Yau's uniformization conjecture

Theorem 1.1: if a connected noncompact complex manifold of dimension $n\ge1$ carries a smooth complete Kahler metric with pointwise strictly positive holomorphic bisectional curvature, then it is biholomorphic to $\mathbb C^n$; no curvature bounds, volume growth, noncollapsing or topological hypotheses are assumed. The proof runs a Kahler-Ricci flow built from exhaustions, proves Harnack inequalities by a disc-minimization argument, obtains shrinking holomorphic charts and assembles them by polynomial-automorphism dynamics into a biholomorphism onto all of $\mathbb C^n$. It does not say the metric is flat, and it does not treat nonnegative curvature.

Theorem 1.1: if a connected noncompact complex manifold of dimension $n\ge1$ carries a smooth complete Kahler metric with pointwise strictly positive holomorphic bisectional curvature, then it is biholomorphic to $\mathbb C^n$; no curvature bounds, volume growth, noncollapsing or topological hypotheses are assumed. The proof runs a Kahler-Ricci flow built from exhaustions, proves Harnack inequalities by a disc-minimization argument, obtains shrinking holomorphic charts and assembles them by polynomial-automorphism dynamics into a biholomorphism onto all of $\mathbb C^n$. It does not say the metric is flat, and it does not treat nonnegative curvature.