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.

Figures (2)

• Figure 1: Illustration of the first of four cases in the proof of \ref{['lem:admissibility']}. The blue region represents the metric band $A_{\alpha^{n - 1},3\alpha^{n - 1}/2}^\bullet(y; D_\Phi)$, with the green metric band $A_{\alpha^{n - 1} + 2\alpha^{n - 1 + \zeta},\alpha^{n - 1} + 4\alpha^{n - 1 + \zeta}}^\bullet(y; D_\Phi)$ overlaid upon it. The sequence of red metric balls, $B_{4\alpha^n}(x_{i_1}; D_\Phi), \cdots, B_{4\alpha^n}(x_{j_1}; D_\Phi)$, forms a minimal subpath crossing between the inner and outer boundaries of the green band. This figure depicts the scenario where $B_{3\alpha^{n - 1}/2}^z(y; D_\Phi)$ is bounded and $y \notin B_{\alpha^{n - 1 + \zeta}}^\bullet(x_j; D_\Phi)$ for all $j \in [i_1, j_1]_\mathbf Z$. Consequently, the subpath satisfies condition \ref{['eq:admissibility-illustration-0']} with $[a, b] = [i_1, j_1]$. Note that the holes of the red metric balls are omitted for clarity.
• Figure 2: Illustration of the second of four cases in the proof of \ref{['lem:admissibility']}. This figure depicts the scenario where $B_{3\alpha^{n - 1}/2}^z(y; D_\Phi)$ is bounded, but unlike the first case, $y$ is contained in at least one of the filled metric balls forming the minimal subpath (highlighted here as the central red ball). In this situation, we utilize the event $\widetilde{F}(x_{k_1}, n)$ to identify a suitable metric band $A$, represented in green. We then find a minimal subpath crossing the inner and outer boundaries of $A_{3\alpha^{n - 1 + \zeta},5\alpha^{n - 1 + \zeta}}^O(I; D_\Phi)$, shown in yellow, which satisfies condition \ref{['eq:admissibility-illustration-0']}.

Theorems & Definitions (60)

• Theorem 1.1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5