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 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.
Related
BibTeX
@misc{cho2026statistical,
title = {{Statistical Uniformization of Surfaces by Heat-Kernel Fisher Information}},
author = {Gunhee Cho and Hyun Chul Jang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37520},
eprint = {2610.37520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37520},
}
RIS
TY - UNPB
TI - Statistical Uniformization of Surfaces by Heat-Kernel Fisher Information
AU - Gunhee Cho
AU - Hyun Chul Jang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37520
UR - https://arxiv.org/abs/2610.37520
AB - 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 ∫ 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≤θ<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… 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^−sys^2/(16t), which selects the hexagonal lattice as t→0 but not at all times.
ER -