Flexibility and rigidity of conformal embeddings in Lorentzian manifolds

We prove that for any Riemannian metric $g$ on a closed orientable surface $\mathrm{\Sigma }$ and any spacelike embedding $f:\mathrm{\Sigma }\to M$ 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 $S{O}^{\circ }(2,1)$, we show that the set of negatively curved metrics on $\mathrm{\Sigma }$ that admit isometric embeddings in $M$ projects into a relatively compact set in the Teichmüller space.