Good metric spaces without good parameterizations

Abstract

A classical problem in geometric topology is to recognize when a topological space is a topological manifold. This paper addresses the question of when a metric space admits a quasisymmetric parametrization by providing examples of spaces with many Eucledian-like properties which are nonetheless substantially different from Euclidean geometry. These examples are geometrically self-similar versions of classical topologically self-similar examples from geometric topology, and they can be realized as codimension 1 subsets of Euclidean spaces. Unlike earlier examples going back to Rickman, these sets enjoy good bounds on their geodesic distance functions and good mass bounds (Ahlfors regularity). They are also smooth except for reasonably tame degenerations near small sets, they are uniformly rectifiable, and they have good properties in terms of analysis (like Sobolev and Poincare inequalities). The construction also produces uniform domains which have many nice properties but which are not quasiconformally equivalent to balls.