{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37520",
  "author_total": 2,
  "authors": [
    {
      "name": "Gunhee Cho"
    },
    {
      "name": "Hyun Chul Jang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1msbg9a8260v3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37520",
  "fulltext_url": null,
  "id": "1msbg9a8260v3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37520",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7224118840",
  "pdf_url": "https://arxiv.org/pdf/2610.37520",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1msbg9a8260v3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37520",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37520"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Statistical Uniformization of Surfaces by Heat-Kernel Fisher Information",
  "title_text": "Statistical Uniformization of Surfaces by Heat-Kernel Fisher Information",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/1msbg9a8260v3",
  "venue": "arXiv",
  "versions": []
}