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Creators/Authors contains: "Hoehn, L"

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  1. ; ; (, Canadian Journal of Mathematics)
    Abstract Let $$\Omega $$ be a connected open set in the plane and $$\gamma : [0,1] \to \overline {\Omega }$$ a path such that $$\gamma ((0,1)) \subset \Omega $$ . We show that the path $$\gamma $$ can be “pulled tight” to a unique shortest path which is homotopic to $$\gamma $$ , via a homotopy h with endpoints fixed whose intermediate paths $$h_t$$ , for $$t \in [0,1)$$ , satisfy $$h_t((0,1)) \subset \Omega $$ . We prove this result even in the case when there is no path of finite Euclidean length homotopic to $$\gamma $$ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a “shortest” path. This work generalizes previous results for simply connected domains with simple closed curve boundaries. 
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  2. ; ; (, Houston journal of mathematics)
    null (Ed.)
    Accepted Manuscript Full Text Available
  3. ; (, Transactions of the American Mathematical Society)
    Abstract. It is known that a holomorphic motion (an analytic version of an isotopy) of a set X in the complex plane C always extends to a holomorphic motion of the entire plane. In the topological category, it was recently shown that an isotopy h : X  [0; 1] -> C, starting at the identity, of a plane continuum X also always extends to an isotopy of the entire plane. Easy examples show that this result does not generalize to all plane compacta. In this paper we will provide a characterization of isotopies of uniformly perfect plane compacta X which extend to an isotopy of the entire plane. Using this characterization, we prove that such an extension is always possible provided the diameters of all components of X are uniformly bounded away from zero. 
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    Accepted Manuscript Full Text Available