Statistical Uniformization of Surfaces by Heat-Kernel Fisher Information

A Brownian particle observed at time t carries information about where it started. On a closed surface this makes the surface itself a statistical model, whose Fisher information pulls back to a heat-smoothed metric hₜ=2tJₜ; its volume Vₜ is a normalized Jeffreys volume, and log(Vₜ/2t) is the model-dependent constant in Rissanen's minimum-description-length complexity. We ask whether maximizing this statistical size among metrics of fixed area in a conformal class recovers the uniformizing metric. First, 9(A-2πχt-Vₜ)/t² converges to the curvature energy ∫ K²\,dA (the Calabi energy up to a topological constant), uniformly on classes of bounded geometry; hence, within a fixed-area conformal class of bounded geometry containing the uniformizer, approximate maximizers of Vₜ converge to the constant-curvature metric (on the sphere, to the round metrics). Second, near every flat torus we prove a sharp finite-scale version: for small perturbations whose oscillations are resolved by the diffusion, t|k|²≤θ<τ_*, the flat-subtracted defect is comparable to \|Δf\|₂², uniformly in the frequency cutoff and over noncollapsing lattices. The explicit threshold τ_*=4.408… is optimal, and it is exactly the observation time at which the information about one curvature mode is maximal. Third, the flat structure is seen only at an exponential scale: the volume loss of a flat torus measures the ambiguity of winding and is comparable to t-1/2e-sys²/(16t), which selects the hexagonal lattice as t→0 but not at all times.