CATALOGUE 338
Yau's uniformization conjecture
The official subject is Differential geometry. This single-manuscript catalogue concerns biholomorphic uniformization of complete noncompact Kähler manifolds with positive holomorphic bisectional curvature. The article follows the analytic and coordinate constructions used without additional geometric hypotheses.
1. Paper in this catalogue
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | Uniformization of complete Kähler manifolds with positive bisectional curvature →Short name: Positive bisectional curvature uniformizationSeptember 23, 2026 · 77 pagesOriginal paper (GitHub) ↗ | A connected noncompact complex n-manifold, n≥1, admitting a smooth complete Kähler metric with pointwise strictly positive holomorphic bisectional curvature is biholomorphic to C^n, without extra upper-curvature, volume-growth or topological assumptions. Theorem 1.1 · p. 2 ↗ | Connects localized flow and scalar comparison, a cotangent disc envelope, shrinking inverse charts and forward polynomial degrees. Bilingual proof overview with three diagrams and limited input comparisons. Read the proof overview → |
2. Guide to the article
Uniformization of complete Kähler manifolds with positive bisectional curvature connects flow construction, scalar comparison before completeness, the disc envelope, shrinking charts and surjectivity. The conclusion is biholomorphism of complex manifolds, not an isometric classification of the initial metric.
[OpenAI26YU, Theorem 1.1, p. 2]
Weighted ∂̄ solvability, short-time Chern–Ricci flow, boundary matrix factorization and shears are listed as external inputs in the article. Within-catalogue inter-paper dependence is inapplicable to this single-paper catalogue; this does not mean absence of external inputs.
3. Sources and checking scope
Official entry 338 is on PDF p.37. Differential geometry spans entries 333–361 and starts on p.36. The inventory records the formal title, official order, September 23, 2026 version, 77 pages, fixed commit and file hash.
OpenAI Research Catalog · Catalogue 338; Differential geometry · p. 37
The article includes a self-check comparing principal proof passages and statements/uses in six external references. This is not complete independent verification or expert review. Remaining scope concerning Hodge heat, disc deformations and nonuniform polynomial models is stated at the end of the article.