The harmonic coordinates of a point and the harmonic metric (Q2782476)

Let \(D\) be a simply connected Jordan domain in the plane with three distinguished boundary points \(z_1\), \(z_2\), \(z_3\). \(D(z_1,z_2,z_3)\) denotes this trilateral. There are harmonic measures \(\omega_D((z_i,z_j),z)\) given with the boundary values \(1\) on the boundary arc between \(z_i\) and \(z_j\) and \(0\) otherwise. The author introduces the harmonic center, this is the unique determined point \(s\in D\) such that \(\omega_D((z_1,z_2),s)= \omega_D((z_2,z_3), s)= 1/3\) holds and the harmonic coordinates \(\xi_1\), \(\xi_2\) for a point \(w\) with respect to the trilateral \(D(z_1,z_2,z_3)\): NEWLINE\[NEWLINE\xi_1= \omega_D((z_2, z_3), w)- 1/3,\quad \xi_2= \omega_D((z_3, z_1), w)- 1/3.NEWLINE\]NEWLINE Basing on this a characterization of \(K\)-quasiconformal mappings is given which used the well-known Grötzsch function and a metric is introduced.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00021].