Flexibility and rigidity of conformal embeddings in Lorentzian manifolds

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.