Convergence of Kähler-ricci Flow

In this paper, we prove a theorem on convergence of Kähler-Ricci flow on a compact Kähler manifold M which admits a Kähler-Ricci soliton. A Kähler metric h is called a Kähler-Ricci soliton if its Kähler form ωh satisfies equation Ric(ωh)− ωh = LXωh, where Ric(ωh) is the Ricci form of h and LXωh denotes the Lie derivative of ωh along a holomorphic vector field X on M . As usual, we denote a Kähler-Ricci soliton by a pair (gKS , X). According to [TZ1], X should lie in the center of a reductive Lie subalgebra ηr(M) of η(M), which consists of all holomorphic vector fields on M . If X = 0, ωh is just a Kähler-Einstein metric. Since ωh is d-closed, we may write LXωh = √ −1 2π ∂∂θ for some real-valued smooth function θ. It follows that the first Chern class c1(M) is positive and it is represented by ωh. The Ricci flow was first introduced by R. Hamilton in [Ha]. If the underlying manifold M is Kähler with positive first Chern class, it is more natural to study the following Kähler-Ricci flow (normalized):