# Translating Solitons in a Lorentzian Setting, Submersions and Cohomogeneity
One Actions

Marie-Amélie Lawn
Department of Mathematics, Faculty of Natural Sciences,
Imperial College London, 180 Queen‘s gate London SW72AZ (UK)
<EMAIL_ADDRESS>
Miguel Ortega
Department of Geometry and Topology, Institute of Mathematics
Universidad de Granada, 18071 Granada (Spain)
<EMAIL_ADDRESS>ORCID 0000-0002-1390-9980

###### Abstract

We study new examples of translating solitons of the mean curvature flow,
especially in Minkowski space. We consider for this purpose manifolds
admitting submersions and cohomegeneity one actions by isometries on suitable
open subsets. This general setting also covers the classical Euclidean
examples. As an application, we completely classify timelike, invariant
translating solitons by rotations and boosts in Minkowski space.

Keywords: Timelike Translating soliton, submersions, pseudo-Riemannian
manifolds, Lie group, Cohomogeneity one action.

MSC[2010] Classification: 53C44, 53C21, 53C42, 53C50.

##  1 Introduction

The evolution by mean curvature flow is classicaly studied for hypersurfaces
in the Euclidean space $\mathbb{R}^{n+1}$. One of the approaches is to
consider particular solutions, the so-called translating solitons or
translators, which are invariant by a subgroup of translations of the ambient
space. It is well-known that they admit a forever flow, characterized by a
constant (unit) vector $v\in\mathbb{R}^{n+1}$, $n\geq 1$, in such a way that
the mean curvature flow equation simplifies to

$H=v^{\perp}.$

Translators have been widely studied in literature ([2, 5, 12, 10],…, and
references therein). For instance, they naturally appear in the study of
solutions of the mean curvature flow with a certain type of singularities (see
for example [6]) and are equivalent to minimal surfaces for a conformally
modified metric [7]. There are other studies for translators in other ambients
spaces, such as $\mathbb{R}\times M$ [11], in $\mathbb{H}^{2}\times\mathbb{R}$
[4], a solvable group [14], the Heisenberg 3-group [15], etc.

One of the first analytical point of view was to see which of them were
rotationally invariant [5]. This approach can be revisited from a geometrical
point of view, in the sense that they are invariant by the Lie group
$SO(n-1)$. Thus, the PDE is rapidly reduced to an ODE, which is simpler to
study. Translating solitons were studied from an analytical point of view in
the Lorentz-Minkowski space, [9], focusing on the rotationally invariant case
and only for spacelike hypersurfaces. Again, the underlying geometrical
structure is the use of the Lie group $SO(n)$ as a subgroup of isometries.

In this paper, we are interested in studying graphical translating solitons in
Minkowski space by the action of some subgroups of isometries. From a pure
analytical point of view, we seek to simplify the PDE to an ODE by a group of
transformations. However, adding a geometrical sauce provides an extra layer
of flavour. To do so, we wished to use quotients, although we realized that
suitable tools are submersions, for almost the same effort. Those who wish to
stick to the analytical taste can jump directly to Sections 5 and 6.

Typical examples are cohomogeneity one actions by Lie groups. If the action of
a group $\Sigma$ is proper and free, the orbit space becomes a smooth manifold
(with boundary), and the natural projection is a submersion. This holds when
$M$ is Riemannian and $\Sigma$ is compact, acting by isometries. However, the
situation becomes much more complicated when the action is not proper.
Fortunately, it is still possible to work on some good open subsets $\Omega$
of $M$ quite often, because the orbit space $\Omega/\Sigma$ is diffeomorphic
to an open interval, even when $M/\Sigma$ is not Hausdorff.

Since our starting point was to study translators in Lorentz-Minkowski space,
the base manifolds of the quotients have to be either Riemannian or
Lorentzian. To unify the notations, we are forced to consider pseudo-
Riemannian submersions. The advantage is that we can make a quite more general
study, although we will return to our origins in the last sections.

We will mainly focus on graphical translating solitons. Namely, given $u\in
C^{2}(M)$, we construct its graph map $F:M\rightarrow M\times\mathbb{R}$,
$F(x)=(x,u(x))$, and we assume that its induced metric is not degenerate. Let
$\nu$ be the upward normal vector along $F$ with
$\varepsilon=\left\langle\nu,\nu\right\rangle=\pm 1$. We recall that
$\varepsilon=-1$ when $F(M)$ is spacelike, and $\varepsilon=+1$ if timelike.
In Section 2, we show that the mean curvature flow is characterized by the
following partial differential equation, which is similar to the one obtained
in the classical case (see [5])

$\mathrm{div}\left(\frac{\nabla
u}{\sqrt{\varepsilon\big{(}\varepsilon^{\prime}+|\nabla
u|_{g}^{2}\big{)}}}\right)=\frac{1}{\sqrt{\varepsilon\big{(}\varepsilon^{\prime}+|\nabla
u|_{g}^{2}\big{)}}}=H.$

As in [4, 5, 11], we show in Section 3 that it is possible to simplify this
PDE to an ODE under reasonable assumptions on the geometry of the manifold
$M$, namely when $M$ admits a submersion
$\pi:(M,g)\rightarrow(I,\tilde{\varepsilon}ds^{2})$ whose fibers $\pi^{-1}(s)$
have constant mean curvature $h(s)$, where $I$ is an interval,
$\tilde{\varepsilon}=\pm 1$. In Theorem 3.1, we prove that a function $f\in
C^{2}(M)$ is solution to the ODE

$f^{\prime\prime}(s)=(\tilde{\varepsilon}+\varepsilon^{\prime}f^{\prime}(s)^{2})(1-f^{\prime}(s)h(s)),$

if, and only if, the graph map $F:M\to M\times\mathbb{R}$,
$F(x)=(x,(f\circ\pi)(x))$, is a translating soliton. Next, we will give some
results about the behavior of translating solitons within this context.
Corollary 3.1 generalizes the well-known Wing-like translating soliton in
$\mathbb{R}^{m+1}$, [5], whose tangent planes at some points are vertical, and
thus, they are not graphical at such points. The standard idea is to rewrite
the ODE by putting $f$ as the inverse function of a given $\alpha$. Corollary
3.2 is an existence result for globally defined graphical solutions.

Section 4 is devoted to analytical computations, which are necessary for the
final sections.

Section 5 is devoted to applying all these ideas to rotationally invariant
translating solitons in Minkowski space $\mathbb{L}^{n+1}$, $n\geq 2$, that is
to say, those invariant by the action of $SO(n)$ as a subgroup of isometries
of $\mathbb{L}^{n+1}$. Also, in Section 6, we consider the Lie group
$SO^{\uparrow}(n-1,1)$ acting on $\mathbb{L}^{n+1}$, $n\geq 2$. We focus on
the timelike cases, because the spacelike cases were already considered in [9]
(although without the Geometrical topping). Probably, the most surprising
result is the fact that there is an example which crosses more than one
fundamental region. Or in other words, we need more than one profile curve to
obtain the whole surface, by gluing up four pieces lying in continuous open
subsets. Probably, this unusual and slippery technique seems to prevent many
people from studying timelike examples, but makes the Lorentzian setting
surprisingly different from the classical Euclidean case.

##  2 Setup

Let $(M,g)$ be a connected pseudo-Riemannian manifold. Consider $u\in
C^{2}(M,\mathbb{R})$, and let $F:M\rightarrow M\times\mathbb{R}=:\bar{M}$,
$F(p)=(p,u(p))$ be its graph map. We denote $(p,t)\in M\times\mathbb{R}$. We
denote by $|w|_{g}^{2}={g(w,w)}$ the squared $g$-norm of any tangent vector
$w$ to $M$. Assume that
$F:(M,\gamma=F^{*}\left\langle,\right\rangle)\rightarrow(\bar{M},\left\langle,\right\rangle=g+\varepsilon^{\prime}dt^{2})$
is a non-degenerate hypersurface, where $\varepsilon^{\prime}=\pm 1$. Under
the usual identifications, for each $X\in TM$, we have

$dF(X)=(X,du(X))=(X,g(\nabla u,X)),$

where $\nabla u$ is the $g$-gradient of $u$. We consider the metric
$\gamma=F^{*}\left\langle,\right\rangle$ on $M$. By assumption, $\gamma$ is
non-degenerate. The upward normal vector field is

$\nu=\frac{1}{W}(-\varepsilon^{\prime}\nabla u,1),\quad
W=+\sqrt{\varepsilon\big{(}\varepsilon^{\prime}+|\nabla u|_{g}^{2}\big{)}},$
(1)

where we are assuming that
$\varepsilon:=\mathrm{sign}\big{(}\varepsilon^{\prime}+|\nabla
u|_{g}^{2}\big{)}=\pm 1$ is a constant function on the whole $M$. Note that
$\left\langle\nu,\nu\right\rangle=\varepsilon$. It is important to remark our
definitions of the mean curvature vector and function. In our setting, if
$II_{F}$ is the second fundamental form, the mean curvature vector is

$\vec{H}_{F}=\mathrm{trace}_{g_{M}}(II_{F})=\varepsilon H\nu_{F},$

where $H$ is the mean curvature function. Similar notations will be used for
other hypersurfaces along the paper.

The following proposition is well-known in the Euclidean setting (see for
example [10]). The proof is similar in the pseudo-Riemannian context, but we
include it for completeness and discuss the necessary modifications.

######  Proposition 2.1.

Under the previous setting, $F$ is a (vertical) translating soliton if, and
only if, function $u$ satisfies

$\displaystyle\mathrm{div}\left(\frac{\nabla
u}{\sqrt{\varepsilon\big{(}\varepsilon^{\prime}+|\nabla
u|_{g}^{2}\big{)}}}\right)=\frac{1}{\sqrt{\varepsilon\big{(}\varepsilon^{\prime}+|\nabla
u|_{g}^{2}\big{)}}}=H.$ (2)

###### Proof.

Take a local $g$-orthonormal frame $B=(e_{1},\ldots,e_{n})$ on $M$, such that
$g(e_{i},e_{j})=\varepsilon_{i}\delta_{ij}$ for any $i,j=1,\dots n$, with
$\varepsilon_{i}=\pm 1$ depending on the signature of $g$. We denote
$u_{i}=du(e_{i})$, $i=1,\ldots,n$. For this frame, we compute the induced
metric $\gamma=F^{*}\left\langle,\right\rangle$, so that the coefficients of
the Gram matrix and of its inverse are

$\gamma_{ij}=\gamma(e_{i},e_{j})=\varepsilon_{i}\delta_{ij}+\varepsilon^{\prime}u_{i}u_{j},\quad\gamma^{ij}=\varepsilon_{i}\delta_{ij}-\frac{\varepsilon}{W^{2}}\varepsilon_{i}\varepsilon_{j}u_{i}u_{j}.$

Let $\bar{\nabla}$ the Levi-Civita connection of
$(\bar{M},\left\langle,\right\rangle)$. We recall O’Neill’s book [13], and its
equations for the Levi-Civita connection of a (warped) product. Then,

$\displaystyle
H=\left\langle\vec{H}_{F},\nu\right\rangle=\left\langle\partial_{t}^{\perp},\nu\right\rangle=\left\langle\partial_{t},\nu\right\rangle=\left\langle\nu,\mathrm{tr}_{\gamma}(II_{F})\right\rangle=\sum_{i,j}\gamma^{ij}\left\langle\nu,\bar{\nabla}_{dF(e_{i})}dF(e_{j})\right\rangle$
$\displaystyle=\sum_{i,j}\gamma^{ij}\left\langle\nu,\big{(}\nabla_{e_{i}}e_{j},0\big{)}+e_{i}(u_{j})\partial_{t}\right\rangle=\sum_{i,j}\gamma^{ij}e_{i}(u_{j})\left\langle\nu,\partial_{t}\right\rangle,$

so that

$\displaystyle
1=\sum_{i,j}\gamma^{ij}e_{i}(u_{j})=\sum_{i,j}\left(\varepsilon_{i}\delta_{ij}-\frac{\varepsilon_{i}\varepsilon_{j}u_{i}u_{j}}{\varepsilon
W^{2}}\right)e_{i}(u_{j})=\mathrm{div}(\nabla
u)-\frac{\varepsilon}{W^{2}}\sum_{i,j}\varepsilon_{i}\varepsilon_{j}u_{i}u_{j}e_{i}(u_{j}).$

By recalling that $W^{2}=\varepsilon(\varepsilon^{\prime}+g(\nabla u,\nabla
u))$, we compute

$\displaystyle 2W\left\langle\nabla W,\nabla
u\right\rangle=\left\langle\nabla(W^{2}),\nabla
u\right\rangle=\varepsilon\sum_{i}\varepsilon_{i}u_{i}e_{i}\left(g(\nabla
u,\nabla u)\right)$
$\displaystyle=\varepsilon\sum_{i}\varepsilon_{i}u_{i}\sum_{j}\varepsilon_{j}e_{i}(u_{j}^{2})=2\varepsilon\sum_{i,j}\varepsilon_{i}\varepsilon_{j}u_{i}u_{j}e_{i}(u_{j}).$

Coming back

$\displaystyle 1=\mathrm{div}(\nabla u)-\frac{1}{W}\left\langle\nabla W,\nabla
u\right\rangle,$ $\displaystyle\frac{1}{W}=\frac{\mathrm{div}(\nabla
u)}{W}-\frac{1}{W^{2}}\left\langle\nabla W,\nabla
u\right\rangle=\frac{\mathrm{div}(\nabla
u)}{W}+\left\langle\nabla\big{(}1/W\big{)},\nabla
u\right\rangle=\mathrm{div}\left(\frac{\nabla u}{W}\right).$

∎

######  Corollary 2.1.

Let $(M,g)$ be a compact orientable manifold without boundary. Then, $M$ does
not admit any globally defined, non-degenerate, graphical, spacelike
Translating Soliton
$F:M\rightarrow(M\times\mathbb{R},g+\varepsilon^{\prime}dt^{2})$.

###### Proof.

Assume that there exists a globally defined graphical Translating Soliton on
$M$. Then, for some function $u\in C^{2}(M)$, equation (2) holds true. By
using the volume form $d\mu_{g}$, we obtain

$0=\int_{M}\mathrm{div}\left(\frac{\nabla
u}{\sqrt{\varepsilon(\varepsilon^{\prime}+|\nabla
u|^{2}_{g})}}\right)d\mu_{g}=\int_{M}\frac{1}{\sqrt{\varepsilon(\varepsilon^{\prime}+|\nabla
u|^{2}_{g})}}d\mu_{g}>0.$

This is a contradiction. ∎

##  3 Submersions and Lie Groups

Our next target consists of studying when the equation of the mean curvature
flow can be reduced to a particular ODE. The main geometrical technique
consists of using a Lie group acting by isomorphisms whose orbits are
codimension one sumbanifolds, namely cohomogenity one actions. We choose the
case when the image of the natural projection is going to be identified with
an open interval. This is not a big deal, since one can remove the non-regular
orbits, but later it might be necessary to see the possible extension. Before
starting, we need the following technical lemma.

######  Lemma 3.1.

Take $\varepsilon^{\prime},\tilde{\varepsilon}\in\\{\pm 1\\}$, $I$ an open
interval and $\pi:(M,g_{{}_{M}})\rightarrow(I,\tilde{\varepsilon}ds^{2})$ a
pseudo-Riemannian submersion such that each fiber $\pi^{-1}(s)$ has constant
mean curvature $h(s)$ w.r.t. $-\nabla\pi$. Given $f\in C^{2}(I)$, define
$u=f\circ\pi\in C^{2}(M)$. Assume that
$\varepsilon=\mathrm{sign}(\varepsilon^{\prime}+\tilde{\varepsilon}(f^{\prime}\circ\pi)^{2})=\pm
1$ is a constant function.

  1. 1.

$h:I\rightarrow\mathbb{R}$ is a smooth function, and such that
$\mathrm{div}(\nabla\pi)=h\circ\pi$.

  2. 2.

$\nabla u=(f^{\prime}\circ\pi)\nabla\pi$, $|\nabla
u|_{g}^{2}=\tilde{\varepsilon}(f^{\prime}\circ\pi)^{2}$, $\mathrm{div}(\nabla
u)=\tilde{\varepsilon}(f^{\prime\prime}\circ\pi)+(f^{\prime}\circ\pi)(h\circ\pi)$.

  3. 3.

If
$W=+\sqrt{\varepsilon\big{(}\varepsilon^{\prime}+\tilde{\varepsilon}(f^{\prime}(\pi))^{2}\big{)}}\,>0$,
then $\nabla
W=\displaystyle\frac{\varepsilon\tilde{\varepsilon}f^{\prime}(\pi)f^{\prime\prime}(\pi)}{W}\nabla\pi.$

###### Proof.

First, given a local orthonormal frame $(e_{1},\ldots,e_{n})$ such that
$e_{n}=\nabla\pi$ and the vector fields $\\{e_{i}\\}_{i=1}^{n-1}$ are tangent
to the fibers, with $\varepsilon_{i}=g(e_{i},e_{i})$, then

$\mathrm{div}(\nabla\pi)=\sum_{i=1}^{n}\varepsilon_{i}g(\nabla_{e_{i}}\nabla\pi,e_{i})=\sum_{i=1}^{n-1}\varepsilon_{i}g(\nabla_{e_{i}}\nabla\pi,e_{i})=h\circ\pi.$

For simpleness, we write $f\circ\pi=f(\pi)$, etc. For the second statement,
given $p\in M$ and $X\in T_{p}M$, we compute

$g_{p}(\nabla(f\circ\pi),X)=d(f\circ\pi)_{p}X=df_{\pi(p)}(d\pi_{p}(X))=f^{\prime}(\pi(p))g_{p}(\nabla\pi,X),$

and we point out that $|\nabla\pi|_{g}^{2}=\tilde{\varepsilon}$. Next,

$\mathrm{div}(\nabla
u)=\mathrm{div}(f^{\prime}(\pi)\nabla\pi)=g(\nabla(f^{\prime}\circ\pi),\nabla\pi)+f^{\prime}(\pi)\mathrm{div}(\nabla\pi)=\tilde{\varepsilon}f^{\prime\prime}(\pi)+f^{\prime}(\pi)h(\pi).$

Finally, $2W\nabla W=\nabla(W^{2})=2\tilde{\varepsilon}\varepsilon
f^{\prime}(\pi)f^{\prime\prime}(\pi)\nabla\pi.$ ∎

######  Proposition 3.1.

Let $(M,g)$ be a connected pseudo-Riemannian manifold and $I$ an open
interval. Let $\Sigma$ be a Lie group acting by isometries on $M$ and
$\pi:M\rightarrow I$ be a submersion such that the fibers of $\pi$ are orbits
of the action, and $\nabla\pi$ is never zero or lightlike. Then, there exist a
constant $\tilde{\varepsilon}=\pm 1$ and a smooth bijective map
$v:I\rightarrow J\subset\mathbb{R}$, $J$ an interval, such that
$v\circ\pi:(M,g)\rightarrow(J,\tilde{\varepsilon}ds^{2})$ is a pseudo-
Riemannian submersion with constant mean curvature fibers.

###### Proof.

Since $\pi$ is a submersion, and $|\nabla\pi|_{g}^{2}\neq 0$, there exists a
smooth function $z:I\rightarrow\mathbb{R}\backslash\\{0\\}$ such that
$(z\circ\pi)^{2}=\tilde{\varepsilon}|\nabla\pi|_{g}^{2}$, for a constant
$\tilde{\varepsilon}=\pm 1$. Next, we consider a function
$v:I\rightarrow\mathbb{R}$ such that $v^{\prime}=1/z$. As $v^{\prime}$ has
constant sign, $v$ is injective, and we restrict it to its image,
$v:I\rightarrow J$. By Lemma 3.1,
$|\nabla(v\circ\pi)|^{2}_{g}=\tilde{\varepsilon}$. In particular,
$v\circ\pi:(M,g)\rightarrow(J,\tilde{\varepsilon}ds^{2})$ is a pseudo-
Riemannian submersion. Note that the fibers of $v\circ\pi$ are the same as the
fibers of $\pi$, which are the orbits of $\Sigma$. As it acts by isometries,
they have constant mean curvature w.r.t. $-\nabla(v\circ\pi)$. ∎

From now on, we assume that the action of the Lie group $\Sigma$ on $(M,g)$ by
isometries is proper, and that at least one of the orbits is of codimension
one. These are the well-known cohomogeneity one $\Sigma$-manifolds. According
to [1] (and references therein), there is another Riemannian metric $g_{R}$
such that $\Sigma$ acts on $(M,g_{R})$ by isometries, and the quotient
$M/\Sigma$ is diffeomorphic to an interval or to
$\mathbb{S}^{1}\equiv\mathbb{R}/\mathbb{Z}$. By removing the possible singular
orbits (if any, up to two), we can assume that $M/\Sigma=:I$ is (diffeomorphic
to) an open interval. This works at differentiable level, so we can return to
our original metric $g$, and the quotient is still an open interval. Let
$\pi:M\to I\equiv M/\Sigma$ be the projection. We need $\nabla\pi$ not to be
zero or lightlike. By Lemma 3.1, we can assume that
$g_{{}_{M}}(\nabla\pi,\nabla\pi)=\pm 1=:\tilde{\varepsilon}$, making $\pi$ a
pseudo-Riemannian submersion. Each fiber of $\pi$ will be an orbit of
$\Sigma$. And because it acts by isometries, then each orbit $\pi^{-1}\\{s\\}$
will be of constant mean curvature $h(s)$ w.r.t $-\nabla\pi$. Thus, we have an
associated smooth function $h:I\to\mathbb{R}$. We need to bear in mind all
this setting, so that we introduce the following definition.

######  Definition 3.1.

Under the previous setting, let $\Sigma_{o}$ one of the stabilizers of
$\Sigma$. Let $\beta:I\to M$ be a unit curve orthogonal to all orbits of
$\Sigma$, such that the map

$\phi:I\times(\Sigma/\Sigma_{o})\to
M,\quad\phi(s,[\sigma])=\sigma\cdot\beta(s),$ (3)

is a diffeomorphism and $\pi\circ\phi(s,[\sigma])=s$. We will say that (3) is
a decomposition of $M$ by $\Sigma$, or that $M$ is decomposable by $\Sigma$,
with associated function $h:I\to\mathbb{R}$. We cannot forget the submersion
$\pi:(M,g)\to(I,\tilde{\varepsilon}ds^{2})$.

######  Remark 3.1.

For the sake of simpleness, sometimes is it better to use
$\phi:I\times\Sigma\to M$. But then, $\phi(s,-):\Sigma\to M$ might not be an
immersion, although the image is again an orbit.$\Box$

We recall that a function $u:M\rightarrow\mathbb{R}$ is called invariant by
the Lie group $\Sigma$, or also $\Sigma$-invariant, if it satisfies

$u:M\rightarrow\mathbb{R},\quad u(x)=u(\sigma\cdot x),\ \forall x\in M,\
\forall\sigma\in\Sigma.$ (4)

Accordingly, we say that a graphical translating soliton is invariant by the
Lie group $\Sigma$, or also $\Sigma$-invariant, when its graph map is
invariant by $\Sigma$. Similarly, a translating soliton $F:\Omega\subset M\to
M\times\mathbb{R}$ is called $\Sigma$-invariant when $\sigma\cdot F(x)\in
F(\Omega)$ for each $x\in\Omega$ and each $\sigma\in\Sigma$.

######  Theorem 3.1.

Let $(M,g)$ be a pseudo-Riemannian manifold, decomposable by $\Sigma$, as in
Definition 3.1. Given $\varepsilon^{\prime}=\pm 1$, $f:I\to\mathbb{R}$ a
smooth function, construct the graph map
$F:M\rightarrow(M\times\mathbb{R},\left\langle,\right\rangle=g_{{}_{M}}+\varepsilon^{\prime}dt^{2})$,
$F(x)=(x,u(x))$, $u=f\circ\pi$. Then, $F$ is a $\Sigma$-invariant translating
soliton if, and only if, $f$ is a solution to

$f^{\prime\prime}(s)=\big{(}\tilde{\varepsilon}+\varepsilon^{\prime}(f^{\prime}(s))^{2}\big{)}\big{(}1-f^{\prime}(s)h(s)\big{)}.$
(5)

###### Proof of Theorem 3.1.

We denote $u=f\circ\pi$. Using Lemma 3.1 we get directly from equation (2.1),

$\displaystyle\frac{1}{W}=\mathrm{div}\left(\frac{\nabla
u}{W}\right)=\frac{\mathrm{div}(\nabla u)}{W}-\frac{g(\nabla W,\nabla
u)}{W^{2}}$
$\displaystyle=\frac{\tilde{\varepsilon}f^{\prime\prime}(\pi)+f^{\prime}(\pi)h(\pi)}{W}-\frac{1}{W^{2}}g\left(\frac{\tilde{\varepsilon}\varepsilon
f^{\prime}(\pi)f^{\prime\prime}(\pi)}{W}\nabla\pi,f^{\prime}(\pi)\nabla\pi\right)$
$\displaystyle=\frac{\tilde{\varepsilon}f^{\prime\prime}(\pi)+f^{\prime}(\pi)h(\pi)}{W}-\frac{\tilde{\varepsilon}\varepsilon
f^{\prime}(\pi)^{2}f^{\prime\prime}(\pi)}{W^{3}}g\left(\nabla\pi,\nabla\pi\right)$
$\displaystyle=\frac{1}{W}\left(f^{\prime}(\pi)h(\pi)+f^{\prime\prime}(\pi)\frac{\varepsilon^{\prime}\tilde{\varepsilon}}{\varepsilon^{\prime}+\tilde{\varepsilon}f^{\prime}(\pi)^{2}}\right).$

Therefore, we conclude that (2) reduces to

$f^{\prime}(\pi)h(\pi)+\frac{1}{\tilde{\varepsilon}+\varepsilon^{\prime}f^{\prime}(\pi)^{2}}f^{\prime\prime}(\pi)=1.$

In other words, function $f$ is a solution to (5). ∎

######  Remark 3.2.

Another approach is to start with a $\Sigma$-invariant function
$u:M\to\mathbb{R}$. But then, we immediately obtain another function
$f:I\to\mathbb{R}$ such that $u=f\circ\pi$.

On the other hand, we can extend the action of $\Sigma$ to $M\times\mathbb{R}$
as follows:

$\displaystyle\bar{M}=M\times\mathbb{R},\left\langle\,,\right\rangle=g+\varepsilon\mathrm{d}t^{2},\
\varepsilon=\pm 1,\quad\Sigma\times\bar{M}\to\bar{M},\
(\sigma,(x,t))\mapsto(\sigma\cdot x,t).$

In fact, given any curve $\delta:J\subset I\to I\times\mathbb{R}$,
$\delta=(\delta_{1},\delta_{2})$, we can construct a map whose image is
$\Sigma$-invariant, namely

$F:J\times(\Sigma/\Sigma_{o})\to M\times\mathbb{R},\quad
F(s,[\sigma])=\big{(}\phi(\delta_{1}(s),[\sigma]),\delta_{2}(s)\big{)}.$ (6)

We will now give some general results about the behavior of solutions. This
next result states the existence of the so-called wing-like solutions.

######  Corollary 3.1.

Assume there $M$ is decomposable as in Definition 3.1, where $h$ is the mean
curvature of the orbits of the action. Consider $s_{o}\in I$ such that either
$h(s_{o})\neq 0$, or $s_{o}$ is an isolated zero of $h$. Then, for each
$y_{o}\in\mathbb{R}$, there exist a real number $\rho>0$ and a translating
soliton $F:(y_{o}-\rho,y_{o}+\rho)\times(\Sigma/\Sigma_{o})\rightarrow\bar{M}$
such that it is the union of two graphical translating solitons.

###### Proof.

Choose $y_{o}\in\mathbb{R}$. We consider the following IVP:

$\alpha^{\prime\prime}(y)=\left(\varepsilon^{\prime}+\tilde{\varepsilon}\alpha^{\prime}(y)^{2}\right)\left(h(\alpha(y))-\alpha^{\prime}(y)\right),\quad\alpha^{\prime}(y_{o})=0,\
\alpha(y_{o})=s_{o}\in I.$

As usual, there exists a smooth solution
$\alpha:(y_{o}-\rho,y_{o}+\rho)\rightarrow\mathbb{R}$. Note that $y_{o}$ is a
critical point of $\alpha$ and
$\alpha^{\prime\prime}(y_{o})=\varepsilon^{\prime}h(s_{o}).$
Case $h(s_{o})\neq 0$: Then, $y_{o}$ is an extremum of $\alpha$. The
restrictions $\alpha_{+}=\alpha|_{(y_{o},y_{o}+\rho)}$ and
$\alpha_{-}=\alpha|_{(y_{o}-\rho,y_{o})}$ will be injective, by reducing
$\rho$ if necessary. Construct their inverse functions $f_{+}=\alpha_{+}^{-1}$
and $f_{-}=\alpha_{-}^{-1}$. We just have to show that $f_{+}$ and $f_{-}$
satisfy (5). To do so, we put $f_{+}(\alpha(y))=y$, and therefore

$\displaystyle 1=f_{+}^{\prime}(\alpha(y))\,\alpha^{\prime}(y),\quad
0=f_{+}^{\prime\prime}(\alpha(y))\,\alpha^{\prime}(y)^{2}+f_{+}^{\prime}(\alpha(y))\,\alpha^{\prime\prime}(y),$
$\displaystyle
f_{+}^{\prime\prime}(\alpha(y))\,\alpha^{\prime}(y)^{2}=-f_{+}^{\prime}(\alpha(y))\big{(}\varepsilon^{\prime}+\tilde{\varepsilon}\alpha^{\prime}(y)^{2}\big{)}\big{(}h(\alpha(y)-\alpha^{\prime}(y)\big{)}.$

Next, we change $s=\alpha(y)$, and then
$\alpha^{\prime}(y)=1/f^{\prime}_{+}(s)$, so that

$\displaystyle\frac{f_{+}^{\prime\prime}(s)}{f_{+}^{\prime}(s)^{2}}=-f_{+}^{\prime}(s)\left(\varepsilon^{\prime}+\frac{\tilde{\varepsilon}}{f_{+}^{\prime}(s)^{2}}\right)\left(h(s)-\frac{1}{f_{+}^{\prime}(s)}\right)=\frac{1}{f_{+}^{\prime}(s)^{2}}\big{(}\tilde{\varepsilon}+\varepsilon^{\prime}f_{+}(s)^{2}\big{)}\big{(}1-h(s)f_{+}(s)\big{)}.$

A similar computation holds for $f_{-}$. The union of the corresponding
graphical translating solitons and their common boundary provide a smooth
translating soliton, because $\alpha$ is a smooth map and $f_{+}$, $f_{-}$ are
tools to reparametrize its graph.

Case $s_{o}$ is an isolated zero of $h$: By shrinking $\rho$ if necessary,
then $\alpha^{\prime\prime}(y)\neq 0$ for any $y\neq y_{o}$,
$y\in(y_{o}-\rho,y_{o}+\rho)$. The restriction
$\alpha^{\prime}|_{[y_{o},y_{o}+\rho)}$ will be injective, and therefore,
$\alpha^{\prime}(y)\neq 0$ for any $y\in(y_{o},y_{o}+\rho)$. This makes
$\alpha|_{(y_{o},y_{o}+\rho)}$ also injective. Similarly,
$\alpha|_{(y_{o}-\rho,y_{o})}$ is injective. We continue as in the previous
case $(h(s_{o})\neq 0)$.

In either case, it is possible to obtain a smooth curve
$\tau:(y_{o}-\rho,y_{o}+\rho)\to I\times\mathbb{R}$,
$\tau=(\tau_{1},\tau_{2})$, which is the union of both graphs. Then, our
translating soliton is constructed by using (6). ∎

######  Remark 3.3.

By (3), $\bar{M}$ is diffeomorphic to
$I\times(\Sigma/\Sigma_{o})\times\mathbb{R}$, so the profile curve $\tau$ of
Corollary 3.1 can be embedded into $I\times\\{[\sigma]\\}\times\mathbb{R}$ for
some $[\sigma]\in(\Sigma/\Sigma_{o})$. At a certain point, $\partial_{t}$
($t\in\mathbb{R}$) will be tangent to the image of $\tau$ (as in the proof of
previous lemma). The causal character of the product will therefore influence
the causal character of the translating soliton. Moreover the causal character
of the submersion (i.e. the sign of $\tilde{\varepsilon}$) as well as the sign
of $\varepsilon^{\prime}+\tilde{\varepsilon}f^{\prime 2}$ will determine the
causal character of the winglike soliton. For example, when $M$ is Riemannian,
the product $M\times\mathbb{R}$ is Lorentzian (i.e.
$\varepsilon^{\prime}=-1$), and the submersion is Riemannian (as in the
rotationally invariant example of Section 5), the wing-like translating
solitons have to be timelike, since for spacelike solitons,
$\alpha^{\prime}(y_{0})$ cannot be $0$ for any $y_{0}$. $\Box$

######  Corollary 3.2.

Assume that $M$ is decomposable as in Definition 3.1. Take
$\tilde{\varepsilon},\varepsilon^{\prime}\in\\{\pm 1\\}$ such that
$\tilde{\varepsilon}\varepsilon^{\prime}=-1$. Given $f$ a local solution to
(5) and $s_{o}\in I$, such that $(f^{\prime}(s_{o}))^{2}<1$, then $f$ can be
globally extended to $f:I\to\mathbb{R}$. In particular, the associated
$\Sigma$-invariant translating soliton is graphical and it can be globally
defined on $M$.

###### Proof.

By Theorem 3.1, any $\Sigma$-invariant translating soliton will be constructed
by a solution $f$ to (5). We make the change $w=f^{\prime}$, so it reduces to
$w^{\prime}(s)=(\tilde{\varepsilon}+\varepsilon^{\prime}w(s)^{2})(1-w(s)h(s))$.
Since $\tilde{\varepsilon}\varepsilon^{\prime}=-1$, the constant functions
$w(s)=\pm 1$ are solutions to this differential equation. Then, given an
initial condition $(s_{o},f_{1})\in I\times(-1,1)$, there exists a local
solution $w:(s_{o}-\rho,s_{o}+\rho)\rightarrow\mathbb{R}$ such that
$w(s_{o})=f_{1}$ and $|w|<1$. By the uniqueness of solutions to IVP, $w$
cannot reach the values $\pm 1$, and so, it can be globally extended to
$w:I\to(-1,1)$. We define $f(s)=f_{o}+\int_{s_{o}}^{s}w(x)dx$ for some
$f_{o}\in\mathbb{R}$. By using the curve $\delta:I\to I\times\mathbb{R}$,
$\delta(s)=(s,f(s))$, we can (re)construct our $\Sigma$-invariant translating
soliton by (6). ∎

######  Remark 3.4.

This section generalizes several classical results in a general geometric
context. A well-known example is the case of a rotationally invariant
translating soliton in $\mathbb{R}^{n+1}$. We consider the Lie group $SO(n)$
acting by isometries on $\mathbb{R}^{n}$ which gives rise to the Riemannian
submersion
$\pi:\mathbb{R}^{n}\backslash\\{0\\}\rightarrow\mathbb{R},\,\pi(x)=\|x\|$, and
the Riemannian product
$\bar{M}=(\mathbb{R}^{n}\times\mathbb{R},g^{\mathbb{R}^{n}}+dt^{2})$. An easy
computation shows that the mean curvature of the fiber $\pi^{-1}(s)$ is
$h(s)=(n-1)/s$. Using Theorem 3.1, we recover the well-known ODE for
rotationally invariant translating solitons
$f^{\prime\prime}=(1+f^{2})(1-\frac{n-1}{s}f^{\prime})$, studied in [5], where
such solitons are classified. In that paper, it is shown that there exist only
two types: a globally defined graphical soliton, the so-called bowl soliton or
translating paraboloid, which can be seen as a particular application of our
Corollary 3.2, and a family of non-graphical wing-like translators or
translating catenoids, obtained by gluing two graphical solutions, which are a
special case of Corollary 3.1. $\Box$

##  4 All solutions to an ODE

In this section, we will need some tools which can be found in the book [17].
Our targent is to find all local solutions to the following equation, for
$s>0$,

$f^{\prime\prime}(s)=\big{(}1-f^{\prime}(s)^{2}\big{)}\Big{(}1-\frac{n-1}{s}f^{\prime}(s)\Big{)}.$
(7)

or rather

$w=f^{\prime},\quad
w^{\prime}=\big{(}1-w^{2}\big{)}\Big{(}1-\frac{n-1}{s}w\Big{)}.$ (8)

where $n\geq 2$ is a natural number. Clearly, there are two degenerate
examples, namely, $\hat{f}_{\pm}:(0,\infty)\rightarrow\mathbb{R}$,
$\hat{f}_{\pm}(s)=\pm s+f_{o}$, $f_{o}\in\mathbb{R}$, that is,
$\hat{w}_{\pm}(s)=\pm 1$.

###  4.1 The case when $|x|<1$

The following particular example was studied in [8].

######  Lemma 4.1.

There exists a unique function $f_{B}:\mathbb{R}\to\mathbb{R}$ such that $f$
is even, analytical, and such that its derivative $w_{B}=f^{\prime}$ is the
unique solution to the boundary problem

$w^{\prime}=\big{(}1-w^{2}\big{)}\Big{(}1-\frac{n-1}{s}w\Big{)},\quad w(0)=0.$

We consider the following open domains

$\displaystyle\Omega=\\{(s,x)\in\mathbb{R}^{2}:s>0,\ |x|<1\\},$
$\displaystyle\Omega_{1}=\\{(s,x)\in\Omega:w_{B}(s)>x\\},\quad\Omega_{2}=\\{(s,x)\in\Omega:w_{B}(s)<x\\}.$

######  Lemma 4.2.

Any unextendable solution $w$ inside $\Omega$ is globally defined,
$w:(0,\infty)\to\mathbb{R}$, and it is one of the following:

Case I:

The solution $w_{B}=f^{\prime}:[0,\infty)\to\mathbb{R}$ of Lemma 4.1;

Case II:

$w>w_{B}$ everywhere, with one critical point $s_{1}>0$, such that
$w(s_{1})=s_{1}/(n-1)$, and $\lim_{s\to 0}w(s)=1$.

Case III:

$w<w_{B}$ everywhere, without critical points, strictly increasing, with no
critical points, and $\lim_{s\to 0}w(s)=-1$.

In addition, all of them (also $w_{B}$) satisfy $\lim_{s\to\infty}w(s)=1$.

###### Proof.

Take $(s_{o},w_{o})\in\Omega$ initial conditions for (8), $w(s_{o})=w_{o}$,
and let $w$ be the solution. By the solutions $\hat{w}_{\pm}$, it will be
globally defined $w:(0,\infty)\rightarrow\mathbb{R}$ and its graph will remain
inside $\Omega$. Then, $w^{\prime}(s)=0$ for some point $s\in(0,\infty)$ if,
and only if, either $w(s)=\pm 1$ (excluded) or $w(s)=s/(n-1)$, at some point
$s\in(0,\infty)$. In other words, $w$ will admit a critical point if its graph
intersects the line $r$ of equation $x=s/(n-1)$ horizontally. As the line $r$
is a monotonic curve, the solution can only intersects it at most once due to
our ODE.

Case I: We already know the solution
$w_{B}=f^{\prime}:[0,\infty)\to\mathbb{R}$ of Lemma 4.1.

Case II: Take initial conditions $(s_{o},w_{o})\in\Omega_{1}$. The associated
solution $w:(0,\infty)\rightarrow\mathbb{R}$ will remain in $\Omega_{1}$, due
to the uniqueness of solutions to IVP. As $w(s)<w_{B}(s)<s/(n-1)$ for any
$s>0$, then $w$ has no critical points, and $w^{\prime}(s_{o})>0$, so it is
strictly increasing for any $s>0$.

Since, $w(s)<1$ for any $s>0$, there exists
$\lim_{s\to+\infty}w(s)=w_{1}\in(-1,1]$ and
$\lim_{s\to+\infty}w^{\prime}(s)=0$. However,
$0=\lim_{s\to+\infty}w^{\prime}(s)=\lim_{s\to+\infty}(1-w(s)^{2})(1-(n-1)w(s)/s)$
$=1-w_{1}^{2}$. Therefore, $\lim_{s\to+\infty}w(s)=1.$

As $w_{B}(0)=0$, and $w(s)<w_{B}(s)$, there exists $s_{1}>0$ such that
$w(s_{1})=0$. We want to compute the limit of $w(s)$ and of $w^{\prime}(s)$
when $s\to 0$. As $-1<w(s)$ and $w$ is strictly increasing, then $\lim_{s\to
0}w(s)=w_{2}\in[-1,0)$. Define now $z:\mathbb{R}\rightarrow\mathbb{R}$,
$z(t)=w(e^{t})$. Then, $z$ is still strictly increasing, and
$z^{\prime}(t)=(1-z(t)^{2})(e^{t}-(n-1)z(t))$. Clearly,
$w_{2}=\lim_{t\to-\infty}z(t)$ and this implies
$0=\lim_{t\to-\infty}z^{\prime}(t)=\lim_{t\to-\infty}(1-z(t)^{2})(e^{t}-(n-1)z(t))=-(n-1)(1-w_{2}^{2})w_{2}$.
This means that $w_{2}=-1$. To summarize,

$\lim_{s\to 0}w(s)=-1,\quad\lim_{s\to 0}w^{\prime}(s)=0.$

Again, by the uniqueness of solutions to IVP, we can parametrize this family
by considering the initial values, $(0,+\infty)\ni s_{0}$, that is
$(s_{0},0)\in\Omega_{1}$, and the union of all the graphs of the solutions
will foliate $\Omega_{1}$.

Case III: Take initial conditions $(s_{o},w_{o})\in\Omega_{2}$. The associated
solution $w:(0,\infty)\rightarrow\mathbb{R}$ will remain in $\Omega_{2}$,
similarly to Case II. If $w^{\prime}(s)\neq 0$ for all $s$, its graph cannot
cross the line $r$ of equation $x=s/(n-1)$ and it is strictly above the graph
of $w_{B}=f_{B}^{\prime}$. But then, either $w(s)>s/(n-1)$ for any $s>0$,
which is impossible since $w$ is bounded by $1$ and defined for all $s>0$, and
the line $r$ crosses the horizontal line $x=+1$; or $s/(n-1)>w(s)>w_{B}(s)$
for any $s>0$, which shows that $\lim_{s\to 0}w(s)=0$. Then, we found a second
solution to the problem (8) with boundary value $w(0)=0$. By the uniqueness of
the solution to this boundary problem, we get to a contradiction. Thus, there
exists a unique $s_{1}>0$ such that $w(s_{1})=s_{1}/(n-1)$, which is the only
critical point. In this way, we parametrize this family by the open segment
$r\cap\Omega_{2}$. It is easy to compute that

$w^{\prime\prime}(s_{1})=\frac{-1}{s_{1}}\left(1-\frac{s_{1}^{2}}{(n-1)^{2}}\right)<0.$

Therefore, $s_{1}$ is a absolute minimum of $w$. This means that $w$ is
strictly decreasing when $0<s<s_{1}$ and strictly increasing when $s>s_{1}$.
For big $s>0$, we know $w_{B}(s)<w(s)<1$, which shows
$\lim_{s\to+\infty}w(s)=1.$ By similar computations to Case II, $\lim_{s\to
0}w(s)=1$. ∎

The following picture illustrates the solutions to (8) such that
$-1<w=f^{\prime}<1$, for $n=3$, made with wxMaxima©.

###  4.2 The case when $|x|>1$

For the following study, we define the open domains

$\Gamma_{-}=\\{(s,x)\in\mathbb{R}^{2}:s>0,\
x<-1\\},\quad\Gamma_{+}=\\{(s,x)\in\mathbb{R}^{2}:s>0,\ x>1\\}.$

######  Lemma 4.3.

(Case IV) Any unextendable solution $w$ contained in $\Gamma_{-}$ is defined
on $w:(0,s_{1})\to\mathbb{R}$, for some $s_{1}>0$, it is strictly increasing,
with $\lim_{s\to 0}w(s)=1$ and having a finite time blow-up at $s_{1}$.

###### Proof.

Let $(s_{0},w_{o})\in\Gamma_{-}$ initial solutions for (8),
$w(s_{o})=w_{o}<-1$. A local solution
$w:(s_{o}-\delta,s_{o}+\delta)\to\mathbb{R}$ stays inside of $\Gamma_{-}$.
Since the line $r$ does not intersects $\Gamma_{-}$, $w$ cannot have any
critical point. And due to

$w^{\prime}(s_{o})=(1-w_{o}^{2})\left(1-\frac{(n-1)w_{o}}{s_{o}}\right)<0,$

then $w$ is always strictly decreasing. As it is bounded by the constant
solution $\hat{w}_{-1}(s)=-1$, $w$ can be extended to $(0,s_{o}+\delta)$.
Also, a simple computation shows $\lim_{s\to 0}w(s)=-1$. Next, since
$1-(n-1)w_{o}/s_{o}>1$, for any $s\geq s_{o}$ we have,

$(1-w(s)^{2})\left(1-\frac{(n-1)w(s)}{s}\right)\leq 1-w(s)^{2}<0.$

Therefore, the ODE (8) is dominated by the ODE $z^{\prime}(s)=1-z(s)^{2}$,
whose general solution is
$z(s)=\coth(s+\rho)=\frac{e^{2(s+\rho)}+1}{e^{2(s+\rho)}-1}$,
$\rho\in\mathbb{R}$. We choose $\rho$ such that $z(s_{o})=w_{o}=w(s_{o})<-1$.
From here, $e^{2(s_{o}+\rho)}-1<0$ and $\rho<0$, so that $z$ can be extended
at most to $(0,-\rho)$. But as $w^{\prime}(s)\leq z^{\prime}(s)<0$, then $w$
has a finite-time blow-up at some $s_{1}<-\rho$. ∎

######  Lemma 4.4.

Any unextandable solution to (8) included in $\Gamma_{+}$ is one of the
following:

Case V:

There exists a unique $\mathbf{w}:(0,+\infty)\to\mathbb{R}$ which is strictly
increasing, asymptotic to the line $r$ of equation $x=s/(n-1)$ at infinity,
and $\mathbf{w}(s)>s/(n-1)$ for any $s>0$.

Case VI:

$w:(0,+\infty)\to\mathbb{R}$ is globally defined, with just one critical point
$s_{1}>s_{o}$, $\lim_{s\to+\infty}w(s)=1$, $w(s)<\mathbf{w}(s)$ for any $s>0$.

Case VII:

$w:(0,s_{1})\to\mathbb{R}$ has a finite time blow up at certain $s_{1}>s_{o}$,
without critical points, and $w(s)>\mathbf{w}(s)$ for any $s<s_{1}$.

In addition, all of them satisfy (also $\mathbf{w}$), $\lim_{s\to 0}w(s)=1$
and $\lim_{s\to 0}w^{\prime}(s)=0$.

###### Proof.

Take $(s_{o},w_{o})\in\Gamma_{+}$ initial conditions for (8), and let $w$ be
the solution such that $w(s_{o})=w_{o}$. Its graph will remain inside
$\Gamma_{+}$. Also, $w$ will have a critical point if its graph intersects the
line $r$ horizontally ($w^{\prime}(s_{1})=0$ for some $s_{1}>0$), i. e,
$w(s_{1})=s_{1}/(n-1)$. For any point $s>s_{1}$, $w^{\prime}(s)<0$, whereas
for any $s<s_{1}$, $w^{\prime}(s)>0$ holds. In such case, $s_{1}$ will be the
only critical point, and its absolute maximum.

Step 1: Consider $(s_{o},w_{o})\in\Gamma_{+}$ such that $w_{o}=s_{o}/(n-1)$,
and a solution to (8) around $s_{o}$, namely
$w:(s_{o}-\delta,s_{o}+\delta)\rightarrow\mathbb{R}$, with
$w(s_{o})=w_{o}=s_{o}/(n-1)$. By (8), $w_{o}^{\prime}(s_{0})=0$, and it is its
unique critical point.

Then, $w$ will always be decreasing for $s>s_{0}$. By the barrier solution
$w_{1}(s)=+1$, $w$ can be extended to
$w:(s_{o}-\delta,+\infty)\rightarrow\mathbb{R}$, with
$\lim_{s\to+\infty}w(s)=w_{1}\geq 1$, so that
$0=\lim_{s\to+\infty}w^{\prime}(s)=\lim_{s\to+\infty}(1-w(s)^{2})\big{(}1-(n-1)w(s)/s\big{)}=1-w_{1}^{2}$.
This means

$\lim_{s\to+\infty}w(s)=+1.$

On the other hand, $w$ will always be increasing for $s<s_{o}$. Hence we can
extend $w:(0,+\infty)\to\mathbb{R}$, $w$ increasing on $(0,s_{o})$, and
$w(0)=w_{2}\geq 1$. As in Case II, by the change of variable $s=e^{t}$, we get
immediately that $\lim_{s\to 0}w(s)=+1.$

Step 2: As in Corollary 3.1, we get for each $s_{0}>0$ a solution
$w=f_{-}^{\prime}$ such that $\lim_{s\to s_{0}}w(s)=+\infty$, i.e. with finite
time blow up in $s_{0}$. As in step 1, $w$ can be extended to $(0,s_{0})$ and
similarly $\lim_{s\to 0}w(s)=+1.$

Step 3: We want to obtain the first type solution as explained above. We fix
now $s_{o}=n-1$. Define

$J=\\{w_{o}\in\mathbb{R}:w_{o}>1,\ \exists
w:(0,+\infty)\rightarrow\mathbb{R},\ \mathrm{solution\ to}\
\eqref{unordenmenos},\ w(n-1)=w_{o}\\}.$

By step 1, $J\neq\emptyset$. By step 2 we have a solution
$w_{n}:(0,n)\to\mathbb{R}$, such that $\lim_{s\to n}w_{n}(s)=+\infty$. Then,
we have that $j\leq w_{n}(n-1)$ for all $j\in J$. Thus, we can define $A:=\sup
J<w_{n}(n-1)$.

Now define $\mathbf{w}:(0,n-1+\delta)\to\mathbb{R}$ the solution to (8) such
that $\mathbf{w}(n-1)=A$. Assume that $\lim_{s\to s_{1}}\mathbf{w}=+\infty$
for some $s_{1}>n-1$. But then let $s_{2}=s_{1}+\varpi$, $\varpi>0$. By step
2, there exists a blow up solution $w_{2}$, such that $\lim_{s\to
s_{2}}w_{2}(s)=\infty$. Therefore $\mathbf{w}>w_{2}$ for all $s\in(0,s_{1}]$.
This is a contradiction with $A=\sup J$. This means that $\mathbf{w}$ is
globally defined.

Moreover, $\mathbf{w}$ has no critical points. Indeed, assume that there is a
$s_{c}>0$, such that $\mathbf{w}^{\prime}(s_{c})=0$. As before ,
$(s_{c},w(s_{c}))\in r$. Then, by step 1 there exists another globally defined
solution $w$, such that $w^{\prime}(s_{c}+1)=0$, but then by uniqueness of
solutions $w>\mathbf{w}$ for all s, which is a contraction to $A=\sup J$.
Therefore $\mathbf{w}$ is strictly increasing and bounded below by the line
$r$, so that

$\lim_{s\to+\infty}\mathbf{w}(s)=+\infty.$

We are going to show that $\mathbf{w}$ is asymptotic to the line $r$. Let
$K>1$. Suppose that for large enough $s_{1}>0$,
$\mathbf{w}^{\prime}(s)>K/(n-1)$ for any $s\geq s_{1}$. Then, there exists
$c\in\mathbb{R}$ such that for all $s\geq s_{1}$, $\mathbf{w}(s)>Ks/(n-1)+c$.
But now, there exists $c_{o}>0$ such that
$\frac{(n-1)\mathbf{w}(s)}{s}-1>\frac{n-1}{s}\left(\frac{Ks}{n-1}+c\right)-1=K-1+\frac{(n-1)c}{s}>c_{0}>0$
for large enough $s\geq s_{2}\geq s_{1}$. And so,
$\mathbf{w}^{\prime}(s)>c_{o}\left(\mathbf{w}(s)^{2}-1\right)$ for any $s\geq
s_{2}$. Then, (8) dominates the ODE $g^{\prime}=c_{o}(g^{2}-1)$. The solution
to this equation such that $g(s_{2})=w_{o}$ is given by

$g(s)=\frac{e^{-2c_{o}(s-s_{2})}+a}{e^{-2c_{o}(s-s_{2})}-a},\quad
a=\frac{w_{o}-1}{w_{o}+1}<1.$

Since $0<a<1$, $g$ is well defined and positive. Therefore, for any $s\geq
s_{2}$, it holds $\mathbf{w}(s)\geq g(s)$. However, $g$ has a blow up at
$s_{3}=s_{2}-\ln(a)/(2c_{o})$. This shows that $\mathbf{w}$ has a blow up at
$s_{4}<s_{3}$. This is a contradiction. Therefore, since $\mathbf{w}$ cannot
cross $r$, we get using L’Hospital that

$\lim_{s\to+\infty}\mathbf{w}^{\prime}(s)=\frac{1}{n-1}.$

Now we notice that since $\mathbf{w}>1$, (8) is equivalent to the expression

$(n-1)\mathbf{w}(s)-s=\frac{s\mathbf{w}^{\prime}(s)}{\mathbf{w}(s)^{2}-1}.$

But now, as $\mathbf{w}$ is defined until $+\infty$, and without critical
points,

$\displaystyle\lim_{s\to+\infty}[(n-1)\mathbf{w}(s)-s]=\lim_{s\to+\infty}\frac{s\,\mathbf{w}^{\prime}(s)}{\mathbf{w}(s)^{2}-1}=\frac{1}{n-1}\lim_{s\to\infty}\frac{s}{\mathbf{w}(s)^{2}-1}$
$\displaystyle=\frac{1}{n-1}\lim_{s\to+\infty}\frac{1}{2\mathbf{w}(s)\mathbf{w}^{\prime}(s)}=0.$

Summing up, the solution $\mathbf{w}$ is asymptotic to the line $r$.

Since $A=\sup J$, take any other globally defined solution
$w_{B}:(0,+\infty)\to\mathbb{R}$ such that $w_{B}(n-1)<\mathbf{w}(n-1)$ and
$w_{B}<\mathbf{w}$ everywhere. Assume that $w_{B}$ has no critical points. By
similar computations as above, $w_{B}$ will also be asymptotic to the line
$r$. Then, the difference $h=\mathbf{w}-w_{B}$ will satisfy $h>0$ and
$\lim_{s\to+\infty}h(s)=0$. By (8), $\mathbf{w}^{\prime}>w_{B}^{\prime}$
everywhere. But then, $h^{\prime}>0$, that is to say, $h$ is strictly
increasing everywhere. This is a contradition with $\lim_{s\to+\infty}h(s)=0$.

In other words, there is a unique solution $\mathbf{w}$ globally defined on
$(0,+\infty)$, with no critical points. We will keep this notation for the
rest of the paper.

Finally, given $(s_{o},w_{o})\in\Gamma_{+}$, such that
$\mathbf{w}(s_{o})<w_{o}$, consider the solution to (8) such that
$w(s_{o})=w_{0}$. As before, there exists $w:(0,s_{o}+\delta)\to\mathbb{R}$
for some (small) $\delta$. Assume that $w$ is globally defined on
$(0,+\infty)$. Then, $w(n-1)>\mathbf{w}(n-1)=A$. This is a contradiction.
Therefore, any other solution over $\mathbf{w}$ admits a finite-time blow-up.
∎

The following picture illustrates this lemma, for $n=3$, made with wxMaxima©.

##  5 The Action of the Group $SO(n)$

We consider $\mathbb{R}^{n}$ with usual coordinates $x=(x_{1},\ldots,x_{n})$.
To obtain a smooth Riemannian submersion, we restrict it to
$\pi:M=\mathbb{R}^{n}\backslash\\{0\\}\to(0,+\infty)$,
$\pi(x)=\sqrt{\sum_{i}x_{i}^{2}}$. Then,

$\nabla\pi(x)=\sum_{i}\frac{x_{i}}{\pi(x)}\partial_{i}|_{x},\quad|\nabla\pi(x)|^{2}=\sum_{i}\frac{x_{i}^{2}}{\pi(x)^{2}}=1,\quad\mathrm{div}(\nabla\pi)(x)=\frac{n-1}{\pi(x)},$

for any $x\in M$. Note that our map $\pi$ is invariant by the standard action
of $SO(n)$ on $\mathbb{R}^{n}$. Consider now the Minkowski space
$\mathbb{L}^{n+1}=\mathbb{R}^{n}\times\mathbb{R}\ni(x_{1},\ldots,x_{n+1})$
with standard flat metric $g=\sum_{i=1}^{n}dx_{i}^{2}-dx_{n+1}^{2}$. In order
to obtain a graphical $SO(n)$-invariant translating soliton, we consider
$u=f\circ\pi:M\rightarrow\mathbb{R}$, where
$f:I\subset(0,+\infty)\rightarrow\mathbb{R}$ is the desired function, which
will be a solution to the ODE (5), which turns out to be

$f^{\prime\prime}(s)=\big{(}1-f^{\prime}(s)^{2}\big{)}\Big{(}1-\frac{n-1}{s}f^{\prime}(s)\Big{)}.$
(9)

Then, we resort to Section 4, and make a geometrical interpretation. Clearly,
the submersion $\pi$ is Riemannian, hence $\tilde{\varepsilon}=1$; the product
is Lorentzian, and therefore $\varepsilon^{\prime}=-1$. Moreover,
$\varepsilon=\mathrm{sign}(-1+|\nabla
u|^{2})=\mathrm{sign}(-1+(f^{\prime}(\pi))^{2})=\pm 1$. If $\varepsilon=-1$,
the rotational surface is spacelike, whereas $\varepsilon=+1$ gives rise to a
timelike surface. The spacelike examples are characterized by
$(f^{\prime})^{2}<1$, since $\varepsilon=-1$, but the timelike examples are
those coming from $(f^{\prime})^{2}>1$.

######  Example 5.1.

From the above section, from each (inextendible) solution $w$ to (5), we
consider a primitive $f=\int w$, which will be a solution to (7). Then, the
hypersurface

$\Sigma_{f}:\Omega\subset\mathbb{R}^{n}\to\mathbb{L}^{n+1},\quad\phi(x)=(x,f(\pi(x)))$

is a $SO(n)$-invariant translating soliton. We are extending continously $\pi$
to the rotation axis such that $\pi(x)=0$ when $x$ is a point of the axis. By
the limits of $w$ at $0$ (Lemmatta 4.3 and 4.4), all these solutions $f$ are
asymptotic at the axis to a light-like straight line of equation $x=\pm
s+s_{2}$. Thus, all $\Sigma_{f}$ will be asymptotic at the axis of rotation to
a light-like cone.

  * •

The Ogival Paraboloid: $\Omega=\mathbb{R}^{n}$. Case V in Lemma 4.4, the
function $f=\int\mathbf{w}$ is asymptotic at infinity to an Euclidean parabola

  * •

Timelike Calyx: $\Omega=\mathbb{R}^{n}$. Case VI of Lemma 4.4, since $w$
satisfies $1=\lim_{s\to 0}w(s)=\lim_{s\to\infty}w(s)$, then $\Sigma_{f}$ will
be asymptotic to two upper light-like cones, one at the origin and one at
infinity.

######  Example 5.2.

The spindle: We consider an example of Lemma 4.3, Case IV, $w$ and $f_{-}=\int
w$. Since $w<-1$, then $f_{-}$ is injective, so we can consider its inverse
$\alpha=f_{-}^{-1}$. Both functions are strictly decreasing. As $f_{-}$ is
defined on $(0,s_{o})$, there exists $\alpha:(a,b)\to\mathbb{R}$ such that
$\lim_{y\to a}\alpha(y)=s_{o}$ and $\lim_{y\to b}\alpha(y)=0$. We are going to
use the proof of Corollary 3.1. Then, $\alpha$ is a solution to the ODE

$\alpha^{\prime\prime}(y)=\left(-1+\alpha^{\prime}(y)^{2}\right)\left(\frac{n-1}{\alpha(y)}-\alpha^{\prime}(y)\right).$

Assume $a=-\infty$. Since $\lim_{s\to s_{o}}w^{\prime}(s)=-\infty$, then
$\lim_{y\to-\infty}\alpha^{\prime}(y)=0$, so that it also holds
$\lim_{y\to-\infty}\alpha^{\prime\prime}(y)=0$. By inserting this in the ODE,
$0=\lim_{y\to-\infty}\alpha^{\prime\prime}(y)=(1-n)/s_{o}$. This is a
contradiction.

Therefore, $a\in\mathbb{R}$. Thus, we can extend $\alpha$ a little,
$\alpha:(a-\epsilon,b)\to\mathbb{R}$. But at $a$, $\alpha^{\prime}(a)=0$, and
$\alpha(a)=s_{1}\in(0,s_{o})$. By the proof of Corollary 3.1, $\alpha$ can be
split in two inverse functions, one after $a$ (which is the original $f_{-}$)
and one before $a$, which we call $f_{+}$. Now, it is quite clear that
$w_{+}={f}_{+}^{\prime}>1$ provides an example of Case VII in Lemma 4.4. This
means that the union of these two examples make a new type of translating
soliton. This hypersurface is a (topological) sphere with two lightlike-conic
singularities, namely, asymptotic to two different light-like cones, one in
the upper half and one in the lower half, with both vertices on the rotation
axis. We will call it a fusiform hypersurface or a spindle. A similar
symmetric reasoning holds if we start with an example of Case VII in Lemma
4.4, arriving to an example of Lemma 4.3.

######  Theorem 5.1.

Up to isometries, any timelike, $SO(n)$-invariant translating soliton in
$\mathbb{L}^{n+1}$, $n\geq 2$, is an open subset of either an Ogival
Paraboloid, or a Timelike Calyx, or a Spindle.

The wing-like technique used in the previous theorem makes no sense for
spacelike hypersurfaces, because at some point, the hypersurfase would be
parallel to the timelike rotation axis.

######  Remark 5.1.

In [9], independently, the author obtained the spacelike examples, obtaining
three types. In Proposition 3.2, the profile curve of type 2 gives rise to the
Bowl. However, from our point of view, the geometrical reconstruction of the
two new surfaces can be slightly improved. Indeed, a profile curve of type 1
in Proposition 3.2 will hit the axis by a $-\pi/4$ angle. However, if one
tries to extend it beyond the axis, by rotating the curve, we must arrive
again to one of the three cases of Proposition 3.2. But now, the angle will be
$\pi/4$, so this extension has to be a profile curve of type 3. This reasoning
can be reversed, starting from a profile curve of type 3, and ending up with a
profile curve of type 1. Moreover, the timelike examples were not considered
in this paper.

######  Theorem 5.2.

Up to isometries, any $SO(n)$-invariant, spacelike, translating soliton in
$\mathbb{L}^{n+1}$, $n\geq 2$, is an open subset of a hypersurface generated
by the graph of any of the three cases in Lemma 4.2.

Similarly to the Euclidean case, we call the hypersurface generated by case I,
the Bowl.

######  Corollary 5.1.

The bowl is the only entire, $SO(n)$-invariant, graphical translating soliton
in $\mathbb{L}^{n+1}$, $n\geq 2$, up to isometries.

######  Remark 5.2.

Timelike translating solitons do not satisfy a tangency principle. Indeed, we
consider two different spindles. At the points where the distance to the axis
is maximal, the tangent planes contain directions parallel to the rotation
axis. Due to the rotation action of $SO(n-1)$, the study of the relative
position of the hypersurfaces will reduce to the study of the curves from
Corollary 3.1. We recall that the differential equation is

$\alpha^{\prime\prime}(y)=\left(-1+\alpha^{\prime}(y)^{2}\right)\left(\frac{n-1}{\alpha(y)}-\alpha^{\prime}(y)\right),$

with initial conditions $\alpha_{i}(y_{0})=s_{i}$, $\alpha^{\prime}(y_{0})=0$,
for $i=1,2$. We can assume $0<s_{1}<s_{2}$. Then,

$\alpha_{1}^{\prime\prime}(y_{o})=\frac{1-n}{s_{1}}<\frac{1-n}{s_{2}}=\alpha_{2}^{\prime\prime}(y_{o})<0.$

To compare their graphs, we use
$\beta=\alpha_{1}+\alpha_{2}(y_{o})-\alpha_{1}(y_{o})$. Clearly,
$\beta^{\prime}=\alpha_{1}^{\prime}$ and
$\beta^{\prime\prime}=\alpha_{1}^{\prime\prime}$. But now,
$\beta(y_{0})=\alpha_{2}(y_{o})$,
$\beta^{\prime}(y_{o})=0=\alpha_{2}^{\prime}(y_{o})$ and
$\beta^{\prime\prime}(y_{o})<\alpha_{2}^{\prime\prime}(y_{o})$. Therefore,
$\beta(y)<\alpha_{2}(y)$ holds for any $y\neq y_{o}$ in a neighborhood of
$y_{o}$. Geometrically, this is the same as moving one of the spindles in such
a way that their tangent planes coincide (parallel to the rotation axis), but
one spindle is at one side of the other spindle. $\Box$

The following pictures were made with wxMaxima© and Gnuplot©.

Spacelike examples
---
|  |
The bowl (Case I.) | Case II. | Case III.
Timelike examples
|  |
A Timelike Calyx | The Ogival Paraboloid | A Spindle

##  6 The action of the group $SO^{\uparrow}(n-1,1)$

Let $\mathbb{L}^{n}$, $n\geq 2$, be the Minkowski (linear) space with the
standard metric $g(X,Y)=X_{1}Y_{1}+\ldots+X_{n-1}Y_{n-1}-X_{n}Y_{n}$,
$X=(X_{1},\ldots,X_{n})^{t},Y=(Y_{1},\ldots,Y_{n})^{t}\in\mathbb{L}^{n}$. We
consider the important subgroup of linear isometries

$\displaystyle SO^{\uparrow}(n-1,1)=$
$\displaystyle\left\\{A\in\mathcal{M}_{n}(\mathbb{R}):\det(A)=+1,\
A^{t}I_{n-1,1}A=I_{n-1},\right.$ $\displaystyle\quad\left.A\mbox{ preserves
the time orientation}\right\\},$

where $I_{n-1,1}=\mathrm{Diagonal}(1,\ldots,1,-1)$, $A^{t}$ is the transpose
of $A$. The light cone $\mathcal{C}=\\{p\in\mathbb{L}^{n}:g(p,p)=0\\}$ can be
split into the zero, the future part and the past part,
$\mathcal{C}^{\uparrow}=\\{p\in\mathcal{C}:p_{n}>0\\}$,
$\mathcal{C}^{\downarrow}=\\{p\in\mathcal{C}:p_{n}<0\\}.$ We call
$\mathbf{T}^{\uparrow}$ the open subset of all future pointing time-like
vectors, whereas $\mathbf{T}^{\downarrow}$ is the open subset of all past
pointing time-like vectors. Let $\mathcal{S}$ be the open subset of all space-
like vectors, which is connected when $n\geq 3$, and it has two connected
components when $n=2$. Then, we have the disjoint union
$\mathbb{L}^{n}=\\{0\\}\cup\mathcal{C}^{\uparrow}\cup\mathcal{C}^{\downarrow}\cup\mathbf{T}^{\uparrow}\cup\mathbf{T}^{\downarrow}\cup\mathcal{S}$.
Each of them are invariant by $SO^{\uparrow}(n-1,1)$.

Firstly, we study the case $n=2$. Take the Minkowski plane $\mathbb{L}^{2}$
with standard flat metric $g=dx^{2}-dy^{2}$. The Boost Lie group is the set of
matrices

$SO^{\uparrow}(1,1)=\mathbf{B}=\left\\{A_{\theta}=\begin{pmatrix}\cosh(\theta)&\sinh(\theta)\\\
\sinh(\theta)&\cosh(\theta)\end{pmatrix}:\theta\in\mathbb{R}\right\\},$

which acts on $\mathbb{L}^{2}$ by isometries. If we remove the origin $(0,0)$,
the action is free, but it is not proper, because the quotient space
$(\mathbb{L}^{2}\backslash\\{0\\})/\mathbf{B}$ is not Haussdorf. Indeed, the
light cone projects to four points if $n=1$ which cannot be separated by
neighbourhoods. We split the plane in four regions whose boundaries are made
of two light-like geodesics, namely

$\displaystyle\Omega_{1}=\\{(x,y)\in\mathbb{L}^{2}:y^{2}<x^{2},\ 0<x\\},$
$\displaystyle\quad\Omega_{2}=\\{(x,y)\in\mathbb{L}^{2}:y^{2}>x^{2},\ 0<y\\},$
$\displaystyle\Omega_{3}=\\{(x,y)\in\mathbb{L}^{2}:y^{2}<x^{2},\ 0>x\\},$
$\displaystyle\quad\Omega_{4}=\\{(x,y)\in\mathbb{L}^{2}:y^{2}>x^{2},\ 0>y\\}.$

We will use the globally defined orthonormal frame
$\\{\partial_{x},\partial_{y}\\}$. The action of $\mathbf{B}$ works very well
on each $\Omega_{k}$, $k=1,2,3,4$, because they admit decompositions as in (3)
(see below). According to [3], first we will obtain the fundamental examples
included in the fundamental regions
$\Omega_{k}\times\mathbb{R}\subset\mathbb{L}^{3}$, $k=1,2,3,4$. In
$\Omega_{1}\cup\Omega_{3}$, the orbits of $\mathbf{B}$ are timelike. In
$\Omega_{2}\cup\Omega_{4}$, the orbits are spacelike. When $n=2$,
$\mathcal{S}=\Omega_{1}\cup\Omega_{3}$, $\mathbf{T}^{\uparrow}=\Omega_{2}$,
$\mathbf{T}^{\downarrow}=\Omega_{4}$.

We come back to the general case. To simplify computations and notations, we
call $\chi:\Omega\subset\mathbb{R}^{k}\to\mathbb{R}^{k}$ the position vector.
Define the continuous map

$\pi:\mathbb{L}^{n}\to[0,+\infty),\quad\pi(x)=\sqrt{|g(x,x)|},\ \textrm{also}\
\pi=\sqrt{|g(\chi,\chi)|}.$

Outside the light cone, $\pi$ is smooth. Given
$x\in\mathbb{L}^{n}\backslash\mathcal{C}$, if we put
$\tilde{\varepsilon}=\mathrm{sign}(g(x,x))=\pm 1$,

$\nabla\pi(x)=\tilde{\varepsilon}\frac{\chi(x)}{\pi(x)},\quad|\nabla\pi(x)|^{2}=\tilde{\varepsilon},\quad\mathrm{div}(\nabla\pi)(x)=\frac{\tilde{\varepsilon}}{\pi(x)}.$
(10)

This projection can be restricted to $\Omega_{i}$, $\mathcal{S}$,
$\mathbf{T}^{\uparrow}$, $\mathbf{T}^{\downarrow}$, obtaining the necessary
pseudo-Riemannian submersions onto $(0,+\infty)$. We explain the curves and
diffeomorphisms of (3) in some cases, and the other ones are left to the
reader.

$\star$ Decomposition of $\mathbf{T}^{\uparrow}$. The curve is
$\beta:(0,+\infty)\to\mathbf{T}^{\uparrow}$, $\beta(s)=(0,\ldots,0,s)$, and
the map is $\phi:(0,+\infty)\times SO(n-1,1)\to\mathcal{C}^{\uparrow}$,
$\phi(s,A)=\beta(s)A$. Also, $\tilde{\varepsilon}=-1$.

$\star$ Decomposition of $\mathcal{S}$. The curve is now
$\beta:(0,+\infty)\to\mathcal{S}$, $\beta(s)=(s,0,\ldots,0)$, and the map is
$\phi:(0,+\infty)\times SO(n-1,1)\to\mathcal{C}^{\uparrow}$,
$\phi(s,A)=\beta(s)A$. Also, $\tilde{\varepsilon}=+1$.

######  Remark 6.1.

Given a domain (open and connected)
$\Omega\subset\mathbb{L}^{n}\backslash\mathcal{C}$, such that any of the above
decompositions make sense, then $\pi:\Omega\to(0,+\infty)$. By (10), function
$h:(0,+\infty)\to\mathbb{R}$ is $h(s)=\tilde{\varepsilon}/s$.

It is important to remember that we are studying
$\mathbb{L}^{n+1}=\mathbb{L}^{n}\times\mathbb{R}$, so that
$\varepsilon^{\prime}=+1$. Therefore, with our previous considerations,
equation (5) reduces to

$f^{\prime\prime}(s)=\big{(}\tilde{\varepsilon}+f^{\prime}(s)^{2}\big{)}\left(1-\frac{f^{\prime}(s)}{s}\right).$
(11)

where $\tilde{\varepsilon}$ depends of the type of the orbits of B, i.e. of
the region we are considering

######  Example 6.1.

Assume $\tilde{\varepsilon}=1$, i.e. we consider solutions in the domain $S$.
Then (5) becomes

$f^{\prime\prime}(s)=\big{(}1+f^{\prime}(s)^{2}\big{)}\left(1-\frac{f^{\prime}(s)}{s}\right).$
(12)

This is the very same ODE as in the rotationally symmetric case in
$\mathbb{R}^{3}$. By [5], there are two types of solutions.
$\bullet$ Type Z: Firstly, we recall the only solution $f_{1}$ such that
$f_{1}(0)=0$ and $f_{1}^{\prime}(0)=0$. The traslating soliton will be the
corresponding graph

$\Phi:\Omega\to\Omega\times\mathbb{R}\subset\mathbb{L}^{n+1},\quad\Phi(x)=\big{(}x,f(\pi(x))\big{)}.$

When $n=2$, there will be two twin surfaces constructed like this, one in
$\Omega_{1}$ and one in $\Omega_{3}$. For $n\geq 3$, since
$\Omega=\mathcal{S}$ is connected, there is just one hypersurface.

$\bullet$ Wing-like: Secondly, as in [5], (also, recall Corollary 3.1), there
is a family of unextendable, spacelike profile curves
$\alpha:\mathbb{R}\to\\{(x,y,t)\in\mathbb{L}^{3}:y=0,x>0\\}$. By denoting the
$3\times 3$ matrix $\hat{A}_{\theta}=\left(\begin{smallmatrix}A_{\theta}&0\\\
0&1\end{smallmatrix}\right)$, the translating soliton is then

$\Phi:\mathbb{R}^{2}\to\mathbb{L}^{3},\quad\Phi(\theta,s)=\alpha(s)\hat{A}_{\theta}.$

$\Box$

######  Example 6.2.

For $\varepsilon^{\prime}=1$ and $\tilde{\varepsilon}=-1$, (5) becomes

$f^{\prime\prime}(s)=\big{(}-1+f^{\prime}(s)^{2}\big{)}\left(1+\frac{f^{\prime}(s)}{s}\right).$
(13)

By the easy change $q(s)=-f(s)$, we transform this problem in

$q^{\prime\prime}(s)=\big{(}1-q^{\prime}(s)^{2}\big{)}\left(1-\frac{q^{\prime}(s)}{s}\right).$

By Section 4, there are $7$ types of solutions to this differential equation.
Once we have a solution $(f=-q)$, we construct our profile curve and the
associated translating soliton. As $f^{\prime}=-q^{\prime}$, the solutions
producing timelike translating solitons are those satisfying
$|f^{\prime}(s)|<1$. So, for each solution as in Lemma 4.2, we construct a
timelike translating soliton.

######  Example 6.3.

The Hybrid Translator: $\bullet$ Case $n=2$. We call $f_{1}$ the solution to
(12) in $\mathbb{R}$, and $f_{2}$ the solution to (13) in $\mathbb{R}$ such
that $f_{i}(0)=0$ and $f_{i}^{\prime}(0)=0$, $i=1,2$. Recall that they are
even functions and analytical. As such, their derivatives of odd order are
$f_{i}^{(2k-1)}(0)=0$, $k\geq 1$. As a result,
$f_{1}(s)=\sum_{k=1}^{\infty}\frac{f_{1}^{(2k)}(0)}{(2k)!}s^{2k}$. Then, it
makes sense to define $\hat{f}:\mathbb{R}\to\mathbb{R}$,
$\hat{f}(s)=f_{1}(is)$, where $i=\sqrt{-1}$. By simple computations,

$\hat{f}^{\prime\prime}(s)=-\big{(}1-(\hat{f}^{\prime}(s))^{2}\big{)}\Big{(}1+\frac{\hat{f}^{\prime}(s)}{s}\Big{)}.$

Hence, $\hat{f}$ is a solution to equation (13), and consequently
$\hat{f}=f_{2}$. By comparing the derivatives of $f_{1}$ and $f_{2}$,
$f_{1}^{(k)}(0)=f_{2}^{(k)}(0)=0$, if $k$ is odd, since $f_{1}$ and $f_{2}$
are even, and $f_{1}^{(4k+2)}(0)=-f_{2}^{(4k+2)}(0)$ and
$f_{1}^{(4k)}(0)=f_{2}^{(4k)}(0)$ for all $k\geq 0$. Next, we can define

$u:\mathbb{R}^{2}\rightarrow\mathbb{R},\quad
u(x,y)=\left\\{\begin{array}[]{cl}f_{1}\left(\sqrt{x^{2}-y^{2}}\right),&(x,y)\in\Omega_{1}\cup\Omega_{3},\\\
0,&(x,y)\in\partial\Omega_{i},i=1,2,3,4,\\\
f_{2}\left(\sqrt{y^{2}-x^{2}}\right),&(x,y)\in\Omega_{2}\cup\Omega_{4}.\end{array}\right.$

Note that we get immediately $u\in C^{0}(\mathbb{R}^{2})\cap
C^{\infty}(\bigcup_{k=1}^{4}\Omega_{k})$. We want to prove that $u\in
C^{\infty}(\mathbb{R}^{2})$. To do so, we now just need to prove the following

######  Lemma 6.1.

Let $f_{1}$, $f_{2}$ be functions in $C^{2m}(\mathbb{R})$, such that
$f_{1}^{(k)}(0)=f_{2}^{(k)}(0)=0$, if $k$ is odd,
$f_{1}^{(k)}(0)=(-1)^{\frac{k}{2}}f_{2}^{(k)}(0)$, if $k$ is even. Then the
function $u$ defined as above is in $C^{m}(\mathbb{R}^{2})$.

###### Proof.

We prove the statement by induction over $m$. The case $m=0$ is trivially
satisfied, since $f_{1}(0)=f_{2}(0)$. Moreover let
$g_{1}(z)=\frac{f^{\prime}_{1}(z)}{z}$,
$g_{2}(z)=\frac{f^{\prime}_{2}(z)}{z}$. We have

$\displaystyle\partial_{x}f_{1}(\sqrt{x^{2}-y^{2}})=xg_{1}(\sqrt{x^{2}-y^{2}}),\quad\partial_{x}f_{2}(\sqrt{y^{2}-x^{2}})=-xg_{2}(\sqrt{y^{2}-x^{2}}),$
$\displaystyle\partial_{y}f_{1}(\sqrt{x^{2}-y^{2}})=-yg_{1}(\sqrt{x^{2}-y^{2}}),\quad\partial_{y}f_{2}(\sqrt{y^{2}-x^{2}})=yg_{2}(\sqrt{y^{2}-x^{2}}).$

Obviously $g_{1},g_{2}\in C^{2m-2}$ since
$f_{1}^{\prime}(0)=f^{\prime}_{2}(0)=0$, and
$g_{1}^{(4k+2)}(0)=-g_{2}^{(4k+2)}(0)$ and $g_{1}^{(4k)}(0)=g_{2}^{(4k)}(0)$
for all $k\geq 0$. Then by the induction hypothesis, the function

$v:\mathbb{R}^{2}\rightarrow\mathbb{R},\quad
v(x,y)=\left\\{\begin{array}[]{cl}g_{1}\left(\sqrt{x^{2}-y^{2}}\right),&(x,y)\in\Omega_{1}\cup\Omega_{3},\\\
0,&(x,y)\in\partial\Omega_{i},i=1,2,3,4,\\\
g_{2}\left(\sqrt{y^{2}-x^{2}}\right),&(x,y)\in\Omega_{2}\cup\Omega_{4}.\end{array}\right.$

is in $C^{m-1}(\mathbb{R}^{2})$. Hence $\partial_{x}u$ and $\partial_{y}u$
extend to $C^{m-1}(\mathbb{R}^{2})$, and finally $u\in C^{m}(\mathbb{R}^{2})$.
∎

We point out that the curve joining each two adjacent pieces is a lightlike
straight line. Thus, it is possible to consider two or three contiguous
pieces, and their gluing straight lines. We cannot consider the other straight
half-lines, since the boundary would be lightlike. In other words, we can
choose to glue either two, three or four adjacent pieces, to obtain
translating solitons.

$\bullet$ Case $n\geq 3$. $\mathbb{L}^{n}$ is isometric to the Lorentzian
product $\mathbb{R}^{n-1}\times_{-1}\mathbb{R}$, $n\geq 3$, with the standard
metric as at the beginning of this section. If we call $g_{o}$ the standard
Riemannian metric on $\mathbb{R}^{n-1}$, its norm is
$\|x\|=\sqrt{g_{o}(x,x)}$, for any $x\in\mathbb{R}^{n-1}$. The function $u$ of
the Hybrid Translator can be easily extended to $\mathbb{L}^{n}$ as follows:

$\tilde{u}:\mathbb{R}^{n-1}\times_{-1}\mathbb{R}=\mathbb{L}^{n}\to\mathbb{L}^{n+1},\quad(x,y)\mapsto\tilde{u}(x,y):=u(\|x\|,y).$

There is no confusion if we also call its graph the Hybrid Translator.

In addition, take $A\in SO^{\uparrow}(n-1,1)$, and decompose it as
$A=\left(\begin{smallmatrix}B&d^{t}\\\ c&a\end{smallmatrix}\right)$ for
suitable $B\in\mathcal{M}_{n-1}(\mathbb{R})$, $c,d\in\mathbb{R}^{n-1}$,
$a\in\mathbb{R}$. With this, given
$(x,y)\in\mathbb{R}^{n-1}\times_{-1}\mathbb{R}=\mathbb{L}^{n}$, taking
$(z,t)=(z,y)A$, then $\|z\|^{2}-t^{2}=\|x\|^{2}-y^{2}$. This means that
$\tilde{u}$ is invariant by $SO^{\uparrow}(n-1,1)$.

Sprunk and Xiao, [16], proved that any entire translating soliton in
$\mathbb{R}^{3}$ must be convex, but this is not the case for our hybrid
example, because $f_{1}f_{2}<0$. $\Box$

######  Theorem 6.1.

Let $M$ be a $SO^{\uparrow}(n-1,1)$ invariant, timelike, translating soliton
in $\mathbb{L}^{n+1}$, $n\geq 2$. Then, up to isometries, $M$ is an open
subset of one of the following examples:

  1. 1.

A translating soliton of type $Z$ or Wing-like.

  2. 2.

Given $w$ any of the three types of solutions in Lemma 4.2, take $f=-\int w$.

  3. 3.

The Hybrid Translator.

###### Proof.

We will use Theorem 3.1. As we regard
$\mathbb{L}^{n+1}=\mathbb{L}^{n}\times\mathbb{R}$, we need
$\varepsilon^{\prime}=+1$. Recall
$\mathbb{L}^{n}=\mathcal{C}\cup\mathcal{S}\cup\mathbf{T}^{\uparrow}\cup\mathcal{C}^{\downarrow}$,
and each subset is invariant by the action of $SO^{\uparrow}(n-1,1)$ by
isometries. Only on
$\mathcal{S}\cup\mathbf{T}^{\uparrow}\cup\mathcal{C}^{\downarrow}$ the action
is proper, and we can obtain decompositions as in (3). The value of
$\tilde{\varepsilon}$ depends on the chosen domain. The orbits of the action
are well-known. First, one finds the lightcone. For $n=2$, they are
(Euclidean) hyperbolas. For $n\geq 3$, they are hypersurfaces with non-zero
constant sectional curvature (namely, real hyperbolic spaces for any $n\geq 3$
and Anti-de Sitter spaces for $n\geq 4$).

$\bullet$ For $\mathbf{T}^{\uparrow}$ (resp. $\mathbf{T}^{\downarrow}$), the
curve $\beta:(0,+\infty)\to\mathbf{T}^{\uparrow}$, $\beta(s)=(0,\ldots,0,s)$
is timelike (resp. $\beta(s)=(0,\ldots,0,-s)$). Then, $\tilde{\varepsilon}=-1$
and therefore we recall Example 6.2, obtaining another 3 timelike translating
solitons.

$\bullet$ For $\Omega=\mathcal{S}$, the curve
$\beta:(0,+\infty)\to\mathcal{S}$, $\beta(s)=(s,0,\ldots,0)$ is spacelike.
Then, $\tilde{\varepsilon}=+1$. The differential equation (5) becomes

$f^{\prime\prime}(s)=\big{(}1+f^{\prime}(s)^{2}\big{)}\left(1-\frac{f^{\prime}(s)}{s}\right).$
(14)

Now, we recall Examples 6.1. One should bear in mind that for $n=2$, there
will two twin surfaces, in $\Omega_{1}\times\mathbb{R}$ and in
$\Omega_{3}\times\mathbb{R}$, since $\mathcal{S}$ is not connected.

$\bullet$ It remains to study whether the solutions touching the lightcone can
be extended smoothly. Since the lightcone has a degenerate metric, it has to
be strictly contained in the hypersurface. In other words, we wonder if it is
possible to glue two solutions, one in $\mathbf{T}^{\uparrow}$ and one in
$\mathcal{S}$ (alternatively, in $\mathbf{T}^{\downarrow}$). By regarding the
extended action of $SO^{\uparrow}(n-1,1)$ on $\mathbb{L}^{n+1}$, if any, the
tangent plane at the origing must be
$\\{x\in\mathbb{L}^{n+1}=\mathbb{L}^{n}\times\mathbb{R}:x_{n+1}=0\\}$. This
means that the derivatives of the functions must be zero. Then, we finish the
proof by recalling the Hybrid Translator. ∎

By another method, in [9], the author obtained (essentially) the three
spacelike types, but not the timelike cases. Again, we also recover the
following result from [9].

######  Theorem 6.2.

Up to isometries, any $SO^{\uparrow}(n-1,1)$ invariant, spacelike, translating
soliton in $\mathbb{L}^{n+1}$, $n\geq 2$, is an open subset of a hypersurface
generated by the graph of any of Cases V, VI or VII in Lemma 4.4.

######  Remark 6.2.

As in the proof of Theorem 5.1, Type IV from Lemma 4.3 and Type VII can be
joined to provide a single profile curve.

######  Corollary 6.1.

The only $SO^{\uparrow}(n-1,1)$-invariant, entire translating soliton in
$\mathbb{L}^{n+1}$, $n\geq 2$, is the hybrid translator.

## Acknowledgements

The second author has been partially financed by the Spanish Ministry of
Economy and Competitiveness and European Regional Development Fund (ERDF),
project MTM2016-78807-C2-1-P. The second author also belongs to the Excellence
Scientific Unit ’Science in the Alhambra’ of Granada University, ref. UCE-
PP2018-01. Both authors would like to thank the London Mathematical Society
and Imperial College London, since this was partially supported by a Scheme 4
Grant of the LMS.

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