A legendary mathematical mystery about the 'perfect' shape of complex surfaces has finally been solved after decades of uncertainty.
April 14, 2026
Original Paper
Uniformisation of complete Kähler surfaces with positive sectional curvature
arXiv · 2604.11220
The Takeaway
The paper resolves a major form of Yau's uniformisation conjecture, proving that specific complex spaces have a remarkably simple and predictable structure. This simplifies our understanding of the fundamental geometry that governs higher-dimensional manifolds in physics.
From the abstract
We prove that any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to $\mathbb{C}^2$, establishing the two dimensional case of the weaker form of Yau's uniformisation conjecture. In contrast to all previous results, no assumptions are made on the geometry at infinity.The proof introduces a new approach towards Yau-type uniformisation problems, based on uniformly Lipschitz plurisubharmonic weight functions with finite Monge-Ampère mass, and weighted $L^p$ hol