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

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 $Ω_1\times\cdots\timesΩ_m,$ equipped with positive constant multiples of their canonical Kähler--Einstein metrics. We prove that every n