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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