Holomorphic maps from the complex unit ball to Type IV classical domains

The first part of this paper is devoted to establish new rigidity results for proper holomorphic maps from the complex unit ball to higher rank bounded symmetric domains. The rigidity properties have been extensively studied in the past decades for proper holomorphic maps F : Ω1 → Ω2, between bounded symmetric domains Ω1,Ω2. The pioneer works are due to Poincaré [P] and later to Alexander [Al] when Ω1,Ω2 are complex unit balls. In particular, any proper holomorphic self-map of the unit ball in C is an automorphism if n ≥ 2 [Al]. It is well-known that the rigidity properties fail dramatically for proper holomorphic maps between balls of different dimensions. For this type of results, see [HS], [L], [Fo1],[Gl], [St], [Do], [D1] and etc. However, the rigidity properties can still be expected if certain boundary regularity of the map is assumed. See [W], [Fa], [CS], [Hu1], [Hu2], [HJ], [HJY], [Eb] and etc. The lists above are by no means to be complete. On the other hand, it is a widely open problem to understand proper holomorphic maps F : Ω1 → Ω2 between bounded symmetric domains Ω1,Ω2 of higher rank. When rank(Ω1) ≥ rank(Ω2) ≥ 2 and Ω1 is irreducible, it was proved by Tsai [Ts] that F must be a totally geodesic isometric embedding with respect to Bergman metrics. Tu [Tu1] proved that the proper holomorphic map between equal dimensional irreducible bounded symmetric domains must be an automorphism. When rank(Ω2) ≥ rank(Ω1), the studies are mainly focused on the Type I classical domains and many interesting results have been established (cf. [Tu2], [Ng4], [KZ1], [KZ2] et al). Note that the total geodesy of F fails in general although it is believed that F should take certain special forms module automorphisms (cf. [Ng4] [KZ2]). In this paper, we prove new rigidity results for proper holomorphic maps from the unit ball in C to