本项目旨在研究Fano流形上Kaehler-Ricci流的Hamilton-Tian猜想及相关问题。在紧致Fano流形上，Kaehler类为第一陈类的Kaehler-Ricci流长期存在；Hamilton-Tian猜想这种Kaehler-Ricci流在Gromov-Hausdorff拓扑下收敛到有奇异集合的收缩Kaehler-Ricci孤立子，奇异集合的Hausdorff余维数大于等于4，在奇异集外度量光滑收敛。证明要点包含两个方面：（1）Gromov-Hausdorff极限空间正则性部分的刻画和奇异集合的估计，（2）Gromov-Hausdorff极限空间的唯一性。作为应用，Hamilton-Tian暗示Fano流形上Kaehler-Einstein度量存在性的Yau-Tian-Donaldson猜想。

The project aims to study the Hamilton-Tian conjecture for Kaehler-Ricci flow on Fano manifolds. It is known that the Kaehler-Ricci flow in the anti-canonical class on a compact Fano manifold exists for all time. Hamilton-Tian conjectures that such a solution always converge in the Gromov-Hausdorff topology to a shrinking Kaehler-Ricci soliton, maybe with mild singularities of Hausdorff codimension greater than or equal to 4, and the convergence takes place smoothly on the regular set. There are two main integedients contained in the proof of the conjecture: (1) the characterization of the regular points and the estimation of the singular set, (2) the uniqueness of the Gromov-Hausdorff limit of the Kaehler-Ricci flow. As one application, the Hamilton-Tian conjecture implies the Yau-Tian-Donaldson conjecture for the existence problem of Kaehler-Einstein metrics on Fano manifolds.