2023-1 Math Colloquium | Bergman metrics and Kähler-Einstein metrics on projective manifolds

A basic problem in differential geometry is to find canonical, or best, metrics on a given manifold. By the famous uniformization theorem, every Riemann surface admits a Riemannian metric of constant curvature surfaces. As a higher dimensional generalization of this result, we will introduce two canonical Kähler metrics for Kähler manifolds, especially for projective manifolds: Bergman metrics and Kähler-Einstein metrics. Both metrics can be considered as higher dimensional generalizations of the Poincaré metirc. In this talk, we will study several results related to the Yau-Tian-Donaldson conjecture and Kazhdan's Theorem, which tell us that these two metrics are deeply related to each other.