October 30, 2006
Xiuxiong Chen (Madison): The existence of C^{1,1} solution and its applications
Abstract:
According to Mabuchi, Semmes and Donaldson, one can define
a Weil Peterson type metric in the space of Kähler potentials.
According to Semmes, the geodesic equation in this metric is a
homogenous complex MA equation over a product manifold (Riem surface x
Kähler manifold).
One can solve this by continuous method to derive a C^{1,1} solution.
Using this existence, one can prove
a) it is a metric space;
b) it is non-positively curve in the sense of Alexandrov;
c) the Calabi flow is ideally a distance contracting flow;
d) the cscK metric is unique in each Kähler class if C_1 <=0.
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