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Rigidity for local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ up to conformal factors

Yuan Yuan Johns Hopkins University Yuan Zhang University of California at San Diego

Differential Geometry mathscidoc:1609.10255

Journal of Differential Geometry, 90, (2), 329-349, 2012
In this article, we study local holomorphic isometric embeddings from Bn into BN1 × · · · × BNm with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok, and Mok and Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.
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@inproceedings{yuan2012rigidity,
  title={Rigidity for local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ up to conformal factors},
  author={Yuan Yuan, and Yuan Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913223953816570917},
  booktitle={Journal of Differential Geometry},
  volume={90},
  number={2},
  pages={329-349},
  year={2012},
}
Yuan Yuan, and Yuan Zhang. Rigidity for local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ up to conformal factors. 2012. Vol. 90. In Journal of Differential Geometry. pp.329-349. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913223953816570917.
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