Assouad-Nagata dimension addresses both large and small scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space $M$ is a minor-closed metric if there exists an (edge-)weighted graph $G$ in a fixed minor-closed family such that the underlying space of $M$ is the vertex-set of $G$, and the metric of $M$ is the distance function in $G$. Minor-closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge-deletion and edge-contraction. In this paper, we determine the Assouad-Nagata dimension of every minor-closed metric. It is a common generalization of known results for the asymptotic dimension of $H$-minor free unweighted graphs and the Assouad-Nagata dimension of some 2-dimensional continuous spaces (e.g.\ complete Riemannian surfaces with finite Euler genus) and their corollaries.

翻译：Assouad-Nagata维数同时刻画度量空间的大尺度与小尺度行为，是Gromov渐近维数的精细化概念。若存在固定极小闭族中的（边）加权图$G$，使得度量空间$M$的底层空间为$G$的顶点集，且$M$的度量为$G$中的距离函数，则称$M$为极小闭度量空间。这类度量空间自然产生于通过边删除与边收缩操作移除底层图中的冗余边之际。本文确定了所有极小闭度量空间的Assouad-Nagata维数。该结果统一推广了关于无$H$-极小闭非加权图渐近维数的已知结论，以及若干二维连续空间（如有限欧拉示性数的完备黎曼曲面）的Assouad-Nagata维数及其推论。