无穷小球面与共形不变度量
Infinitesimally small spheres and conformally invariant metrics
摘要：
域$D\subset \mathbb{R}^n$上的模度量（也称为容量度量）可以被定义为$μ_D(x,y)=\inf\{\mbox{cap}\,(D,γ)\}$，其中${\mbox{cap}}\,(D,γ)$代表电容器$(D,γ)$的容量，并且下确界在包含点$x$和$y$的所有连续体$γ\subset D$上取得。它是由J. Ferrand，G. Martin和M. Vuorinen在1991年证明了模度量中的等距映射是共形映射.本文证实了这一猜想，并证明了度量空间$(D,μ_D)$中球面曲面的新的几何性质。
英文摘要：
The modulus metric (also called the capacity metric) on a domain $D\subset \mathbb{R}^n$ can be defined as $μ_D(x,y)=\inf\{\mbox{cap}\,(D,γ)\}$, where ${\mbox{cap}}\,(D,γ)$ stands for the capacity of the condenser $(D,γ)$ and the infimum is taken over all continua $γ\subset D$ 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 $(D,μ_D)$.
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