Discrete Einstein metrics on trees
Differential Geometry
2026-05-25 v2
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.
Cite
@article{arxiv.2604.22449,
title = {Discrete Einstein metrics on trees},
author = {Shuliang Bai and Haoxuan Cheng and Bobo Hua},
journal= {arXiv preprint arXiv:2604.22449},
year = {2026}
}
Comments
32 pages