Geometric rectifiability in metric measure spaces

We extend several results on the geometric rectifiability of subsets in Euclidean spaces to metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincar´e inequality). Specifically, we provide a strengthened version of the classical Denjoy-Riesz theorem, generalize MacManus’ result on the quasisymmetric rectifiability of uniformly disconnected subsets, and extend the work of David and Semmes on bi-Lipschitz rectifiability to metric measure spaces.

For the Denjoy-Riesz theorem, we prove that compact, totally disconnected subsets of metric measure spaces with controlled geometry can be captured by an arc that is locally bi-Lipschitz away from the totally disconnected set. In the case of MacManus’ result, we show that compact, uniformly disconnected subsets in these spaces can be captured by the image of a quasisymmetric embedding, quantitatively. For David and Semmes’ work, we provide sharp conditions on a metric measure space such that a subset which is bi-Lipschitz homeomorphic to a subset of the real line is captured by a bi-Lipschitz arc, quantitatively.

Finally, we establish several results concerning the co-uniformity of subsets in these spaces under various conditions, including bounds on Assouad dimension and uniform disconnectedness.