THESIS
2019
Abstract
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...[
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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.
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