Rigidity for local holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains

Abstract

Let $U$ be a connected open subset of the complex unit ball $B^n$, $n\ge 2$, and let $F_i:U\to {\Omega }_i$ be non-constant holomorphic maps into irreducible bounded symmetric domains. We prove that if $F=(F_1,\cdots ,F_m)$ is a holomorphic isometric map from $U$ to the product of ${\Omega }_i$, then every $F_i$ extends to a proper holomorphic isometric embedding up to a positive constant from $B^n$ into ${\Omega }_i$.