Discrete Einstein metrics on trees

Shuliang Bai; Haoxuan Cheng; Bobo Hua

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arXiv · 2604.22449

Discrete Einstein metrics on trees

Shuliang Bai·Haoxuan Cheng·Bobo Hua

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Abstract

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.

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Shuliang Bai, Haoxuan Cheng, Bobo Hua. 2026-04-24. Discrete Einstein metrics on trees. https://arxiv.org/abs/2604.22449

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