Title:“Collapsing Behavior of the Kähler-Ricci flow and its Singularity Analysis”
Speaker: Frederick Tsz-Ho Fong, Stanford University
Date: Tuesday, January 24, 2012
Time: 02:30pm – 03:30pm
Place: Seminar Room 313, Simons Center
Abstract:
In this talk, I will discuss my recent works on the collapsing behavior of the Kähler-Ricci flow. The first work studies the Kähler-Ricci flow on P^1-bundles over Kähler-Einstein manifolds. We proved that if the initial Kähler metric is constructed by the Calabi’s Ansatz and is in the suitable Kähler 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 Kähler 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.
***
Reference for my first work
http://arxiv.org/pdf/1104.3924v2