Kaehler-Ricci flow on Fano G-manifolds

I will talk about a recent work jointly with Tian on Kaehler-Ricci flow on Fano G-manifolds. We prove that on a Fano G-manifold, the Gromov-Hausdorff limit of Kaehler-Ricci flow with initial metric in $2\pi c_1(M)$ must be a Q-Fano horosymmetric variety which admits a singular Keahler-Ricci soliton. Moreover, we show that the complex structure of limit variety can be constructed by a $C^*$-degeneration induced by an element in the Cartan torus Lie algebra of G. A similar result can be also proved for Kaehler-Ricci flows on any Fano horosymmetric manifolds.