Pure Maths Colloquium: Convergence of the Calabi flow
24 October 2016 16:00 in CM221
In the 1950s, E. Calabi proposed a program in Kähler geometry and then introduced the Calabi flow, aiming to find the constant scalar curvature Kähler (cscK) metrics. When the first Chern class is zero, the cscK metric reduces to Ricci flat Kähler metric. The problem to find such metrics is called the Calabi conjecture and its resolution was S.T. Yau’s seminal work. For general Kähler class, it is known as the Yau-Tian-Donaldson conjecture.
On Riemann surfaces, the global existence and the convergence of the Calabi flow have been proved by X.X. Chen, P.T. Chrusciel and M. Struwe by different methods. However, much less is known in high dimension, due to the fourth order of the flow and the lack of a maximum principle. In this talk, I will present our recent progress on Donaldson's conjectural picture on the asymptotic behavior of the Calabi flow, i.e. the results which partially confirm this conjectural picture in complex dimension 2. I will also discuss similar results in higher dimension with an extra assumption that the scalar curvature is uniformly bounded.
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