On stability of Möbius transformations in the class of mappings with bounded distortion (Q1335960)

Let \(f: U\to \mathbb{R}^ n\) be a mapping with bounded distortion, \(U\subset \mathbb{R}^ n\). Denote the operator norm of the formal derivative of \(f\) by \(| f'(x)|\), the Jacobian by \(J(x,f)\), \[ K(x,f)= | f'(x)|/| J(x,f)|,\quad K(f)= \text{ess sup}_{x\in U} K(x,f). \] For \(n\geq 3\) every mapping \(f\) with \(K(f)= 1\) is a Möbius transformation [the second author, Stability theorems in geometry and analysis [in Russian] (1982; Zbl 0523.53025)]. M. A. Lavrent'ev raised the following question: Is the mapping \(f\) close to some Möbius transformation, if the function \(K(x,f)\) is close to 1? In the case when the discrepancy between \(K(x,f)\) and 1 is measured in the norm of \(L_ \infty(U)\) the question has a positive answer. The aim of the article is to prove that this result remains true in the case when the function \(K(x,f)\) is close to 1 only in some integral sense.