I am trying to read the paper [Boost invariant marginally trapped surfaces in Minkowski $4$-space](http://iopscience.iop.org/article/10.1088/0264-9381/24/22/009/pdf), by Haesen and Ortega. Basically, they work with metric signature $(-+++)$ in $\Bbb R^4\_1$, consider the group of boosts $$G = \{ R^h\_\theta\oplus {\rm Id}\_2 \mid \theta \in \Bbb R \},$$where $R^h\_\theta \in {\rm O}\_1^{+\uparrow}(2,\Bbb R)$ denotes hyperbolic rotation by $\theta$, and they claim that if $S \subseteq \Bbb R^4\_1$ is a **connected** spacelike surface which is invariant by the action of $G$ in $\Bbb R^4\_1$, then $S$ admits a parametrization $X\colon I \times \Bbb R \to \Sigma\_\alpha \subseteq S$ of the form $$X(s,\theta) = B\_\theta(\alpha(s))$$for some open interval $I\subseteq \Bbb R$, *where the image $\Sigma\_\alpha$ is an open dense subset of $S$*.

> 
> I can't check that such a parametrization is always available, let alone prove that the image is dense in $S$.
> 
> 
> 

They mention that the set of fixed points of the action is $\{(0,0)\} \times \Bbb R^2$ very near of that claim, so I think this should be useful. Please help.

---

**Edit 1**: I think I can convince myself why the parametrization has that form. If you take $(x,y,z,w) \in S$, since $S$ is invariant, we can write $(x(s),y(s)) = (\alpha\_1(s)\sinh \theta, \alpha\_1(s)\cosh\theta)$, assuming that $S \subseteq \mathcal{R}$. Since $\dim S = 2$, $z$ and $w$ can only depend on one parameter, which obviously cannot be $\theta$. Meaning $(z(s),w(s))=(\alpha\_3(s),\alpha\_4(s))$. So every point has the form $B\_\theta(\alpha(s))$. But I think this is a bit too much handwaving and tells me nothing about the image being dense.