Classification of Rational Holomorphic Maps from B 2 into B N with Degree 2

Denote by Prop(Bn,BN ) the space of proper holomorphic maps from the unit ball Bn ⊂ Cn into the unit ball BN ⊂ CN , Propk(B,B ) := Prop(Bn,BN ) ∩ Ck(Bn) and Rat(Bn,BN ) := Prop(Bn,BN ) ∩ {rational maps}. We recall that F,G ∈ Rat(Bn ,BN ) are said to be equivalent if there are automorphisms σ ∈ Aut(Bn) and τ ∈ Aut(BN ) such that F = τ ◦ G ◦ σ. In this paper, we study the classification problem for elements in Rat(B2 ,BN ) with degree two. For an element F in Rat(B2 ,BN ), there is a naturally associated invariant RkF 6 1, called the geometric rank of the map (for the definition, see §2). Since F is linear if and only if its geometric rank RkF = 0, we only need to consider maps with geometric rank RkF = 1. By using Cayley transformation ρk : H k → Bk where Hk is the Siegel upper-half space (see § 2), studying Rat(B2 ,BN ) is equivalent to studying Rat(H2,HN ). Making use of results obtained in the previous work [8] [1], we give a complete description for the modular space for maps in Rat(B2 ,BN ) with degree 6 2 under the above mentioned equivalence relation. Our main result is the following Theorem 1.1. Notice that when N = 3, Rat(B2 ,B3) has been classified by Faran [4]; and when N = 4, a complete list of monomial maps in Rat(B2 ,B4) has been given by D’Angelo [3]. Theorem 1.1. (i) Any nonlinear map in Rat(B2,BN ) with degree 2 is equivalent to a map (F, 0) where F ∈ Rat(B2 ,B5) is of one of the following forms: (I): F = (Gt, 0) where Gt ∈ Rat(B2 ,B4) is defined by Gt(z,w) = (z , p