  We study how the existence of a negatively pinched K\"ahler 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 K\"ahler 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 domain. 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 $C^{2,\alpha}$ boundary is
strongly pseudoconvex if and only if it admits a complete K\"ahler metric with
sufficiently tight pinched negative holomorphic sectional curvature outside a
compact set.
