{
  "schema_version": 1,
  "problem_number": "OWR-15208-008",
  "title": "The Conformal Parametrization of Nonnegatively Curved Surfaces Is Not a Homeomorphism at Integer Smoothness",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let M_0 = ℂ and M_1 = S², with complete metrics g_κ of constant curvature κ ∈ {0, 1}. Every complete smooth metric of nonnegative curvature on M_κ can be written uniquely as φ*(e^{−2u} g_κ), where φ is an orientation-preserving diffeomorphism fixing 0 and 1 (and ∞ if κ = 1) and u is a smooth function. Work of Belegradek, Hu and Banakh shows that the map (u, φ) ↦ φ*(e^{−2u} g_κ) is a homeomorphism when metrics and functions carry the topology of C^{k+α} convergence on compact sets and diffeomorphisms that of C^{k+1+α} convergence, with 0 < α < 1. Belegradek asked whether this remains true for α = 0, and expected that it does not. We show that it fails for every integer k ≥ 0 and on both surfaces: the map is a continuous bijection whose inverse is discontinuous at every point, also on the subspace of positively curved metrics. For k = 0 this follows from an elementary spiral construction. For k ≥ 1 we use diffeomorphisms φ_ε = z + ε z^{k+1} H_ε(|z|²) whose Beltrami coefficient is approximately ε z^{k+1}/z̄, regularized at the scale e^{−1/ε}. They converge to the identity in C^k, while their (k+1)-st derivative at 0 tends to −2(k+1)!. The main work is to choose the conformal factors so that the metrics converge in C^k and keep nonnegative curvature; for k = 1 this needs an additional C¹-small conformal correction. The conformal factors then fail to converge in C^k as well. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.DG",
    "math.CV"
  ],
  "keywords": [
    "nonnegative curvature",
    "spaces of Riemannian metrics",
    "uniformization",
    "Beltrami equation",
    "Hölder topology",
    "integer smoothness",
    "conformal parametrization",
    "Oberwolfach Reports",
    "OWR-15208-008",
    "math.DG",
    "math.CV",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15208-008/",
  "pdf_url": "https://eulersolve.org/papers/owr-15208-008/paper.pdf?v=d9a2c57a8a26",
  "doi": "10.5281/zenodo.23062867",
  "zenodo_record_url": "https://zenodo.org/records/23062867",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Belegradek's question (OWR 3/2017) negatively for every integer k ≥ 0, on ℂ and on S², also in the report's pushforward form. RP² and the spaces of metrics with curvature bounded below by λ ≠ 0 are not treated; the mechanism of the failure is classical, and the homeomorphism results for non-integral exponents are due to Belegradek, Hu and Banakh.",
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    "source.zip": {
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    "verification_report.md": {
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  "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."
}