Discrete Einstein metrics on trees
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914589636034560 |
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| author | Bai, Shuliang Cheng, Haoxuan Hua, Bobo |
| author_facet | Bai, Shuliang Cheng, Haoxuan Hua, Bobo |
| contents | We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. We establish a sharp upper bound for the largest eigenvalue of the associated Ricci matrix in terms of the maximum degree. Turning to structural properties, notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22449 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Discrete Einstein metrics on trees Bai, Shuliang Cheng, Haoxuan Hua, Bobo Differential Geometry 53C21 We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. We establish a sharp upper bound for the largest eigenvalue of the associated Ricci matrix in terms of the maximum degree. Turning to structural properties, notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge. |
| title | Discrete Einstein metrics on trees |
| topic | Differential Geometry 53C21 |
| url | https://arxiv.org/abs/2604.22449 |