Flexibility and rigidity of conformal embeddings in Lorentzian manifolds

TL;DR

This paper investigates the flexibility of conformal embeddings of surfaces into Lorentzian manifolds, showing approximation results and properties of negatively curved metrics in specific spacetime models.

Contribution

It establishes that any spacelike embedding can be approximated by smooth conformal embeddings and analyzes the space of negatively curved metrics admitting isometric embeddings in certain Lorentzian manifolds.

Findings

Any spacelike embedding can be approximated by a smooth conformal embedding.

The set of negatively curved metrics admitting isometric embeddings projects into a relatively compact set in Teichmüller space.

Results apply to embeddings in quotients of the 2+1-dimensional solid timelike cone.

Abstract

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.