Flexibility and Rigidity of Conformal Embeddings in Lorentzian Manifolds

We prove that for any Riemannian metric g on a closed orientable surface and any spacelike embedding f: 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^ (2, 1), we show that the set of negatively curved metrics on that admit isometric embeddings in M projects into a relatively compact set in the Teichm\"uller space.