Conformality in the sense of Gromov and a generalized Liouville theorem

2021/08/02 by Zorich, V. A.
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.00945

Abstract

M.Gromov extended the concepts of conformal and quasiconformal mapping to the mappings acting between the manifolds of different dimensions. For instance, any entire holomorphic function f: \Cn → \mathbb C defines a mapping conformal in the sense of Gromov. In this connection Gromov addressed a natural question: which facts of the classical theory apply to these mappings? In particular is it true that \em If the mapping F: \R^{n + 1} → \Rⁿ is conformal and bounded, then it is a constant mapping, provided that n ≥ 2 ~? We present arguments confirming the validity of such a Liouville-type theorem.

Related

• A generalization of the Picard theorem
• Liouville-type theorems for conformal mappings and their application
• Conformality of quasiconformal mappings at a point, revisited
• On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry
• A Commentary on Teichmüller's paper "Untersuchungen über konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) (to appear in Vol. VII of the Handbook of Teichmüller theory