Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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RIGIDITY OF PROPER HOLOMORPHIC MAPS FROM Bn+1 TO B3n-1

  • Wang, Sung-Ho (DEPARTMENT OF MATHEMATICS KOREA INSTITUTE FOR ADVANCED STUDY)
  • Published : 2009.09.01
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Abstract

Let $B^{n+1}$ be the unit ball in the complex vector space $\mathbb{C}^{n+1}$ with the standard Hermitian metric. Let ${\Sigma}^n={\partial}B^{n+1}=S^{2n+1}$ 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 $B^{n+1}$ to $B^{2n+2}$ by Hamada to the classification of the rational proper holomorphic maps from $B^{n+1}$ to $B^{3n+1}$.

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  2. Flatness of CR submanifolds in a sphere vol.53, pp.3, 2010, https://doi.org/10.1007/s11425-010-0052-4