Isometries for the modulus metric in higher dimensions are conformal mappings

Abstract

for a proper subdomain D of ${{\overline {\mathbb{R}}} ^n}$ and for all x,y ∈ D define

$${\mu _D}\left({x,\,y} \right) = \mathop {\inf}\limits_{{C_{xy}}} \,\,{\rm{Cap}}\left({D,\,{C_{xy}}} \right),$$

where the infimum is taken over all curves C_xy = γ[0, 1] in D with γ(0) = x and γ(1) = y, and Cap denotes the conformal capacity of condensers. The quantity μ_D is a metric if and only if the domain D has a boundary of positive conformal capacity. If Cap(∂D) > 0, we call μ_D 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.