Rigidity for proper holomorphic ball maps with degenerate CR Gauss map

Abstract

Let $n$ and $N$ be integers with $2\leq n<N$, and let $F:\mathbb{B}^n\to\mathbb{B}^N$ be a proper holomorphic map that admits a $C^{N-n+1}$-smooth extension to the boundary. We prove that if the CR Gauss map of its boundary restriction is generically degenerate, then $F$ is holomorphically equivalent, under composition with automorphisms of the source and target balls, to $V_m\oplus 0$ for some positive integer $m$, where $ V_m(z)=\big(\sqrt{\frac{m!}{\alpha!}} z^\alpha\big)_{|\alpha|=m} $ is the degree-$m$ Veronese map. This resolves a problem posed by Xiaojun Huang.