Regular domains in the theory of mappings on Riemann manifolds (Q2879998)

The authors investigate mappings \(f:D\rightarrow D_*\), where the domains \(D\) and \(D_*\) belong to some Riemann manifolds \({\mathbb M}^n\) and \(\mathbb M_*^n,\) respectively, and \(\dim{\mathbb M}^n=\dim \mathbb M_*^n=n\). Some problems regarding the boundary behavior of such mappings are considered. The maps investigated here are the so-called \(Q\)-homeomorphisms which generalize quasiconformal mappings. The main results of the paper are the following.NEWLINENEWLINE Every convex, quasiconvex, Lipschitz, uniform or QED-domain of a smooth Riemann manifold has a weakly flat boundary. A domain \(D\) is said to be regular if and only if it satisfies at least one of the properties mentioned above. The authors state the possibility of continuous boundary extension of every inverse mapping \(f^{-1}:D_*\rightarrow D\) for a ring \(Q\)-homeomorphism \(f:D\rightarrow D_*\) between the regular domains \(D\subset{\mathbb M}^n\) and \(D_*\subset\mathbb M_*^n\), where \(Q\) is integrable and the closure \(\overline{D}\) is assumed to be compact. For regular domains \(D\) and \(D_*\) having compact closures and an integrable function \(Q\), the possibility of continuous extension of the \(Q\)-homeomorphism \(f:D\rightarrow D_*\) to the boundary of \(D\) when \(Q\) has finite mean oscillation at the boundary is proved. Another condition for the continuous extension of the mapping \(f\) consists in assuming some integral divergence type for \(Q\).