Invariant translators of the Solvable group

TL;DR

This paper classifies invariant translators of the mean curvature flow in the solvable group $Sol_3$, revealing new examples of graphical translators and establishing non-existence results, thus advancing understanding of geometric flows in non-Euclidean spaces.

Contribution

It provides a classification of invariant translators in $Sol_3$, including existence of graphical solutions and non-existence results, which was previously unknown.

Findings

$Sol_3$ admits graphical translators on a half-plane.

Non-existence results for certain invariant translators.

Contrast with Euclidean space rigidity results.

Abstract

We classify the translators to the mean curvature flow in the three-dimensional solvable group $Sol_3$ that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular we show that $Sol_3$ admits graphical translators defined on a half-plane, in contrast with a rigidity result of Shahriyari for translators in the Euclidean space. Moreover we exhibit some non-existence results.

Equations

$${\mathcal{M}}_t={\phi }_t\left(\mathcal{M}\right).$$

$$H=\bar{g}(\nu ,V),$$

$$(x_1,y_1,z_1)\star (x_2,y_2,z_2)=\left(x_1+e^{-z_1}{x}_2,y_1+e^{z_1}{y}_2,z_1+z_2\right),$$

$$\bar{g}=e^{2z}d{x}^2+e^{-2z}d{y}^2+d{z}^2.$$

$$E_1=e^{-z}\frac{\partial }{\partial x},E_2=e^z\frac{\partial }{\partial y},E_3=\frac{\partial }{\partial z}.$$

$$F_1=\frac{\partial }{\partial x},F_2=\frac{\partial }{\partial y},F_3=-x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+\frac{\partial }{\partial z}.$$

$$y=y_0,z=z_0,$$

$$z=\log (y_0\pm y)+z_0,$$

$$z=\frac{1}{2}\log (y_0\pm y)+z_0,$$

$$E_1=e^{-z}\frac{\partial }{\partial x},E_2=e^z\frac{\partial }{\partial y},E_3=\frac{\partial }{\partial z}.$$

$$\begin{array}[]{lll}\bar{\nabla}_{E_{1}}E_{1}=-E_{3}&\bar{\nabla}_{E_{2}}E_{1}=0&\bar{\nabla}_{E_{3}}E_{1}=0\\ \bar{\nabla}_{E_{1}}E_{2}=0&\bar{\nabla}_{E_{2}}E_{2}=E_{3}&\bar{\nabla}_{E_{3}}E_{2}=0\\ \bar{\nabla}_{E_{1}}E_{3}=E_{1}&\bar{\nabla}_{E_{2}}E_{3}=-E_{2}&\bar{\nabla}_{E_{3}}E_{3}=0\end{array}$$

$$\begin{array}[]{ll}\left\{\left.L_{\left(\frac{a}{c}(1-e^{-ct}),\frac{b}{c}(e^{ct}-1),ct\right)}\right|t\in\mathbb{R}\right\},&\text{if}\ c\neq 0;\\ &\\ \Bigl{\{}\left.L_{\left(at,bt,0\right)}\right|t\in\mathbb{R}\Bigr{\}},&\text{if}\ c=0.\end{array}$$

$$\begin{array}[]{l}\left(A(t),B(t),C(t)\right)\star\left(A(s),B(s),C(s)\right)\\ \phantom{aaaa}=\left(A(t)+e^{-C(t)}A(s),B(t)+e^{C(t)}B(s),C(t)+C(s)\right)\\ \phantom{aaaa}=\left(A(t+s),B(t+s),C(t+s)\right).\end{array}$$

$$A^{\prime }(t)$$

$$B^{\prime }(t)$$

$$\mathcal{M}(u,s)=(u,0,0)\star \gamma (s)=\left(u,y(s),z(s)\right).$$

$$\left\{\begin{array}[]{rcl}y^{\prime}&=&e^{z}\cos\vartheta;\\ z^{\prime}&=&\sin\vartheta;\\ \vartheta^{\prime}&=&H;\end{array}\right.$$

$$V=\lambda e^{-z}{E}_2+\mu \left(-x{e}^z{E}_1+y{e}^{-z}{E}_2+E_3\right).$$

$$H=-\lambda e^{-z}\sin \vartheta +\mu \left(\cos \vartheta -y{e}^{-z}\sin \vartheta \right).$$

$${\vartheta }^{\prime }$$

$$=$$

$$\vartheta -{\vartheta }_0+\lambda =e^{-z}(\lambda +\mu y).$$

$$\left\{\begin{array}[]{rcl}\vartheta^{\prime}&=&f(\vartheta):=\mu\cos\vartheta-(\vartheta-\vartheta_{0}+\lambda)\sin\vartheta,\\ \vartheta(0)&=&\vartheta_{0}.\end{array}\right.$$

$$f(k\pi )f((k+1)\pi )=-{\mu }^2<0.$$

$$\underset{s\to +\infty }{\lim }\vartheta (s)={\vartheta }_2,\underset{s\to -\infty }{\lim }\vartheta (s)={\vartheta }_1;$$

$$\underset{s\to +\infty }{\lim }\vartheta (s)={\vartheta }_1,\underset{s\to -\infty }{\lim }\vartheta (s)={\vartheta }_2.$$

$$\vartheta -{\vartheta }_0+\lambda =\lambda e^{-z}.$$

$$y^{\prime }=e^z\cos \vartheta =\frac{\lambda \cos \vartheta }{\vartheta -{\vartheta }_0+\lambda }.$$

$$\underset{s\to +\infty }{\lim }\frac{\cos \vartheta (s)}{\vartheta (s)-{\vartheta }_0+\lambda }=\underset{\vartheta \to {\vartheta }_0-\lambda }{\lim }\frac{\cos \vartheta }{\vartheta -{\vartheta }_0+\lambda }=\pm 1.$$

$$\det (A{g}^{-1})=-({\vartheta }^{\prime }+\cos \vartheta )\cos \vartheta ,K=-{\sin }^2\vartheta -\cos \vartheta {\vartheta }^{\prime }.$$

$$K$$

Full text

Invariant translators of the Solvable group

Giuseppe Pipoli

Abstract: We classify the translators to the mean curvature flow in the three-dimensional solvable group $Sol_{3}$ that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular we show that $Sol_{3}$ admits graphical translators defined on a half-plane, in contrast with a rigidity result of Shahriyari [Sh] for translators in the Euclidean space. Moreover we exhibit some non-existence results.

Keywords: Translators, mean curvature flow, Sol group

MSC 2010 subject classification: 53A10, 53C42, 53C44.

1 Introduction

The aim of this paper is to classify all the translators of the three-dimensional solvable group $Sol_{3}$ that are invariant under the action of a one-parameter group of symmetries.

A hypersurface ${\cal M}$ in a general ambient space is said a soliton to the mean curvature flow if it evolves without changing its shape. More precisely it means that there exists $G=\left\{\left.\varphi_{t}\ \right|\ t\in\mathbb{R}\right\}$ a one-parameter group of isometries of the ambient space such that the evolution by mean curvature flow of ${\cal M}$ is given by the family of hypersurfaces

[TABLE]

When the ambient space is a Lie group equipped with a left invariant metric, then we can choose $G$ consisting of left translations. In this case we say that ${\cal M}$ is a translator.

Translators provide interesting examples of explicit eternal solutions of the flow and, as shown by Huisken and Sinestrari [HS], they have a fundamental role in the analysis of type II singularities.

It is well known that the property of being a translator can be view as a prescribed mean curvature problem: let $V$ be the Killing vector field associated to $G$, then ${\cal M}$ evolves moving according to (1.1) if and only if

[TABLE]

where $\nu$ and $H$ are respectively the unit vector field and the mean curvature of ${\cal M}$, while $\bar{g}$ is the metric in the ambient space. In this case we will say that ${\cal M}$ is a translator in the direction of $V$. A proof of (1.2) can be found in [HuSm].

The ambient space considered in this paper is the three-dimensional solvable group $Sol_{3}$. It is one of the eight geometries of Thurston, the one with the least number of isometries. It can be seen as a Riemannian Lie group defined in the following way: on $\mathbb{R}^{3}$ equipped with the usual coordinates we consider the group operation,

[TABLE]

and the left invariant Riemannian metric

[TABLE]

It follows that an orthonormal basis of left invariant vector fields is

[TABLE]

The geometry of this manifold is well known, in particular its isometry group is generated by the left translation, hence it is $3$-dimensional. Moreover any Killing vector field is a linear combination with constant coefficients of

[TABLE]

Every time we wish to study hypersurfaces satisfying some curvature condition (like minimal, CMC, or constant Gaussian curvature), a successful strategy for finding examples is to look for the symmetrical ones. This idea was considered in $Sol_{3}$ [Lo, LM1, LM2], but also in many other ambient spaces: without completeness claims, we mention [FMP] for CMC in the Heisenberg group, [On, MO1] in $\mathbb{H}^{2}\times\mathbb{R}$ and [MO2] for more general ambient spaces.

Since the mean curvature flow preserves the symmetries (see [Pi1] for a proof), the same strategy can be applied for equation like (1.2). In fact the first known examples of translators in the Euclidean space are the tilted grim reapers cylinders (invariant by translation), the bowl solution and the translating catenoids [CSS] (rotationally symmetric) and the translating helicoids [Ha]. The author of the present paper described in [Pi2] the analogous of these examples in the Heisenberg group. It is not difficult to imagine that many other non-symmetric examples exist. Without completeness claim we mention some recent literature about the construction of interesting families of translators in the Euclidean space: [BLT, HIMW, HMW].

The goal of the present paper is to classify the translators in $Sol_{3}$ that are invariant under the action of a one-parameter group of isometries. Let $X$ and $V$ be two non-vanishing Killing vector field. With an abuse of notation we denote with $X$ also the one-parameter group of isometries associated to the vector field $X$. We want to describe the $X$-invariant translators in the direction of $V$. Since our ambient manifold is homogeneous we can always suppose that $(0,0,0)$ belongs to the surface considered. On the hoter hand, $Sol_{3}$ is not isotropic, then the result depends strongly on the interaction between $X$ and $V$.

The most interesting examples appear in the very special case $X=F_{1}$. We point out that the minimal surfaces with this kind of symmetry are classified by Lopez and Munteanu in [LM1]: they are the planes

[TABLE]

and the logarithmic surfaces

[TABLE]

where $y_{0},z_{0}\in\mathbb{R}$. Clearly, by (1.2), these surface can be seen as trivial examples of translators when $V$ is tangent to them. For later use we introduce a further family of $F_{1}$-invariant surfaces that we can call of half-logarithmic type:

[TABLE]

where once again $y_{0},z_{0}\in\mathbb{R}$.

The whole family of $F_{1}$-invariant translators are described by our first result.

Theorem 1.1

Let $V=\eta F_{1}+\lambda F_{2}+\mu F_{3}$ be a non-zero Killing vector field and let ${\cal M}$ be a $F_{1}$-invariant translator in the direction of $V$. Then we have one of the following possibilities.

${\cal M}$* is one of the minimal surfaces (1.3), (1.4) if and only if $V$ is tangent to it.*

• 2)

If $\mu=0$ and $\lambda\neq 0$, then ${\cal M}$ is a complete graph $y=F(x,z)$, for some function $F$ defined on a slab or on an half-plane. The lower half surface ${\cal M}^{-}={\cal M}\cap\left\{z<0\right\}$ is always asymptotic to a horizontal plane, while the upper half surface ${\cal M}^{+}={\cal M}\cap\left\{z>0\right\}$ is asymptotic to one of the following: a horizontal plane, a surface of logarithmic type (1.4), a surface of half-logarithmic type (1.5) or a generic $F_{1}$-invariant plane $y=C_{1}z+C_{0}$, for some constants $C_{0},C_{1}$.

• 3)

if $\mu\neq 0$, there is a line $l=\left\{(s,y_{0},z_{0})|s\in\mathbb{R}\right\}$ such that ${\cal M}\backslash l$ has two connected components. Each one of them is asymptotic to the special vertical plane $y=-\frac{\lambda}{\mu}$ or to a minimal surface of logarithmic type (1.4).

This Theorem shows that in $Sol_{3}$ there are some graphical translators that are defined on a half-plane. This property cannot be satisfied for translators in the Euclidean space as proved in [Sh]. Moreover some of them are not convex in contrast with another rigidity property of translators in the Euclidean space [SX]. See Remark 3.5 for a more precise statement. This phenomenon has been observed in the Heisenberg group too [Pi2]. We point out that for almost all the cases considered in part $2)$ of Theorem 1.1 ${\cal M}^{+}$ is asymptotic to a minimal surface. The latter two possibilities appear only in very special cases. We refer to Lemma 3.6 for more details. Some numerical simulations of the translators described in Theorem 1.1 can be seen in figures from 1 to 3 for part $2)$ and figures from 4 to 7 for part $3)$.

For a more general kind of invariance we do not have a precise description but we still have some structural properties.

Theorem 1.2

Let $X=aF_{1}+bF_{2}$, let $V$ any Killing vector field and let ${\cal M}$ be a $X$-invariant translator in the direction of $V$. If $ab\neq 0$, then we have two possibilities:

either ${\cal M}$ is the horizontal plane $z=0$ (hence it is minimal) and $V$ is tangent to it,

• 2)

or ${\cal M}$ is a complete graph $y=F(x,z)$ for some function $F$.

We complete the classification with the most rigid case.

Theorem 1.3

Let $X=aF_{1}+bF_{2}+cF_{3}$, let $V$ be any Killing vector field and let ${\cal M}$ be a $X$-invariant surface. If $c\neq 0$ the following are equivalent:

${\cal M}$* is a translator in the direction of $V$,*

• 2)

${\cal M}$* is minimal and $V$ is tangent to ${\cal M}$.*

As a consequence of this Theorem we have that for suitable combinations of $X$ and $V$, there are no $X$-invariant translator in the direction of $V$. For example there are no such surfaces if $a=b=0$, $c=1$ and $\eta\lambda\neq 0$. See Remark 4.3 for the proof. We mention that the $F_{3}$-invariant minimal surfaces are studied by Lopez in [Lo].

Finally we point out that Theorems 1.1, 1.2 and 1.3 exhaust all the possibilities, in fact the case $a=c=0$ is left, but it can be easily traced back to Theorem 1.1 applying to the whole ambient space the isometry $\phi:(x,y,z)\mapsto(x,y,-z)$.

The paper is organized as follows. In Section 2 we collect some basic facts about the geometry of $Sol_{3}$, in particular we describe all the one-parameter group of isometries of this space. In Section 3 we start proving Theorem 1.1 and we exhibit some numerical simulations of these surfaces. We finish the paper with the proof of Theorem 1.2 and 1.3 in Section 4.

2 Preliminaries

The geometry of $Sol_{3}$ is well known, here we write only what is strictly necessary for this paper. Other information about this space can be found for example in [Tr]. We racall that a left-invariant orthonormal basis is

[TABLE]

The Levi-Civita connection of $\bar{g}$ is determined by

[TABLE]

For every $p\in Sol_{3}$, we denote by $L_{p}:q\in Sol_{3}\mapsto L_{p}(q)=p\star q$ the left translation by $p$. The following result probably was already known, but we give the proof for completeness.

Lemma 2.1

Let $X=aF_{1}+bF_{2}+cF_{3}$ be a non vanishing Killing vector field, then the one-parameter group of isometries of $Sol_{3}$ generated by $X$ is

[TABLE]

*Proof. *Let $A,B,C$ be three real functions such that $\left\{\left.L_{\left(A(t),B(t),C(t)\right)}\right|t\in\mathbb{R}\right\}$ is a one-parameter group of isometries of $Sol_{3}$. Then $A(0)=B(0)=C(0)=0$ and for every $t,s\in\mathbb{R}$ we have

[TABLE]

Looking at the third components of (2.7), we deduce that $C$ is a linear function, hence there is a constant $c$ such that $C(t)=ct$. If $c=0$, then we can see that $A$ and $B$ are linear too. Otherwise we can compute the derivative with respect to $s$ in $s=0$ of each member of the equality (2.7) and we get that for every $t$

[TABLE]

The two ODEs can be easily solved. Finally, deriving with respect to $t$, we can see that $X$ is the Killing vector field associated to the group. $\square$

3 $F_{1}$-Invariant translators

The goal of this section is to prove Theorem 1.1 describing the $F_{1}$-invariant translators. We recall that, as noticed in [LM1], it is equivalent to study the $F_{2}$-invariant translators: in fact it is sufficient to apply to the whole ambient space the isometry $\phi(x,y,z)=(x,y,-z)$ to transform a surface of the first kind to one with the other direction of symmetry.

Let ${\cal M}$ be a $F_{1}$-invariant surface, then there is a planar curve $\gamma(s)=(0,y(s),z(s))$ parametrized by arc length such that the surface is

[TABLE]

Following the idea of [LM1], we define $\vartheta(s)$ as the angle between $\gamma^{\prime}$ and $E_{2}$, hence we have the following ODE system.

[TABLE]

where the derivatives are rispect to $s$ and $H$ is the mean curvature of ${\cal M}$. Note that, because ${\cal M}$ is invariant, $H$ depends only on $s$. Moreover, unlike [LM1], we prefer to define the mean curvature as the sum of the principal curvatures (and not as the mean of the pricipal curvatures). Let $V=\eta F_{1}+\lambda F_{2}+\mu F_{3}$ be a Killing vector field. Note that $F_{1}$ is tangent to ${\cal M}$, therefore by the definition of translators (1.2) we can consider $\eta=0$ without loss of generality. As shown in [LM1], $\nu=-\sin\vartheta E_{2}+\cos\vartheta E_{3}$, while the Killing vector field can be rewritten as

[TABLE]

Therefore, by (1.2), ${\cal M}$ is a translator in the direction of $V$ if and only if

[TABLE]

By (3.9) we get

[TABLE]

Obviously the solution of (3.9) is uniquely determined fixing the initial conditions. Since $Sol_{3}$ is homogeneous, we can always suppose without loss of generality that $y(0)=z(0)=0$, then for every fixed $\lambda$ and $\mu$, we have just one degree of freedom given by $\vartheta_{0}:=\vartheta(0)$. Once the initial conditions have been chosen, by (3.12) we get

[TABLE]

Putting (3.13) into (3) we get that the third equation of the system (3.9) becomes an autonomous independent equation:

[TABLE]

The crucial part of the present paper is a careful estimation of the solution of the ODE (3.14). After that the behaviour of $y$ and $z$ can be easily understood by (3.9) and (3.13). By periodicity it would be sufficient to consider $\vartheta_{0}\in\left[0,2\pi\right]$, but we prefer to let $\vartheta_{0}$ vary in the whole $\mathbb{R}$. Particular attention will be given to the zeros of the function $f$.

Lemma 3.1

For any $\lambda,\mu,\vartheta_{0}\in\mathbb{R}$ the set of the zeros of $f$ is unbounded from above and from below. Moreover either $f(\vartheta_{0})=0$ or there are $\vartheta_{1},\vartheta_{2}\in\mathbb{R}$ such that $\vartheta_{0}\in\left(\vartheta_{1},\vartheta_{2}\right)$, $f(\vartheta_{1})=f(\vartheta_{2})=0$ and $f$ has a sign in $\left(\vartheta_{1},\vartheta_{2}\right)$. In particular we have that $\vartheta_{2}-\vartheta_{1}\leq\pi$ if $\mu=0$ and $\vartheta_{2}-\vartheta_{1}<2\pi$ otherwise.

*Proof. *When $\mu=0$ the equation $f=0$ is very easy to solve and the thesis is trivial. Let us consider the case $\mu\neq 0$. We have that for any $k\in\mathbb{Z}$

[TABLE]

Hence the thesis follows because $f$ is a continuous function. $\square$

As an immediate consequence we have the following result.

Corollary 3.2

The function $\vartheta$, maximal solution of (3.14), is defined for any $s\in\mathbb{R}$, is monotone and bounded. More precisely it evolves according one of the following mutually exclusive possibilities:

if $f(\vartheta_{0})=0$ then $\vartheta(s)=\vartheta_{0}$ for every $s$;

• 2)

if $f(\vartheta_{0})>0$ then $\vartheta$ is strictly increasing and there exist the limits

[TABLE]

• 3)

if $f(\vartheta_{0})<0$ then $\vartheta$ is strictly decreasing and there exist the limits

[TABLE]

Since $\vartheta$ is defined for any $s$, by (3.9) also $y$ and $z$ are defined on the whole real line too. Therefore we get that the curve $\gamma$ and the associated surface ${\cal M}$ are complete. Moreover if $f(\vartheta_{0})=0$, then $H=\vartheta^{\prime}=0$ and ${\cal M}$ is a $F_{1}$-invariant minimal surface. In this case equation (1.2) is satisfied if and only if $V$ is tangent to ${\cal M}$. Such kind of minimal surfaces are classified in [LM1]: they are planes (1.3) or logarithmic surfaces (1.4). From now on we will consider always the case $f(\vartheta_{0})\neq 0$. As a consequence of these first results we have some restrictions on the behavior of the coordinate functions $y$ and $z$.

Lemma 3.3

If $\mu=0$, the function $z$ is strictly monotone, while $y$ has at most one critical point. If $\mu\neq 0$, the function $z$ has at most one critical point, while $y$ has at most two critical points. It follows that for any value of $\mu$, both the functions have a limit (finite or not) as $s$ goes to $\pm\infty$.

*Proof. *We proved in Corollary 3.2 that $\vartheta$ is strictly monotone for any value of the parameters $\lambda$, $\mu$ and $\vartheta_{0}$. Let us consider firs the case $\mu=0$: Lemma 3.1 says that $\vartheta$ varies in an interval of length smaller than $\pi$, then, by (3.9), we get that each one of the two functions can have at most one critical point. Moreover, since $\mu=0$, if $z^{\prime}=0$ then $f=0$ too. Therefore $z$ cannot have a critical point in the open interval $(\vartheta_{1},\vartheta_{2})$ by definition of $\vartheta_{1}$ and $\vartheta_{2}$. When $\mu\neq 0$, Lemma 3.1 says that $\vartheta$ varies in an interval of length smaller than $2\pi$, therefore the critical points of the two functions are at most two for each. Finally, as shown in the proof of Lemma 3.1, at least one zero of $f$ lies between $k\pi$ and $(k+1)\pi$ for any $k\in\mathbb{Z}$. Therefore $z^{\prime}=\sin\vartheta=0$ can have at most one solution in the open interval $(\vartheta_{1},\vartheta_{2})$. $\square$

3.1 $F_{1}$-invariant translators in the direction of $F_{2}$

In this section we describe the properties of $\gamma(s)$ when $\mu=0$ and $\lambda\neq 0$. As a consequence we will prove part $2)$ of Theorem 1.1. At first we can notice that, by Lemma 3.3, $z$ is strictly monotone, hence the curve $\gamma$ and the surface ${\cal M}$ are embedded and ${\cal M}$ is a graph $y=F(x,z)$ for some function $F$. The following results describe the asymptotic behavior of $\gamma$.

Lemma 3.4

If $\mu=0$, for every $\vartheta_{0}$ and $\lambda\neq 0$ we have that, as $|s|$ diverges, the function $y(s)$ diverges while we have two possibilities for $z(s)$:

$z(s)$* diverges to $+\infty$ if and only if $\lim_{|s|\rightarrow+\infty}\vartheta(s)=\vartheta_{0}-\lambda$;*

• 2)

$z(s)$* converges to a finite constant otherwise.*

In particular $z$ in bounded from below.

*Proof. *For simplicity we consider only the case of $s\rightarrow+\infty$, the other one is analogous. Since $\mu=0$, (3.13) assume the simpler form

[TABLE]

By Corollary 3.2 we know that the left hand term of (3.15) converges to a finite constant. It follows that $z$ cannot be arbitrary negative. Furthermore $z$ converges to a finite constant if and only if the left hand term does not converge to zero, that is if $\lim_{s\rightarrow+\infty}\vartheta(s)\neq\vartheta_{0}-\lambda$.

Once we know the behavior of $z$, we can describe the evolution of $y$ too. Recalling that $y^{\prime}=e^{z}\cos\vartheta$, if $\lim_{s\rightarrow+\infty}\cos\vartheta(s)\neq 0$, then $y^{\prime}$ is bounded away from zero at least for $s$ big enough, hence $y$ blows up. Finally we have to study the special case of $\lim_{s\rightarrow+\infty}\cos\vartheta(s)=0$. We claim that, under this hypothesis, $z$ diverges. In fact by Corollary 3.2 we have that $\lim_{s\rightarrow+\infty}\vartheta(s)$ is a zero of the function $f(\vartheta)=(\vartheta_{0}-\lambda-\vartheta)\sin\vartheta$. Therefore if $\cos\vartheta$ converges to zero, necessarily $\lim_{s\rightarrow+\infty}\vartheta(s)=\vartheta_{0}-\lambda$ and, by what we proved so far, $z$ diverges. By (3.15) we get that

[TABLE]

Since $\cos(\vartheta_{0}-\lambda)=0$,we have

[TABLE]

Therefore by (3.16) $y^{\prime}$ is bounded away from zero with a sign depending on $\vartheta_{0}$ and $\lambda$, hence $y$ blows up in this case too. $\square$

Remark 3.5

As a consequence of the previous result at least one of the two limits $\lim_{s\rightarrow\pm\infty}z(s)$ is finite, in fact at least one of the two limits $\lim_{s\rightarrow\pm\infty}\vartheta(s)\neq\vartheta_{0}-\lambda$.

• 2)

Since $z$ is strictly monotone in this case, ${\cal M}$ can be globally parametrized as a graph $y=F(z)$ for some function $F$. It follows that in $Sol_{3}$ there are graphical translators that are defined on a half-plane, see Figures 2 and 3 for examples. This phenomenon cannot occur in the Euclidean space as showed by Shahriyari in **[Sh]**.

• 3)

If $z$ converges to a finite constant in both directions, then by Theorem 3.4 there is a $k\in\mathbb{Z}$ such that $\vartheta$ varies in $(k\pi,(k+1)\pi)$. Therefore by (3.9) $y$ has a critical point and it is unique by Lemma 3.3. It follows that in this case $y$ has a global minimum (resp. maximum) and diverges to $+\infty$ (resp. $-\infty$) in both directions. These examples can be thought as the analogous in $Sol_{3}$ of the tilted grim-reaper cylinders in the Euclidean space. See figure 1 for an example.

• 4)

The Gaussian and the intrinsic sectional curvature of a $F_{1}$-invariant surface in $Sol_{3}$ have been computed in **[LM2]**:

[TABLE]

By (3.14) we have that

[TABLE]

Hence we can see that both the Gaussian curvature and $K$ can change their sign on ${\cal M}$. The same phenomenon holds, for example, in the Heisenberg group **[Pi2]**. It is in contrast with the rigidity of the translators in the Euclidean space: in **[SX]** the authors proved that any mean-convex and complete graphical translator in $\mathbb{R}^{3}$ is convex.

We can to improve Lemma 3.4 describing better the asymptotic shape of $\gamma$ and ${\cal M}$. The curve $\gamma$ has two ends: let $\gamma^{+}=\gamma\cap\left\{z>0\right\}$ and let $\gamma^{-}=\gamma\cap\left\{z<0\right\}$. Analogously we define ${\cal M}^{+}={\cal M}\cap\left\{z>0\right\}$ and let ${\cal M}^{-}={\cal M}\cap\left\{z<0\right\}$. By Remark 3.5 we have that $\gamma^{-}$ is always asymptotic to a horizontal line, hence ${\cal M}^{-}$ is asymptotic to a horizontal plane $z=z_{0}$ for some constant $z_{0}<0$. For the upper part we have different possibilities collected in the following result.

Lemma 3.6

Fix $\mu=0$, $\lambda\neq 0$ and $\vartheta_{0}\in\mathbb{R}$.

If $-\pi<\lambda<\pi$ and $\vartheta_{0}=\lambda+k\pi$ for some $k\in\mathbb{Z}$, then ${\cal M}^{+}$ is asymptotic to a half-logarithmic surface 1.5.

• 2)

If $-\frac{\pi}{2}<\lambda<\frac{\pi}{2}$ and $\vartheta_{0}=\lambda+\frac{\pi}{2}+k\pi$ for some $k\in\mathbb{Z}$, then ${\cal M}^{+}$ is asymptotic to a plane $y=C_{1}z+C_{0}$ for some constant $C_{0},C_{1}$.

• 3)

In all other cases ${\cal M}^{+}$ is asymptotic to a horizontal plane or a minimal logarithmic surface (1.4).

*Proof. *We will focus on the curve $\gamma$, after that by $F_{1}$-invariance we can recover the analogous properties for ${\cal M}$.

Arguing by periodicity and changing the orientation of $\gamma$ if necessary, (3.14) can be reduced to

[TABLE]

Let us consider the case $\vartheta_{0}>0$, the other one is analogous. In this case we have that $\lim_{s\rightarrow\infty}\vartheta(s)=0$. Therefore for $s$ big enough we have that

[TABLE]

Hence

[TABLE]

Combining the asymptotic expansion of $y$ and $z$ it is evident that ${\cal M}^{+}$ is asymptotic to a half-logarithmic surface (1.5).

• 2)

By periodicity and changing the sign of $\vartheta$ if $k$ is odd, we can reduce (3.14) to

[TABLE]

We get the end $\gamma^{+}$ when $\lim_{|s|\rightarrow+\infty}\vartheta(s)=\frac{\pi}{2}$. It follows that

[TABLE]

Therefore

[TABLE]

for some constant $C$. Hence

[TABLE]

Then both $y$ and $z$ grow linearly and ${\cal M}^{+}$ is asymptotic to a plane.

• 3)

Up to reverse the orientation of $\gamma$, we can suppose that $\gamma^{+}=\left\{\gamma(s)\ |\ s>0\right\}$. Suppose now that $z$ is unbounded, then by Lemma 3.4 and (3.15) we have that

[TABLE]

Let us define $\sigma_{1}:=\sin(\vartheta_{0}-\lambda)$ and $\sigma_{2}:=\cos(\vartheta_{0}-\lambda)$. Under our hypothesis $\sigma_{1}\in(0,1)$. Therefore for $s$ big enough we get

[TABLE]

and

[TABLE]

The thesis follows easily.

$\square$

We conclude this section with some numerical simulations.

3.2 $F_{1}$-invariant translators in a general direction

In this section we complete the proof of Theorem 1.1 describing the behavior of $\gamma$ for a general direction of evolution.

Lemma 3.7

For any value of $\mu\neq 0$, $\lambda$ and $\vartheta_{0}$ we have that, as $|s|$ diverges, the function $z$ diverges. In particular if $z\rightarrow-\infty$, then $y$ converges to the constant $-\frac{\lambda}{\mu}$, if $z\rightarrow+\infty$ $y$ can either converge to the constant $-\frac{\lambda}{\mu}$ or diverge. When both $y$ and $z$ diverge, $\gamma$ is asymptotic to a logarithmic minimal surface (1.4).

*Proof. *We consider the case of $s\rightarrow+\infty$, the other one is analogous. We know by Corollary 3.2 that $\vartheta$ is converging to a zero of $f=\mu\cos\vartheta+(\vartheta_{0}-\lambda-\vartheta)\sin\vartheta$. Since $\mu\neq 0$, then $\sin\vartheta$ is converging to a non zero constant. By (3.9) $z^{\prime}=\sin\vartheta$, we get that $z$ diverges linearly for any choice of $V$ with $\mu\neq 0$. We recall the formula (3.13) that links the angle with the coordinate funtions:

[TABLE]

As in the proof of Theorem 3.4, we can deduce a lot of information from the fact that the left hand side of the previous equation is converging to a finite constant $k$. It is easily to see that, if $z$ is going to $-\infty$, then $y$ is converging to the constant $-\frac{\lambda}{\mu}$. On the other hand, when $z$ goes to $+\infty$ and $k\neq 0$, then $y$ necessarily blows up exponentially with a sign that depends on $k$ and $\mu$. Finally suppose that $z\rightarrow+\infty$ and $k=0$. The condition $k=0$ is satisfied if and only if $f(\vartheta_{0}-\lambda)=0$, therefore we can deduce that $\cos(\vartheta_{0}-\lambda)=0$. Moreover in this case $\lambda\neq 0$ otherwise $f(\vartheta_{0})=0$. By (3.13) we have

[TABLE]

Note that under our hypothesis $z^{\prime}=\sin\vartheta>0$ at least for $s$ big enough, then $\frac{\cos\vartheta}{\vartheta-\vartheta_{0}+\lambda}$ necessarily converges to $-1$. It follows that for $|s|$ big enough

[TABLE]

Therefore in this case too $y$ can converge to $-\frac{\lambda}{\mu}$ or it can blow up exponentially with a sign that depends on $\lambda$ and $\mu$. $\square$

We conclude this section showing some numerical simulations.

4 Translator invariant with respect to a generic direction

Let us consider a generic Killing vector field $X=aF_{1}+bF_{2}+cF_{3}$. As usual we denote with $X$ also the one-parameter group of isometries generated by this vector field. We want to study the $X$-invariant translators: their properties are very different depending whether $c=0$ or not. We start considering $c=0$ proving Theorem 1.2. We suppose that $ab\neq 0$, otherwise we are in the situation described in Theorem 1.1. Moreover it is not restrictive to assume $a=1$. Analogously to what we have seen in (3.8), we can parametrize ${\cal M}$ as

[TABLE]

As before we can introduce the angle function $\vartheta$, hence we have

[TABLE]

and, since $Sol_{3}$ is homogeneous, we can suppose that $y(0)=z(0)=0$. Since $b\neq 0$, the equation for $\vartheta$ is more complicated than in [LM1], in particular we are not able to reproduce the strategy that we used to transform (3) into (3.14). In order to compute $\vartheta^{\prime}$, first we need to know the mean curvature of ${\cal M}$.

Lemma 4.1

Let ${\cal M}$ as in (4.17), then its mean curvature is

[TABLE]

*Proof. *From (4.17), it is easy to see that a basis of tangent vector field is

[TABLE]

Therefore the induced metric on ${\cal M}$ and a unit normal vector field are respectively

[TABLE]

[TABLE]

By (2.6) we have

[TABLE]

It follows that the second fundamental form of ${\cal M}$ is

[TABLE]

Since the mean curvature is the trace of $Ag^{-1}$, after some straightforward computations we get the thesis. $\square$

Now we have all the ingredients for describing the main properties of the functions $\vartheta$, $y$ and $z$. Therefore by (4.17) we recover the properties of ${\cal M}$ proving Theorem 1.2.

Lemma 4.2

Let ${\cal M}$ defined in (4.17), and let $V$ be a Killing vector field. If ${\cal M}$ is a translator in the direction of $V$, then we have two possibilities:

the function $\vartheta$ is constant, in this case ${\cal M}$ is the horizontal plane $z=0$, hence it is minimal and $V$ is tangent to ${\cal M}$,

• 2)

the function $\vartheta$ is non-constant, but $\sup\vartheta-\inf\vartheta\leq\pi$. In this case ${\cal M}$ is complete and embedded, $z$ is strictly monotone and $y$ can have at most one critical point. It follows that both $y$ and $z$ have a limit (finite or not) as $|s|$ diverges.

*Proof. *Let as usual $V=\eta F_{1}+\lambda F_{2}+\mu F_{3}$ be a Killing vector field. Without loss of generality we can assume $\eta=0$, in fact if it is not the case we have that

[TABLE]

where $\tilde{\lambda}=\lambda-b\eta$. In (4.20) we computed the expression of the normal $\nu$, therefore we get that ${\cal M}$ is a translator in the direction of $V$ if and only if

[TABLE]

Because of the symmetry of ${\cal M}$, the last term in the equalities above cannot depend on $u$, it follows that either $\mu=0$ or $\sin\vartheta\equiv 0$. In the latter case, by (4.18), we have that $z\equiv 0$ and it is well known that this surface is minimal in $Sol_{3}$. We conclude this proof by considering the case $\mu=0$. By Lemma 4.1 and (4.22), after some algebraic manipulation we have that

[TABLE]

Unfortunately we are unable to release this equation from the others, as was done in Section 3, but (4.23) is enough to deduce some information about ${\cal M}$. Let $\vartheta_{0}=\vartheta(0)$ and suppose that $\sin\vartheta_{0}\neq 0$. First of all we can notice that $\vartheta$ is bounded: in fact for every $k\in\mathbb{Z}$ the constant function $\vartheta(s)=k\pi$ is a solution of (4.23) and it acts as a barrier, then for every $\vartheta_{0}$, there is a $k\in\mathbb{Z}$ such that for every $s$

[TABLE]

In particular we have that $\vartheta$ is defined for every $s\in\mathbb{R}$. By (4.18), the same holds for $y$ and $z$. Hence ${\cal M}$ is complete. Finally by (4.24) and (4.18), we can see that $z$ has no critical points, hence ${\cal M}$ is embedded, and $y$ has at most one critical point. $\square$

Finally we prove Theorem 1.3.

Proof of Theorem 1.3. Let $X=aF_{1}+bF_{2}+cF_{3}$ with $c\neq 0$. Without loss of generalization we can suppose $c=1$. Let ${\cal M}$ be a $X$-invariant surface, therefore by Lemma 2.1 there is a curve $\gamma(s)=(x(s),y(s),0)$ parametrized by arc-length such that ${\cal M}$ can be parametrized as follows

[TABLE]

A basis of tangent vector field is

[TABLE]

It follows that a normal vector field is

[TABLE]

Let $V=\eta F_{1}+\lambda F_{2}+\mu F_{3}$ be a Killing vector field. Note that we can always suppose that $\mu=0$ because, since $c\neq 0$ we can write

[TABLE]

where $\tilde{\eta}=\eta-a$ and $\tilde{\lambda}=\lambda-b$. Therefore ${\cal M}$ is a translator in the direction of $V$ if and only if

[TABLE]

After some straightforward calculation we have

[TABLE]

Because of the symmetry of ${\cal M}$, the left hand term in the equation above depends only on $s$ and not on $u$, therefore the only possibility is that the right hand term vanishes, that is $H=0$ and $V$ is tangent to ${\cal M}$. Finally, if ${\cal M}$ satisfies hypothesis $2)$, then it is trivial that it is a translator in the direction of $V$. $\square$

Remark 4.3

By Theorem 1.3 and equation (4.25), we have that when $c\neq 0$, ${\cal M}$ is a $X$-invariant translator in the direction of $V$ if and only if

[TABLE]

Clearly $x^{\prime}$ and $y^{\prime}$ cannot be both zero. Therefore we deduce that such ${\cal M}$ exists if and only if $\tilde{\lambda}\tilde{\eta}=0$. Moreover if in this case $\tilde{\lambda}\neq 0$ (resp. $\tilde{\eta}\neq 0$) then $x^{\prime}\equiv 0$ (resp. $y^{\prime}\equiv 0$). It the special case $a=b=0$ this means that the only $F_{3}$-invariant translators are the planes $x\equiv c_{0}$ and $y\equiv c_{0}$ for some constant $c_{0}$.