In this thesis, we study the Kähler-Ricci flow on compact Kähler manifolds. We will begin with reviewing the estimates and known results of the Kähler-Ricci flow, then giving the proofs to the two main results on finite-time singularity of solutions to the Kähler-Ricci flow on a compact Kähler manifold with the maximal existence time T.

For the first result, we prove that uniformly equivalent solutions to the Kähler-Ricci flow on a compact Kähler manifold develop the same singularity type (Type I or Type IIa) at the maximal time T.

The second result, we assume there are uniformly equivalent solutions to the Kähler-Ricci flow, on a compact Kähler manifold where the eigenvalues of metrics admit a natural lower bound T-t/C . The solutions develop the same singularity ty...[ Read more ]

In this thesis, we study the Kähler-Ricci flow on compact Kähler manifolds. We will begin with reviewing the estimates and known results of the Kähler-Ricci flow, then giving the proofs to the two main results on finite-time singularity of solutions to the Kähler-Ricci flow on a compact Kähler manifold with the maximal existence time T.

For the first result, we prove that uniformly equivalent solutions to the Kähler-Ricci flow on a compact Kähler manifold develop the same singularity type (Type I or Type IIa) at the maximal time T.

The second result, we assume there are uniformly equivalent solutions to the Kähler-Ricci flow, on a compact Kähler manifold where the eigenvalues of metrics admit a natural lower bound T-t/C . The solutions develop the same singularity type locally. This implies the solutions admit the same Type I singularity set, which is the open set where the solutions converge locally to shrinking solitons after rescaling the metric by 1/T-t.