12. Monge-Ampère equation on compact Kähler
manifolds
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Polish supervisor:
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Zbigniew B³ocki
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Cooperating partners:
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Takeo Ohsawa (Nagoya University)
Duong Phong (Columbia University)
Ahmed Zeriahi (Universite Paul Sabatier, Toulouse)
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The proof of the Calabi conjecture by Yau in 1976 was a major
breakthrough in mathematics which also had a lot of important
applications in physics. It boiled down to solving nondegenerate
complex Monge-Ampère equation on compact Kähler manifolds and
had big impact in such areas of mathematics as algebraic geometry,
partial differential equations, pluripotential theory. The subject
is still very much alive today, for example related on one hand to
Hamilton-Perelman's theory of Kähler-Ricci flow, and on the
other to the Donaldson programme of existence of geodesics in the
space of Kähler metrics and constant scalar curvature metrics.
The conjecture posed by Donaldson and many related questions seem
to be a particularly promising area of research. It boils down to
solving a degenerate homogeneous Monge-Ampère equation with
specified boundary values in the context of compact Kähler
manifolds, and so far only C1,1-regularity has been proved.
There is also a series of open problems related to various a
priori estimates for solutions of Monge-Ampère and more general
Hessian equations. Another interesting link is with the theory of
the Bergman kernel through a so called Tian-Yau-Zelditch
approximation of Kähler metrics.
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