Statistical Uniformization of Surfaces by Heat-Kernel Fisher Information

Abstract

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 $h_t=2tJ_t$; its volume $V_t$ is a normalized Jeffreys volume, and $\log(V_t/2t)$ 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, $9(A-2πχt-V_t)/t^2$ converges to the curvature energy $\int K^2\,dA$ (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 $V_t$ 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, $t|k|^2\leθ<τ_*$, the flat-subtracted defect is comparable to $\|Δf\|_2^2$, uniformly in the frequency cutoff and over noncollapsing lattices. The explicit threshold $τ_*=4.408\ldots$ 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 $t^{-1/2}e^{-\operatorname{sys}^2/(16t)}$, which selects the hexagonal lattice as $t\to0$ but not at all times.