Colloquium Series Seminar: Intrinsic and extrinsic geometries in several complex variables

A bounded domain in complex Euclidean space, despite being one of the simplest types of manifolds, has a number of interesting geometric structures. For instance, on the interior there is the Kobayashi metric, the Bergman metric, and the complete Kaehler-Einstein metric constructed by Cheng-Yau. These are all intrinsic to the domain (viewed as a complex manifold). When the domain has smooth boundary, there is also a natural extrinsic structure: the CR-geometry of the boundary. In this talk, I will define all these structures and then describe connections between them. Then, I will discuss how these connections can lead to new analytic results. No background knowledge of the topic is assumed.