Reduced $p$-modulus, $p$-harmonic radius and $p$-harmonic Green's mappings

We consider the definitions and properties of the metric characteristics of the spatial domains previously introduced by the author, and their connection with the class of mappings, the particular case of which are the harmonic Green's mappings introduced by A. I. Janushauskas. In determining these mappings, the role of the harmonic Green's function is played by the $p$-harmonic Green's function of the $n$-dimensional region ($1<p<\infty$), the existence and properties of which are established by S. Kichenassamy and L. Veron. The properties of $p$-harmonic Green mappings established in the general case are analogous to the properties of harmonic Green's mappings ($p = 2$, $n = 3$). In particular, it is proved that the $p$-harmonic radius of the spatial domain has a geometric meaning analogous to the conformal radius of a plane domain.