Rigidity for local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ up to conformal factors

TL;DR

This paper investigates local holomorphic isometric embeddings from complex balls into products of balls, establishing conditions under which components are either constant or proper maps, and proving total geodesy of non-constant components.

Contribution

It introduces a new approach combining algebraic extension, holomorphic continuation, and a linearity criterion to classify embeddings up to conformal factors.

Findings

Components are either constant or proper holomorphic maps.

Non-constant components are totally geodesic.

Embeddings are characterized by algebraic and analytic methods.

Abstract

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ 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-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.