Yau uniformization conjecture
For every complete, noncompact Kähler manifold with positive holomorphic bisectional curvature, i.e. for all points and all nonzero vectors , the manifold is biholomorphic to complex Euclidean space .
References
Primary source
Additional references
- An m-Hessian approach to Yau uniformization conjecture — arXiv — Truong Dinh Dat
Progress summary
A new 2026 paper gives a useful analytic step toward the conjecture, but does not prove it, so the general question remains open.
Proposed by Yau in 1974, the conjecture says that every complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space.
Known results
- Maximal volume growth: Gang Liu, 2016, proved the conclusion under nonnegative bisectional curvature and maximal volume growth.
- Maximal-volume-growth extensions are also attributed to Liu and to Lee–Tam.
- A recent preprint proves a weak two-dimensional version under positive sectional curvature, without an upper curvature bound or assumptions at infinity.
September 2026 analytic advance
On September 16, 2026, Truong Dinh Dat’s preprint An m-Hessian approach to Yau uniformization conjecture reported Hessian-capacity estimates producing a proper Lipschitz plurisubharmonic exhaustion with finite top-degree Monge–Ampère mass in dimensions at least three. This is a claimed analytic ingredient toward uniformization, not a proof of the full conjecture.
Current status (as of September 2026): Special cases are established and a new higher-dimensional analytic mechanism is claimed, but the unrestricted Yau uniformization conjecture remains open.
Solutions 0
No solutions have been posted yet.