Lipschitz and quasiconformal approximation of homeomorphism pairs
This paper deals with manifolds in the categories LIP of locally bi-Lipschitz homeomorphisms and LQC of locally quasiconformal (or equivalently locally quasisymmetric) homeomorphisms. The following theorem consists of representative special cases of the main results of the paper. Let CAT be LIP or LQC, let \(0\leq q\leq n\) with \(n,q\neq 4\), let \(M\) be a topological \(n\)-manifold without boundary, and let \(Q\) be a closed (as a subset) locally flat \(q\)-submanifold of \(M\) without boundary. Then the pair \((M,Q)\) can be given an essentially unique locally CAT flat CAT manifold pair structure as follows.NEWLINENEWLINENEWLINE(1) There is a CAT structure on \(M\) for which \(Q\) is a locally CAT flat submanifold of \(M\).NEWLINENEWLINENEWLINE(2) If \({\mathcal A}\) and \({\mathcal B}\) are two CAT structures on \(M\) satisfying (1), then there is an arbitrarily small isotopy of \((M,Q)\) from the identity to a CAT homeomorphism \(f:(M_{\mathcal A},Q)\to(M_{\mathcal B},Q)\). NEWLINENEWLINENEWLINE(3) If \({\mathcal A}\) and \({\mathcal B}\) are as in (2), then every two CAT homeomorphisms \(f,g:(M_{\mathcal A},Q) \to(M_{\mathcal B},Q)\) sufficiently close to the identity can be joined by an arbitrarily small CAT isotopy of \((M_{\mathcal A},Q)\) onto \((M_{\mathcal B},Q)\). NEWLINENEWLINENEWLINEFor the manifold \(M\) itself, with \(Q=\emptyset\) (or equivalently with \(q=n)\), this theorem is due to \textit{D. Sullivan} [Geometric topology, Proc. Conf., Athens/Ga 1977, 543-555 (1979; Zbl 0478.57007)]. The main results allow manifolds with boundary, and they include the generalization, due to \textit{P. Tukia} and \textit{J. Väisälä} [Ann. Acad. Sci. Fenn., Ser A I Math. 6, 303-342 (1981; Zbl 0448.30021)], of Sullivan's results (1) and (2) to this case. In the case \(q=n-1\) the author gave a proof for (1) and (2) in [ibid. 8, 107-138 (1983; Zbl 0488.57005)].
scientific article; zbMATH DE number 1566038
scientific article