The range of holomorphic maps at boundary points

TL;DR

This paper establishes a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains, showing that near certain boundary points, the image of the map contains all admissible regions with a vertex at the boundary image.

Contribution

It proves a boundary open mapping theorem for holomorphic maps at boundary points of strongly pseudoconvex domains, extending classical results to boundary behavior.

Findings

Local images near boundary points contain all admissible regions.

Jacobian bounded away from zero along normal directions is crucial.

Results apply to boundary regular contact points.

Abstract

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map $f:D\to D'$ close to a boundary regular contact point $p\in \de D$ where the Jacobian is bounded from zero along normal non-tangential directions has to eventually contain every cone (and more generally every admissible region) with vertex at $f(p)$.