The geometry of domains with negatively pinched K\"ahler metrics

TL;DR

This paper investigates how negatively pinched K"ahler metrics influence the boundary geometry of convex domains in complex Euclidean space, linking curvature conditions to boundary regularity and pseudoconvexity.

Contribution

It establishes new restrictions on boundary subvarieties and characterizes strong pseudoconvexity using curvature conditions on convex domains.

Findings

Boundaries of convex domains with such metrics lack positive-dimensional complex subvarieties.

Smooth boundaries are of finite D'Angelo type under these conditions.

Strong pseudoconvexity is characterized by the existence of tightly pinched negatively curved K"ahler metrics.

Abstract

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.