A generalization of the Picard theorem

TL;DR

This paper extends classical conformal and quasiconformal mappings to Gromov's sense, proving that such mappings from higher-dimensional spaces to the plane that omit multiple values must be constant.

Contribution

It introduces a generalized notion of quasiconformal mappings in Gromov's framework and proves a Picard-type theorem for these mappings.

Findings

Quasiconformal Gromov mappings omitting more than one value are constant.

Extension of classical Picard theorem to higher dimensions and Gromov's quasiconformal mappings.

Establishes a new link between Gromov's geometric analysis and classical complex analysis.

Abstract

We recall the notions of conformal and quasiconformal mappings \textit{in the sense of Gromov}, extending the classical notions of conformal and quasiconformal mappings, and prove the following theorem. {\em If the mapping $ F: \mathbb{R}^{n} \to \mathbb{R}^{2} $, where $ n \geq 2 $, quasiconformal in the sense of Gromov, omits more than one value on the plane $\mathbb{R}^{2} $, then it is a constant mapping.}