## Journal of Differential Geometry

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

### 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 and Yuan Zhang

#### 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

**Permanent link to this document**

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