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

Title & Authors
RIGIDITY OF PROPER HOLOMORPHIC MAPS FROM Bn+1 TO B3n-1
Wang, Sung-Ho;

Abstract
Let $\small{B^{n+1}}$ be the unit ball in the complex vector space $\small{\mathbb{C}^{n+1}}$ with the standard Hermitian metric. Let $\small{{\Sigma}^n={\partial}B^{n+1}=S^{2n+1}}$ be the boundary sphere with the induced CR structure. Let f : $\small{{\Sigma}^n{\hookrightarrow}{\Sigma}^N}$ 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 $\small{{\Sigma}^n}$ 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 : $\small{{\Sigma}^n{\hookrightarrow}{\Sigma}^N}$, N $\small{\leq}$ 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 $\small{B^{n+1}}$ to $\small{B^{2n+2}}$ by Hamada to the classification of the rational proper holomorphic maps from $\small{B^{n+1}}$ to $\small{B^{3n+1}}$.
Keywords
CR immersion;proper holomorphic map;rigidity;
Language
English
Cited by
1.
Linearity and Second Fundamental Forms for Proper Holomorphic Maps from $\mathbb{B}^{n+1}$ to $\mathbb{B}^{4n-3}$, Journal of Geometric Analysis, 2012, 22, 4, 977
2.
Flatness of CR submanifolds in a sphere, Science China Mathematics, 2010, 53, 3, 701
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