On proper holomorphic maps between bounded symmetric domains

Abstract

We study proper holomorphic maps between bounded symmetric domains D D and Ω \Omega . In particular, when D D and Ω \Omega are of the same rank ≥ 2 \ge 2 such that all irreducible factors of D D are of rank ≥ 2 \ge 2 , we prove that any proper holomorphic map from D D to Ω \Omega is a totally geodesic holomorphic isometric embedding with respect to certain canonical Kähler metrics of D D and Ω \Omega . We also obtain some results regarding holomorphic maps F : D → Ω F:D\to \Omega which map minimal disks of D D properly into rank- 1 1 characteristic symmetric subspaces of Ω \Omega . On the other hand, we obtain new rigidity results regarding semi-product proper holomorphic maps between D D and Ω \Omega under a certain rank condition on D D and Ω \Omega .