Quasiregular mappings and metrics on the n-sphere with punctures
A Schottky-type theorem is established for quasimeromorphic mappings: \(\sigma\) being the spherical metric, the unit sphere \(S^ n\) with \(q\geq p_ 0(n,K)\) punctures \(a_ 1,...,a_ q\) admits a conformal metric \(d\tau =\lambda d\sigma\) with \(\lambda (y)=O(| \sigma (y,a_ i)\log \sigma (y,a_ i)|^{-1})\) near the punctures, with the property that all K-quasiregular mappings \(f: B^ n\to S^ n\) omitting the values \(a_ 1,...,a_ q\) are distance decreasing for large distances, \(\tau (f(x),f(y))\leq \max (\rho (x,y),\delta (n,K,a_ 1,...,a_ q))\) for all \(x,y\in B^ n\) with respect to the hyperbolic metric \(\rho\) of the unit ball \(B^ n\). Quite as in the classical case, this implies the big Picard theorem: an n-dimensional K-quasimeromorphic mapping cannot omit \(q_ 0(n,K)\) values in any neighbourhood of an essential singularity. Actually, the proof of the above Schottky theorem is based on techniques developed for proving the quasiregular Picard theorem [cf. the author in J. Anal. Math. 37, 100-117 (1980; Zbl 0451.30012); Lect. Notes Math. 981, 220-245 (1983; reviewed above)].
scientific article; zbMATH DE number 3886224