A gap rigidity for proper holomorphic maps from $ \B^{n+1}$ to $   \B^{3n-1}$
  Let $ \B^{n+1} \subset \C^{n+1}$ be the unit ball in a complex Euclidean space, and let $ \Sigma^n = \partial \B^{n+1} = S^{2n+1}$. Let $ f: \Sigma^n \hook \Sigma^{N}$ be a local CR immersion.If $ N-n<2n-1$, the asymptotic vectors of the second fundamental form of $ f$ at each point form a subspace of the holomorphic tangent space of $ \Sigma^n$ of codimension at most 1. We exploit the successive derivatives of this relation and show that a linearly full local CR immersion $ f: \Sigma^n \hook \Sigma^{N}$, $ N \leq 3n-2$, can only occur when $ N = n, 2n$, or $ 2n+1$. Together with the recent classification of the rational proper holomorphic maps from $ \B^{n+1}$ to $ \B^{2n+2}$ by Hamada, this gives a classification of the rational proper holomorphic maps from $ \B^{n+1}$ to $ \B^{3n-1}$ for $ n \geq 3$. 
Seungho Wang