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Title:
Collapsing Behavior of the Kähler-Ricci flow and its Singularity Analysis
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Abstract:
In this talk, I will discuss my recent works on the collapsing behavior of the K\"ahler-Ricci flow. The first work studies the K\"ahler-Ricci flow on P^1-bundles over K\"ahler-Einstein manifolds. We proved that if the initial K\"ahler metric is constructed by the Calabi's Ansatz and is in the suitable K\"ahler class, the flow must develop Type I singularity and the singularity model is P^1 X C^n. It is an extension of Song-Weinkove's work on Hirzebruch surfaces. The second work discusses the collapsing behavior in a more general setting without any symmetry assumption. We showed that if the limiting K\"ahler class of the flow is given by a holomorphic submersion and the Ricci curvature is uniformly bounded from above with respect to the initial metric, then the fibers will collapse in an optimal rate, i.e. diam ~(T-t)^{1/2}. It gives a partial affirmative answer to a conjecture stated in Song-Szekelyhidi-Weinkove's work on projective bundles.
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