The geometry of domains with negatively pinched Kähler metrics

We study how the existence of a negatively pinched Kähler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete Kähler metric, with pinched negative holomorphic bisectional curvature outside a compact set, then the boundary of the domain does not contain any complex subvariety of positive dimension. Moreover, if the boundary of the domain is smooth, then it is of finite type in the sense of D’Angelo. We also use curvature to provide a characterization of strong pseudoconvexity amongst convex domains. In particular, we show that a convex domain with Cˆ2,\alpha boundary is strongly pseudoconvex if and only if it admits a complete Kähler metric with sufficiently tight pinched negative holomorphic sectional curvature outside a compact set.