After Yau suggested the problem of approximations of Kähler-Einstein metrics by Bergman type metrics, various types of Bergman metrics have been developed and studied by Tian, Donaldson, Tsuji, etc. They showed that if a polarized manifold admits a Kähler-Einstein metric, there exist a sequence of Bergman type metrics which converges to the Kähler-Einstein metric. In this talk, we will survey their results and discuss some remaining problems in this subject.
Sungmin Yoo, Fiberwise Kähler-Ricci Flows on Families of Strongly Pseudoconvex Domains
A study on the positive variation of Kähler-Einstein metrics is first developed by Schumacher. More precisely, he has proved that the fiberwise Kähler-Einstein metrics on a family of canonically polarized compact Kähler manifolds is positive-definite on the total space. Later, Berman constructed the fiberwise Kähler-Ricci flow which converges to the fiberwise Kähler-Einstein metrics and showed that the positivity is preserved under this flow. In this talk, I will explain how to generalize these to a family of bounded strongly pseudoconvex domains, which is an important examples of non-compact complete Kähler manifolds. This is joint work with Young-Jun Choi.