Flexibility and Rigidity of Conformal Embeddings in Lorentzian Manifolds

We prove that for any Riemannian metric g on a closed orientable surface Σ and any spacelike embedding f:Σ → M in a pseudo-Riemannian manifold $(M,h)$, the embedding f can be C⁰-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 Σ that admit isometric embeddings in M projects into a relatively compact set in the Teichm\"uller space.