A proof of the Muir–Suffridge conjecture for convex maps of the unit ball in Cn

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let F : B-n -> C-n be a univalent map from the unit ball whose image D is convex. Let S Bn be the set of points. such that limz.. F(z) = 8. Then we prove that S is either empty, or contains one or two points and F extends as a homeomorphism (F) over tilde : (B) over barn S -> (D) over bar. Moreover, S = empty set if D is bounded, S has one point if D has one connected component at infinity and S has two points if D has two connected components at infinity 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.