The Weyl problem for unbounded convex domains in $\HH^3$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912076858916864 |
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| author | Schlenker, Jean-Marc |
| author_facet | Schlenker, Jean-Marc |
| contents | Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. The induced metric on $\partial K$ then has curvature $K>-1$. It was proved by Alexandrov that if $K$ is bounded, then it is uniquely determined by the induced metric on the boundary, and any smooth metric with curvature $K>-1$ can be obtained.
We propose here an extension of the existence part of this result to unbounded convex domains in $\HH^3$. The induced metric on $\partial K$ is then clearly not sufficient to determine $K$. However one can consider a richer data on the boundary including the ideal boundary of $K$. Specifically, we consider the data composed of the full conformal structure on the boundary of $K$ (in the Poincaré model of $\HH^3$), together with the induced metric on $\partial K$. We show that a wide range of "reasonable" data of this type, satisfying mild curvature conditions, can be realized on the boundary of a convex subset in $\HH^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_02101 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Weyl problem for unbounded convex domains in $\HH^3$ Schlenker, Jean-Marc Differential Geometry Geometric Topology Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. The induced metric on $\partial K$ then has curvature $K>-1$. It was proved by Alexandrov that if $K$ is bounded, then it is uniquely determined by the induced metric on the boundary, and any smooth metric with curvature $K>-1$ can be obtained. We propose here an extension of the existence part of this result to unbounded convex domains in $\HH^3$. The induced metric on $\partial K$ is then clearly not sufficient to determine $K$. However one can consider a richer data on the boundary including the ideal boundary of $K$. Specifically, we consider the data composed of the full conformal structure on the boundary of $K$ (in the Poincaré model of $\HH^3$), together with the induced metric on $\partial K$. We show that a wide range of "reasonable" data of this type, satisfying mild curvature conditions, can be realized on the boundary of a convex subset in $\HH^3$. |
| title | The Weyl problem for unbounded convex domains in $\HH^3$ |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2106.02101 |