A proof of the Muir–Suffridge conjecture for convex maps of the unit ball in ${\mathbb {C}}^n$

Abstract

We prove (and improve) the Muir–Suffridge conjecture for holomorphic convex maps. Namely, let $F:{\mathbb {B}}^n\rightarrow {\mathbb {C}}^n$ 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 $\lim _{z\rightarrow \xi }\Vert F(z)\Vert =\infty$. Then we prove that ${\mathcal {S}}$ is either empty, or contains one or two points and F extends as a homeomorphism $\tilde{F}:\overline{{\mathbb {B}}^n}{\setminus } {\mathcal {S}}\rightarrow \overline{D}$. 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 automorphism of the ball and renormalization, F is an extension of the strip map in the plane to higher dimension.