Geometry and Topology Seminar

Every bounded pseudoconvex domain in C^n has a natural complete Kahler metric: the Kahler-Einstein metric constructed by Cheng-Yau. In this talk I will describe how the curvature of this metric restricts the CR-geometry of the boundary. In particular, I will sketch the proofs of the following two theorems: First, if a smoothly bounded convex domain has a complete Kahler metric with pinched negatively curved bisectional curvature, then the boundary of the domain has finite type in the sense of D'Angelo. Second, if a smoothly bounded convex domain has a complete Kahler metric with sufficiently tight pinched negatively curved holomorphic sectional curvature, then the boundary of the domain is strongly pseudoconvex. The proofs use recent results of Wu-Yau, classical results of Shi on the Ricci flow, and ideas from Benoist's work in real projective geometry. This is joint work with F. Bracci and H. Gaussier.