ON PROPER HOLOMORPHIC MAPS BETWEEN BOUNDED SYMMETRIC DOMAINS

We study proper holomorphic maps between bounded symmetric domains D and Omega. In particular, when D and Omega are of the same rank >= 2 such that all irreducible factors of D are of rank >= 2, we prove that any proper holomorphic map from D to Omega is a totally geodesic holomorphic isometric embedding with respect to certain canonical Kahler metrics of D and Omega. We also obtain some results regarding holomorphic maps F : D -> Q 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.