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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.05056 |
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| _version_ | 1866913191453261824 |
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| author | Xu, Xu Zheng, Chao |
| author_facet | Xu, Xu Zheng, Chao |
| contents | In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial $α$-curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures, we introduce the combinatorial $α$-Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial $α$-Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial $α$-Ricci flow with surgery. As an application of the combinatorial $α$-Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. We further introduce the combinatorial $α$-Calabi flow with surgery and study its longtime behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces Xu, Xu Zheng, Chao Differential Geometry (2020): 52C26 In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial $α$-curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures, we introduce the combinatorial $α$-Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial $α$-Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial $α$-Ricci flow with surgery. As an application of the combinatorial $α$-Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. We further introduce the combinatorial $α$-Calabi flow with surgery and study its longtime behavior. |
| title | A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces |
| topic | Differential Geometry (2020): 52C26 |
| url | https://arxiv.org/abs/2401.05056 |