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 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,α 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.
B(g)(X, Yand the holomorphic sectional curvature of a non-zero X ∈ T p M is given byWe say that (M, g) has pinched negative holomorphic bisectional curvature if there exists a, b > 0 such that −a ≤ B(g)(X, Y ) ≤ −b for all p ∈ M and non-zero vectors X, Y ∈ T p M . Likewise, we say that (M, g) has pinched negative holomorphic sectional curvature if there exists a, b > 0 such that −a ≤ H(g)(X) ≤ −b