Advanced Graduate Lecture Course (VO 260130)
March 11 - 27, and April 14 - May 8, 2015: Wednesday and Friday 14:00 - 15:30.
The goal of this course is to highlight the role of weak solutions of the complex Monge-Ampère equation in finding singular canonical metrics in Kahler geometry or, equivalently, the limits of the Kähler-Ricci flow. The methods include: PDE a priori estimates and the continuity method, pluripotential theoretic approach based on the notion of the positive current and properties of plurisubharmonic functions, and the parabolic maximum principle for the Ricci flow.
Part I. Preliminaries:
- Some basic facts from complex analysis, especially on subharmonic and plurisubharmonic functions. Positive differential forms.
- Basic notions of theory of distributions.
- Elliptic and parabolic differential operators. Maximum principles. Schauder estimates.
- Introduction to Kähler geometry. Ricci curvature. Canonical bundle. Kähler-Einstain metrics.
Part II. The complex Monge-Ampère equation:
- Calabi-Yau theorem. Continuity method and a priori estimates.
- Introduction to pluripotential theory: positive currents, capacities.
- Weak solutions to the complex Monge-Ampère equation. Stability of those solutions.
Part III. The Kähler-Ricci flow on compact Kähler manifolds:
- The evolution of geometric quantities along the flow.
- Long time existence.
- Convergence of the flow for definite Chern classes.
- Singular Kähler-Einstain metrics.
- The current research activity in the area. Similar equations and their geometrical context.
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