Cambridge Journal of Mathematics

Volume 7 (2019)

Number 1–2

On Yau’s uniformization conjecture

Pages: 33 – 70



Gang Liu (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)


Let $M^n$ be a complete noncompact Kähler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $\mathbb{C}^n$. This confirms the uniformization conjecture of Yau when $M$ has maximal volume growth.


Kähler manifold, uniformization conjecture

2010 Mathematics Subject Classification

Primary 53C55. Secondary 32H02.

The author was partially supported by NSF grant DMS-1709894 and the Alfred P. Sloan Fellowship.

Received 26 October 2017

Published 26 February 2019