Isometries for the modulus metric in higher dimensions are conformal mappings  

作　　者：Xiaohui Zhang 

机构地区：[1]Department of Mathematical Sciences,Zhejiang Sci-Tech University,Hangzhou,310018,China

出　　处：《Science China Mathematics》2021年第9期1951-1958,共8页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11771400 and 11911530457);Science Foundation of Zhejiang Sci-Tech University(Grant No.16062023Y)。

摘　　要：for a proper subdomain D of R^(n) and for all x,y∈D defineμD(x,y)=infC_(xy)Cap(D,C_(xy)),where the infimum is taken over all curves Cxy=γ[0,1]in D withγ(0)=x andγ(1)=y,and Cap denotes the conformal capacity of condensers.The quantityμD is a metric if and only if the domain D has a boundary of positive conformal capacity.If Cap(∂D)>0,we callμD the modulus metric of D.Ferrand et al.(1991)have conjectured that isometries for the modulus metric are conformal mappings.Very recently,this conjecture has been proved for n=2 by Betsakos and Pouliasis(2019).In this paper,we prove that the conjecture is also true in higher dimensions n⩾3.

关 键 词：capacity modulus metric ISOMETRY M?bius transformation 

分 类 号：O186.12[理学—数学]