Sébastien Gontard

Given a domain $D$ in $\mathbf{C}^n$ (or more generally a complex manifold without boundary) it is natural to study the relationships between quantities containing information about the complex geometry of $D$. For instance one can ask whether the curvature of the boundary of $D$ is related to the curvatures of objects defined on $D$, such as complete Kähler metrics. We expect that the behaviour of the curvatures of a complete Kähler metric is influenced by the geometry of the boundary $\partial D$ of $D$, at least when we look at points of $D$ close to $\partial D$. More formally, given a bounded pseudoconvex domain $D$ and a boundary point $p$ of finite D'Angelo 1-type we study the following question:
(Q): "Does there exists a complete Kähler metric on $D$ whose holomorphic bisectional curvatures are negatively pinched in a neighbourhood of $q$?"

Since we study this question for a general pseudoconvex domain with no explicit description it is reasonable to pick a canonical metric and study its curvatures. In our case we work with the Bergman metric and with the Kähler-Einstein metric of negative Ricci curvature.

The situation is well understood if $\partial D$ "looks like the boundary of a ball $B$" near a given point p of the boundary (for instance if $\partial D$ is strictly pseudoconvex at $p$): in this case D is metrically "curved like $B$" when we look at points in $D$ near $q$. Especially, the answer to question (Q) is affirmative.
Apart from the above situation, very few is known.

This talk is divided in four parts. The first part provides with basic ideas regarding metrics, curvatures (both geometric and metric) and motivations. In the second part we study the behaviour of the Kähler-Einstein metric and its curvatures at "ball-like" boundary points. Then we explain what problems we encounter when dealing with weakly pseudoconvex boundary points and what results we have for the moment. Finally we end this talk with some more general questions arising when we study the boundary behaviour of metrics and their curvatures.