The volume of conformally flat manifolds as hypersurfaces in the light-cone
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910418732056576 |
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| author | Kishida, Riku |
| author_facet | Kishida, Riku |
| contents | In this paper, we focus on a conformally flat Riemannian manifold $(M^n,g)$ of dimension $n$ isometrically immersed into the $(n+1)$-dimensional light-cone $Λ^{n+1}$ as a hypersurface. We compute the first and the second variational formulas on the volume of such hypersurfaces. Such a hypersurface $M^n$ is not only immersed in $Λ^{n+1}$ but also isometrically realized as a hypersurface of a certain null hypersurface $N^{n+1}$ in the Minkowski spacetime, which is different from $Λ^{n+1}$. Moreover, $M^n$ has a volume-maximizing property in $N^{n+1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The volume of conformally flat manifolds as hypersurfaces in the light-cone Kishida, Riku Differential Geometry Primary 53C42, Secondly 53B30, 53C50 In this paper, we focus on a conformally flat Riemannian manifold $(M^n,g)$ of dimension $n$ isometrically immersed into the $(n+1)$-dimensional light-cone $Λ^{n+1}$ as a hypersurface. We compute the first and the second variational formulas on the volume of such hypersurfaces. Such a hypersurface $M^n$ is not only immersed in $Λ^{n+1}$ but also isometrically realized as a hypersurface of a certain null hypersurface $N^{n+1}$ in the Minkowski spacetime, which is different from $Λ^{n+1}$. Moreover, $M^n$ has a volume-maximizing property in $N^{n+1}$. |
| title | The volume of conformally flat manifolds as hypersurfaces in the light-cone |
| topic | Differential Geometry Primary 53C42, Secondly 53B30, 53C50 |
| url | https://arxiv.org/abs/2404.14761 |