Columbia Informal Complex Geometry
and PDE Seminar
Spring 2012
Organizers: D.H. Phong, V. Tosatti
The seminar meets on Thursdays from 2.40pm to 4.00pm in 520 Mathematics.
Schedule
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January 26
Organizational Meeting 2:40pm-3:00pm
Frederick T.-H. Fong (Stanford) - Collapsing Behavior of the Kähler-Ricci flow and its Singularity Analysis
- References 1 2 - 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 P1-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 P1 X Cn. 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.
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February 2
Ben Weinkove (UCSD) - The evolution of a Hermitian metric by its Chern-Ricci form
- Reference - Abstract
I will discuss the evolution of a Hermitian metric on a
compact complex manifold by its Chern-Ricci form. This is an evolution
equation first studied by M. Gill, and coincides with the Kähler-Ricci
flow if the initial metric is Kähler. I will describe the maximal
existence time for the flow in terms of the initial data. I will
discuss the behavior of the flow on complex surfaces when the initial
metric is Gauduchon, on complex manifolds with negative first Chern
class, and on some Hopf manifolds. This is a joint work with
Valentino Tosatti.
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February 9
Yu Wang - C2,α regularity for the complex Monge-Ampère equation - Reference
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February 16
Tristan Collins - TBA
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February 23
Sławomir Dinew - Hölder continuous solutions to Monge-Ampère equations - Reference
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March 1
Valentino Tosatti - Strominger's system - Reference
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March 8
Valentino Tosatti - Strominger's system II - Reference
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March 22
Jian Song - Form-type Calabi-Yau equations - References 1 2
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March 29
Xiaowei Wang - TBA
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April 5
TBA
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April 12
Michael Siepmann (ETH Zürich) - TBA
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April 16 - Special Date
Martine Klughertz (Toulouse) - TBA
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April 20 - Special Date
Ahmed Zeriahi (Toulouse) - TBA
Schedule for the past years:
2009/2010 -
2010/2011 -
Fall 2011