A zoo of translating solitons on a parallel light-like direction in Minkowski 3-space

Abstract

We deal with solitons of the mean curvature flow. The definition of translating solitons on a light-like direction in Minkowski 3-space is introduced. Firstly, we classify those which are graphical, translation surfaces, obtaining space-like and time-like, entire and not entire, complete and incomplete examples. Among them, all our time-like examples are incomplete. The second family consists of those which are invariant by a 1-dimensional subgroup of parabolic motions, i.e., with light-like axis. The classification result implies that all examples of this second family have singularities.

Introduction

Hypersurfaces in Euclidean space which evolve along the mean curvature flow have been widely studied. Of particular interest are those called translating solitons, which are those whose mean curvature vector satisfies the following equation

$$\overrightarrow{H}={\overrightarrow{K}}^{\perp }$$

where ⊥ and $\overrightarrow{K}$ denote projection on the normal bundle and a unit vector field, respectively. A much more general, but very weak, definition can be found in [1], where virtually no restriction on $\overrightarrow{K}$ is set (see also [5]). Among many papers, we can select [1], [5], [10], which study translating solitons in Riemannian manifolds. However, the theory seems to be less developed in Lorentzian geometry, although relevant results can be found in [6] and [14]. According to them, translating solitons in Minkowski space are mainly studied when the vector $\overrightarrow{K}$ is parallel and time-like.

In this paper, we wish to introduce a new family, namely translating solitons of the mean curvature flow such that $\overrightarrow{K}$ is a light-like parallel vector. More precisely.

Definition 1.1

Given a parallel light-like vector $\overrightarrow{K}$ in Minkowski 3-space ${\mathbb{L}}^3$, a non-degenerate immersion $\psi :M\to {\mathbb{L}}^3$ will be called a translating soliton on the light-like direction $\overrightarrow{K}$ if its mean curvature vector $\overrightarrow{H}$ satisfies $\overrightarrow{H}={\overrightarrow{K}}^{\perp }$, where ⊥ means the orthogonal projection on the normal bundle.

This definition makes sense because the induced metric is not degenerate, which implies that $\overrightarrow{H}$ will not be light-like at any point.

We write the standard flat metric in Minkowski 3-space in a suitable way for our needs, namely $〈,〉=-2dxdy+d{z}^2$. That is to say, we are considering a basis $\mathcal{B}=\{\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}\}$ such that $\overrightarrow{x}$, $\overrightarrow{y}$ are light-like, future pointing, and satisfying the normalizing condition $〈\overrightarrow{x},\overrightarrow{y}〉=-1$. It is important to recall that any two parallel light-like vectors are linked by an isometry of ${\mathbb{L}}^3$, because the light cone is invariant by rotations, boosts and a few reflections. This means that we can reduce to the case $\overrightarrow{K}=\overrightarrow{x}$. We will study two families.

Firstly, Section 3 is devoted to studying graphical surfaces, that is to say, the ones that admit a parametrization $\psi :\mathrm{\Omega }\subset {\mathbb{R}}^2\to {\mathbb{L}}^3$, $\psi (y,z)=(u(y,z),y,z)$, where $u:\mathrm{\Omega }\to \mathbb{R}$. In Theorem 3, we will classify those which, in addition, are translation surfaces, i.e., for some smooth functions a and b, then $u(y,z)=a(y)+b(z)$. Translation surfaces in Euclidean space were introduced by S. Lie (see [4], also [12]). Needless to say, the same definition can be easily set in Minkowski space. We study space-like and time-like surfaces, obtain four types, which are flat by chance. Note that [6] and [14] paid attention to entire examples. We later study the completeness of the four cases, showing entire and not entire, complete and incomplete surfaces, along Corollary 3.2, Corollary 3.3, Corollary 3.4, Corollary 3.5. We should remark that the standard techniques to show the completeness of space-like surfaces in Minkowski space do not work in our setting (see Remark 3.1).

Secondly, there are 1-dimensional subgroups of parabolic isometries of ${\mathbb{L}}^3$, whose axis is light-like. We will reduce to the well-adapted case when the rotation axis is spanned by the vector $\overrightarrow{x}$ (see Section 4 for more details.) That is to say, we study surfaces obtained by letting this subgroup of isometries act on a suitable profile curve. We classify in Theorem 4.2 those translating solitons which are invariant by this subgroup of isometries.