On boundary behavior of generalized quasi-isometries

By D. Kovtonyuk, V. Ryazanov. 2010

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We establish a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in $$\overline {{^n}} = {^{^n}} \cup \{ \infty \} ,n \ge 2$$, under integral constraints of the type ∫ Φ(Qn−1(x))dm(x) < ∞ with a convex non-decreasing function Φ: [0,∞]→[0,∞]. Integral conditions on Φ are found that are necessary and sufficient for a continuous extension of f to the boundary. Our results are applied to finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries as well as quasi-isometries in ℝn. In particular, a generalization and strengthening of the well-known theorem of Gehring-Martio on homeomorphic extension to boundaries of quasi-conformal mappings between QED (quasi-extremal distance) domains is obtained.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5b51bee3faa670aa88377b0680b7e43989afe411