Conditional Uniformization of K\"ahler Surfaces

Abstract

We prove that a complete noncompact K\"ahler surface with nonnegative Ricci and nonnegative quadratic orthogonal bisectional curvature is contractible, and hence homeomorphic to $\mathbb{R}^4$, if it is simply connected at infinity. Under positive bisectional curvature, this removes the contractibility assumption from the conditional uniformization theorem of Datar--Pingali--Seshadri: strong Steinness and simple connectivity at infinity suffice to identify the surface biholomorphically with $\mathbb{C}^2$. We also derive bounded-gradient strictly plurisubharmonic exhaustions and uniform holomorphic kernel estimates for complete $U(n)$-invariant K\"ahler metrics on $\mathbb{C}^n$ with nonnegative bisectional curvature.