发表机构
Beijing Yanqi Lake Institute of Mathematical Sciences and Applications（北京燕京数学科学应用研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究单圈图上离散爱因斯坦度量，将有限树相关理论扩展至此。确定树状情形持续条件及范围，超出范围问题呈分段线性，有新现象。证明正则太阳图的爱因斯坦度量存在且唯一。
AI 中文摘要
在早期与程和华的合作中，我们证明了在有限树上，林 - 陆 - 丘曲率的离散爱因斯坦度量是边索引里奇矩阵的佩龙特征向量。我们将此理论扩展到单圈图。我们精确确定树状情形何时持续存在——即平衡状态，此时谱变为周期性而非狄利克雷型——并以封闭形式计算裸圈和正则太阳图（带悬垂叶的圈）的情况；对于长圈上的单个装饰顶点，它持续存在直至明确的黄金比例阈值。超出此范围，问题是分段线性的，并且出现了树上不可能出现的现象：爱因斯坦度量可能不唯一或不存在——带悬垂叶的三角形就没有。对于正则太阳图，我们证明它存在且唯一。
英文摘要
In earlier work with Cheng and Hua we showed that on a finite tree the Lin--Lu--Yau discrete Einstein equation is a single linear eigenvalue problem: the metric is the Perron eigenvector of an edge-indexed Ricci matrix. We extend this theory to unicyclic graphs. For geodesic cycle weightings of girth at least four, we characterize exactly when the same local matrix computes the true curvature; this is the balanced regime. Schur complementation of the pendant trees then reduces the eigenvalue problem to a periodic transfer equation on the cycle. We obtain closed formulas for bare cycles and regular suns, prove a sharp maximum-degree bound, and show that the spectral top of a fixed localized defect on a growing cycle converges to the top bound state of the corresponding infinite-line operator. This yields an explicit golden-ratio threshold for balance. Outside the balanced regime the active shortest paths and optimal transport constraints must also be tracked, and phenomena impossible on trees occur: the bare four-cycle has a one-parameter family of Einstein metrics, whereas a triangle with one pendant leaf has none. For every regular sun we construct the symmetric Einstein metric and prove that it is the unique Einstein metric, up to scale, among all positive weightings. As long as a normalized Ricci-flow trajectory remains in the balanced cone, its projective dynamics is generated by the fixed Ricci matrix.
