The Monge-Ampère Equations and the Bergman Kernel - Lecture 1

Explore complex analysis in higher dimensions through this lecture on the Monge-Ampère equations and the Bergman kernel. Delve into the geometry of domains in Cn, Riemann's mapping theorem, and Fefferman's extension theorem. Examine Cauchy-Riemann (CR) structures, the CR version of Hartogs extension theorem, and biholomorphic geometry. Learn about the Bergman kernel and its properties through lemmas and examples. Gain insights into the L2-theory of the ∂¯-problem and its applications in complex geometry, partial differential equations, and operator theory.

Start
Geometry of domains in Cn
Riemann's mapping theorem
Example
Poincore, Reinhardt
We assume omega is strictly pseudo convex
Fetterman's extension theorem
Cauchy RiemannCR structure
Corollary Fetterman's Theorem
CR version of Hartogs extension theorem
CF: Hartogs theorem
Biholic geometry and CR geometry
Reference
The Bergman Kernel
Lemma 1
Proof
Lemma 2
Example