A Note on the Uniformization of Gradient Kähler Ricci Solitons

Applying a well known result for attracting fixed points of biholomorphisms [4, 6], 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. We will show the following: Theorem 1. If (M, g) is a complete non-compact gradient KählerRicci 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 C. Recall that a Kähler manifold (M, gij̄(x)) is said to be a KählerRicci soliton if there is a family of biholomorphisms φt on M , given by a holomorphic vector field V , such that gij(x, t) = φ ∗ t (gij(x)) is a solution of the Kähler-Ricci flow: ∂ ∂t gij̄ = −Rij̄ − 2ρgij̄ gij̄(x, 0) = gij̄(x) (0.1) for 0 ≤ t < ∞, where Rij̄ denotes the Ricci tensor at time t and ρ is a constant. If ρ = 0, then the Kähler-Ricci soliton is said to be of steady type and if ρ > 0 then the Kähler-Ricci soliton is said to be of expanding type. We always assume that g is complete and M is non-compact. If in addition, the holomorphic vector field is given by the gradient of a real valued function f , then it is called a gradient Kähler-Ricci soliton. Date: April 2004 (Revised in July 2004). 2000 Mathematics Subject Classification. Primary 53C44; Secondary 58J37, 35B35. Research partially supported by The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong, China. Research partially supported by Earmarked Grant of Hong Kong #CUHK4032/02P.