x_{\rm
max}=+\infty.$

Let us prove that $\phi$ is bounded above. To do so, set

$F(x,\phi,\phi^{\prime}):=-(1+(\phi^{\prime})^{2})\left(\frac{2x}{\phi^{2}}+\frac{1}{x}\right)$

and observe that (11), together with the equality
$(\log(\phi^{\prime}))^{\prime}=\phi^{\prime\prime}/\phi^{\prime},$ yields

(12)
$\phi^{\prime}(x)=\lambda\exp\left(\int_{x_{0}}^{x}F(t,\phi(t),\phi^{\prime}(t))dt\right).$

Clearly, $F(x,\phi(x),\phi^{\prime}(x))<-1/x$ for all $x\in I_{\rm max}.$
Thus,

$0<\phi^{\prime}(x)\leq\lambda\exp\left(\int_{x_{0}}^{x}-\frac{1}{t}dt\right)=\frac{\lambda
x_{0}}{x}\,,$

which implies that

(13) $\lim_{x\to+\infty}\phi^{\prime}(x)=0.$

Now, notice that the equality

$\lim_{x\to+\infty}\frac{x}{\phi(x)}=+\infty$

holds regardless of $\phi$ being bounded or unbounded. Indeed, in the first
case, the equality is trivial, and in the latter case, it follows from (13)
and the l’Hôpital rule. In particular, there exists $x_{1}>x_{0}$ such that
${x^{2}}/{\phi^{2}(x)}>1/2\ \forall x\geq x_{1},$ which yields

$\frac{2x}{\phi^{2}(x)}+\frac{1}{x}>\frac{2}{x}\quad\forall x\in
I_{1}:=[x_{1},+\infty).$

From this last inequality, we have that $F(x,\phi(x),\phi^{\prime}(x))<-2/x$
for all $x\in I_{1}.$ Set

$\log(\varLambda)=\int_{x_{0}}^{x_{1}}F(t,\phi(t),\phi^{\prime}(t))dt.$

Then, considering (12) once more, we obtain, $\forall x\in I_{1}$

$\displaystyle\phi^{\prime}(x)$ $\displaystyle=$
$\displaystyle\lambda\exp\left(\int_{x_{0}}^{x_{1}}F(t,\phi(t),\phi^{\prime}(t))dt+\int_{x_{1}}^{x}F(t,\phi(t),\phi^{\prime}(t))dt\right)$
$\displaystyle\leq$
$\displaystyle\lambda\varLambda\exp\left(\int_{x_{1}}^{x}-\frac{2}{t}dt\right)$
$\displaystyle=$ $\displaystyle\frac{\lambda\varLambda x_{1}^{2}}{x^{2}}\,.$

By integrating both sides on $[x_{1},x]\subset I_{1}$, we finally get

$\phi(x)-\phi(x_{1})\leq\lambda\varLambda{x_{1}}^{2}\left(\frac{1}{x_{1}}-\frac{1}{x}\right)<\lambda\varLambda
x_{1}\quad\forall x\in I_{1},$

which implies that $\phi$ is bounded above. This proves (ii).

To prove (iii), we can argue as in the proof of (ii) to conclude that $\phi$
is decreasing and convex in $I_{\rm max}$ if $\lambda<0.$ We claim that

$\lim_{x\to x_{\rm max}}\phi(x)>0,$

which, by the definition of $\Psi$, implies that $x_{\rm max}=+\infty$.

Assume, by contradiction, that $\lim_{x\to x_{\rm max}}\phi(x)=0.$ Then, one
has

(14) $\lim_{x\to x_{\rm max}}\phi^{\prime\prime}(x)=+\infty.$

Indeed, equality (14) follows directly from (11) if $\lim_{x\to x_{\rm
max}}\phi^{\prime}(x)\neq 0.$ If, instead, $\lim_{x\to x_{\rm
max}}\phi^{\prime}(x)=0,$ then

$\lim_{x\to x_{\rm max}}\frac{\phi^{\prime}(x)}{\phi^{2}(x)}=\lim_{x\to x_{\rm
max}}\frac{\phi^{\prime\prime}(x)}{2\phi(x)\phi^{\prime}(x)}\cdot$

So, if $\phi^{\prime\prime}$ were bounded, the above limit would be infinite.
But then, from (11), we would have $\lim_{x\to x_{\rm
max}}\phi^{\prime\prime}(x)=+\infty,$ which would be a contradiction. Hence,
(14) holds.

Now, we compute $\phi^{\prime\prime\prime}$ from equality (11), obtaining

$\displaystyle\phi^{\prime\prime\prime}$ $\displaystyle=$
$\displaystyle-\phi^{\prime\prime}(1+(\phi^{\prime})^{2})\left(\frac{2x}{\phi^{2}}+\frac{1}{x}\right)-2(\phi^{\prime})^{2}\phi^{\prime\prime}\left(\frac{2x}{\phi^{2}}+\frac{1}{x}\right)$
$\displaystyle-\phi^{\prime}(1+(\phi^{\prime})^{2})\left(\frac{2}{\phi^{2}}-\frac{4x\phi^{\prime}}{\phi^{3}}-\frac{1}{x^{2}}\right).$

Therefore, setting

$\chi:=-\phi^{\prime\prime}(1+(\phi^{\prime})^{2})\frac{1}{x}-2(\phi^{\prime})^{2}\phi^{\prime\prime}\frac{1}{x}+\phi^{\prime}(1+(\phi^{\prime})^{2})\frac{1}{x^{2}},$

we have $\chi<0$ in $I_{\rm max}$ and

(15) $\displaystyle\phi^{\prime\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{2x(1+(\phi^{\prime})^{2})}{\phi^{2}}\left(\frac{2(\phi^{\prime})^{2}}{\phi}-\phi^{\prime\prime}\right)-\frac{2\phi^{\prime}(1+(\phi^{\prime})^{2})}{\phi^{2}}-\frac{4x(\phi^{\prime})^{2}\phi^{\prime\prime}}{\phi^{2}}+\chi$
$\displaystyle=$
$\displaystyle\frac{2x(\phi^{\prime})^{2}(1+(\phi^{\prime})^{2})}{\phi^{3}}\left(2-\frac{\phi\phi^{\prime\prime}}{(\phi^{\prime})^{2}}\right)-\frac{2\phi^{\prime}}{\phi^{2}}(1+(\phi^{\prime})^{2}+2x\phi^{\prime}\phi^{\prime\prime})+\chi.$

However, from (11), one has

$\displaystyle\lim_{x\to x_{\rm
max}}\frac{\phi(x)\phi^{\prime\prime}(x)}{(\phi^{\prime}(x))^{2}}$
$\displaystyle=$ $\displaystyle\lim_{x\to x_{\rm
max}}-\frac{1+(\phi^{\prime}(x))^{2}}{\phi^{\prime}(x)}\left(\frac{2x}{\phi(x)}+\frac{\phi(x)}{x}\right)$
$\displaystyle\geq$ $\displaystyle\lim_{x\to x_{\rm
max}}-\frac{2x}{\phi(x)\phi^{\prime}(x)}=+\infty,$

and

$\displaystyle\lim_{x\to x_{\rm max}}[\phi^{\prime}(x)\phi^{\prime\prime}(x)]$
$\displaystyle=$ $\displaystyle\lim_{x\to x_{\rm
max}}\left[-(\phi^{\prime}(x))^{2}(1+(\phi^{\prime}(x))^{2})\left(\frac{2x}{\phi^{2}(x)}+\frac{1}{x}\right)\right]$
$\displaystyle\leq$ $\displaystyle\lim_{x\to x_{\rm
max}}-\frac{2x(\phi^{\prime}(x))^{2}}{\phi^{2}(x)}=-\infty.$

In the last limit, we used the fact that

$\lim_{x\to x_{\rm max}}\frac{\phi^{\prime}(x)}{\phi(x)}=-\infty,$

which is immediate if $\lim_{x\to x_{\rm max}}\phi^{\prime}(x)<0.$ Otherwise,
it follows easily from the l’Hôpital rule.

From the above limits and (15), we conclude that
$\phi^{\prime\prime\prime}(x)<0$ for all sufficiently large $x\in I_{\rm
max},$ which contradicts (14). This finishes the proof of (iii), and so of the
lemma. ∎

Lemmas 3.6 and 3.7 already imply the existence of rotational translators.
However, to improve the description of these examples, we next consider
rotational surfaces which are also horizontal graphs. More precisely, given a
rotational surface $\varSigma\subset\mathbb{H}^{3}$ with axis
$\ell:=\\{(0,0)\\}\times(0,+\infty)$, let us consider
$\gamma=\varSigma\cap\\{x_{1}=0\\}$ as the profile curve of $\varSigma$ and
assume that the tangent plane of $\varSigma$ at a given point $p\in\gamma$ is
not orthogonal to $\ell$. If we let $d$ denote the Euclidean distance function
from $\gamma$ to $\ell$ on $\mathbb{R}_{+}^{3}$ and let $v$ parameterize
$\gamma$, then, in a neighborhood of $p,$ $\varSigma$ can be parameterized as

$X(u,v):=(u,\sqrt{d^{2}(v)-u^{2}},v),\,\,\,(u,v)\in
U\subset\mathbb{R}\times(0,+\infty).$

We shall call $X(U)$ the _horizontal rotational graph determined by $d$._

######  Lemma 3.8.

A horizontal rotational graph determined by a smooth function $d$ is a
translator to MCF in $\mathbb{H}^{3}$ if and only if the function $d$ is a
solution to the ODE:

$y^{\prime\prime}=\left(\frac{2y^{2}}{x^{2}}+1\right)\frac{1+(y^{\prime})^{2}}{y}\cdot$

In particular, such a solution $d$ is strictly convex.

###### Proof.

Writing $\varphi(u,v):=\sqrt{d^{2}(v)-u^{2}},$ we have that a Euclidean unit
normal to $\varSigma$ is

$\bar{\eta}:=(\bar{\eta}_{1},\bar{\eta}_{2},\bar{\eta}_{3})=\varrho(-\varphi_{u},1,-\varphi_{v}),\quad\varrho:=\frac{1}{\sqrt{1+\varphi_{u}^{2}+\varphi_{v}^{2}}},$

and the corresponding Euclidean mean curvature is

$\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern
1.5mu(X(u,v))=\frac{\varrho^{3}(u,v)}{2}\varLambda(u,v),$

where $\varLambda$ is the function

$\varLambda:=\varphi_{uu}(1+\varphi_{v}^{2})-2\varphi_{uv}\varphi_{u}\varphi_{v}+\varphi_{vv}(1+\varphi_{u}^{2}).$

Hence, the hyperbolic mean curvature $H$ of $\varSigma$ is

(16) $H=v\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern
1.5mu+\bar{\eta}_{3}=\varrho\left(\frac{v\varrho^{2}}{2}\varLambda-\varphi_{v}\right),$

and its hyperbolic unit normal is $\eta:=v\bar{\eta},$ so that

(17) $\langle
X,\eta\rangle=\frac{\varrho}{v}(\varphi-u\varphi_{u}-v\varphi_{v}).$

From (16) and (17), after noticing that $\varphi_{u}=\frac{-u}{\varphi}$, we
have that the translating soliton equation $\langle X,\eta\rangle=H$ for
$\varSigma$ is equivalent to

(18) $\varLambda=\frac{2d^{2}}{v^{2}\varphi\varrho^{2}}\cdot$

After taking all first and second order partial derivatives of $\varphi$ and
applying to $\varLambda,$ we get from a direct and long calculation that

(19)
$\varLambda=\frac{d^{2}}{\varphi^{3}}(dd^{\prime\prime}-(d^{\prime})^{2}-1).$

Finally, observing that

$\frac{\varphi^{2}}{\varrho^{2}}=\varphi^{2}(1+\varphi_{u}^{2}+\varphi_{v}^{2})=\varphi^{2}\frac{u^{2}+(dd^{\prime})^{2}+\varphi^{2}}{\varphi^{2}}=d^{2}(1+(d^{\prime})^{2}),$

it follows from (18) and (19) that

$d^{\prime\prime}=\left(\frac{2d^{2}}{v^{2}}+1\right)\frac{1+(d^{\prime})^{2}}{d}\,,$

as we wished to prove. ∎

Now, we are in position to prove the existence of properly embedded annular
translators to MCF in $\mathbb{H}^{3},$ which we shall call _translating
catenoids_ , see Figure 3.

Figure 3. The profile curve of a translating catenoid $\varSigma_{r}$ in
$\mathbb{H}^{3}$ bounded by (and asymptotic to) two horospheres
$\mathscr{H}_{r^{-}}$ and $\mathscr{H}_{r^{+}}$.
$\varSigma_{r}\setminus\mathscr{H}$ decomposes as two vertical graphs
$\varSigma_{r}^{-}$ and $\varSigma_{r}^{+}$ over the complement of the
Euclidean disk of radius $r$ centered at the rotation axis $x_{3}$ in the
horosphere $\mathscr{H}$.

######  Theorem 3.9.

There exists a one-parameter family $\mathscr{C}:=\\{\varSigma_{r}\mid r>0\\}$
of noncongruent, properly embedded rotational annular translators in
$\mathbb{H}^{3}$ (to be called translating catenoids). For each $r>0$, the
surface $\varSigma_{r}\in\mathscr{C}$ satisfies:

  * i)

$\varSigma_{r}$ is contained in a slab determined by two horospheres
$\mathscr{H}_{r^{-}}$ and $\mathscr{H}_{r^{+}}.$ In particular, the asymptotic
boundary of $\varSigma_{r}$ is the point at infinity of the horosphere
$\mathscr{H}$ at height $1$.

  * ii)

$\varSigma_{r}$ is the union of two vertical graphs $\varSigma_{r}^{-}$ and
$\varSigma_{r}^{+}$ over the complement of the Euclidean $r$-disk
$\mathcal{D}_{r}$ centered at the rotation axis in the horosphere
$\mathscr{H}$.

  * iii)

The graphs $\varSigma_{r}^{-}$ and $\varSigma_{r}^{+}$ lie in distinct
connected components of $\mathbb{H}^{3}-\mathscr{H}$ with common boundary the
$r$-circle that bounds $\mathcal{D}_{r}$ in $\mathscr{H},$ being
$\varSigma_{r}^{-}$ asymptotic to $\mathscr{H}_{r^{-}}$ and
$\varSigma_{r}^{+}$ asymptotic to $\mathscr{H}_{r^{+}}.$

In addition, the limiting behaviour of $\varSigma_{r}$ is as follows:

  * iv)

As $r\to 0,$ $\varSigma_{r}$ converges (on the $C^{2,\alpha}$-norm, on compact
sets outside $(0,0,1)$) to a double copy of $\mathscr{H}.$

  * v)

As $r\to+\infty,$ the distance between $\varSigma_{r}$ and the $x_{3}$-axis
becomes unbounded; in particular, $\varSigma_{r}$ escapes to infinity.

###### Proof.

Given $r>0,$ let $d_{r}:(1-\delta,1+\delta)\rightarrow(0,+\infty)$ be the
local solution to the following initial value problem:

(20)
$\left\\{\begin{array}[]{l}y^{\prime\prime}=\left(\frac{2y^{2}}{x^{2}}+1\right)\frac{1+(y^{\prime})^{2}}{y}\,,\\\\[4.30554pt]
y(1)=r,\\\\[4.30554pt] y^{\prime}(1)=0.\end{array}\right.$

By Lemma 3.8, the rotational horizontal graph $\varSigma_{r}$ determined by
$d_{r}$ is a translator to MCF in $\mathbb{H}^{3}.$ Since $d_{r}$ is strictly
convex, $x=1$ is a strict local minimum of $d_{r}$ and
$\varSigma_{r}-\mathscr{H}$ is the union of two disjoint rotational vertical
graphs $\varSigma_{r}^{-}$ and $\varSigma_{r}^{+}$ over an open set contained
in $\mathscr{H}-\mathcal{D}_{r}.$ Let us index $\varSigma_{r}^{+}$ as being
the component contained in the horoball $\\{x_{3}>1\\}$. Then, Lemma 3.6
applies to $\varSigma_{r}^{+}$, which corresponds to an increasing solution of
(10) (i.e., one for which the initial condition $\lambda$ is positive). By
Lemma 3.7, such a solution is defined in an interval $[x_{0},+\infty)$ and is
bounded above. Therefore, $\varSigma_{r}^{+}$ can be continued indefinitely,
being asymptotic to a horosphere $\mathscr{H}_{r^{+}}$ of $\mathbb{H}^{3}.$ In
particular, $\varSigma_{r}^{+}$ is a graph over $\mathscr{H}-\mathcal{D}_{r}.$

Analogously, Lemmas 3.6 and 3.7 give that $\varSigma_{r}^{-}$ can be continued
indefinitely and is asymptotic to a horosphere $\mathscr{H}_{r^{-}}$ of
$\mathbb{H}^{3}.$ Since $\varSigma_{r}={\rm
closure}(\varSigma_{r}^{-})\cup{\rm closure}(\varSigma_{r}^{+}),$ we have that
$\varSigma_{r}$ is an annular properly embedded translator to MCF in
$\mathbb{H}^{3}.$ This proves assertions (i)–(iii).

In the proof of assertions (iv) and (v), we will use the rotational symmetry
of $\varSigma_{r}$ to work uniquely with the curves $G_{r}$, defined as the
graphs of the solutions $d_{r}$ of (20), parameterized by

$\alpha_{r}(s):=(0,d_{r}(s),s),\,\,\,s\in(r^{-},r^{+}).$

Furthermore, let $G_{r}^{+}=\\{\alpha_{r}(s)\mid s\in[1,r^{+})\\}$ and
$G_{r}^{-}=\\{\alpha_{r}(s)\mid s\in(r^{-},1])\\}$ be the respective
intersections of $G_{r}$ with $\\{x_{3}>1\\}$ and $\\{x_{3}<1\\}$, and set
$p_{r}=\alpha_{r}(1)=(0,r,1)$. We get directly from (20) that, with the
induced Euclidean metric, the curvature of $G_{r}$ at $p_{r}$ is
$k_{r}(1)=d_{r}^{\prime\prime}(1)=2r+1/r$, which gives (see Fig. 4):

(21) ${\lim_{r\to 0}k_{r}(1)=\lim_{r\to+\infty}k_{r}(1)=+\infty.}$

Figure 4. In the proof of Theorem 3.9, the curvature of the horizontal graph
$G_{r}$ at the point of minimal Euclidean distance to the $x_{3}$-axis goes to
infinity as $r$ goes to either $0$ or $+\infty.$

Now, set $r_{*}$ for either $0$ or $+\infty$. Then, equation (21), together
with the fact that $d_{r}$ is strictly convex, increasing in $[1,r^{+})$ and
decreasing in $(r^{-},1]$ allows us to choose points
$p_{r}^{-}:=\alpha_{r}(s_{r}^{-})\in G_{r}^{-}$ and
$p_{r}^{+}:=\alpha_{r}(s_{r}^{+})\in G_{r}^{+}$ satisfying (with respect to
the Euclidean norm $\|\cdot\|$)

(22) $\lim_{r\to r_{*}}\|p_{r}^{-}-p_{r}\|=\lim_{r\to
r_{*}}\|p_{r}^{+}-p_{r}\|=0,$

with unit tangent vectors so that

(23) $\lim_{r\to
r_{*}}\frac{\alpha_{r}^{\prime}(s_{r}^{+})}{\|\alpha_{r}^{\prime}(s_{r}^{+})\|}=-\lim_{r\to
r_{*}}\frac{\alpha_{r}^{\prime}(s_{r}^{-})}{\|\alpha_{r}^{\prime}(s_{r}^{-})\|}=\partial_{x_{2}}.$

Let $\ell_{r}^{+}$ and $\ell_{r}^{-}$ denote the tangent lines to $G_{r}$ at
the points $p_{r}^{+}$ and $p_{r}^{-}$, respectively. Then, by convexity, it
follows that $G_{r}$ lies in the region limited between $\ell_{r}^{+}$ and
$\ell_{r}^{-}$ that contains $p_{r}$. Since (22) implies that

$\lim_{r\to r_{*}}s_{r}^{-}=\lim_{r\to r_{*}}s_{r}^{+}=1,$

it follows from (23) that both $\ell_{r}^{+}$ and $\ell_{r}^{-}$ converge to
the horizontal line $\\{(0,s,1)\mid s\in\mathbb{R}\\}$, when $r\to r_{*}$,
which immediately proves item (iv).

To prove item (v), we use that equidistant curves to the $x_{3}$-axis in the
$x_{2}x_{3}$ halfplane are half-lines issuing from $(0,0,0)$; our next
argument is to prove that there is no such curve that intersects $G_{r}$ for
any $r>0$ arbitrarily large. Fix some $\lambda>0$ and consider the half-line
$\ell=\\{(0,x_{2},\lambda x_{2})\mid x_{2}>0\\}$. Let $r_{1}>0$ be such that
the slope of $\ell_{r}^{+}$ is less then $\lambda$ for all $r>r_{1}$ and let
$r_{0}={\rm max}\\{r_{1},1/\lambda\\}$ (see Fig. 5). Then, if $r>r_{0}$, in
the region $\\{x_{2}\geq r\\}$, the line $\ell$ is contained in
$\\{x_{3}>1\\}$ and the line $\ell_{r}^{+}$ is below $\ell$. Since
$G_{r}\subset\\{x_{2}\geq r\\}$, staying in one side of $\ell_{r}^{+}$, this
yields $G_{r}\cap\ell=\emptyset$ for all $r>r_{0}$, as we wished to prove. ∎

Figure 5. In the proof of Theorem 3.9, the equidistant halfline $\ell$ has no
intersection with $G_{r}$ if $r>0$ is sufficiently large.

Let $\varSigma$ be a connected rotational translator in $\mathbb{H}^{3}$ with
(possibly empty) boundary. If $\varSigma$ is not a horosphere, Lemmas 3.6 and
3.8, together with the uniqueness of solutions of ODE’s with given initial
conditions, imply that the profile curve of $\varSigma$ coincides, up to its
boundary, with the profile curve of some translating catenoid $\varSigma_{r}$
obtained in Theorem 3.9. Therefore, we have the following uniqueness result.

######  Theorem 3.10.

Any connected rotational translator of $\mathbb{H}^{3}$ is either an open
subset of a horosphere or of some translating catenoid.

###  3.2. Parabolic translators.

Having considered rotational translators in the previous section, we now look
at translators which are invariant by a 1-parameter group of parabolic
isometries of $\mathbb{H}^{3}$, i.e., isometries of $\mathbb{H}^{3}$ that fix
parallel families of horospheres. Horizontal cylinders over curves on vertical
totally geodesic planes of $\mathbb{H}^{3}$ (to be called _parabolic
cylinders_) are the simplest examples of surfaces which are invariant by
parabolic translations. When these generating curves are graphs on the whole
of $\mathbb{R},$ such a surface can be parameterized by a map
$X\colon\mathbb{R}^{2}\to\mathbb{R}_{+}^{3}$ defined by

$X(u,v)=(u,v,\phi(v)),\,\,(u,v)\in\mathbb{R}^{2},$

where $\phi$ is a smooth positive function on $\mathbb{R}.$ We shall call
$\varSigma:=X(\mathbb{R}^{2})$ the _parabolic cylinder determined by_ $\phi.$

Defining $\varrho(v):=(1+(\phi^{\prime}(v))^{2})^{-1/2},$ we have that

$\bar{\eta}:=\varrho(0,-\phi^{\prime},1)$

is a unit normal to $\varSigma$ with respect to the induced Euclidean metric
of $\mathbb{R}_{+}^{3}.$ With this orientation, the Euclidean mean curvature
$\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu$ of $\varSigma$
is

$\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern
1.5mu=\frac{\varrho^{3}\phi^{\prime\prime}}{2}\,\cdot$

From this last equality and (1), we have that the hyperbolic mean curvature
$H$ of $\varSigma$ with respect to the orientation $\eta:=\phi\bar{\eta}$ is

$H=\varrho\left(\frac{\varrho^{2}\phi\phi^{\prime\prime}}{2}+1\right).$

Since $\langle\eta,X\rangle=\varrho(\phi-v\phi^{\prime})/\phi,$ we also have
that the identity (4) for the parabolic cylinder $\varSigma=X(\mathbb{R}^{2})$
is equivalent to the following second order ODE:

$\phi^{\prime\prime}=-\phi^{\prime}(1+(\phi^{\prime})^{2})\frac{2v}{\phi^{2}}\cdot$

The above considerations yield

######  Lemma 3.11.

A parabolic cylinder determined by a smooth function $\phi$ is a translator to
MCF in $\mathbb{H}^{3}$ if and only if $\phi$ is a solution to the second
order ODE:

(24) $y^{\prime\prime}=-y^{\prime}(1+(y^{\prime})^{2})\frac{2x}{y^{2}}\cdot$

The solutions of (24) are all increasing on $\mathbb{R}$ and their graphs are
“S-shaped”, as attested by the following

######  Lemma 3.12.

Given $\lambda\geq 0,$ the initial value problem

(25)
$\left\\{\begin{array}[]{l}y^{\prime\prime}=-y^{\prime}(1+(y^{\prime})^{2})\frac{2x}{y^{2}}\\\\[4.30554pt]
y(0)=1\\\\[4.30554pt] y^{\prime}(0)=\lambda\end{array}\right.$

has a unique smooth solution $\phi:\mathbb{R}\to\mathbb{R}$ which has the
following properties:

  * i)

$\phi$ is constant if $\lambda=0.$

  * ii)

$\phi$ is increasing, convex in $(-\infty,0),$ and concave in $(0,+\infty)$ if
$\lambda>0.$

  * iii)

$\phi$ is bounded above and below by positive constants.

###### Proof.

Assertion (i) is immediate. So, assume $\lambda>0.$ Proceeding as in the proof
of Lemma 3.7, we get from the equality

(26)
$\phi^{\prime\prime}=-\phi^{\prime}(1+(\phi^{\prime})^{2})\frac{2x}{\phi^{2}}$

that $\phi$ necessarily satisfies:

$\phi^{\prime}(x)=\lambda\exp\left(\int_{0}^{x}F(t,\phi(t),\phi^{\prime}(t))\right)dt,$

where $F$ is given by

$F(x,y,z):=-\frac{2x}{y^{2}}(1+z^{2}).$

Hence, $\phi$ is increasing. This, together with (26), implies that $\phi$ is
defined in a maximal interval $I_{\rm max}:=(x_{\rm min},+\infty)$ with
$-\infty\leq x_{\rm min}<0.$ It also follows from (26) that $\phi$ is convex
in $(x_{\rm min},0),$ and concave in $(0,+\infty).$

Next, we prove that the solution $\phi$ is bounded above. Since $\phi$ is
concave in $(0,+\infty),$ we have that
$\lambda=\phi^{\prime}(0)>\phi^{\prime}(x)$ for all $x>0.$ Then, integration
on both sides of this last inequality yields

$\phi(x)\leq\lambda x+1\,\,\,\forall x\geq 0,$

which implies that

(27)
$\lim_{x\to+\infty}\frac{x}{\phi(x)}\geq\lim_{x\to+\infty}\frac{x}{\lambda
x+1}=\frac{1}{\lambda}>0.$

Now, choose a small $\epsilon>0$ such that $C:=\lambda^{-1}-\epsilon$ is
positive. It follows from (27) that, for a sufficiently large $x_{1}>0,$ one
has $x/\phi(x)>C$ for all $x\geq x_{1},$ so that

$\frac{2x}{\phi^{2}(x)}\geq\frac{2C^{2}}{x},\,\,\,\,\forall x\geq x_{1},$

from which we obtain

(28) $F(x,\phi(x),\phi^{\prime}(x))\leq-\frac{2C^{2}}{x},\,\,\,\forall x\geq
x_{1}.$

Now, for any given $x_{0}>0$, consider the following initial value problem:

(29)
$\left\\{\begin{array}[]{l}y^{\prime\prime}=-y^{\prime}(1+(y^{\prime})^{2})\frac{2x}{y^{2}}\\\\[4.30554pt]
y(x_{0})=\phi(x_{0})\\\\[4.30554pt]
y^{\prime}(x_{0})=\phi^{\prime}(x_{0}).\end{array}\right.$

By uniqueness, $\phi$ is a solution to (29), and once again we may write

(30)
$\phi^{\prime}(x)=\lambda_{0}\exp\left(\int_{x_{0}}^{x}F(t,\phi(t),\phi^{\prime}(t))\right)dt,\,\,\,\,\forall
x\geq x_{0},$

where $\lambda_{0}:=\phi^{\prime}(x_{0})>0.$ In particular, defining
$\lambda_{1}=\phi^{\prime}(x_{1})$, we obtain

(31)
$\phi^{\prime}(x)=\lambda_{1}\exp\left(\int_{x_{1}}^{x}F(t,\phi(t),\phi^{\prime}(t))\right)dt,\,\,\,\,\forall
x\geq x_{1}$

and it follows from (28) and (31) that

$\phi^{\prime}(x)\leq\lambda_{1}\left(\frac{x_{1}}{x}\right)^{2C^{2}},\,\,\,\forall
x\geq x_{1},$

so that $\phi^{\prime}(x)\to 0$ as $x\to+\infty.$ Therefore, we have

$\lim_{x\to+\infty}\frac{x}{\phi(x)}=+\infty.$

In particular, there exists $x_{2}\geq x_{1}$ such that the inequality

$\frac{2x}{\phi^{2}(x)}\geq\frac{2}{x}$

holds for all $x\geq x_{2},$ which gives that
$F(x,\phi(x),\phi^{\prime}(x))<-2/x$ for all $x\geq x_{2}.$ Therefore,
applying (30) for $x_{2}$,

$\phi^{\prime}(x)=\lambda_{2}\exp\left(\int_{x_{2}}^{x}F(t,\phi(t),\phi^{\prime}(t))\right)dt,\,\,\,\lambda_{2}=\phi^{\prime}(x_{2})$

and we may proceed just as in the proof of Lemma 3.7 to conclude that

$\phi(x)-\phi(x_{2})\leq\lambda_{2}x_{2}^{2}\left(\frac{1}{x_{2}}-\frac{1}{x}\right)<\lambda_{2}x_{2},\quad\forall
x\geq x_{2},$

which implies that $\phi$ is bounded above.

Next, we show that $\phi$ is bounded below by a positive constant. With this
purpose, assume by contradiction that

(32) $\lim_{x\to x_{\rm min}}\phi(x)=0.$

Reasoning as in the proof of item iii of Lemma 3.7, we obtain from this
assumption that

(33) $\lim_{x\to x_{\rm min}}\phi^{\prime\prime}(x)=+\infty,$

which, in turn, implies that

(34) $\lim_{x\to x_{\rm min}}\frac{\phi^{\prime}(x)}{\phi(x)}=+\infty.$

Equality (33) gives that $\phi^{\prime\prime}$ is necessarily decreasing in a
neighborhood of any sufficiently small $x\in(x_{\rm min},0).$ However, by
computing $\phi^{\prime\prime\prime}$ from (26), we get

(35)
$\phi^{\prime\prime\prime}=-\phi^{\prime\prime}(1+(\phi^{\prime})^{2})\frac{2x}{\phi^{2}}-4(\phi^{\prime})^{2}\phi^{\prime\prime}\frac{x}{\phi^{2}}-\phi^{\prime}(1+(\phi^{\prime})^{2})\left(\frac{2}{\phi^{2}}-\frac{4x\phi^{\prime}}{\phi^{3}}\right).$

Then, considering the equality
$-\phi^{\prime}(1+(\phi^{\prime})^{2})=\phi^{2}\phi^{\prime\prime}/(2x),$
which we get from (26), and applying it to the last summand of (35), we obtain

$\phi^{\prime\prime\prime}=\phi^{\prime\prime}\left[-\frac{2x}{\phi^{2}}(1+3(\phi^{\prime})^{2})+\frac{1}{x}-\frac{2\phi^{\prime}}{\phi}\right]=\phi^{\prime\prime}\left[-\frac{2x}{\phi^{2}}-2\frac{\phi^{\prime}}{\phi}\left(1+3x\frac{\phi^{\prime}}{\phi}\right)+\frac{1}{x}\right].$

This last equality, together with equations (32)–(34), clearly yields

$\lim_{x\to x_{\rm min}}\phi^{\prime\prime\prime}(x)=+\infty,$

which contradicts (33). Therefore, we have

$\lim_{x\to x_{\rm min}}\phi(x)>0,$

from which we conclude that $\phi$ is bounded from below by a positive
constant. In particular, we must have $x_{\min}=-\infty.$ This finishes the
proof. ∎

Lemmas 3.11 and 3.12 immediately give the following result (see Fig. 6).

Figure 6. The profile curve of a hyperbolic grim reaper in $\mathbb{H}^{3}$
lying between two horospheres $\mathscr{H}_{\lambda^{-}}$ and
$\mathscr{H}_{\lambda^{+}}$.

######  Theorem 3.13.

In hyperbolic space $\mathbb{H}^{3},$ there exists a one-parameter family

$\mathscr{G}:=\\{\varSigma_{\lambda}\mid\lambda\in[0,+\infty)\\}$

of noncongruent, complete translators (to be called _grim reapers_) which are
horizontal parabolic cylinders generated by the solutions of (25).
$\varSigma_{0}$ is the horosphere $\mathscr{H}\subset\mathbb{H}^{3}$ at height
one, and for $\lambda>0,$ each $\varSigma_{\lambda}\in\mathscr{G}$ is an
entire graph over $\mathbb{R}^{2}$ which is contained in a slab determined by
two horospheres $\mathscr{H}_{\lambda^{-}}$ and $\mathscr{H}_{\lambda^{+}}.$
Furthermore, there exist open sets $\varSigma_{\lambda}^{-}$ and
$\varSigma_{\lambda}^{+}$ of $\varSigma_{\lambda}$ such that
$\varSigma_{\lambda}^{-}$ is asymptotic to $\mathscr{H}_{\lambda^{-}}$,
$\varSigma_{\lambda}^{+}$ is asymptotic to $\mathscr{H}_{\lambda^{+}},$ and
$\varSigma_{\lambda}={\rm closure}\,(\varSigma_{\lambda}^{-})\cup{\rm
closure}\,(\varSigma_{\lambda}^{+}).$

If $\Gamma\subset\mathbb{R}^{2}$ is the graph of the function
$t\in(-\pi/2,\pi/2)\mapsto-\log(\cos t)$, then the cylinder
$\varSigma=\Gamma\times\mathbb{R}\subset\mathbb{R}^{3}$ is a translator to MCF
contained in a slab $\mathcal{S}$ of $\mathbb{R}^{3}$, known as the _grim
reaper cylinder_. This nomenclature is due to the fact that the curve $\Gamma$
provides a solution to the curve shortening flow, called _the grim reaper_ ,
which is given by the translation of $\Gamma$ in $\mathbb{R}^{2}$ in the
$\vec{e}_{2}$-direction. By the avoidance principle, such a solution “kills”
any other solution in the region $(-\pi/2,\pi/2)\times\mathbb{R}$ (see [2,
Chapter 2]). Similarly, two surfaces (one of them compact) in $\mathbb{R}^{3}$
moving under MCF which are initially disjoint remain so until one of them
collapses. Hence, as $\varSigma$ translates under MCF, it “kills” all
solutions to (2) in $\mathcal{S}$ with compact initial condition. An analogous
process occurs in our case: any surface of the family $\mathscr{G}$ in Theorem
3.13 has this “killing” property. Indeed, by [7, Theorem 4], the avoidance
principle applies to surfaces moving under MCF in $\mathbb{H}^{3}.$ For this
reason, we named the elements of $\mathscr{G}$ grim reapers.

######  Remark 3.14.

The symmetry in (24) allows us to extend the family $\mathscr{G}$ in Theorem
3.13 for values $\lambda<0$ by simply defining $\psi(x)=\phi(-x)$ for a given
solution $\phi$ to (25) with positive initial data for $\phi^{\prime}$.
However, the translator generated by $\psi$ correspond to a rotation of $\pi$
around the $x_{3}$-axis of a grim reaper, being therefore congruent to it.

######  Remark 3.15.

At the completion of this manuscript, we became acquainted with the preprint
[17], in which the authors consider solitons to MCF generated by conformal
fields in $\mathbb{H}^{n}$, called _conformal solitons_. There, they obtained
rotational and cylindrical conformal solitons whose initial conditions are
named winglike catenoids and grim reaper cylinder, respectively. However, such
solitons are not related to the ones considered here, since their generating
fields are not Killing.

Analogously to the rotational case, the uniqueness of solutions of ODE’s with
given initial conditions yields the following result.

######  Theorem 3.16.

Any connected translator in $\mathbb{H}^{3}$ which is a parabolic cylinder is,
up to an ambient isometry (see Remark 3.14), an open subset of a grim reaper
or of a totally geodesic plane containing the $x_{3}$-axis.

###  3.3. The tangency principle and applications.

A distinguished property of translators to MCF in $\mathbb{R}^{3}$ is that
they are critical points of a weighted area functional and, therefore, they
become minimal surfaces when changing the ambient metric in a suitable manner
[14]. In particular, the tangency principle applies to them, which allows one
to use translators as barriers (cf. [16]). On the other hand, it is unknown to
us if translators to MCF in $\mathbb{H}^{3}$ can be made minimal in a similar
fashion. Nevertheless, as we establish in the next result, the tangency
principle holds for translators in $\mathbb{H}^{3}$.

######  Theorem 3.17 (tangency principle for translators).

Let $\varSigma_{1}$ and $\varSigma_{2}$ be two translators to MCF in
$\mathbb{H}^{3}$ which are tangent at a point $p\in{\rm
int}\,\varSigma_{1}\cap{\rm int}\,\varSigma_{2}.$ If $\varSigma_{1}$ lies on
one side of $\varSigma_{2}$ in a neighborhood of $p$ in $\mathbb{H}^{3},$ then
$\varSigma_{1}$ and $\varSigma_{2}$ coincide in a neighborhood of $p$ in
$\varSigma_{1}\cap\varSigma_{2}.$ Moreover, if $\varSigma_{1}$ and
$\varSigma_{2}$ are both complete and connected, then
$\varSigma_{1}=\varSigma_{2}.$

###### Proof.

Let $\varSigma_{1}$ and $\varSigma_{2}$ be two translators to MCF in
$\mathbb{H}^{3}$, tangent at a point $p\in\varSigma_{1}\cap\varSigma_{2}$, and
such that $\varSigma_{1}$ stays locally on one side of $\varSigma_{2}$. If
$T_{p}\varSigma_{1}$ is not vertical, there exists a domain
$\Omega\subset\mathbb{R}^{2}$ and positive functions
$u_{1},\,u_{2}\colon\Omega\to\mathbb{R}$ such that neighborhoods
$U_{1}\subset\varSigma_{1}$ and $U_{2}\subset\varSigma_{2}$ containing $p$ are
respectively parameterized by

$U_{1}=\\{(x,y,u_{1}(x,y))\mid(x,y)\in\Omega\\},\quad
U_{2}=\\{(x,y,u_{2}(x,y))\mid(x,y)\in\Omega\\}.$

Furthermore, after reindexing we may assume that $u_{1}\geq u_{2}$ in
$\Omega$.

Let $\varSigma_{1}$ and $\varSigma_{2}$ be oriented with respect to vector
fields $\eta_{1}$ and $\eta_{2}$ so that $\eta_{1}(p)=\eta_{2}(p)$ points
upwards. Thus, if $Q$ is the quasilinear elliptic operator

(36) $Q(u)=u_{xx}(1+u_{y}^{2})+u_{yy}(1+u_{x}^{2})-2u_{xy}u_{x}u_{y},$

it follows from (1) that the mean curvature functions $H_{1},\,H_{2}$ of
$U_{1}$ and $U_{2}$ satisfy

$H_{i}=u_{i}\frac{Q(u_{i})}{2(1+(u_{i})_{x}^{2}+(u_{i})_{y}^{2})^{\frac{3}{2}}}+\frac{1}{(1+(u_{i})_{x}^{2}+(u_{i})_{y}^{2})^{\frac{1}{2}}},\quad
i\in\\{1,2\\}.$

Then, after setting $B(x,y,u,Du)=2(1+u_{x}^{2}+u_{y}^{2})(xu_{x}+yu_{y})$,
where $Du$ denotes the (Euclidean) gradient of $u$, it follows from (4) that

(37) $(u_{i})^{2}Q(u_{i})+B(x,y,u_{i},Du_{i})=0,\quad i\in\\{1,2\\}.$

But the operator $u^{2}Q(u)+B(x,y,u,Du)$ in (37) satisfies the hypothesis of
the tangency principle for quasilinear operators [19, Theorem 2.2.2], thus
$u_{1}=u_{2}$, which implies $U_{1}=U_{2}$.

The case where $T_{p}\varSigma_{1}$ is vertical can be treated analogously:
after a rotation about the $x_{3}$-axis (which preserves the property of being
a translator to MCF), locally, both $\varSigma_{1}$ and $\varSigma_{2}$ can be
parameterized as horizontal graphs

$\\{(x,u_{1}(x,z),z)\mid(x,z)\in\widehat{\Omega}\\}\,\,\,\text{and}\,\,\,\\{(x,u_{2}(x,z),z)\mid(x,z)\in\widehat{\Omega}\\}$

for some domain $\widehat{\Omega}\subset\mathbb{R}^{2}_{+}$, and both
$u_{1},\,u_{2}$ satisfy

$z^{2}Q(u)+\widehat{B}(x,z,u,Du)=0$

for $\widehat{B}(x,z,u,Du)=2(xu_{x}-u)(1+u_{x}^{2}+u_{z}^{2})$ and $Q$ as in
(36). Once again, we obtain from [19, Theorem 2.22] that $\varSigma_{1}$ and
$\varSigma_{2}$ coincide in a neighborhood of $p$.

At this point, we have shown that if $\varSigma_{1}$ and $\varSigma_{2}$ are
tangent at a point $p$, they must coincide in neighborhoods which are either
horizontal or vertical graphs for $\varSigma_{1}$ and $\varSigma_{2}$. The
proof for the case where $\varSigma_{1}$ and $\varSigma_{2}$ are complete and
connected now follows from covering $\varSigma_{1}$ and $\varSigma_{2}$ with
such (overlapping) neighborhoods. ∎

######  Remark 3.18.

Theorem 3.17 contrasts with the tangency principle for the constant mean
curvature case (see, for instance, [15, Theorem 3.2.4]): two distinct geodesic
spheres in $\mathbb{R}^{3}$ with the same mean curvature can be tangent to
each other without violating the tangency principle. In the setting of
translators, the tangency principle does not require any assumptions on the
orientation of $\varSigma_{1}$ and $\varSigma_{2}$ because, from (4), if
$\varSigma_{1}$ and $\varSigma_{2}$ are translators to MCF which are tangent
at a point $p$, then necessarily their mean curvature vectors $\mathbf{H}_{1}$
and $\mathbf{H}_{2}$ must agree at $p$, which defines a coinciding, standard
(local) orientation for both $\varSigma_{1}$ and $\varSigma_{2}$.

On the remainder of the section we will apply the tangency principle to
establish some classification results concerning translators in
$\mathbb{H}^{3}$. First, we show that properly immersed translators in
$\mathbb{H}^{3}$ are never cylindrically bounded, and next we prove that any
horoconvex translator which is complete or transversal to the $x_{3}$-axis is
necessarily an open set of a horizontal horosphere.

Recall that a circular cone in
$\mathbb{R}_{+}^{3}:=\mathbb{R}^{2}\times(0,+\infty)$ with vertex at
$p\in\mathbb{R}^{2}$ and axis $\gamma_{p}:=\\{p\\}\times(0,+\infty)$
constitutes a _cylinder_ $\mathscr{C}$ in $\mathbb{H}^{3},$ that is, the set
of points of $\mathbb{H}^{3}$ at a fixed distance to the vertical geodesic
$\gamma_{p}.$ The convex side of $\mathscr{C}$ is the component of
$\mathbb{H}^{3}-\mathscr{C}$ which contains $\gamma_{p}.$

######  Theorem 3.19.

There is no properly immersed translator to MCF in $\mathbb{H}^{3}$ which is
contained in the convex side of a cylinder with vertex at $p=(0,0).$ In
particular, there is no closed (i.e., compact without boundary) translator to
MCF in $\mathbb{H}^{3}$.

###### Proof.

Suppose, by contradiction, that there exists a properly immersed translator
$\varSigma$ to MCF in $\mathbb{H}^{3}$ which is contained in the convex side
$\Omega$ of a cylinder $\mathcal{C}$ with vertex at $p=(0,0).$ Clearly, the
property of being a translator is invariant by the translations
$\Gamma_{t}(p):=e^{t}p,\,t\in\mathbb{R}.$ Therefore, we can assume without
loss of generality that $\varSigma$ intersects the horosphere $\mathscr{H}$ of
height $1.$

Under the above conditions, we have from item (v) of Theorem 3.9 that there
exists $R>0$ such that, for any $r>R,$ the translating catenoid
$\varSigma_{r}$ of the family $\mathscr{C}$ is disjoint from $\mathcal{C},$
and so from $\varSigma.$ On the other hand, for a sufficiently small $r>0,$
$\varSigma_{r}$ and $\varSigma$ have nonempty intersection. Taking into
account the asymptotic behavior of $\varSigma_{r}$, together with the
hypothesis that $\varSigma$ is contained in $\Omega,$, as $r$ decreases from
$R$ to zero, a standard argument shows that there will be a first value
$r_{*}$ such that $\varSigma_{r_{*}}$ is the element of $\mathscr{C}$ that
first establishes a contact with $\varSigma$ at a point
$p\in\varSigma\cap\varSigma_{r_{*}}$, as in Figure 7. Then, $\varSigma$ and
$\varSigma_{r_{*}}$ are tangent at $p$ with $\varSigma$ on one side of
$\varSigma_{r}$, and the tangency principle (Theorem 3.17) applies to show
that $\varSigma=\varSigma_{r_{*}},$ which is a contradiction, since
$\varSigma$ is contained in $\Omega$ and $\varSigma_{r_{*}}$ is not. ∎

Figure 7. In the proof of Theorem 3.19, there exists a smallest $r_{*}>0$ such
that $\varSigma_{r_{*}}$ intersects $\varSigma$ tangentially, with $\varSigma$
in one of the two regions of $\mathbb{H}^{3}$ defined by $\varSigma_{r_{*}}$.

An oriented surface $\varSigma$ of $\mathbb{H}^{3}$ is called _horoconvex_ if
its principal curvatures $k_{1},k_{2}$ satisfy $k_{i}\geq 1$, $i=1,2.$ It is
well known that every point $p$ of a horoconvex surface
$\varSigma\subset\mathbb{H}^{3}$ is locally supported by a horosphere
$\mathcal{H}$, meaning that $\mathcal{H}$ is tangent to $\varSigma$ at $p$,
and there exists a neighborhood of $p$ in $\varSigma$ which lies in the
horoball bounded by $\mathcal{H}$ (see, e.g., [4, Lemma 1]). Since the
tangency principle immediately implies that no translator can attain a local
maximum or local minimum for its height function, we obtain that _any
horoconvex translator whose tangent plane is horizontal at one of its points
is necessarily an open set of a horizontal horosphere_. With these
considerations, we are in position to establish the following uniqueness
result.

######  Theorem 3.20.

Let $\varSigma$ be a horoconvex translator in $\mathbb{H}^{3}$ and assume that
one of the following occurs:

  1. i)

$\varSigma$ is complete.

  2. ii)

$\varSigma$ intersects the $x_{3}$-axis transversely.

Then, $\varSigma$ is an open set of a horizontal horosphere.

###### Proof.

Let $\varSigma$ be as stated. By the comments above, in order to prove that
$\varSigma$ is an open set of a horizontal horosphere, it suffices to show
that the tangent plane of $\varSigma$ is horizontal at one of its points.

First, assume that item i) holds, so Theorem 3.19 implies that $\varSigma$ is
noncompact. The main results in [4] show that horospheres are the unique
complete and noncompact horoconvex surfaces of $\mathbb{H}^{3}$, so
$\varSigma$ is a horosphere. In particular, there is a point $p\in\varSigma$
where $T_{p}\varSigma$ is horizontal, so $\varSigma$ is a horizontal
horosphere.

Assume now that $\varSigma$ intersects the $x_{3}$-axis transversally at a
point. Then, there is an open neighborhood $U$ of $\varSigma$ which is a graph
of a smooth function $u$ over an open ball $\Omega\subset\mathbb{R}^{2}$
centered at the origin $(0,0)$. Since $\varSigma$ is a translator, $u$
satisfies (see the proof of Theorem 3.17),

(38) $u^{2}Q(u)+B(x,y,u,Du)=0,$

where $Q$ is defined by (36) and
$B(x,y,u,Du)=2(1+u_{x}^{2}+u_{y}^{2})(xu_{x}+yu_{y})$. In addition, it follows
from the horoconvexity of $\varSigma$ that its mean curvature $\mkern
1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu$ with respect to the
induced Euclidean metric cannot change sign, being either everywhere
nonpositive or everywhere nonnegative, which, when orienting $U$ with respect
to the upwards pointing normal, correspond to the respective cases $Q(u)\geq
0$ or $Q(u)\leq 0$.

First, assume that $Q(u)\geq 0$. Then, it follows from (38) that

(39) $xu_{x}(x,y)+yu_{y}(x,y)\leq 0\,\,\,\forall(x,y)\in\Omega.$

Let $\theta\in[0,\pi]$ be given and consider
$u_{\theta}(t)=u(t\cos(\theta),t\sin(\theta))$, the restriction of $u$ to the
line through the origin $L_{\theta}=\\{(t\cos(\theta),t\sin(\theta))\mid
t\in\mathbb{R}\\}\cap\Omega$. Then, (39) implies that

$tu_{\theta}^{\prime}(t)=t\cos(\theta)u_{x}(t\cos(\theta),t\sin(\theta))+t\sin(\theta)u_{y}(t\cos(\theta),t\sin(\theta))\leq
0.$

In particular, $u_{\theta}^{\prime}(t)\geq 0$ when $t<0$ and
$u_{\theta}^{\prime}(t)\leq 0$ when $t>0$, therefore $t=0$ is a point where
$u_{\theta}$ assumes a local maximal value. Since $\theta$ is any given value
in the compact set $[0,\pi]$, this implies that the tangent plane of
$\varSigma$ at $p=(0,0,u(0,0))$ is horizontal, as we wished to prove.

The case when $Q(u)\leq 0$ (i.e., when the mean curvature vector of
$\varSigma$ points downwards) can be treated analogously. ∎

######  Remark 3.21.

The horoconvexity hypothesis in Theorem 3.20 is necessary to the conclusion,
since translating catenoids and grim reapers are complete, and any grim reaper
is transversal to the $x_{3}$-axis. Furthermore, as we proved in Theorem
3.9-(iv), horizontal horospheres are limits of hyperbolic translating
catenoids, so that they constitute the analogues of the bowl soliton of
$\mathbb{R}^{3}$ (cf. [6]). From this point of view, Theorem 3.20 relates to
certain uniqueness results for the bowl soliton, such as the celebrated
theorem by Wang [21], which asserts that the bowl soliton is the only convex
entire translator of $\mathbb{R}^{3}$.

##  4\. Rotators to MCF in $\mathbb{H}^{3}$

Let us consider now the one-parameter group $\mathcal{G}\subset{\rm
Iso}(\mathbb{H}^{3})$ of rotations $\Gamma_{t}$ of
$\mathbb{H}^{3}=(\mathbb{R}_{+}^{3},ds^{2})$ about the $x_{3}$-axis.
Considering the decomposition
$\mathbb{R}_{+}^{3}=\mathbb{R}^{2}\times(0,+\infty),$ we have that

$\Gamma_{t}=\left[\begin{array}[]{cc}e^{tJ}&\\\ &1\end{array}\right],\quad
J=\begin{bmatrix}0&-1\\\ 1&\phantom{-}0\end{bmatrix}.$

In this setting, an initial condition of a $\mathcal{G}$-soliton will be
called a _rotating soliton_ or simply a _rotator_. The (horizontal) Killing
field associated to $\mathcal{G}$ is $\xi(p)=J\pi(p),$ $p\in\mathbb{H}^{3},$
where $\pi$ denotes the projection over
$\\{(0,0,1)\\}^{\perp}\subset\mathbb{R}^{3}$, i.e.,
$\pi(x_{1},x_{2},x_{3})=x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}$. Hence, a
surface $\varSigma$ of hyperbolic space $\mathbb{H}^{3}$ is a rotator to MCF
if and only if

(40) $H(p)=\langle J\pi(p),\eta(p)\rangle\,\,\,\forall p\in\varSigma.$

Clearly, a minimal surface of $\mathbb{H}^{3}$ is a (stationary) rotator if
and only if it is invariant by rotations about the $x_{3}$-axis. These minimal
surfaces were classified in [5], and they include, of course, the one-
parameter family of totally geodesic planes which intersect the $x_{3}$-axis
orthogonally, as well as the hyperbolic catenoid obtained by Mori [18]. On the
other hand, it is clear that no horosphere is a rotator in $\mathbb{H}^{3}$.
These facts, together with the considerations of Remark 3.5, yield

######  Theorem 4.1.

The only rotators of constant mean curvature in $\mathbb{H}^{3}$ are the
minimal surfaces of revolution with axis $\\{(0,0)\\}\times(0,+\infty).$

We shall seek for rotators in $\mathbb{H}^{3}$ in the class of _helicoidal
surfaces_ , which are described as follows. Choose a smooth curve with trace
contained in the horosphere $\mathscr{H}:=\mathbb{R}^{2}\times\\{1\\}$ of
height $1:$

$s\in\mathbb{R}\mapsto(\alpha(s),1)\in\mathscr{H},$

where $\alpha\colon\mathbb{R}\rightarrow\mathbb{R}^{2}$ is a regular curve
parameterized by arc length. Given a constant $h>0,$ we call a parameterized
surface $\varSigma=X(\mathbb{R}^{2})\subset\mathbb{H}^{3}$ a _helicoidal
surface_ generated by $\alpha$ with _pitch_ $h$ if the parameterization
$X:\mathbb{R}^{2}\rightarrow\mathbb{H}^{3}$ writes as

(41) $X(u,v)=e^{hv}(e^{vJ}\alpha(u),1),\,\,\,(u,v)\in\mathbb{R}^{2}.$

Considering a parameterization of $\alpha$ by arc length,
$\alpha(s)=(u(s),v(s),0),\,s\in\mathbb{R},$ and writing

$T(s)=u^{\prime}(s)\partial_{x}+v^{\prime}(s)\partial_{y},\quad
N(s)=-v^{\prime}(s)\partial_{x}+u^{\prime}(s)\partial_{y},$

the curvature of $\alpha$ is given by

${k(s)=\langle\alpha^{\prime\prime}(s),N(s)\rangle_{e}=-u^{\prime\prime}(s)v^{\prime}(s)+v^{\prime\prime}(s)u^{\prime}(s),}$

where $\langle\,,\,\rangle_{e}$ stands for the Euclidean metric of
$\mathbb{R}^{2}.$ Furthermore, by the well known Frenet-Serret equations, one
has

${T^{\prime}=kN,\quad N^{\prime}=-kT.}$

In this setting, if we define the functions

(42)
$\uptau:=\langle\alpha,T\rangle_{{e}}\quad\text{and}\quad\mu:=\langle\alpha,N\rangle_{{e}},$

we get from a direct computation that

(43) $\mkern 1.5mu\overline{\mkern-1.5mu\eta\mkern-1.5mu}\mkern
1.5mu=\varrho(e^{vJ}N,-(\uptau+h\mu)/h),\,\,\,\varrho:=h(h^{2}+(\uptau+h\mu)^{2})^{-1/2},$

is an Euclidean unit normal to the helicoidal surface $\varSigma,$ and that
its Euclidean mean curvature in this orientation is

$\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern
1.5mu=e^{-hv}\varrho\frac{k((h^{2}+1)r^{2}+h^{2})-(h\uptau-\mu)}{2(h^{2}+(\uptau+h\mu)^{2})},$

where $r^{2}:=\uptau^{2}+\mu^{2}.$ From this equality and (1), we have that
the hyperbolic mean curvature $H$ of $\varSigma$ is

(44)
$H=\frac{\varrho}{h}\left(h\frac{k((h^{2}+1)r^{2}+h^{2})-(h\uptau-\mu)}{2(h^{2}+(\uptau+h\mu)^{2})}-(\uptau+h\mu)\right).$

These considerations yield the following existence result, which brings [10,
Theorem 3.1] to $\mathbb{H}^{3}$.

######  Theorem 4.2.

For any smooth function $\Psi\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ and
any constant $h>0,$ there exists a one-parameter family of complete helicoidal
surfaces of pitch $h$ in $\mathbb{H}^{3}$ each of them with mean curvature
function $H$ satisfying

$H(X(u,v))=\Psi(\uptau(u),\mu(u)),$

where $X$ is the parameterization given in (41) and $\uptau$ and $\mu$ are as
in (42).

###### Proof.

Considering equality (44) for the given function $H=H(\uptau,\mu)$ and solving
for $k,$ we have that $k=k(\uptau,\mu)$ is a smooth function of
$(\uptau,\mu)\in\mathbb{R}^{2}.$ However, by [10, Lemma 3.2], there exists a
one-parameter family of plane curves
$\alpha:\mathbb{R}\rightarrow\mathbb{R}^{2},$ each of them with curvature $k.$
Therefore, for such an $\alpha,$ and for a given $h>0,$ the helicoidal surface
of $\mathbb{H}^{3}$ with pitch $h$ whose generating curve is $\alpha$ has mean
curvature function $H=H(\uptau,\mu),$ as we wished to prove. ∎

Now, we verify the conditions under which a helicoidal surface
$\varSigma=X(\mathbb{R}^{2})$ of $\mathbb{H}^{3}$ is a rotator to MCF. By
(41)–(43),

$\displaystyle\langle J\pi(X),\eta(X)\rangle$ $\displaystyle=$
$\displaystyle{\langle J(e^{hv}e^{vJ}\alpha),e^{hv}\mkern
1.5mu\overline{\mkern-1.5mu\eta\mkern-1.5mu}\mkern 1.5mu(X)\rangle}={\langle
Je^{vJ}\alpha,\mkern 1.5mu\overline{\mkern-1.5mu\eta\mkern-1.5mu}\mkern
1.5mu(X)\rangle_{e}}$ $\displaystyle=$ $\displaystyle{\langle
e^{vJ}J\alpha,\varrho e^{vJ}N\rangle_{e}=\varrho\langle J\alpha,N\rangle_{e}}$
$\displaystyle=$
$\displaystyle{-\varrho\langle\alpha,JN\rangle_{e}=\varrho\langle\alpha,T\rangle_{e}}$
$\displaystyle=$ $\displaystyle\varrho\uptau,$

which, together with (40), implies the following result.

######  Lemma 4.3.

A helicoidal surface $\varSigma=X(\mathbb{R}^{2})$ of pitch $h>0$
parameterized as in (41) is a rotator to MCF in $\mathbb{H}^{3}$ if and only
if its mean curvature function $H=H(\uptau,\mu)$ satisfies

(45) $H=\frac{h\uptau}{\sqrt{h^{2}+(h\mu+\uptau)^{2}}}.$

In what follows, we prove the main result of this section, which provides the
existence of complete rotators in $\mathbb{H}^{3}$ by means of helicoidal
surfaces, and completely describe the topology of the corresponding generating
curves (see Figure 8).

Figure 8. The generating curve of a helicoidal rotator in $\mathbb{H}^{3}$.

######  Theorem 4.4.

For any $h>0,$ there exists a one-parameter family of complete rotators to MCF
in $\mathbb{H}^{3}$ whose elements are all helicoidal surfaces of pitch $h.$
For each such surface, the trace of the generating curve
$\alpha\colon\mathbb{R}\to\mathbb{R}^{2}$ consists of two unbounded properly
embedded arms centered at the point of $\alpha$ which is closest to the origin
$o\in\mathbb{R}^{2},$ with each arm spiraling around $o.$

###### Proof.

The existence part of the statement follows directly from Lemma 4.3 and
Theorem 4.2. So, it remains to prove that the generating curve $\alpha$ of any
such helicoidal surface has the asserted geometric properties.

Keeping the above notation, we first observe that, from equalities (44) and
(45), the curvature $k$ of $\alpha$ satisfies:

(46)
$k=\frac{2(h^{2}+(\uptau+h\mu)^{2})((h+1)\uptau+h\mu)+h(h\uptau-\mu)}{h((h^{2}+1)r^{2}+h^{2})}\cdot$

Also, from (42) and the Frenet-Serret equations, one has

(47) $\uptau^{\prime}=1+k\mu\quad\text{and}\quad\mu^{\prime}=-k\uptau,$

which, together with (46), yields the ODE system (see Fig. 9)

(48) $\left\\{\begin{array}[]{ccl}\uptau^{\prime}&=&\displaystyle
1+\frac{2(h^{2}+(\uptau+h\mu)^{2})((h+1)\uptau\mu+h\mu^{2})+h^{2}\uptau\mu-h\mu^{2}}{h((h^{2}+1)r^{2}+h^{2})},\\\\[12.91663pt]
\mu^{\prime}&=&\displaystyle-\frac{2(h^{2}+(\uptau+h\mu)^{2})((h+1)\uptau^{2}+h\uptau\mu)+h^{2}\uptau^{2}-h\uptau\mu}{h((h^{2}+1)r^{2}+h^{2})}\cdot\end{array}\right.$

Figure 9. Phase portrait of system (48) for $h=1.$

Now, we establish the properties of $\alpha=\uptau T+\mu N$ through the
following claims.

######  Claim 4.5.

The ODE system (48) has no constant solutions, and all solutions are defined
on $\mathbb{R}.$

###### Proof of Claim 4.5.

Assume, by contradiction, that there exists a constant solution
$\psi(s)=(\uptau_{0},\mu_{0}),\,s\in\mathbb{R}.$ Since
$\uptau^{\prime}=\mu^{\prime}=0,$ we have from (47) that
$k_{0}:=k(\uptau_{0},\mu_{0})$ satisfies $k_{0}\mu_{0}=-1$ and
$k_{0}\uptau_{0}=0,$ which yields $\uptau_{0}=0$ and $\mu_{0}\neq 0.$ However,
from the first equation in (48), one has

$\uptau^{\prime}=1+\frac{2h(1+\mu_{0}^{2})\mu_{0}^{2}-\mu_{0}^{2}}{(h^{2}+1)\mu_{0}^{2}+h^{2}}=\frac{h^{2}(\mu_{0}^{2}+1)+2h(1+\mu_{0}^{2})\mu_{0}^{2}}{(h^{2}+1)\mu_{0}^{2}+h^{2}}>0,$

which is a contradiction. Therefore, (48) has no constant solutions. From this
fact, and since $k=k(\uptau,\mu)$ is defined on $\mathbb{R}^{2},$ we conclude
that any solution of (48) is defined on $\mathbb{R}.$ ∎

######  Claim 4.6.

Suppose that any integral curve $\psi(s):=(\uptau(s),\mu(s))$ of (48)
satisfies that the limit $\lim_{s\to+\infty}\uptau(s)$ (resp.
$\lim_{s\to+\infty}\mu(s)$) exists. Then, $\lim_{s\to-\infty}\uptau(s)$ (resp.
$\lim_{s\to-\infty}\mu(s)$) also exists. Furthermore, if there exists some
$L\in[-\infty,+\infty]$ with the property that for any integral curve

$\lim_{s\to+\infty}\uptau(s)=L\,\,\,(\text{resp.}\,\,\lim_{s\to+\infty}\mu(s)=L),$

then it also holds that any integral curve satisfies

$\lim_{s\to-\infty}\uptau(s)=-L\,\,\,(\text{resp.}\,\,\lim_{s\to-\infty}\mu(s)=-L).$

###### Proof of Claim 4.6.

Let $\psi(s):=(\uptau(s),\mu(s))$ be an integral curve of the system (48).
Then, it is easily checked that $\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu(s):=-\psi(-s)$ is
also an integral curve of that system. Setting $\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu=(\mkern
1.5mu\overline{\mkern-1.5mu\uptau\mkern-1.5mu}\mkern 1.5mu,\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu),$ we have that
$\mkern 1.5mu\overline{\mkern-1.5mu\uptau\mkern-1.5mu}\mkern
1.5mu(s)=-\uptau(-s)$ and $\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu(s)=-\mu(-s).$ By
hypothesis, $\lim_{s\to+\infty}\mkern
1.5mu\overline{\mkern-1.5mu\uptau\mkern-1.5mu}\mkern 1.5mu(s)$ exists and the
first part of the claim follows from observing that
$\lim_{s\to-\infty}\mu(s)=-\lim_{s\to+\infty}\mkern
1.5mu\overline{\mkern-1.5mu\mu\mkern-1.5mu}\mkern 1.5mu(s)$. The remainder of
the proof is argued analogously and will be omitted. ∎

######  Claim 4.7.

The function $\uptau$ has precisely one zero $s_{0}$ and $\uptau$ is negative
in $(-\infty,s_{0})$ and positive in $(s_{0},+\infty).$ As a consequence, the
function $r^{2}=\uptau^{2}+\mu^{2}$ has a global minimum and satisfies
$\lim_{s\rightarrow\pm\infty}r^{2}=+\infty.$

###### Proof of Claim 4.7.

First, observe that the equalities (47) yield

$(r^{2})^{\prime}=2(\uptau\uptau^{\prime}+\mu\mu^{\prime})=2(\uptau(1+k\mu)+\mu(-k\uptau))=2\uptau,$

which implies that the zeroes of $\uptau$ are the critical points of $r^{2}$.
Also, as seen in the first part of the proof of Claim 4.5, if
$\uptau(s_{0})=0$ for some $s_{0},$ then $\uptau^{\prime}(s_{0})>0,$ which
gives that $\uptau$ has at most one zero $s_{0},$ in which case $\uptau$ is
negative in $(-\infty,s_{0}),$ and positive in $(s_{0},+\infty).$

Next, we argue by contradiction and assume that $\uptau$ has no zeroes. We
will also assume that $\uptau>0$ on $\mathbb{R},$ since the complementary case
$\uptau<0$ can be treated analogously. Under this assumption, the function
$r^{2}$ is strictly increasing. So, there exists $\delta\geq 0$ such that

$\lim_{s\to-\infty}r^{2}(s)=\delta.$

In particular, since $\uptau=\frac{(r^{2})^{\prime}}{2}$, we also have that

(49) $\lim_{s\to-\infty}\uptau(s)=0,$

which implies that $\mu^{2}\rightarrow\delta$ as $s\rightarrow-\infty.$
However, the first equality in (48) yields
$\lim_{s\rightarrow-\infty}\uptau^{\prime}(s)>0,$ which contradicts (49),
proving that $\uptau$ has exactly one zero and that $r^{2}$ has only one
critical point. Consequently, both the limits of $r^{2}$ as
$s\rightarrow\pm\infty$ exist in $[0,+\infty]$.

To finish the proof of the claim, just note that if either
$\lim_{s\to-\infty}r^{2}=\delta$ or $\lim_{s\to+\infty}r^{2}=\delta$ for some
$\delta>0,$ the same arguments as before lead to a contradiction, thus
$\lim_{s\to\pm\infty}r^{2}(s)=+\infty$. ∎

######  Claim 4.8.

The limits of $\uptau$ and $\mu$ as $s\rightarrow\pm\infty$ exist (possibly
being infinite).

###### Proof of Claim 4.8.

First, we show that $k$ has at most one zero in $\mathbb{R}.$ Assume that
$k(s_{0})=0$ for some $s_{0}\in\mathbb{R}.$ We have from (46) that, at
$s_{0},$

(50) $2(h^{2}+(\uptau+h\mu)^{2})((h+1)\uptau+h\mu)+h(h\uptau-\mu)=0.$

Also, by (47), $\uptau^{\prime}(s_{0})=1$ and $\mu^{\prime}(s_{0})=0.$ This,
together with (50), gives that, at $s_{0},$

(51)
$k^{\prime}=\frac{2(h+1)(h^{2}+(\uptau+h\mu)^{2})+4((h+1)\uptau+h\mu)(\uptau+h\mu)+h^{2}}{h((h^{2}+1)r^{2}+h^{2})}\cdot$

If $\uptau(s_{0})\mu(s_{0})\geq 0,$ we have from (51) that
$k^{\prime}(s_{0})>0.$ Assume then $\uptau(s_{0})\mu(s_{0})<0$ and notice
that, by (50), one has

(52) ${\rm sign}((h+1)\uptau(s_{0})+h\mu(s_{0}))={\rm
sign}(\mu(s_{0})-h\uptau(s_{0})).$

If $\uptau(s_{0})<0<\mu(s_{0}),$ then both signs in (52) are positive. In
addition,

$\uptau(s_{0})+h\mu(s_{0})=(h+1)\uptau(s_{0})+h\mu(s_{0})-h\uptau(s_{0})>0,$

and then (51) yields $k^{\prime}(s_{0})>0.$ Analogously,
$\mu(s_{0})<0<\uptau(s_{0})$ implies $k^{\prime}(s_{0})>0.$

It follows from the above that $k$ has at most one zero $s_{0}\in\mathbb{R}$
and, if so, $k$ is negative in $(-\infty,s_{0})$ and positive in
$(s_{0},+\infty).$ Since, by Claim 4.7, $\uptau$ has exactly one zero, we have
that $\mu^{\prime}=-k\uptau$ has at most two zeros, which implies that $\mu$
has at most two critical points. In particular, the limits
$\lim_{s\to\pm\infty}\mu(s)$ exist.

To finish the proof of the claim, let us assume, by contradiction, that the
limit of $\uptau$ as $s\rightarrow+\infty$ does not exist. In this case, for
some $\uptau_{0}>0,$ there exists a strictly increasing sequence
$(s_{n})_{n\in\mathbb{N}}$ diverging to $+\infty$ and such that (see Fig. 10)

$\uptau(s_{n})=\uptau_{0}\quad\text{and}\quad\uptau^{\prime}(s_{n})\uptau^{\prime}(s_{n+1})<0\quad\forall
n\in\mathbb{N}.$

Figure 10. Graph of $\uptau.$

Claim 4.7 implies that $\lim r^{2}(s_{n})=+\infty$, then we must have
$\lim\mu^{2}(s_{n})=+\infty$. In this case, our previous arguments show that
either $\lim\mu(s_{n})=+\infty$ or $\lim\mu(s_{n})=-\infty$. In any case, we
have from (46) that

$\lim_{n\to+\infty}(k(s_{n})\mu(s_{n}))=\lim_{n\to+\infty}\frac{2h^{3}\mu(s_{n})^{4}}{h(h^{2}+1)\mu(s_{n})^{2}}=+\infty.$

In particular, for any sufficiently large $n\in\mathbb{N},$
$\uptau^{\prime}(s_{n})=1+k(s_{n})\mu(s_{n})>0$. This, however, contradicts
the fact that $(\uptau^{\prime}(s_{n}))_{n\in\mathbb{N}}$ is an alternating
sequence. Therefore, $\lim_{s\to+\infty}\uptau(s)$ exists. Since
$(\uptau,\mu)$ is an arbitrary integral curve of (48), Claim 4.6 implies that
$\lim_{s\to-\infty}\uptau(s)$ also exists, thereby finishing the proof of the
claim. ∎

######  Claim 4.9.

$\lim_{s\to\pm\infty}\uptau(s)=\pm\infty$ and
$\lim_{s\to\pm\infty}\mu(s)=\mp\infty.$

###### Proof of Claim 4.9.

By Claim 4.8, all the limits above exist and, arguing by contradiction, we
first treat the case when $\lim_{s\to+\infty}\mu(s)=L\in\mathbb{R}$. Under
this assumption, we have from Claims 4.7 and 4.8 that
$\lim_{s\to+\infty}\uptau(s)=+\infty$. Then, it follows from the second
equation in (48) that $\lim_{s\to+\infty}\mu^{\prime}(s)=-\infty$, which
contradicts the assumed fact $L\in\mathbb{R}$.

Suppose now that $\lim_{s\to+\infty}\mu(s)=+\infty$. From this assumption and
Claim 4.7, we have that $h^{2}+(\uptau(s)+h\mu(s))^{2}>{1/2}$ for all
sufficiently large $s>0.$ Applying this last inequality to (46) yields

$k(s)>\frac{(h+1)\uptau(s)+h\mu(s)+h(h\uptau(s)-\mu(s))}{h((h^{2}+1)r^{2}+h^{2})}=\frac{(h^{2}+h+1)\uptau(s)}{h((h^{2}+1)r^{2}+h^{2})}>0.$

However, for such values of $s,$ $\mu^{\prime}(s)=-k(s)\uptau(s)<0,$ which is
a contradiction. Therefore, $\lim_{s\to+\infty}\mu(s)=-\infty$. Since
$(\uptau,\mu)$ is an arbitrary integral curve, Claim 4.6 applies to show that
$\lim_{s\to-\infty}\mu(s)=+\infty.$

To finish the proof, assume, by contradiction, that
$\lim_{s\to+\infty}\uptau(s)$ is finite. Then, since
$\lim_{s\to+\infty}\mu(s)=-\infty,$ we have from (46) that
$\lim_{s\to+\infty}k(s)=-\infty.$ From this, we have
$\lim_{s\to+\infty}\uptau^{\prime}(s)=\lim_{s\to+\infty}(1+k(s)\mu(s))=+\infty,$
which is a contradiction. Therefore, Claim 4.7 gives that
$\lim_{s\to+\infty}\uptau(s)=+\infty$. Once again,
$\lim_{s\to-\infty}\uptau(s)=-\infty$ follows from Claim 4.6. ∎

######  Claim 4.10.

The function $\nu:=-\uptau/\mu$ is bounded outside of a compact interval.

###### Proof of Claim 4.10.

It follows from Claim 4.9 that $\nu$ is well defined and positive at all
points outside of a compact interval of $\mathbb{R}.$ Assume by contradiction
that there exists a sequence $(s_{n})_{n\in\mathbb{N}}$ in $\mathbb{R}$ which
diverges to infinity, and such that $\lim\nu(s_{n})=+\infty,$ i.e.,
$\lim(-\mu(s_{n})/\uptau(s_{n}))=0.$

From (46), one has

$k\uptau=\frac{2(h^{2}+(\uptau+h\mu)^{2})(h+1+h\mu/\uptau)+h^{2}-h\mu/\uptau}{h((h^{2}+1)(1+\mu^{2}/\uptau^{2})+h^{2}/\uptau^{2})}\,\cdot$

Hence, after passing to a subsequence, we can assume that
$k(s_{n})\uptau(s_{n})$ is positive and bounded away from zero for all
$n\in\mathbb{N}.$ However,

$+\infty=\lim\nu(s_{n})=\lim\frac{\phantom{-}\uptau(s_{n})}{-\mu(s_{n})}=\lim\frac{\phantom{-}\uptau^{\prime}(s_{n})}{-\mu^{\prime}(s_{n})}=\lim\left(\frac{1}{k(s_{n})\uptau(s_{n})}+\frac{\mu(s_{n})}{\uptau(s_{n})}\right)<+\infty,$

which is a contradiction.

Analogously, we derive a contradiction by assuming that there exists
$s_{n}\rightarrow-\infty$ satisfying $\lim\nu(s_{n})=+\infty.$ This proves
Claim 4.10. ∎

In what follows, we shall denote by $\omega=\omega(s)$ the angle function of
$\alpha,$ that is,

$\alpha=r(\cos\omega,\sin\omega).$

It then follows from (47) that the equality

(53) $T=\frac{\uptau}{r^{2}}\alpha+\omega^{\prime}J\alpha$

holds at all points where $r\neq 0.$

######  Claim 4.11.

$\omega(s)\rightarrow+\infty$ as $s\rightarrow\pm\infty.$

###### Proof of Claim 4.11.

Considering (53) and the equality $(r^{2})^{\prime}=2\uptau,$ we have that

$r^{\prime}=\frac{\uptau}{r}\quad\text{and}\quad\omega^{\prime}=-\frac{\mu}{r^{2}}\,\cdot$

So, given a differentiable function $\varphi=\varphi(r),$ $r\in(0,+\infty),$
one has

(54)
$\frac{d\varphi}{d\omega}=\frac{d\varphi}{dr}\frac{dr}{ds}\frac{ds}{d\omega}=-r\varphi^{\prime}(r)\frac{\uptau}{\mu}\cdot$

Now, define $\varphi(r)=\log(\log r).$ Then, $\varphi(r)\rightarrow+\infty$ as
$r\rightarrow+\infty$ and

(55) $r\varphi^{\prime}(r)=\frac{1}{\log r}\rightarrow
0\,\,\,\text{as}\,\,\,r\rightarrow+\infty.$

Since, by Claim 4.10, $-\uptau/\mu$ is bounded outside of a compact interval,
it follows from Claim 4.7 and (54)–(55) that ${d\varphi}/{d\omega}\rightarrow
0$ as $s\rightarrow\pm\infty.$ Consequently,
${d\omega}/d\varphi\rightarrow+\infty$ as $s\rightarrow\pm\infty,$ which
proves Claim 4.11. ∎

It follows from the above that the trace of $\alpha$ has one point $p_{0}$
closest to the origin $o$ (Claim 4.7), and consists of two properly embedded
arms centered at $p_{0}$ (Claim 4.9) which proceed to infinity by spiraling
around $o$ (Claim 4.11). This finishes our proof. ∎

Let $\varSigma=X(\mathbb{R}^{2})$ be a helicoidal surface of pitch $h$ in
$\mathbb{H}^{3}$ as given in (41). Consider the subgroup
$\mathcal{G}=\\{\Gamma_{t}\mid t\in\mathbb{R}\\}\subset{\rm
Iso}(\mathbb{H}^{3})$ of (downward) hyperbolic translations of constant speed
$h,$ i.e., $\Gamma_{t}(p)=e^{-ht}p,$ and notice that the Killing field on
$\mathbb{H}^{3}$ determined by $\mathcal{G}$ is $\xi(p)=-hp.$ Now, recall that
the unit normal to $\varSigma$ is $\eta=e^{hv}\bar{\eta},$ with $\bar{\eta}$
as in (43). From this, we have:

$\langle\xi(X),\eta\rangle=-h\varrho\left(\mu-\frac{\uptau+h\mu}{h}\right)=\varrho\uptau,$

so that $\varSigma$ is a $\mathcal{G}$-soliton if and only if its mean
curvature function is given by

$H=\varrho\uptau=\frac{h\uptau}{\sqrt{h^{2}+(h\mu+\uptau)^{2}}}\cdot$

From this last equality and Lemma 4.3, we have the following result.

######  Proposition 4.12.

Let $\varSigma=X(\mathbb{R}^{2})$ be a helicoidal surface of pitch $h$ in
$\mathbb{H}^{3}.$ Then, the following assertions are equivalent:

  * i)

$\varSigma$ is a rotator to MCF.

  * ii)

$\varSigma$ is a translator to MCF with respect to the Killing field
$\xi(p)=-hp.$

##  5\. The Classification of Minimal Translators

In this section, we prove Theorem 3.2, which is a classification of complete,
properly immersed minimal surfaces of $\mathbb{H}^{3}$ invariant under the
1-parameter group $\\{\Gamma_{t}\\}_{t\in\mathbb{R}}$ of hyperbolic isometries
of $\mathbb{H}^{3}$ defined (in the half-space model) by

${(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}_{+}\mapsto\Gamma_{t}(x_{1},x_{2},x_{3})=(e^{t}x_{1},e^{t}x_{2},e^{t}x_{3}).}$

We let $\overline{\mathbb{H}^{3}}$ be the topological space given by the
compactification of $\mathbb{H}^{3}$ with respect to the so-called cone
topology (as defined in [8]) and let $S^{2}(\infty)$ denote the asymptotic
boundary of $\mathbb{H}^{3}$. In the upper half space model of
$\mathbb{H}^{3},$ $S^{2}(\infty)$ is identified with the one point
compactification of $\mathbb{R}^{2}=\\{x_{3}=0\\}$:

$S^{2}(\infty)=\mathbb{R}^{2}\cup\\{\infty\\}.$

Along the proof, given a surface $\varSigma\subset\mathbb{H}^{3},$ we will
write $\partial_{\infty}\varSigma$ for the asymptotic boundary of $\varSigma,$
that is, $\partial_{\infty}\varSigma:=\mkern
1.5mu\overline{\mkern-1.5mu\varSigma\mkern-1.5mu}\mkern 1.5mu\cap
S^{2}(\infty),$ where $\mkern
1.5mu\overline{\mkern-1.5mu\varSigma\mkern-1.5mu}\mkern 1.5mu$ is the closure
of $\varSigma$ in $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{H}^{3}\mkern-1.5mu}\mkern 1.5mu$.

###### Proof of Theorem 3.2.

Consider $\alpha$ a curve in the horosphere $\mathscr{H}:=\\{x_{3}=1\\}$ and
assume that $\varSigma=\\{e^{t}\alpha\mid t\in\mathbb{R}\\}$ is a complete,
properly immersed minimal surface invariant under the action of
$\\{\Gamma_{t}\\}_{t\in\mathbb{R}}$ and generated by $\alpha$. For the
remainder of the proof, we will assume that $\alpha$ is parameterized by arc
length over a maximal interval $I$.

######  Claim 5.1.

For $\theta\in[0,\pi)$, let $L_{\theta}=\\{(r\cos(\theta),r\sin(\theta),1)\mid
r\in\mathbb{R}\\}$. If $\alpha$ intersects $L_{\theta}$ in two (or more)
points, then $\alpha=L_{\theta}$.

###### Proof.

After a rotation, it suffices to prove the claim for $\theta=0$. Assume that
there are two distinct points $p_{1}=(r_{1},0,1),\,p_{2}=(r_{2},0,1)\in\alpha$
and, arguing by contradiction, assume that $\alpha\neq L_{0}$.

Consider the compact arc $a$ that $\\{p_{1},p_{2}\\}$ bounds in $\alpha$.
Since $\alpha\neq L_{0}$, there exists a point $\widehat{p}$ in the interior
of $a$ where the second coordinate function $x_{2}$ has a local maximum or a
local minimum, and we may rotate $\alpha$ once again to assume it is a local
maximum, attained at $(\widehat{x}_{1},\widehat{x}_{2},1)$ with
$\widehat{x}_{2}>0$. Let $P$ be the tilted plane of $\mathbb{H}^{3}$ that
contains the line $\\{(r,\widehat{x}_{2},1)\mid r\in\mathbb{R}\\}$ and whose
asymptotic boundary contains $(0,0,0)$. Then, $P$ is an equidistant surface to
the totally geodesic plane $\\{x_{2}=0\\}$ and its mean curvature vector
points upwards. Since $\varSigma$ locally stays in the mean convex side of $P$
and intersects $P$ tangentially along the line $\\{e^{t}\widehat{p}\mid
t\in\mathbb{R}\\}$, we obtain a contradiction with the mean curvature
comparison principle. ∎

A straightforward consequence of Claim 5.1 is that the curve $\alpha$ must be
properly embedded, $I=\mathbb{R}$ and, if $(0,0,1)\in\alpha$, $\varSigma$ is a
vertical plane. Thus, assuming that $\varSigma$ is not a vertical plane, we
may parameterize $\alpha$ as

(56) $\alpha(s)=(r(s)\cos(\theta(s)),r(s)\sin(\theta(s)),1),$

where $s$ is the arc length of $\alpha$ and $r(s)>0$ for all $s\in\mathbb{R}$.
Claim 5.1 implies that, after a rotation in $\mathbb{H}^{3}$ (and possibly
reparameterizing on the opposite orientation) the function $\theta$ must
satisfy $\theta^{\prime}(s)\geq 0$ and

(57)
$\lim_{s\to-\infty}\theta(s)=0,\quad\lim_{s\to+\infty}\theta(s)=\theta_{+}>0.$

In fact, $\theta_{+}\in(0,\pi]$. Indeed, arguing by contradiction, assume that
$\theta_{+}>\pi$ and choose $\theta^{*}\in(\pi,\theta_{+})$. Then, Claim 5.1
implies that $\alpha$ intersects $L=L_{\theta^{+}-\pi}$ at most in one point,
so the fact that $(0,0,1)\not\in\alpha$ implies that either
$\theta(s)\in(0,\theta^{*})$ for all $s\in I$ or
$\theta(s)\in(\theta^{*}-\pi,\theta_{+})$ for all $s\in I$, both situations in
contradiction with (57).

For the next claim, the union of two half-lines in $\mathbb{R}^{2}$ issuing
from a point $p$ and making an oriented angle $\theta\in(0,2\pi)$ will be
called a $\theta$-_hinge_ with _vertex_ $p.$

######  Claim 5.2.

$\partial_{\infty}\varSigma\cap\mathbb{R}^{2}$ is a $\theta_{+}$-hinge with
vertex at the origin $\mathbf{0}:=(0,0,0).$

###### Proof.

Using the notation of (56), we may parameterize $\varSigma$ as

$\varSigma=\\{(e^{t}r(s)\cos(\theta(s)),e^{t}r(s)\sin(\theta(s)),e^{t})\mid
t,s\in\mathbb{R}\\}.$

Our next argument is to show that
$\ell_{0}\cup\ell_{\theta_{+}}\cup\\{\mathbf{0}\\}\subset\partial_{\infty}\varSigma$,
where, for $\theta\in[0,2\pi)$,
$\ell_{\theta}=\\{(r\cos(\theta),r\sin(\theta),0)\mid r>0\\}.$

Note that $\lim_{s\to\pm\infty}r(s)=+\infty$, as $\alpha$ is properly embedded
and noncompact. Consider a divergent sequence
$(s_{n})_{n\in\mathbb{N}}\subset\mathbb{R}$, so
$\lim_{n\to\infty}r(s_{n})=+\infty$. Take $r>0$ and let, for $n\in\mathbb{N}$,
$t_{n}=\log(r/r(s_{n}))$ and

$p_{n}=e^{t_{n}}\alpha(s_{n})=\left(r\cos(\theta(s_{n})),r\sin(\theta(s_{n})),\frac{r}{r(s_{n})}\right)\in\varSigma.$

Since $r>0$ and $r(s_{n})\to+\infty$, $\lim_{n\to\infty}\frac{r}{r(s_{n})}=0$.
Furthermore, (57) implies that $\lim_{n\to\infty}\theta(s_{n})=\theta_{+}$,
and it follows that

$\lim_{n\to\infty}p_{n}=(r\cos(\theta_{+}),r\sin(\theta_{+}),0)\in\partial_{\infty}\varSigma.$

Since $r$ is arbitrary, this gives
$\ell_{\theta_{+}}\subset\partial_{\infty}\varSigma$. Analogously, we may
prove that $\\{\mathbf{0}\\}\cup\ell_{0}\subset\partial_{\infty}\varSigma$.

Next, we prove that
$\ell_{0}\cup\ell_{\theta_{+}}\cup\\{\mathbf{0}\\}\supset(\partial_{\infty}\varSigma\cap\mathbb{R}^{2}).$
Choose $\overline{p}\in\partial_{\infty}\varSigma\cap\mathbb{R}^{2}$ and,
assuming that $\overline{p}\neq\mathbf{0}$, write
$\overline{p}=(r\cos(\theta),r\sin(\theta),0)$ for some $r>0$ and
$\theta\in[0,2\pi)$. Let $(p_{n})_{n\in\mathbb{N}}$ be a sequence in
$\varSigma$ such that $p_{n}\to\overline{p}$, so there exist uniquely defined
$s_{n},t_{n}\in\mathbb{R}$ such that

$p_{n}=e^{t_{n}}\alpha(s_{n})=\left(e^{t_{n}}r(s_{n})\cos(\theta(s_{n})),e^{t_{n}}r(s_{n})\sin(\theta(s_{n})),e^{t_{n}}\right).$

The fact that $p_{n}\to\overline{p}$ implies that $e^{t_{n}}\to 0$. Moreover,
$\lim_{n\to\infty}e^{t_{n}}r(s_{n})=r$, thus $r(s_{n})\to+\infty$, and it
follows that either $s_{n}\to+\infty$ (in which case
$\theta(s_{n})\to\theta_{+}$) or $s_{n}\to-\infty$ (and $\theta(s_{n})\to 0$).
In both situations we obtain $\overline{p}\in\ell_{0}\cup\ell_{\theta_{+}}$,
which proves the claim. ∎

At this point, we have shown that a properly immersed minimal surface
$\varSigma\subset\mathbb{H}^{3}$ which is invariant under the group
$\\{\Gamma_{t}\\}_{t\in\mathbb{R}}$ is in fact properly embedded and its
asymptotic boundary $\partial_{\infty}\varSigma\cap\mathbb{R}^{2}$ is a
$\theta_{+}$-hinge with vertex at $\mathbf{0}$. The existence and uniqueness
of such surfaces was proven in [9] (unpublished) and also presented in [20,
Proposition A.1], which finishes the proof of Theorem 3.2. ∎

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