A note on the uniformization of gradient Kähler Ricci solitons

Abstract

Applying a well known result for attracting fixed points of biholomorphisms [5, 7], we observe that one immediately obtains the following result: if (Mn,g) is a complete non-compact gradient Kähler-Ricci soliton which is either steady with positive Ricci curvature so that the scalar curvature attains its maximum at some point, or expanding with non-negative Ricci curvature, then M is biholomorphic to ℂn.