# On proper holomorphic maps between bounded symmetric domains

## TL;DR

This paper investigates the structure and rigidity of proper holomorphic maps between bounded symmetric domains, establishing conditions under which such maps are totally geodesic isometric embeddings and exploring their behavior on minimal disks.

## Contribution

It proves that proper holomorphic maps between certain bounded symmetric domains are totally geodesic isometric embeddings and introduces new rigidity results for semi-product maps.

## Key findings

- Proper maps are totally geodesic isometric embeddings under certain conditions.
- Holomorphic maps can properly map minimal disks into rank-1 characteristic subspaces.
- New rigidity results for semi-product proper holomorphic maps.

## Abstract

We study proper holomorphic maps between bounded symmetric domains $D$ and $\Omega$. In particular, when $D$ and $\Omega$ 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 $\Omega$ is a totally geodesic holomorphic isometric embedding with respect to certain canonical K\"ahler metrics of $D$ and $\Omega$. We also obtain some results regarding holomorphic maps $F:D\to \Omega$ which map minimal disks of $D$ properly into rank-$1$ characteristic symmetric subspaces of $\Omega$. On the other hand, we obtain new rigidity results regarding semi-product proper holomorphic maps between $D$ and $\Omega$ under a certain rank condition on $D$ and $\Omega$.