Uniformisation of complete K\"ahler surfaces with positive sectional curvature

Abstract

We prove that any complete non-compact K\"ahler 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-Amp\`ere mass, and weighted $L^p$ holomorphic functions. A central difficulty is that these weights are neither smooth nor proper. As a consequence of the method, we also obtain B\'ezout-type intersection and multiplicity estimates in considerable generality. In a different direction, we also prove a new obstruction to the existence of complete K\"ahler metrics with non-negative bisectional curvature on non-compact K\"ahler manifolds, and use it to construct new examples admitting no such metrics. We conclude by discussing possible extensions of our methods to higher dimensions and related open problems.