Abstract. A carpet is a metric space homeomorphic to the Sierpiński carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.

The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the disk, the plane, or the sphere, each equipped with a standard conformal structure. We give a similar uniformization for Ahlfors 2‐regular, linearly locally connected metric planes; instead of conformal equivalence, we are concerned with quasisymmetric equivalence.

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected domain in the standard 2-sphere S 2 is quasisymmetrically equivalent to a circle domain in S 2 if and only if X is linearly locally connected and its completion is compact. We also give a counterexample in the countably connected case.

We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces embedded in some Euclidean spaces that are locally bi-Lipschitz equivalent to a ball in the plane.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 624 KEVIN WILDRICK Theorem 1.1. Let (X, d) be a locally Ahlfors 2-regular and locally linearly locally contractible (LLLC) metric space homeomorphic to a surface. Then each point of X has a neighborhood that is quasisymmetrically equivalent to the disk.Our proof is in fact quantitative and provides good bounds on the size of the resulting quasidisk. See Theorem 4.1 for the complete result. An outline of the proof of Theorem 1.1 is as follows. The main task is to construct an Ahlfors 2regular and linearly locally connected (a slightly weaker condition than linear local contractibility) planar neighborhood of a given point. We then apply previously established uniformization results from [32] to produce the desired quasisymmetric mapping. The obstacle is that compact subsets of a locally Ahlfors 2-regular and LLLC metric space need not be Ahlfors 2-regular and linearly locally connected. However, we show that if γ is a quasicircle contained in a planar subset of X, then the closed Jordan domain defined by γ is Ahlfors 2-regular and linearly locally connected. Thus it suffices to construct a quasicircle at a specificed scale that surrounds a given point.In constructing the quasicircle, we first show that (X, d) is locally quasiconvex. This would follow from the result of Semmes [27, Theorem B.10], except that we consider localized conditions. As it is, our methods resemble those employed by Semmes. As we have specialized to two dimensions, our proof is fairly elementary and direct. We also indicate how our proof could be upgraded to give a full local analogue of Semmes' result in two dimensions.A locally compact and locally quasiconvex space is, up to a locally bi-Lipschitz change of metric, locally geodesic. With this simplification, we employ discrete methods to construct a loop surrounding a given point of controlled length in a controlled annulus. We then solve an extremal problem to produce the desired quasicircle.The author would like to thank his advisor Mario Bonk for many contributions to this paper. Also, he thanks Urs Lang and Kai Rajala for useful discussions.This paper is dedicated to Juha Heinonen, who passed away shortly after it was submitted. Juha was a great thinker, a fantastic mentor, and an excellent friend. He is sorely missed.