Quasisymmetric structures on surfaces

Abstract

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.

Quasisymmetric Koebe uniformization

“…See Section 3 for precise definitions. A version of Theorem 1.1 for metric spaces homeomorphic to the plane was derived in [26], and a local version given in [27].…”

Quantitative quasisymmetric uniformization of compact surfaces

“…Here the data of X consists of the topological genus of X, and the constants in the Ahlfors 2regularity and linear local contractibility conditions. Except for the statement of dependence of the distortion function only on the data of X, Theorem 2 follows immediately from the local uniformization theorem of [Wil10] and an elementary local-toglobal result for quasisymmetric mappings due to Tukia and Väisälä [TV80, Theorem 2.23]. While a quantitative proof is significantly more involved, it provides much more information about the "space" of quasisymmetric structures on surfaces.…”

“…Let (X, d) be a metric surface as given in the theorem. By the local uniformization result of [Wil10], (X, d) has an atlas of uniformly quasisymmetric mappings. Our first step is to create a compatible conformal atlas, giving (X, d) the structure of a Riemann surface.…”

Loewner chains and Hölder geometry

“…A more common version of Janiszewski's lemma deals with the case that A has two connected components: if A 1 and A 2 are disjoint, closed subsets of D such that A 1 ∪ A 2 separates x and y in D, then either A 1 or A 2 must separate x and y (cf. [31,Theorem 4.26]). In the terminology of [30,Section II.4], this means that D has the Phragman-Brouwer property.…”