  We prove that for any Riemannian metric $g$ on a closed orientable surface
$\Sigma$ and any spacelike embedding $f:\Sigma \rightarrow M$ in a
pseudo-Riemannian manifold $(M,h)$, the embedding $f$ can be
$C^{0}$-approximated by a smooth conformal embedding for $g$. If in addition,
$M$ is a quotient of the $(2+1)$-dimensional solid timelike cone by a cocompact
lattice of $SO^{\circ}(2,1)$, we show that the set of negatively curved metrics
on $\Sigma$ that admit isometric embeddings in $M$ projects into a relatively
compact set in the Teichm\"uller space.
