Georgia Institute of Technology College of Sciences

The Poincaré metric on the unit disc $\mathbb{D} \subset \mathbb{C}$, known for its invariance under all biholomorphisms (bijective holomorphic maps) of $\mathbb{D}$, is one of the most fundamental Riemannian metrics in differential geometry.

In this presentation, we will first introduce the Bergman metric on a bounded domain in $\mathbb{C}^n$, which can be viewed as a generalization of the Poincaré metric. We will then explore some key theorems that illustrate how the curvature of the Bergman metric characterizes bounded domains in $\mathbb{C}^n$ and more generally, complex manifolds. Finally, I will discuss my recent work related to these concepts.