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On the conformal gauge of a compact metric space. (Sur la jauge conforme d’un espace métrique compact.) (English. French summary) Zbl 1283.30072

For a given compact metric space \((X,d) \) that admits an Ahlfors regular metric in the conformal gauge of \(d\), using sequences of weighted finite coverings of \(X\) satisfying certain geometric conditions, the author provides a combinatorial description of all metrics in the Ahlfors regular conformal gauge up to bi-Lipschitz homeomorphisms. As a consequence, the author obtains Keith-Kleiner’s result relating the conformal dimension of a connected locally connected approximately self-similar metric space to its critical exponent defined via discrete moduli of curve families. Another consequence is Keith-Laakso’s result for \(Q\)-regular metric spaces, where \(Q\) is the Ahlfors regular conformal dimension. It states the existence of weak tangent spaces of such metric spaces that admit curve families of positive \(Q\) modulus.

MSC:

30L10 Quasiconformal mappings in metric spaces
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
51F99 Metric geometry
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