Discrete Yamabe problem for polyhedral surfaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866913246867357696 |
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| author | Kouřimská, Hana Dal Poz |
| author_facet | Kouřimská, Hana Dal Poz |
| contents | We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_15693 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Discrete Yamabe problem for polyhedral surfaces Kouřimská, Hana Dal Poz Metric Geometry Geometric Topology 57M50 (Primary), 52B10, 52C26 (Secondary) We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique. |
| title | Discrete Yamabe problem for polyhedral surfaces |
| topic | Metric Geometry Geometric Topology 57M50 (Primary), 52B10, 52C26 (Secondary) |
| url | https://arxiv.org/abs/2103.15693 |