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Uniformization of complete Kähler manifolds with positive bisectional curvature
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Abstract
We prove that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space. This resolves Yau's uniformization conjecture positively in every complex dimension.
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Possibly relevant literature (found by the agent, not checked)
- Albert Chau and Luen-Fai Tam, A survey on the Kähler-Ricci flow and Yau's uniformization conjecture (arXiv:math/0702257, revised 22 August 2007). A survey of the Kähler–Ricci flow approach to the conjecture, which is the second route discussed in the introduction (flow-generated shrinking charts assembled by complex dynamics); the bibliography cites Chau–Tam's research papers but not this survey.
- Bing-Long Chen and Xi-Ping Zhu, A survey on Yau's uniformization conjecture (Surveys in Differential Geometry XXVI, International Press). A survey with separate sections on the maximal and non-maximal volume growth cases, relevant to the paper's claim that no volume-growth hypothesis is needed; the bibliography cites only the 2005 and 2009 Chen–Zhu papers.
- Gang Liu and Lei Ni, Ricci pinching on complete Kahler manifolds (arXiv:2610.16563, 12 Oct 2026). Uses the Datar–Pingali–Seshadri top Ricci-mass theorem in higher dimensions, without volume-growth or upper curvature assumptions, under positive bisectional and nonnegative sectional curvature; this relates to the finite-mass results surveyed in the paper's introduction. It does not mention the release.
- Hanzhang Yin, Ricci pinching on complete Kähler manifolds with nonnegative bisectional curvature (arXiv:2610.22969, 16 Oct 2026). Constructs Lipschitz plurisubharmonic functions of finite total Monge–Ampère mass under nonnegative bisectional curvature, the kind of weight that the introduction discusses for Datar–Pingali–Seshadri and Dat. It does not mention the release.
Digests and expositions
- Latent Seminar, Kähler uniformization through flow and shrinking charts (7 October 2026). Explains the manuscript's flow construction, cotangent-disc argument, volume clock and polynomial-degree approach to surjectivity, explicitly limiting its scope to exposition rather than independent verification of the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 338
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026,
author = {{OpenAI}},
title = {{Uniformization of complete K{\"a}hler manifolds with positive bisectional curvature}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf}{OAI:Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026}},
year = {2026}
}Family 338: Yau's uniformization conjecture
Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂn.
This is the only paper in this family.