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 $\mathbf S<sup>2$ 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.

• The paper proves that the conformal dimension of the Brownian sphere equals its topological dimension of 2, not minimal in conformal dimension
• The proof employs hyperbolic filling techniques and fine estimates for Liouville quantum gravity
• The authors use estimates for the Brownian plane, including boundary length control, volume bounds, and conformal moduli of bands

Main result

This paper by Miller and Tian proves that the conformal dimension of the Brownian sphere (the Brownian map) is almost surely equal to $2$, its topological dimension (2603.24473). The conformal dimension of a metric space $(X,d)$ is the infimum of Hausdorff dimensions over all metric spaces quasisymmetrically equivalent to $(X,d)$; it is a quasisymmetric invariant lying between the topological and Hausdorff dimensions. Since the Brownian sphere has Hausdorff dimension $4$ (and is homeomorphic to $\mathbb{S}^2$), its conformal dimension was previously known only to lie in $[2,4]$. The theorem shows the infimum is attained at the lower endpoint: the Brownian sphere is not minimal for conformal dimension, in contrast with the graph of one-dimensional Brownian motion, which prior work showed is minimal. This makes the Brownian sphere one of very few random fractals whose conformal dimension has been explicitly determined.

The result also clarifies what the Brownian sphere cannot be: since its Assouad dimension is infinite, it admits no quasisymmetric embedding into any doubling space, so its Ahlfors regular conformal dimension is infinite. The paper's result concerns the unrestricted conformal dimension, which is finite despite this obstruction.

Context and motivation

Conformal dimension was introduced by Pansu in connection with Gromov hyperbolic spaces: quasi-isometries between Gromov hyperbolic spaces induce quasisymmetries on their boundaries, making boundary conformal dimension a quasi-isometric invariant. For boundaries homeomorphic to $\mathbb{S}^2$, Cannon's conjecture asserts that a hyperbolic group with such a boundary acts geometrically on $\mathbb{H}^3$, equivalently that its boundary is quasisymmetrically equivalent to $\mathbb{S}^2$. The Brownian sphere is relevant here because known quasisymmetric uniformization criteria — Bonk–Kleiner's theorem that Ahlfors $2$-regular, linearly locally connected spheres are quasisymmetric to $(X,d)$0 — do not apply: the Brownian sphere is neither Ahlfors regular nor LLC. Prior to this work, the graph of planar Brownian motion was essentially the only random fractal with an explicitly computed conformal dimension.

The authors note structural evidence against minimality: geodesics in the Brownian sphere satisfy the confluence property, and the geodesic frame has Hausdorff dimension $(X,d)$1, suggesting the space lacks the rich family of rectifiable curves required for minimality.

Proof strategy via hyperbolic fillings

The argument adapts the framework relating conformal gauge to combinatorial modulus through hyperbolic fillings. For a compact metric space $(X,d)$2, one builds a Gromov hyperbolic metric graph from nested maximal $(X,d)$3-separated nets $(X,d)$4; its Gromov boundary is quasisymmetrically equivalent to $(X,d)$5. Given an admissible weight function $(X,d)$6 on the vertices (satisfying a crossing condition: any horizontal chain of level-$(X,d)$7 balls crossing an annulus around a parent ball carries total weight at least $(X,d)$8), one constructs a metric on $(X,d)$9 quasisymmetrically equivalent to $(X,d)$0 whose diameters decay like products of weights along vertical paths. Consequently, if for some $(X,d)$1

$(X,d)$2

then the conformal dimension of $(X,d)$3 is at most $(X,d)$4. A notable technical contribution is that the framework is developed without assuming doubling.

The natural candidate weight is the ratio of Euclidean diameter of a small ball to the inradius of a larger filled ball, but verifying the summability condition directly is difficult. Instead the authors truncate: define $(X,d)$5 as this ratio on a carefully designed "good" event $(X,d)$6 and set it to $(X,d)$7 otherwise. The good event occurs with superpolynomially high probability as $(X,d)$8, uniformly across scales, which suffices.

Estimates for the Brownian plane

The analysis is transferred to the Brownian plane ($(X,d)$9-LQG cone), the unbounded analogue of the Brownian sphere, exploiting scale-invariance and independence across scales. Three ingredients are established:

Boundary length control. For the hull process $4$0 (boundary length of filled metric balls), exponential tail bounds show $4$1 does not exceed its mean $4$2 significantly along many scales simultaneously, nor stay uniformly small over long intervals. These follow from explicit Laplace transform computations for the $4$3-stable continuous-state branching process conditioned on its lifetime.

Volume bounds and nets. Metric ball volumes satisfy two-sided bounds $4$4, and conditionally independent samples from the LQG measure form a net of size $4$5 covering a unit ball at scale $4$6 — matching the expected growth $4$7.

Conformal moduli of bands. The key analytic estimate: for each $4$8 there exists $4$9 such that $\mathbb{S}^2$0, where $\mathbb{S}^2$1 is the conformal modulus of the annulus between radii $\mathbb{S}^2$2 and $\mathbb{S}^2$3 in the filled metric ball. The proof combines Teichmüller's extremal ring comparison, Beurling-type estimates controlling eccentricity of high-modulus annuli, Koebe distortion bounds for compactness of test functions, and hitting-time Laplace transforms for drifted Brownian motion applied to the circle-average process. The exponent $\mathbb{S}^2$4 can be taken arbitrarily close to $\mathbb{S}^2$5 from above, which is exactly what is needed to beat the $\mathbb{S}^2$6 growth of the net size.

Constructing the admissible weight

The weight function is defined using a "no bottleneck" event $\mathbb{S}^2$7: a doubly connected LQG surface satisfies $\mathbb{S}^2$8 if no small metric ball disconnects its inner and outer boundaries. The event $\mathbb{S}^2$9 asserts that some thin band near radius $[2,4]$0 around $[2,4]$1 satisfies $[2,4]$2; it holds with conditional probability $[2,4]$3 given any boundary length, proved via the CSBP tail bounds and absolute continuity arguments. A robustified version $[2,4]$4, stable under perturbation of the center, enters the definition

$[2,4]$5

Admissibility is verified by a four-case geometric argument: any horizontal chain of balls crossing an annulus must contain a subchain crossing a width-$[2,4]$6 band, and either the centers avoid the relevant filled balls (giving diameter/inradius summation $[2,4]$7) or the event $[2,4]$8 supplies a good annulus forcing the same conclusion.

Combining the modulus estimate with the CSBP independence lemmas yields the moment bound $[2,4]$9 with $\mathbb{S}^2$0, where $\mathbb{S}^2$1 is the product of weights along a vertical path. Since $\mathbb{S}^2$2 for small $\mathbb{S}^2$3, summing over the $\mathbb{S}^2$4 net points gives expectation decaying like $\mathbb{S}^2$5, and Borel–Cantelli yields the desired vanishing sum almost surely. The passage from the plane to the sphere uses mutual absolute continuity of filled-ball laws given boundary lengths, together with random sampling of centers and radii to cover the whole sphere by finitely many well-behaved metric balls.

Limitations and open questions

The argument relies heavily on the special structure of $\mathbb{S}^2$6-LQG: the equivalence with the Brownian sphere, the exact CSBP description of boundary lengths, and the mating-of-trees framework. The authors conjecture that the conformal dimension of the $\mathbb{S}^2$7-LQG sphere is almost surely $\mathbb{S}^2$8 for all $\mathbb{S}^2$9, but their methods do not extend verbatim, since the general-$\mathbb{H}^3$0 LQG metric lacks the corresponding exact Markovian descriptions used here. Whether the infimum defining the conformal dimension is attained (i.e., whether some quasisymmetrically equivalent space has Hausdorff dimension exactly $\mathbb{H}^3$1) is not addressed. The relationship to Cannon's conjecture — where attainment of the Ahlfors regular conformal dimension is the crux — remains untouched, as the Brownian sphere's Ahlfors regular conformal dimension is infinite.

Conclusion

The paper establishes that the conformal dimension of the Brownian sphere equals its topological dimension $\mathbb{H}^3$2, strictly below its Hausdorff dimension $\mathbb{H}^3$3, thereby showing the Brownian sphere is not minimal for conformal dimension. The proof synthesizes hyperbolic filling techniques with fine estimates for Liouville quantum gravity: CSBP boundary-length tails, volume-controlled random nets, and power-law integrability of conformal moduli of metric bands. It provides the second explicit computation of the conformal dimension of a random fractal, and the first in dimension greater than one, while leaving the extension to general $\mathbb{H}^3$4-LQG surfaces as a concrete open problem.