On a semi-rigidity property for holomorphic maps
Write \(\mathbb{B}^n\) and \(\mathbb{B}^N\) for the unit balls in \(\mathbb{C}^n\) and \(\mathbb{C}^N\), respectively. Let \(F\) be a proper holomorphic map from \(\mathbb{B}^n\) into \(\mathbb{B}^N\), which is said to be \(k\)-linear if for any point \(p\in \mathbf{B}^n\), there is an affine complex subspace \(S_p^k\), that passes through \(p\) and is of dimension \(k\), such that for any affine complex line \(L\) contained in \(S_p^k\), \(F(L\cap\mathbb{B}^n)\) is contained in an affine complex line in \(\mathbb{C}^N\). The author proves that \(F\) is \((n-k+1)\)-linear if \(F\) is \(C^3\)-smooth up to the boundary and if \(1\leq k\leq n-1\) and \(N-n<k(2n-k-1)/2\).
scientific article; zbMATH DE number 2113423