## 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

### Keywords:

Bergman metric; holomorphic isometric embedding
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