A gap theorem for the complex geometry of convex domains

TL;DR

This paper proves a gap theorem linking boundary geometry closeness to the unit ball with strong pseudoconvexity in convex domains, using invariants like the squeezing function and Bergman curvature.

Contribution

It establishes conditions under which near-boundary geometric invariants imply strong pseudoconvexity in convex domains.

Findings

Domains with high squeezing function near boundary are strongly pseudoconvex.

Holomorphic sectional curvature close to -4/(d+1) implies strong pseudoconvexity.

Results hold uniformly across all dimensions.

Abstract

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some $\epsilon > 0$ so that if the squeezing function on a smoothly bounded convex domain is greater than $1-\epsilon$ outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension $d$ there exists some $\epsilon > 0$ so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within $\epsilon$ of $-4/(d+1)$ outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to $-4/(d+1)$ on the boundary).