Flexibility and rigidity of conformal embeddings in Lorentzian manifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909458972540928 |
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| author | Boukholkhal, Alaa |
| author_facet | Boukholkhal, Alaa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flexibility and rigidity of conformal embeddings in Lorentzian manifolds Boukholkhal, Alaa Differential Geometry 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. |
| title | Flexibility and rigidity of conformal embeddings in Lorentzian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2407.10346 |