The geometry of domains with negatively pinched Kähler metrics

Abstract

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

A new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps

“…The condition −c ≤ κ µ ≤ −4 for some constant c > 0 in Theorem 3.2 appears frequently in studying complex geometric properties of domains in C N or complex manifolds, see e.g. [10]. It often goes by the name "negatively pinched".…”

Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

“…In [20,21] it has been proved that Gromov hyperbolicity of convex smooth bounded domains is related to D'Angelo type finiteness of the boundary, while in [15] the same result has been proved in C 2 for pseudoconvex domains. In [9] Gromov hyperbolicity of convex domains is shown to be equivalent to the existence of a negatively pinched metric close to the boundary, giving the idea that Gromov hyperbolicity should be read only by local properties near the boundary.…”