# A proof of the Muir-Suffridge conjecture for convex maps of the unit ball in $\mathbb C^n$

## TL;DR

This paper proves and refines the Muir-Suffridge conjecture for convex holomorphic maps of the unit ball in complex n-space, characterizing boundary behavior and extension properties of such maps.

## Contribution

It establishes the possible boundary point sets where the map tends to infinity and describes the extension of convex maps in higher dimensions, improving previous conjectures.

## Key findings

- The set of boundary points where the map diverges is either empty, one point, or two points.
- The map extends as a homeomorphism outside the divergence set.
- Characterization of the divergence set based on the convexity of the image domain.

## Abstract

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb B^n$ be the set of points $\xi$ such that $\lim_{z\to \xi}\|F(z)\|=\infty$. Then we prove that $\mathcal S$ is either empty, or contains one or two points and $F$ extends as a homeomorphism $\tilde{F}:\overline{\mathbb B^n}\setminus \mathcal S\to \overline{D}$. Moreover, $\mathcal S=\emptyset$ if $D$ is bounded, $\mathcal S$ has one point if $D$ has one connected component at $\infty$ and $\mathcal S$ has two points if $D$ has two connected components at $\infty$ and, up to composition with an affine map, $F$ is an extension of the strip map in the plane to higher dimension.