## Journal of Differential Geometry

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

#### Abstract

In this article, we study local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ 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.

#### Article information

Source
J. Differential Geom., Volume 90, Number 2 (2012), 329-349.

Dates
First available in Project Euclid: 24 April 2012

https://projecteuclid.org/euclid.jdg/1335230850

Digital Object Identifier
doi:10.4310/jdg/1335230850

Mathematical Reviews number (MathSciNet)
MR2899879

Zentralblatt MATH identifier
1248.32008

#### Citation

Yuan, Yuan; Zhang, Yuan. Rigidity for local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ up to conformal factors. J. Differential Geom. 90 (2012), no. 2, 329--349. doi:10.4310/jdg/1335230850. https://projecteuclid.org/euclid.jdg/1335230850