Kähler-ricci Flow on a Toric Manifold with Positive First Chern Class

In this note, we prove that on an n-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial (S)-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manifold with positive first Chern class by using the Kähler-Ricci flow. 0. Introduction. LetM be a compact toric manifold with positive first Chern class. Let T ∼= (S1)n×Rn be a maximal torus which acts onM and K0 ∼= (S) be its maximal compact subgroup. In this note we discuss a Kähler-Ricci flow with a K0-invariant initial metric on M and we shall prove Main Theorem. On a compact toric manifold M with positive first Chern class, the Kähler-Ricci flow with any initial K0-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, it shows that there exists a Kähler-Ricci soliton on any compact toric manifold with positive first Chern class. The existence of Kähler-Ricci solitons on a compact toric manifold with positive first Chern class was proved in [WZ] by using the continuity method. The above theorem gives another proof for the existence of Kähler-Ricci solitons on such a complex manifold by using the Kähler-Ricci flow. We note that a more general convergence theorem of Kähler-Ricci flow on a compact complex manifold which admits a Kähler-Ricci soliton was recently obtained by Tian and the author in [TZ3]. In that paper the assumption of the existence of a Kähler-Ricci soliton plays a crucial role. In the case of Kähler-Einstein manifolds with positive first Chern class the same result was claimed by Perelman ([P2]). In the present paper we do not need any assumption of the existence of Kähler-Ricci solitons or Kähler-Einstein metrics and prove the the convergence of Kähler-Ricci flow. The Ricci flow was first introduced by R. Hamilton in 1982 ([Ha]). Recently G. Perelman has made a major breakthrough in this area for three-dimensional manifolds ([P1]). Our proof of the main theorem is to study certain complex Monge-Ampère flow instead of Kähler-Ricci flow. The flow of this type has been studied before by many people (cf. 1991 Mathematics Subject Classification. Primary: 53C25; Secondary: 32J15, 53C55, 58E11.