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