Properties of close-to-harmonic mappings (Q1594252)

Let \(U\) be an open subset of \(\mathbb{R}^n\). The paper is concerned with continuously differentiable mappings \(f: U\to \mathbb{R}^m\) that satisfy certain Sobolev-type conditions and are close (in terms of local deviation in the \(C^1\)-norm) to harmonic mappings. A large number of theorems are announced, but no proofs are given. Among the announced results are analogues of the classical Liouville theorem for bounded harmonic functions on \(\mathbb{R}^n\) and theorems about the topological properties of certain classes of close-to-harmonic mappings. The paper is related to, and to a certain extent depends upon, an earlier paper of the author [see Zbl 0958.58013].