Conditional Uniformization of Kähler Surfaces

We prove that a complete noncompact Kähler 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 Kähler metrics on ${\mathbb{C}}^n$ with nonnegative bisectional curvature.