SPACES OF SMALL METRIC COTYPE

Naor and Mendel's metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz equivalent to an ultrametric space has infinimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff limits, and use these facts to establish a partial converse of the main result.

Exceptional Sets for Quasiconformal Mappings in General Metric Spaces

A theorem of Balogh, Koskela, and Rogovin states that in Ahlfors Q-regular metric spaces which support a p-Poincare inequality, , an exceptional set of -finite (Q−p)- dimensional Hausdorff measure can be taken in the definition of a quasiconformal mapping while retaining Sobolev regularity analogous to that of the Euclidean setting. Through examples, we show that the assumption of a Poincare inequality cannot be removed.

Quasisymmetric structures on surfaces

We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surtace is locally quasisymmetrically equivalent to tne 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.

Errata to: Exceptional Sets for Quasiconformal Mappings in General Metric Spaces