Componentwise rigidity of holomorphic isometric maps from the complex unit ball to bounded symmetric domains

Abstract

We study holomorphic isometric maps $F=(F_1,\cdots,F_m)$ from the complex unit ball $\mathbb{B}^n$, $n\geq2$, to a product of irreducible bounded symmetric domains $\Omega_1\times\cdots\times\Omega_m,$ equipped with positive constant multiples of their canonical K\"ahler--Einstein metrics. We prove that every nonconstant component $F_i$ is itself a holomorphic isometry up to a constant factor. This proves the componentwise rigidity conjecture formulated by Yuan in 2019.