The modulus metric (also called the capacity metric) on a domain D ⊂ ℝⁿ can be defined as μ_D(x, y) = inf{cap (D, γ)}, where cap (D, γ) stands for the capacity of the condenser (D, γ) and the infimum is taken over all continua γ ⊂ D containing the points x and y. It was conjectured by J. Ferrand, G. Martin and M. Vuorinen in 1991 that every isometry in the modulus metric is a conformal mapping. In this note, we confirm this conjecture and prove new geometric properties of surfaces that are spheres in the metric space (D, γ_D).

中文翻译：

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无限小球体和不变形度量

域上的弹性模量度量（也称为容量度量）d⊂ℝ ^Ñ可以被定义为μ _d（X，Y）= INF {帽（d，γ）}，其中盖（d，γ）表示的能力聚光镜（D，γ）的总和取最小值，取所有包含点x和y的连续体γ⊂D。1991年，J。Ferrand，G。Martin和M. Vuorinen推测，模量度量中的每个等轴测图都是保形映射。在这份说明中，我们证实了这一猜想，证明是在度量空间（球体表面的新的几何特性d，γ _d）。