The conformal dimension of the Brownian sphere is two

Abstract

The conformal dimension of a metric space $(X, d)$ is equal to the infimum of the Hausdorff dimensions among all metric spaces quasisymmetric to $(X, d)$. It is an important quasisymmetric invariant which lies non-strictly between the topological and Hausdorff dimensions of $(X, d)$. We consider the conformal dimension of the Brownian sphere (a.k.a. the Brownian map), whose law can be thought of as the uniform measure on metric measure spaces homeomorphic to the standard sphere $\mathbf S^2$ with unit area. Since the Hausdorff dimension of the Brownian sphere is $4$, its conformal dimension lies in $[2, 4]$. Our main result is that its conformal dimension is equal to $2$, its topological dimension.