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Flexibility and rigidity of conformal embeddings in Lorentzian manifolds

Alaa Boukholkhal

TL;DR

The paper proves that any spacelike embedding of a closed surface into a Lorentzian manifold can be C^0-approximated by a smooth embedding that is conformal for any prescribed Riemannian metric on the surface, using a convex integration via a corrugation process. It then shows that in ambient spaces of the form $\mathcal{T}/\Gamma$, where $\Gamma$ is a cocompact lattice in $SO^\circ(2,1)$, the set of negatively curved metrics admitting isometric embeddings into this spacetime projects to a relatively compact subset of Teichmüller space, highlighting a rigidity for isometric embeddings versus flexibility for conformal embeddings. The argument combines almost-isometric constructions in local patches with global Teichmüller-space analysis, treating genus 1 separately from higher genus, and uses a fixed-point argument to realize the target conformal structure. The results illuminate the geometry of convex Cauchy surfaces in flat maximal globally hyperbolic spacetimes and provide a precise compactness description in Teichmüller theory for negatively curved isometric embeddings.

Abstract

We prove that for any Riemannian metric $g$ on a closed orientable surface $Σ$ and any spacelike embedding $f:Σ\rightarrow M$ 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 $Σ$ that admit isometric embeddings in $M$ projects into a relatively compact set in the Teichmüller space.

Flexibility and rigidity of conformal embeddings in Lorentzian manifolds

TL;DR

The paper proves that any spacelike embedding of a closed surface into a Lorentzian manifold can be C^0-approximated by a smooth embedding that is conformal for any prescribed Riemannian metric on the surface, using a convex integration via a corrugation process. It then shows that in ambient spaces of the form , where is a cocompact lattice in , the set of negatively curved metrics admitting isometric embeddings into this spacetime projects to a relatively compact subset of Teichmüller space, highlighting a rigidity for isometric embeddings versus flexibility for conformal embeddings. The argument combines almost-isometric constructions in local patches with global Teichmüller-space analysis, treating genus 1 separately from higher genus, and uses a fixed-point argument to realize the target conformal structure. The results illuminate the geometry of convex Cauchy surfaces in flat maximal globally hyperbolic spacetimes and provide a precise compactness description in Teichmüller theory for negatively curved isometric embeddings.

Abstract

We prove that for any Riemannian metric on a closed orientable surface and any spacelike embedding in a pseudo-Riemannian manifold , the embedding can be -approximated by a smooth conformal embedding for . If in addition, is a quotient of the -dimensional solid timelike cone by a cocompact lattice of , we show that the set of negatively curved metrics on that admit isometric embeddings in projects into a relatively compact set in the Teichmüller space.
Paper Structure (11 sections, 16 theorems, 79 equations, 1 figure)

This paper contains 11 sections, 16 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgement
  4. Almost isometric maps
  5. Local construction
  6. Global construction
  7. Some Teichmüller theory
  8. Proof of Theorem \ref{['T1']}
  9. Genus one case
  10. Higher genus case
  11. Proof of Theorem \ref{['T2']}

Key Result

Theorem 1.1

Let $(M,h)$ be a pseudo-Riemannian manifold of dimension $\ge 3$ and $f: \Sigma \rightarrow (M,h)$ a spacelike embedding of a closed orientable surface $\Sigma$. For any Riemannian metric $g$ on $\Sigma$, there exists a smooth spacelike embedding $F : \Sigma \rightarrow M$ such that:

Figures (1)

  • Figure 1: Intersection of the tangent plane (in purple) with the level $\mathbb{H}^{2}_\alpha$ (in green) implies that the graph of $u$ (in blue) is not bounded by $\mathbb{H}^{2}_\alpha$

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • ...and 18 more