On proper holomorphic maps between bounded symmetric domains

We study proper holomorphic maps between bounded symmetric domains $D$ and $\mathrm{\Omega }$. In particular, when $D$ and $\mathrm{\Omega }$ are of the same rank $\ge 2$ such that all irreducible factors of $D$ are of rank $\ge 2$, we prove that any proper holomorphic map from $D$ to $\mathrm{\Omega }$ is a totally geodesic holomorphic isometric embedding with respect to certain canonical K"ahler metrics of $D$ and $\mathrm{\Omega }$. We also obtain some results regarding holomorphic maps $F:D\to \Omega$whichmapminimaldisksof$D$properlyintorank-$1$ characteristic symmetric subspaces of $\mathrm{\Omega }$. On the other hand, we obtain new rigidity results regarding semi-product proper holomorphic maps between $D$ and $\mathrm{\Omega }$ under a certain rank condition on $D$ and $\mathrm{\Omega }$.