{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0171",
  "title": "Smooth Convergence of Inverse Gauss Curvature Flow on the Two-Sphere",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We prove smooth convergence of the normalized inverse Gauss curvature flow on the two-sphere for every smooth initial conformal metric of positive Gauss curvature. No smallness or symmetry assumption is imposed, and convergence takes place in the original conformal coordinates. The curvature-weighted distance estimate of Gursky and Streets, together with Darvas's endpoint comparison, controls background energy. Curvature entropy then controls the conformal factor through the spherical Green kernel. Two maximum-principle estimates give time-uniform upper and lower curvature bounds. An exact change of variables to inverse Monge-Ampere flow supplies higher regularity, and distance contraction selects a single round limit.\n\nThis result answers the explicit two-dimensional component of AIM workshop Problem 3.3, recorded as AIM-GEOMETRY-0171 in UnsolvedMath v1.6.0. The source's separate higher-dimensional question is not resolved. Established estimates retain attribution; no absolute priority claim is made. This is an AI-assisted, self-audited and unrefereed preprint, without independent review or formal proof-assistant verification. The author is responsible for the claims.\n\nThe source package includes a portable exact-arithmetic checker and its two matching 20,057-check outputs. These verify finite algebraic identities and boundary examples only; the general convergence theorem is the written analytic proof.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "keywords": [
    "inverse Gauss curvature flow",
    "two-sphere",
    "curvature entropy",
    "Green kernel",
    "inverse Monge-Ampere flow",
    "smooth convergence",
    "AIM-GEOMETRY-0171"
  ],
  "manuscript_version_date": "2026-10-07",
  "publication_date": "2026-10-07",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-07",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0171/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0171/paper.pdf?v=c89660eeffce",
  "doi": "10.5281/zenodo.23216980",
  "zenodo_record_url": "https://zenodo.org/records/23216980",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Every smooth initially positive-curvature conformal metric on the unit two-sphere evolving by u_t=bar(K)/K-1 converges in C-infinity in fixed coordinates to one constant-positive-curvature metric, without smallness or symmetry assumptions. This resolves the explicit two-dimensional component of AIM workshop Problem 3.3 recorded as AIM-GEOMETRY-0171. The separate higher-dimensional question remains unresolved. Gursky--Streets, Darvas and Collins--Hisamoto--Takahashi supply attributed inputs. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "c89660eeffce775ee5eac93dbadd91999fe5daad61585b69e63b77acfb0e9ebe"
    },
    "source.zip": {
      "sha256": "687e8146525f6bb1774dddbd7b5ce209b267ab8904b1da370649afa09af5416c"
    },
    "verification_report.md": {
      "sha256": "7f94d4eade1e8217f4e4f37a45ca2ded77a42770c3c66e1edf0da10c74291a96"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}