RIGIDITY OF PROPER HOLOMORPHIC MAPS FROM Bn+1TO B3n-1

Abstract

Let <TEX>$B^{n+1}$</TEX> be the unit ball in the complex vector space <TEX>$\mathbb{C}^{n+1}$</TEX> with the standard Hermitian metric. Let <TEX>${\Sigma}^n={\partial}B^{n+1}=S^{2n+1}$</TEX> be the boundary sphere with the induced CR structure. Let f : <TEX>${\Sigma}^n{\hookrightarrow}{\Sigma}^N$</TEX> 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 <TEX>${\Sigma}^n$</TEX> 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 : <TEX>${\Sigma}^n{\hookrightarrow}{\Sigma}^N$</TEX>, N <TEX>$\leq$</TEX> 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 <TEX>$B^{n+1}$</TEX> to <TEX>$B^{2n+2}$</TEX> by Hamada to the classification of the rational proper holomorphic maps from <TEX>$B^{n+1}$</TEX> to <TEX>$B^{3n+1}$</TEX>.