##### Department of Mathematics,

University of California San Diego

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### Differential Geometry Seminar

## Frederick Fong

#### Stanford University

## Collapsing Behavior of the Kahler-Ricci flow and its Singularity Analysis.

##### Abstract:

In this talk, I will discuss my recent works on the collapsing behavior of the Kahler-Ricci flow. The first work studies the Kahler-Ricci flow on $P^1$-bundles over Kahler-Einstein manifolds. We proved that if the initial Kahler metric is constructed by the Calabi's Ansatz and is in the suitable Kahler 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 Kahler 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 $\sim (T-t)^{1/2}$. It gives a partial affirmative answer to a conjecture stated in Song-Szekelyhidi-Weinkove's work on projective bundles.

Host: Ben Weinkove

### January 13, 2012

### 1:00 PM

### AP&M 6402

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