A survey on the Kähler-Ricci flow and Yau's uniformization conjecture

Albert Chau, Luen-Fai Tam

Published 2007-02-09, updated 2007-08-22Version 2

Yau's uniformization conjecture states: a complete noncompact K\"ahler manifold with positive holomorphic bisectional curvature is biholomorphic to $\ce^n$. The K\"ahler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this article we present a survey of the K\"ahler-Ricci flow with focus on its application to uniformization. Other interesting methods and results related to the study of Yau's conjecture are also discussed.