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1 ASIAN J. MATH. c 018 International Press Vol., No. 3, pp , June UPPER BOUNDARY POINTS OF THE GAP INTERVALS FOR RATIONAL MAPS BETWEEN BALLS SHANYU JI AND WANKE YIN Dedicated to Professor Ngaiming Mok on the occasion of his 60th birthday Abstract. The paper focuses on the study of rational proper holomorphic maps from B n to B N. We classify these maps when N is the upper boundary point of the gap interval I k, k n and the geometric rank of the map is k. Key words. Moser equation. Proper holomorphic maps, holomorphic classification, geometric rank, Chern- Mathematics Subject Classification. 3H Introduction. Let us denote by B n the unit ball in C n and Rat(B n, B N ) the set of all proper holomorphic rational maps F from B n to B N. We say that f,g Rat(B n, B N )areequivalent if there are σ Aut(B n )andτ Aut(B N )such that f = τ g σ. Let K(n) :=max{t Z + t(t+1) <n}. For any integer k with 1 k K(n), we define the gap interval I k := ( kn, (k +1)n ) k(k +1). (1.1) Let us recall the gap conjecture [HJY09]: Any proper holomorphic rational map F Rat(B n, B N ) (n 3) is equivalent to a map of the form (G, 0 ) where G Rat(B n, B N )wheren <Nif and only if N I k for some 1 k K(n). For the sake of simplicity, we call F is equivalent to G. Recently, P. Ebenfelt proposed a SOS conjecture (i.e., the Sums of Squares of Polynomial conjecture) [E16] and proved that if the SOS conjecture is true, then it implies the gap conjecture. The only if part of the gap conjecture was proved in [HJY09]. For the if part, the cases for I 1, I,andI 3 have been proved by Huang [Hu99], Huang-Ji[HJ01], Hamada [H05], Huang-Ji-Xu[HJX06] and Huang-Ji-Yin [HJY14]. Moreover, if N is the boundary point of the interval I k for k = 1 and, maps in Rat(B n, B N ) have been determined, up to equivalence, as follows. When N = n which is the lower boundary point of I 1 =(n, n 1), it was proved by Alexanda [A77]: any map F Rat(B n, B n ) is an automorphism. When N = which is the upper boundary point of I 1,itwasproved by [HJ01]: any map F Rat(B n, B ) is either the linear map, or the Whitney map W n,1. When N =n which is the lower boundary point of I =(n, 3n 3), it was proved by Hamada [Ha05] that any map F Rat(B n, B n ) is the linear map, or Whitney map, or in the D Angelo map family. Received May 19, 016; accepted for publication June 1, 017. Both authors are supported in part by NSFC , the second author is also supported in part by NSFC Department of Mathematics, University of Houston, Houston, TX 7704, USA (shanyuji@math. uh.edu). School of Mathematics and Statistics, Wuhan University, Hubei 43007, P. R. China (wankeyin@ whu.edu.cn). 493

2 494 S. JI AND W. YIN Recently, when N = 3n 3 which is the upper boundary point of I, it was proved by Andrews-Huang-Ji-Yin [AHJY16] that any map F Rat(B n, B 3n 3 ) is linear, or Whitney map W n,1, or in the D Angelo family, or a generalized Whitney map W n,. For any integer N in the closed interval I k and for any map F Rat(B n, B N ), its geometric rank κ 0 k. Infact,foranyF Rat(B n, B N ) with the geometric rank κ 0, it is known [Hu03] N n + (n κ 0 1)κ 0 =(κ 0 +1)n κ 0(κ 0 +1), (1.) which implies κ 0 k. In this paper, we show that when N is the upper boundary point of I k,wecan determine maps in Rat(B n, B N ) with geometric rank κ 0 = k. This gives a new proof for the above mentioned result in [HJ01] when k = 1 and the above mentioned result in [AHJY16] when k =. Theorem 1.1. Let F Rat(B n, B N ) with n 3. Let N be the upper boundary point of the gap interval I k and k n. Suppose that the geometric rank of F is κ 0 = k. ThenF is equivalent to the generalized Whitney map W n,κ. Notice that in above theorem, N =(k +1)n k(k+1) and that κ 0 = k K(n) by the definition of I k, i.e., κ0(κ0+1) <nholds. Here we recall [HJY09] that a map F Rat(B n, B N ) is called the generalized Whitney map W n,k if N =(k +1)n (k+1)k and W n,k (z) =W n,k (z 1,..., z n )=(z 1 ψ 1,..., z k ψ k,ψ k+1 ) (1.3) where ψ 1 =(z 1, z,..., z k,z k+1,..., z n ), ψ =(z,z z 3,..., z k,z k+1,..., z n ),..., ψ k 1 =(z k 1, z k,z k+1,..., z n ), ψ k =(z k,z k+1,..., z n ), ψ k+1 =(z k+1,..., z n ). (1.4) We may have another interpretation for the above theorem. If N = n + (n κ 0 1)κ 0, from (1.), we say that N is the minimum. Based on the semi-linearity property, the following two problems are formulated in [JX04]. Problem (A): Study and classify maps in F Rat(B n, B N ) with N minimum and 1 κ 0 n. Problem (B): Study and classify maps in F Rat(B n, B N ) with N minimum and κ 0 = n 1. For Problem (B), when n =,itisrat(b, B 3 ) which was solved by Faran s theorem. The next case n =3,Rat(B 3, B 6 ) is unsolved. In general, Problem (B) should be much more difficult than Problem (A) because it lacks of the semi-linearity property (see [Hu03]). For Problem (A), when κ 0 =1,itisRat(B n, B )whichwas solved by [HJ01] that F is the Whitney map W n,1 ;whenκ 0 =,itisrat(b n, B 3n 3 ),

3 RATIONAL MAPS BETWEEN BALLS 495 it is recently solved by [AHJY16] that the map F must be the generalized Whiney map W n,. Then Theorem 1.1 covers the remaining part of Problem (A) under the condition κ0(κ0+1) <n. Theorem 1.. Let F Rat(B n, B N ) be with geometric rank κ 0 n, where N = n + n κ0 1 κ 0 is minimum and κ0(κ0+1) <n. Then F is equivalent to the generalized Whitney map W n,κ0. Note that Theorem 1.1 and Theorem 1. are equivalent, and that the condition κ 0(κ 0+1) <nin both of the above theorems will be used in Lemma.3 below.. Preliminaries. a. The associated maps Fp. Let F =(f,φ,g) =( f,g)=(f 1,, f, φ 1,, φ N n,g) be a non-constant rational CR map from an open subset M of H n into H N with F (0) = 0. For each p M closeto0,wewriteσp 0 Aut(H n)forthe map sending (z,w) to(z + z 0,w+ w 0 +i z,z 0 )andτp F Aut(H N ) by defining τ F p (z,w )=(z f(z 0,w 0 ),w g(z 0,w 0 ) i z, f(z 0,w 0 ) ). Then F is equivalent to F p = τ F p F σ 0 p = (f p,φ p,g p ). Notice that F 0 = F and F p (0) = 0. The following is fundamentally important for the understanding of the geometric properties of F. Let us denote Prop(H n, H N ) := {holomorphic proper maps from H n into H N } and Prop k (H n, H N ):=Prop(H n, H N ) C k (H n ). Lemma.1. ([Hu99]) Let F Prop (H n, H N ) with n N. For each p H n, there is an automorphism τp Aut 0 (H N ) such that Fp := τp F p satisfies the following normalization: f p = z + i a (1) p (z)w + o wt (3), φ p = φ () p (z)+o wt (), gp = w + o wt(4), with z,a (1) p (z) z = φ p () (z). b. Geometric rank. Write A(p) := i( (f p) l z j w 0) 1 j,l (). We call the rank of the (n 1) (n 1) matrix A(p), which we denote by Rk F (p), the geometric rank of F at p. Rk F (p) depends only on p and F, and is a lower semi-continuous function on p, and is independent of the choice of τp (p) [Hu03]. Define the geometric rank of F to be κ 0 (F )=max p Hn Rk F (p). Notice that it always holds that 0 κ 0 n 1. Define the geometric rank of F Prop (B n, B N ) to be the one for the map ρ 1 N F ρ n Prop (H n, H N ). By [Hu03], κ 0 (F ) depends only on the equivalence class of F and when N< n(n+1), the geometric rank κ 0 (F )off is precisely the κ 0 mentioned in the introduction. Under the condition 1 κ 0 n, the following theorem was proved in [Hu03] and [HJX06]. Theorem. ([HJX06]). Suppose that F Prop 3 (H n, H N ) has geometric rank 1 κ 0 n with F (0) = 0. Then there are σ Aut(H n ) and τ Aut(H N ) such that τ F σ takes the following form, which is still denoted by F =(f,φ,g) for convenience of notation:

4 496 S. JI AND W. YIN f l = κ 0 z jflj (z,w); l κ 0 f j = z j, for κ 0 +1 j n 1; φ lk = μ lk z l z k + κ 0 z jφ lkj for (l, k) S 0, φ lk = O wt (3), (l, k) S 1, g = w; flj (z,w) =δj l + iδj l μ l w + b (1) lj (z)w + O wt(4), φ lkj (z,w) =O wt(), (l, k) S 0, φ lk = κ 0 z jφ lkj = O wt(3) for (l, k) S 1 (.1) Here, for 1 κ 0 n, wewrites = S 0 S 1, the index set { for all components of φ, where S 0 = {(j, l) :1 j κ 0, 1 l,j l}, S 1 = (j, l) :j = κ 0 +1,κ 0 +1 } l κ 0 + N n (n κ0 1)κ0,and μ jl = { μj + μ l for j < l κ 0 ; μj if j κ 0 <lorif j= l κ 0. (.) c. A family of affine hyperspaces L ɛ. Let us review some background materials on the semi-linearity properties on Rat(B n, B N ) (cf. [Hu03] and [HJX06]). Let F Rat(B n, B N )with1 κ 0 n. Let E 0 be the proper complex variety consisting of poles and the non-immerse points of F. We define V F := {(Z, S Z ) (C n E 0 ) Gr n,k 0(C)}, F is linear fractional when restricted to S Z + Z}. Here Gr n,k 0(C) is the Grassmannian manifold consisting of all k 0 := n κ 0 - dimensional complex subspaces in C n. Then V F is a complex analytic variety with the projection π : V F C n E 0, (Z, S Z ) Z (.3) is proper holomorphic. There is another proper complex variety E 1 C n E 0 such that for any Z C n E 0 E 1, π has a unique preimage in V F, i.e., for any Z C n E 0 E 1, there is a unique complex subspace S Z of dimension k 0 such that F is linear fractional when restricted to S Z + Z. In particular, if F satisfies the normalization condition as in Theorem., the restriction of F on S Z is affine linear. Write V F = j V (j) F for the irreducible decomposition of V F. Then there is only one irreducible component, say V (1) F, whose projection to Cn E 0 contains a sufficiently small domain inside H n and has a small piece of H n containing 0 as part of its boundary. If necessary, we can assume that 0 E 1 and thus π is biholomorphic near (0,S 0 ) V F. By [HJX06, p. 50], we can assume that for any ɛ =(ɛ 1,ɛ,..., ɛ κ0 )( C κ0 ) 0, there is a unique affine subspace L ɛ of codimension κ 0 defined by equations of the form: z j = i=κ 0+1 a ji (ɛ)z i + a jn (ɛ)w + ɛ j, 1 j κ 0. (.4) such that F is a linear map on L ɛ,wherea ji (ɛ) are holomorphic functions in ɛ near 0 with a ji (0,..., 0) = 0 for all j.

5 RATIONAL MAPS BETWEEN BALLS 497 d. Basic notation. Let F =(f,φ,g) Rat(B n, B N ) be as in Theorem. with geometric rank κ 0. We have N = (f) + (φ) + (g) and (φ) = (S 0 )+ (S 1 ) where we denote by (A) the number of elements of a set A, and (f) =, (g) =1, (S 0 )= ( κ0)κ0 = nκ 0 (κ0+1)κ0, (S 1 )=N n (S 0 ). We denote by P (j,k) (z,w) the polynomial of (z,w) with degree deg(z) = j and degree deg(w) = k and denote P (j,k) (z) the coefficient of w. For example, P (1,1) (z,w) = κ 0 a jz j w = P (1,1) (z)w, P (1,1) (z) = κ 0 a jz j. For any rational holomorphic map H = (P1,...,Pm) Q on C n,wherep j,q are holomorphic polynomials with (P 1,..., P m,q) = 1, the degree of H is defined to be deg(h) :=max{deg(p j ),deg(q), 1 j m}. The part φ. Write φ =(Φ 0, Φ 1 ), Φ 0 =(φ lk ) (l,k) S0 and Φ 1 =(φ lk ) (l,k) S1. Here (Φ 0 )= (S 0 )and (Φ 1 )= (S 1 ). When N = n + n κ0 1 κ 0 and the geometric rank being κ 0,then (Φ 0 )=() + +(n κ 0 )= n κ 0 1 κ 0, (Φ 1 )=N n (Φ 0 )=0 Namely, there is no Φ 1 term. The part f (1,1) j (z). Write f (1,1) (z) =(f (1,1) 1 (z),..., f κ (1,1) 0 Theorem., we have f (1,1) j (z) = iμj (.5) (z), 0,, 0). By z j, μ j > 0for 1 j κ 0. The part Φ (,0) 0 (z). One important portion of Φ 0 is the z-quadric part (see Theorem 3.): Φ (,0) 0 (z) ={φ () jl (z) =μ jlz j z l } (j,l) S0. The part Φ (1,1) 0 (z). Another portion of Φ 0 is Φ (1,1) 0 (z)w which are not mentioned in Theorem.: Φ (1,1) 0 (z) = κ 0 e j z j, e j C (S0). The part f (,1) (z). Write f (,1) (z) =(f (,1) 1 (z),..., f κ (,1) 0 (z), 0,, 0). We see from ([HJY, (3.5)]) that f (,1) j (z) = ξ j. (.6) Here ξ j =Φ (,0) e j = 1 k,l κ 0 φ (,0) kl e j,kl + 1 k κ 0<α φ (,0) kα e j,kα. (.7) e. The components of Φ (3,0) 1. Lemma.3. Let κ 0 and (κ 0 +1)n κ0(κ0+1) N (κ 0 +)n κ 0 (κ 0 + 1) + κ 0. Then ( Φ (3,0) 1 (z) = μj + μ l ( μ j μ l z j ξ l φ (3,0) (z) =4 ( j κ 0 1 μ j ξ j (z) ) z. ) μl ) z l ξ j, 0 μ j 1 j<l κ 0 (.8)

6 498 S. JI AND W. YIN The above result was proved in the third gap paper [HJY14], Corollary 3.4, under the condition (κ 0 +1)n κ 0 N (κ 0 +)n κ 0 (κ 0 +1)+κ 0. By checking the proof in [HJY14], we find that this is still valid when the condition is replaced by (κ 0 +1)n κ 0(κ 0 +1) N (κ 0 +)n κ 0 (κ 0 +1)+κ 0. (.9) Geometrically we notice that the lower bound in (.9) is the right end point of the gap interval I κ0 and the upper bound in (.9) is less than the left end point of the gap interval I κ0+1. When the condition in Theorem 1.1 or Theorem 1. is satisfied, the inequality N =(κ 0 +1)n κ0(κ0+1) (κ 0 +)n κ 0 (κ 0 +1)+κ 0 holds because of the condition κ0(κ0+1) <n. Lemma.3 can be applied to the maps in Theorem Properties of the semi-linear subspace. In this section, we will use the automorphisms of the balls to normalize the semi-linear subspace achieved in [Hu03]. More precisely, by (.4), we will prove the following: Proposition 3.1. Let F Prop 3 (H n, H N ) be as in Theorem., and the semilinear subspace L ɛ be given by z j = α=κ 0+1 a jα (ɛ)z α + a jn (ɛ)w + ɛ j, 1 j κ 0. (3.1) Then there are automorphisms σ Aut(H n ) and τ Aut(H N ) such that ˆF = τ F σ still takes the form (.1), and the semi-linear subspace still has the form (3.1). Moreover, we have a (I1) 1α =0for κ 0 +1 α n 1 and Re(a (I1) 1n ) = 0 (3.) where we denote a (1) jk (ɛ) =a(i1) jk ɛ a (Iκ 0 ) jk ɛ κ0. Proof. Consider the image ˆL ɛ := ˆσ c (L ɛ )givenby Z j = α=κ 0+1 where the inverse of the the automorphism is given by A jα (ɛ)z α + A jn (ɛ)w + ρ j (ɛ), (3.3) ˆσ 1 c (Z, W) := (Z 1,,Z κ0,z κ0+1 + c κ0+1w,..., Z + c W, W ) q c =(z 1,z,..., z,z n ) (3.4) and q c := 1 i c Z +(r i c )W,where c =(0,..., 0,c κ0+1,..., c ). Substituting (3.4) into (3.1), we obtain Z j = α=κ 0+1 a jα (ɛ)(z α + c α W )+a jn (ɛ)w + ɛ j q c,

7 Combining this with (3.3), we get RATIONAL MAPS BETWEEN BALLS 499 = α=κ 0+1 α=κ 0+1 A jα (ɛ)z α + A jn (ɛ)w + ρ j (ɛ) a jα (ɛ)(z α + c α W )+a jn (ɛ)w + ɛ j (1 i c Z +(r i c )W ). (3.5) By considering the coefficients of Z j, κ 0 +1 j n 1andW terms, we obtain A jα (ɛ) = a jα (ɛ) ɛ j (ic α ), A jn (ɛ) = a jα (ɛ)c α + a jn (ɛ) + ɛ j (r i c ). Thus we can choose c and r such that (3.) holds true. From [Hu03] Lemma., there is a corresponding τ Aut(H N ) such that ˆF = τ F σ still takes the form (.1). Next, we give some applications of the semi-linear subspace defined as above, which will be used later: Let H be an affine linear function along L ɛ.write = H H Lɛ ( α=κ 0 +1 a 1αz α + a 1nw + ɛ 1,, Then we must have H Lɛ z α w we infer H Lɛ z α w = κ0 H z α w + + κ 0 i<j,i, α=κ and H Lɛ w H z j w a jα + ) a κ0 αz α + a κ0 nw + ɛ κ0, z κ0 +1,..., z, w. κ 0 0forκ 0 +1 α n 1, from which H z j z α a jn + κ 0 H z i z j (a iα a jn + a jα a in )=0, at (ɛ, 0) H zj a jα a jn (3.6) and H Lɛ w = κ0 H w + + κ 0 i<j,i, H z j w a jn + κ 0 H zj a jn H z i z j a in a jn =0, at (ɛ, 0). (3.7) Choosing H = f h for 1 h κ 0 in (3.6), and collecting ɛ j,1 j κ 0 terms in the above equation, we get κ0 i μ ha (1) hα + f (Ij+Iα+In) h ɛ j =0, at (ɛ, 0). (3.8) This, together with h = 1 in (3.), gives f (I1+Iα+In) 1 (ɛ, 0) = 0. On the other hand, by (.6), we have f (,1) 1 (z) = ξ 1. Recall that ξ (I1+Iα) 1 = μ 1 e 1,1α in (.7). Thus we obtain f (I1+Iα+In) 1 = μ 1 e 1,1α. Hence e 1,1α =0forκ 0 +1 α n 1. (3.9)

8 500 S. JI AND W. YIN Setting H = f j for 1 j κ 0 or φ in (3.7), respectively, we obtain i μ ja (1) jn (ɛ)+f (1,) κ 0 j (ɛ, 0,, 0) = 0, φ (1,) (ɛ, 0,, 0) + e j a (1) jn (ɛ) =0. (3.10) 4. Some applications of the Chern-Moser equation. Let F =(f,φ,g) : H n H N with N = n + n κ0 1 κ 0 and geometric rank κ 0. Moreover, F satisfies the normalization as in Theorem.. We will derive some basic relations from the Chern-Moser equation, which is based on the calculations in Section 4 of [HJY14]. When N = n + n κ0 1 κ 0 and the geometric rank being κ 0, as shown in (.5), we have Φ 1 = 0. In particular, Φ (3,0) 1 = 0, thus (.8) gives μ j z j ξ l = μ l z l ξ j for 1 j, l κ 0. (4.1) Denote e i,jk := e i,kj and φ jk := φ kj when j>k. Then for any j with 1 j κ 0, ξ j =Φ (,0) e j = φ (,0) ik e ej,ik + φ (,0) ik e ej,ik = φ (,0) jk j<k = φ (,0) ji j<i = 1 i φ (,0) (i,k) S 0,i=j, or,k=j e j,jk + 1 i j e j,ji + ij e j,ij + 1 i j φ (,0) ij e j,ij + φ (,0) (i,k) S 0,i,k j ij e j,ij + φ (,0) ik e j,ik. (i,k) S 0,i,k j (i,k) S 0,i,k j (i,k) S 0,i,k j φ (,0) ik e j,ik φ (,0) ik e j,ik Observe that when j l, thetermsz l (i,k) S 0,i,k j φ(,0) ik e j,ik are not divided by z j. Thus (4.1) implies e j,ik =0for(i, k) S 0,i,k j. Now(4.1)isoftheform μ j z j 1 i φ (,0) il e l,il = μ l z l 1 i We can write the above identity as μ j z j φ (,0) li e l,li = μ l z l 1 i 1 i Setting l = 1 and making use of (3.9), we obtain μ j z j 1i e 1,1i = μ 1 z 1 1 i κ 0 φ (,0) 1 i φ (,0) ij e j,ij. (4.) φ (,0) ji e j,ji. φ (,0) ji e j,ji which implies e j,jα =0foranyκ 0 +1 α n 1. Combining this with (.6)-(.7), we know f (Ij+Iα+In) h = 0. Together with (3.8), we obtain a (1) hα =0, 1 h κ 0,κ 0 +1 α n 1. (4.3) The rest relations in (4.) are μ j z j φ (,0) il e l,il = μ l z l φ (,0) ij e j,ij for 1 i, j, l κ 0. Namely, we obtain μ j μ il e l,il = μ l μ ij e j,ij for 1 i, j, l κ 0. (4.4)

9 RATIONAL MAPS BETWEEN BALLS Proof of Theorem 1.1. This section is devoted to the proof of Theorem 1.1. The key point is to prove that the degree of the map in Theorem 1.1 is less than or equals to. Then we can apply Lebl s Theorem [L11] to complete the proof of our main theorem. Lemma 5.1. Keep the notations and assumptions in Theorem 1.1, then deg(f ). Proof. From the basic Chern-Moser equation, we have g(z,w) g(z,w) = f(z,w) f(z,w)+φ(z,w) φ(z,w), Im(w) = z. i By complexification, we write g(z,w) g(χ, η) i z j Applying L j := identity, we obtain L j g(0, 0) i = f l (z,w)f l (χ, η)+ φ t (z,w)φ t (χ, η), l=1 +iχ j w w η i = z χ. for z =0andw = η = 0 to the both sides of the above = L j f l (0, 0)f l (χ, 0) + L j φ t (0, 0)φ t (χ, 0) l=1 and L j L k g(0, 0) = L j L k f l (0, 0)f l (χ, 0) + L j L k φ t (0, 0)φ t (χ, 0). i l=1 In terms of matrix, they take the form χ 1 χ κ0 0 = B. 0 f 1 (χ, 0) f κ0 (χ, 0) φ(χ, 0) where B(F )isa κ0 (n κ 0 +1) κ0 (n κ 0 +1)matrix: B(F ):= L jf h L j φ hl L j φ hα L j L k f h L j L k φ hl L j L k φ hα L j L β f h L j L β φ hl L j L β φ hα (5.1) 1 j,h,k,l κ 0,κ 0+1 α,β (5.) (0,0,χ,0) Write A j = μ1je1,1j μ 1, and let F : C \{1 i κ 0 A jz j =0} C N 1 be defined as follows: f μ (z) =z μ for 1 μ n 1, μ jj zj φ jj (z) = 1 i κ 0 A jz j for 1 j κ 0, μ jk z j z k φ jk (z) = 1 i κ 0 A jz j for 1 j<k κ 0, μ jα z j z α φ jα (z) = 1 i κ 0 A jz j for 1 j κ 0 <α n 1. (5.3)

10 50 S. JI AND W. YIN We claim that F (z,0) = F, which follows from the following identity: χ 1. f 1 (χ) χ κ0 0 = B., (5.4) f κ0 (χ). φ(χ) 0 In fact, once (5.4) is achieved, we infer from (5.1) that f 1 (χ, 0) f 1 (χ) B. f κ0 (χ, 0) f =0. κ0 (χ) φ(χ, 0) φ(χ) Notice that Here B = diag ( 1,, 1,A 1,,A κ0,b 1,,B κ0 ) + O( χ ). A j =( μ j, μ j + μ j+1,, μ j + μ κ0 ) C κ0 j+1, B j =( μ j,, μ j ) C n κ0. Thus B is nonsingular and we derive the claim F (z,0) = F (z). Hence deg(f (z,0)). Replacing F by Fp for any p H n near the origin, we can show deg(fp (z,0)) in a similar manner. By [HJ01, Section 5], we have that deg(f ). The identity (5.4) follows from the following direct computations: Calculate (L h H)(0, 0) with 1 h κ 0 for H = f j,φ jk. At the point (0, 0), we have (L h f j )(0, 0) = δ j h, (L h φ jk )(0, 0) = 0 for (j, k) S 0. Then κ 0 (L h f j )(0, 0) f j (χ)+ (L h φ μν )(0) φ κ 0 μν (χ) = δ j h χ j = χ h. (5.5) (μ,ν) S 0 Calculate (L h H)(0, 0) (1 h κ 0) for H = f j,φ jk. A direct computation shows that L h = +4iχ zh h z h w +(iχ h) w.atthepoint(0, 0), we have (L h f j)(0, 0) = 4iχ h i μ j δ j h = δj h μ jχ j, (L hφ hh )(0, 0) = μ h +4iχ h e h,hh, (L hφ hj )(0, 0) = 4iχ h e h,hj for 1 j n 1, j h, (L hφ jk )(0, 0) = 0 for j, k h.

11 Setting l = 1 in (4.4), we obtain RATIONAL MAPS BETWEEN BALLS 503 μ hj e h,hj = μ h μ 1j e 1,1j = μ h A j. (5.6) μ 1 Thus we get κ 0 (L h f j)(0, 0) f j (χ)+ (L h φ μν)(0, 0) φ μν (χ) (μ,ν) S 0 =( μ h χ h ) χ h + ( ) μ hh χ μ h +4iχ h e h,hh h 1+i κ 0 A jχ j μ hj χ h χ j + 4iχ h e h,hj 1+i κ 0 A jχ j j h,1 j 1 ( = 1+i κ 0 κ 0 ) κ 0 A μ h χ h i A j χ j +4iχ h μ hj e h,hj χ j =0. jχ j (5.7) Calculate (L h L l H)(0, 0) (1 h<l κ 0 ) for H = f j,φ jk. A direct computation shows that L h L l = z h z l +iχ l z h w +iχ h z l w 4χ hχ l w. At the point (0, 0), we have (L h L l f j )(0, 0) = iχ l i μ jδ j h +iχ h i μ jδ j l = μ jχ l δ j h μ lχ h δ j l, (L h L l φ hl )(0, 0) = μ hl +iχ l e h,hl +iχ h e l,lh, (L h L l φ jk )(0, 0) = iχ l e h,jk +iχ h e l,jk for (j, k) (h, l) or(l, h). Combining this with (5.6), we get κ 0 (L h L l f j )(0, 0) f j (χ)+ (L h L l φ μν )(0, 0) φ μν (χ) (μ,ν) S 0 = μ h χ l χ h μ l χ h χ l + ( ) μ hl χ h χ l μ hl +iχ l e h,hl +iχ h e l,lh 1+i κ 0 A jχ j ( ) μ jk χ j χ k + iχl e h,jk +iχ h e l,jk 1+i κ 0 A jχ j = 1 j<k κ 0,(j,k) (h,l) 1 ( 1+i κ 0 A (μ h + μ l )χ h χ l i jχ j ) +iχ h e l,lj μ lj χ j χ l =0. 1 j κ 0 κ 0 A j χ j +iχ l 1 k κ 0 e h,hk μ hk χ h χ k (5.8) Calculate (L h L α H)(0, 0) (1 h κ 0 <α n 1) for H = f i,φ jl. A direct computation shows that L h L α = z h z α +iχ α z h w +iχ h z +iχ α w h iχ α w. At the point (0, 0), we have (L h L α f j )(0, 0) = iχ α i μ jδj h = μ hχ α δj h, (L h L α φ jk )(0, 0) = iχ α e h,jk, (L h L α φ jβ )(0, 0) = μ hα δ j h δβ α.

12 504 S. JI AND W. YIN We get κ 0 (L h L α f j )(0, 0) f j (χ)+ (L h L α φ μν )(0, 0) φ μν (χ) (μ,ν) S 0 = μ h χ α χ h + μ jk χ j χ k iχ α e h,jk 1+i μ hαχ h χ α κ 0 1 j k κ 0 A + μ hα jz j 1+i κ 0 A jz j 1 = 1+i ( κ 0 κ 0 ) κ 0 A μh χ h χ α i A j z j + iχ α e h,jh μ jh χ j χ h =0. jz j (5.9) By all of the above, (5.4) is proved. This also finishes the proof of Lemma 5.1. Now we are in a position to complete the proof of Theorem 1.1. Proof of Theorem 1.1. By Lemma 5.1, the degree of the map in Theorem 1.1 is at most. By Lebl s Theorem [L11, Theorem 1.5], it must have the following form: ( t 1 z 1, t z,..., t n z n, 1 t 1 z 1, 1 t z,..., 1 t n z n, t i t j z i z j ) i j (5.10) where 0 t 1... t n 1, (t 1,t,..., t n ) (1, 1,..., 1). Suppose that t j =0for1 j h, t j =1fork +1 j n and t j (0, 1) for h j k. Here0 h k n and h = 0 means that there is no t j =0. Thenithas the following form: ( th+1 z h+1,, t nz n, 1 t 1z 1,, 1 t k z k, ( t i t jz iz j) 1 i k,1 j n,i<j ) (5.11) Notice that this map is linear on z j = c j for 1 j k and can not be linear on any lower dimensional linear subspace. By [Hu03], the geometric rank of this map is k. Thus we have k = κ 0. By counting the dimension of the map, we have (n h)+k + n(n 1) (n k)(n k 1) = n + n κ 0 1 κ 0. Then we get h = k = κ 0. In this case, (5.11) is exactly the generalized Whitney map defined by (1.3)-(1.4). This completes the proof of Theorem 1.1. REFERENCES [A77] H. Alexander, Proper holomorphic maps in C n, Indiana Univ. Math. Journal, 6 (1997), pp [AHJY16] J. Andrews, X. Huang, S. Ji, and W. Yin, Mapping B n into B 3n 3, Communications in Analysis and Geometry, 4: (016), pp [E16] P. Ebenfelt, On the HJY Gap Conjecture in CR geometry vs. the SOS Conjecture for polynomials, to appear in: Analysis and Geometry in Several Complex Variables. Contemp. Math., Amer. Math. Soc., Providence, RI. [Fa86] J. Faran, The linearity of proper holomorphic maps between balls in the low codimension case, J. Diff. Geom., 4 (1986), pp [FHJZ10] J. Faran, X. Huang, S. Ji, and Y. Zhang, Rational and polynomial maps between balls, Pure and Applied Mathematics Quarterly, 6:3 (010), pp [Fo9] F. Forstneric, A survey on proper holomorphic mappings, Proceeding of Year in SCVs at Mittag-Leffler Institute, Math. Notes 38 (199), Princeton University press, Princeton, N.J. [GK15] D. Grundmeier and J. H. Kacmarcik, An application of Macaulay s estimate to sums of squares problems in several complex variables, Proc. Amer. Math. Soc., 143:4 (014), pp

13 RATIONAL MAPS BETWEEN BALLS 505 [Ha05] H. Hamada, Rational proper holomorphic maps from B n into B n, Math. Ann., 331:3 (005), pp [Hu99] X. Huang, On a linearity of proper holomorphic maps between balls in the complex spaces of different dimensions, J. Diff. Geom, 51 (1999), pp [Hu03] X. Huang, On a semi-rigidity property for holomorphic maps, Asian J. Math, 7 (003), pp [HJ01] X. Huang and S. Ji, Mapping B n into B, Invent. Math., 145 (001), pp [HJX06] X. Huang, S. Ji, and D. Xu, A new gap phenomenon for proper holomorphic mappings from B n to B N, Math Research Letter, 3:4 (006), pp [HJY09] X. Huang, S. Ji, and W. Yin, Recent Progress on Two Problems in Several Complex Variables, Proceedings of International Congress of Chinese Mathematicians 007, Vol. I, pp , International Press, 009. [HJY14] X. Huang, S. Ji, and W. Yin, On the Third Gap for Proper Holomorphic Maps between Balls, Math. Ann., 358 (014), no. 1-, pp [JX04] S. Ji and D. Xu, Maps between B n and B N with geometric rank k 0 n and [L11] minimum N, Asian J. Math, 8 (004), pp J. Lebl, Normal forms, hermitian operators, and CR maps of spheres and hyperquadrics, Michigan Math. J., 60:3 (011), pp [W79] S. Webster, On mapping an (n+1)-ball in the complex space, Pac. J. Math., 81 (1979), pp

14 506 S. JI AND W. YIN