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Rigidity for local holomorphic isometric embeddings from \(\mathbb{B}^n\) into \(\mathbb{B}^{N_1} \times \cdots \times \mathbb{B}^{N_m}\) up to conformal factors. (English) Zbl 1248.32008

This is a beautiful paper on local holomorphic isometric embeddings from the ball \({\mathbb B}^n = \big\{ z \in {\mathbb C}^n : |z| < 1 \big\}\) into a product of balls \({\mathbb B}^{N_1} \times \cdots \times {\mathbb B}^{N_m}\), with respect to the normalized Bergman metrics up to conformal factors. If \(n \geq 2\) and
\[ g^{}_{{\mathbb B}^n} = \sum_{j,k=1}^n \big( 1 - |z|^2 \big)^{-2} \big((1 - |z|^2 ) \delta_{jk} + \overline{z}_j z_k \big) d z_j \odot d \overline{z}_k, \]
let \[ F = (F_1 , \cdots , F_m ) : U \to {\mathbb B}^{N_1} \times \cdots \times {\mathbb B}^{N_m} \] be a holomorphic embedding of an open set \(U \subset {\mathbb B}^n\) such that
\[ \lambda g^{}_{{\mathbb B}^n} = \sum_{\alpha =1}^m \lambda_\alpha \, F_\alpha^\ast g^{}_{{\mathbb B}^{N_\alpha}} \]
for some smooth, positive, Nash algebraic functions \(\lambda\) and \(\lambda_\alpha\) over \({\mathbb C}^n\). The main result in the paper is that for each \(\alpha \in \{ 1, \cdots , m \}\) either \(F_\alpha\) is a constant map or \(F_\alpha\) extends to a totally geodesic holomorphic embedding from \(\big({\mathbb B}^n, g^{}_{{\mathbb B}^n}\big)\) into \(\big({\mathbb B}^{N_\alpha}, g^{}_{{\mathbb B}^{N_\alpha}}\big)\).

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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