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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2309.13533 |
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| _version_ | 1866917060204822528 |
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| author | Chowdhury, Saajid Hu, Hechen Romney, Matthew Tsou, Adam |
| author_facet | Chowdhury, Saajid Hu, Hechen Romney, Matthew Tsou, Adam |
| contents | We study the properties of $\text{CAT}(κ)$ surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the $\text{CAT}(κ)$ condition locally. The main facts about $\text{CAT}(κ)$ surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation. We provide a complete proof that $\text{CAT}(κ)$ surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of $\text{CAT}(κ)$ surfaces. We also show that $\text{CAT}(κ)$ surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most $κ$. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13533 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On CAT($κ$) surfaces Chowdhury, Saajid Hu, Hechen Romney, Matthew Tsou, Adam Metric Geometry Differential Geometry 53C45 We study the properties of $\text{CAT}(κ)$ surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the $\text{CAT}(κ)$ condition locally. The main facts about $\text{CAT}(κ)$ surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation. We provide a complete proof that $\text{CAT}(κ)$ surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of $\text{CAT}(κ)$ surfaces. We also show that $\text{CAT}(κ)$ surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most $κ$. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces. |
| title | On CAT($κ$) surfaces |
| topic | Metric Geometry Differential Geometry 53C45 |
| url | https://arxiv.org/abs/2309.13533 |