Annales AcademiŠ Scientiarum FennicŠ
Mathematica
Volumen 34, 2009, 319-345

A SEWING PROBLEM IN METRIC SPACES

Peter Ha´ssinsky

UniversitÚ de Provence, LATP/CMI
39 rue FrÚdÚric Joliot-Curie, 13453 Marseille Cedex 13, France; phaissin 'at' cmi.univ-mrs.fr

Abstract. This note is devoted to the solution of a sewing problem between metric spaces sharing quasisymmetric copies of a given metric space. It is proved that the sewing yields a well-defined conformal gauge, and we study properties inherited by the new space. It follows from the construction that if Y is a closed uniformly perfect subset of a proper metric space X, then, for any \epsilon > 0, one can find a metric d in the conformal gauge of X so that the Hausdorff dimensions of both (X,d) and (Y,d) are \epsilon-close to their conformal dimension.

2000 Mathematics Subject Classification: Primary 30C65; Secondary 30C35, 54E40, 28A75.

Key words: Sewing problem, quasisymmetric maps, conformal gauge.

Reference to this article: P. Ha´ssinsky: A sewing problem in metric spaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 319-345.

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