Journal of the
Korean Mathematical Society JKMS

Let ${\bf B}^{n+1}$ be the unit ball in the complex vector space $\mathbb C^{n+1}$ with the standard Hermitian metric. Let $\Sigma^n = \partial {\bf B}^{n+1} = S^{2n+1}$ be the boundary sphere with the induced CR structure. Let $f: \Sigma^n \hookrightarrow \Sigma^{N}$ be a local CR immersion. If $N<3n-1$, the asymptotic vectors of the CR second fundamental form of $f$ at each point form a subspace of the CR(horizontal) tangent space of $\Sigma^n$ of codimension at most 1. We study the higher order derivatives of this relation, and we show that a linearly full local CR immersion $f: \Sigma^n \hookrightarrow \Sigma^{N}$, $N \leq 3n-2$, can only occur when $N = n, 2n$, or $2n+1$. As a consequence, it gives an extension of the classification of the rational proper holomorphic maps from ${\bf B}^{n+1}$ to ${\bf B}^{2n+2}$ by Hamada to the classification of the rational proper holomorphic maps from ${\bf B}^{n+1}$ to ${\bf B}^{3n-1}$.

Keywords: CR immersion, proper holomorphic map, rigidity

MSC numbers: 53B25