This paper proves that the Brownian sphere — a random metric space that can be thought of as a “uniform” random surface shaped like the two‑dimensional sphere — has conformal dimension equal to 2. The Brownian sphere is known to be topologically a sphere and to have Hausdorff dimension 4 (a measure of its fractal size). The new result shows that, in the sense of quasisymmetric geometry, its smallest possible Hausdorff dimension is the ordinary topological dimension 2. The statement holds almost surely (with probability 1) for the random Brownian sphere.

Conformal dimension is a way to measure how small the Hausdorff dimension of a space can become after applying a quasisymmetric change of the distances. A quasisymmetric map is a homeomorphism that distorts relative distances in a controlled way. The conformal dimension of a space is the infimum of Hausdorff dimensions among all metric spaces that are quasisymmetrically equivalent to it. For the Brownian sphere this number had to lie between 2 (topological dimension) and 4 (Hausdorff dimension). The authors show the lower bound is attained.

To do this the authors adapt tools from the study of quasisymmetric geometry. The main technique is a construction called a hyperbolic filling, which turns the metric space into a graph with a hyperbolic geometry. On that graph they define a combinatorial object called an admissible weight function. From such a weight one can build a new metric on the sphere that is quasisymmetrically equivalent to the original one and whose small‑scale geometry can be controlled. The paper uses the Brownian sphere’s relation to the p=8/3 Liouville quantum gravity (LQG) sphere, and an unbounded variant called the Brownian plane, to estimate the weight’s moments across scales. These scale‑independence estimates let the authors push the Hausdorff dimension down to 2 in the quasisymmetric class.