j. differential geometry 51 (1999) 13-33

ON A LINEARITY PROBLEM FOR PROPER HOLOMORPHIC MAPS BETWEEN BALLS IN

COMPLEX SPACES OF DIFFERENT DIMENSIONS

XIAOJUN HUANG

1. Introduction

In an important development of several complex variables, Poincaré [26] discovered that any biholomorphic map between two open pieces of the unit sphere in C^{2} is the restriction of a certain automorphism of B_{2}, the unit two-ball in C^{2}. This phenomenon fails obviously in one complex variable and reveals a strong rigidity property of holomorphic mappings in several variables. Later, Tanaka, etc (see [8], [28]) extended this result to any dimensional case. Alexander, in his famous papers [1], [2], further proved that any proper holomorphic selfmapping of the ball in C^{n} (n > 1) is an automorphism, thus finishing off a line of research towards the understanding of proper holomorphic mappings between balls in the same complex space. In 1978, using the Cartan-Chern-Moser [8] theory, Webster [31] took up again the problem of considering a proper holomorphic mapping f from the unit n-ball B_{n} = {z ∈ C^{n} : |z| < 1} into the unit (n + 1)−ball B_{n+1} ⊂ C^{n+1} and showed that f is a totally geodesic embedding when f is C^{3}-smooth up to the boundary and when n > 2. Here, we recall that a proper holomorphic map from B_{n} into B_{N} is called a totally geodesic embedding (or a linear embedding) if there exist automorphisms σ ∈ Aut(B_{n}) and τ ∈ Aut(B_{N}) such that τ ∘ f ∘ σ = (id, 0). In a subsequent paper, Faran [14] classified mappings from B_{2} into B_{3}, that are three times continuously differentiable up to the boundary. In another work [9], Cima-Suffridge studied certain reflection principle for CR mappings between hypersurfaces with codimension one and established the results of Webster and Faran for mappings which are only twice continuously differentiable up to the boundary. In the same paper, they conjectured that any proper holomorphic mapping from B_{n} into B_{N} (n > 1), which is C^{2}-smooth up to the boundary, must be a totally geodesic embedding when N < 2n − 1. Notice that there are many proper polynomial mappings from B_{n} into B_{2n−1} which are not totally geodesic (see [7], in particular, the book by D’Angelo [13], where many related classifications are presented). Indeed, the well-known (non-linear) Whitney map [10] f = (z_{1}^{2}, z_{1}z_{2}, ⋯, z_{1}z_{n}, z_{2}, ⋯, z_{n}) properly sends B_{n} into B_{2n−1}. In [15], Faran verified this conjecture under the assumption that the map extends holomorphically up to the boundary.

Received December 18, 1997, and, in revised form, April 23, 1998. The author was supported in part by NSF DMS-9500881 and an NSF postdoctoral fellowship.

Later, it was shown in the deep work of Forstneric and Cima-Suffridge [17], [18], [10] (see in particular, the recent work in [18]), that any proper holomorphic mapping from B_{n} into B_{N}, that is C^{N−n+1}-regular up to the boundary, extends holomorphically and rationally across the closure of the ball for any N ≥ n ≥ 2. This together with the work of Faran gives a solution to Cima-Suffridge’s problem for mappings which are (N −n)+1 ((codimension)+1) smooth up to the boundary. Meanwhile, in a different direction, the discovery of inner functions over the ball reveals that there are many proper holomorphic mappings from B_{n} into B_{n+1} which are continuous up to \overline{B}_{n} but not linear (see [20], [22], [24], and [27]. In particular, see [19] for a survey on the related topics). This opens up a very interesting but also difficult subject to dig out the minimal boundary regularity for mappings between balls with which the linearity and the reflection principle hold. In particular, it has been an open question for years to obtain results in which the required regularity is independent of the codimension ([9], [6], [19]).

In this paper, we will focus on the linearity portion of the above mentioned problem. We will prove the following theorem, which was first conjectured to be true by Cima-Suffridge in 1983 [9] (see also the question asked in [6] and [18]):

Theorem A. Let M_{1} and M_{2} be two connected open pieces of the boundaries of B_{n} ⊂ C^{n} and B_{N} ⊂ C^{N}, respectively. Let f be a nonconstant twice continuously differentiable CR mapping from M_{1} into M_{2}. Suppose that n > 1, N < 2n − 1. Then f is the restriction of a certain totally geodesic embedding from B_{n} into B_{N}. More precisely, there exist an automorphism σ ∈ Aut(B_{n}) and an automorphism τ ∈ Aut(B_{N}) such that τ ∘ f ∘ σ(z_{1}, ⋯, z_{n}) ≡ (z_{1}, ⋯, z_{n}, 0, ⋯, 0). In particular, f is real analytic over M_{1}.

Corollary B. Let f be a proper holomorphic mapping from B_{n} into B_{N}, that is twice continuously differentiable up to the boundary. Suppose that n > 1, N < 2n − 1. Then there exist σ ∈ Aut(B_{n}) and τ ∈ Aut(B_{N}) such that τ ∘ f ∘ σ(z_{1}, ⋯, z_{n}) ≡ (z_{1}, ⋯, z_{n}, 0, ⋯, 0).

Our approach to the proof of Theorem A is different from the existing ones for the study of mappings between real analytic hypersurfaces in different complex spaces ([31], [14]–[16], [9]–[10], [17]–[19], [21], [5], etc). Namely, we do not use the differential-geometric part of the Cartan-Chern-Moser theory, and we do not take too many times differentiation by CR vector fields along the sphere, neither. This enables us to deal with maps with only C^{2}-smooth regularity. Indeed, our consideration uses only the lower order Moser formal theory and the large automorphism groups on the balls. This method seems also to be very useful for studying other related problems for mappings between balls (such as rationality and degree estimate problems, etc).

Another implication of our theorem is the analyticity result of f. Regularity of CR mappings between hypersurfaces in the same complex space has attracted considerable attentions in recent years and many results have been obtained (see [3] and the survey papers [6], [19]). However, the understanding of regularity of CR mappings between hypersurfaces in different complex spaces is still rather poor (see [16], [11], [18], [21], [5]). We mention that in all previous work, it was always essential to assume that the map under study is at least (codimension + 1)- times continuously differentiable to boost-up higher regularity. It seems that the theorem of this paper also gives the first reflection principle for maps whose initial regularity is independent of the codimension.

Finally, we would like to mention a different but closely related interesting problem ([12], [32]), which stems from problems establishing the Mostow super-rigidity theorem to complex hyperbolic space forms (see [12] for a solution in the small codimensional case and [32] for certain related applications to the study of dynamics): Let f be a proper holomorphic embedding from B_{n} into B_{N} such that f(B_{n}) is stabilized by a certain discrete subgroup Γ of Aut(B_{N}) (N > n > 1) with f(B_{n})/Γ compact. Is f then a totally geodesic embedding?

Acknowledgement. The author thanks J. D’Angelo, Shanyu Ji, S. Krantz, Song-Ying Li, Steve S. T. Yau and Chengbo Yue for their interest to the present work. He appreciates greatly S. Baouendi and L. Rothschild for their very careful reading of the paper and many helpful comments. Also, the author would like to thank S. Webster, from whom he first learnt the beauty of the Moser formal theory.

2. Basic set-ups, a normalization property and a Preliminary Lemma

We let M_{1} ⊂ ∂B_{n} and M_{2} ⊂ ∂B_{N} be two connected open pieces of the unit spheres in C^{n} and C^{N}, respectively. After a linear fractional transformation, we can assume that M_{1}, M_{2} contain the origin and are defined, respectively, by the following equations:

(2.1) Imw = ∑_{j=1}^{n−1} |z_{j}|^{2}, (z, w) ∈ M_{1}; Imw^{∗} = ∑_{j=1}^{N−1} |z_{j}^{∗}|^{2}, (z^{∗}, w^{∗}) ∈ M_{2}.

Write L_{j} = 2i\overline{z_{j}} \frac{∂}{∂w} + \frac{∂}{∂z_{j}} for j = 1, ⋯, n − 1. Then \{L_{1}, ⋯, L_{n−1}\} forms a global basis for the complex tangent bundle T^{(1,0)}M_{1} of M_{1}. Let

F = (f, ϕ, g) = (f_{1}, ⋯, f_{n−1}, ϕ_{1}, ⋯, ϕ_{N−n}, g)

be a non-constant C^{2}-smooth CR map from M_{1} into M_{2} with F (0) = 0. We recall that the CR assumption of F indicates that \overline{L_{j}}F ≡ 0 for each j. By the Lewy extension theorem, it is well-known that F extends holomorphically to a certain pseudoconvex side of M_{1} denoted by Ω. Here, we can also assume that Ω is filled in by holomorphic disks attached to M_{1}. Now, applying the maximum principle and the Hopf lemma to the subharmonic function: −Img + ∑_{j=1}^{n−1} |f_{j}|^{2} + ∑_{j=1}^{N−n} |ϕ_{j}|^{2} over Ω, we conclude that F (Ω) ⊂ H_{N} = {(z^{∗}, w^{∗}) : Im(w^{∗}) > |z^{∗}|^{2}} and \frac{∂(Img)}{∂(Imw)}|_{0} = λ > 0. Since \frac{∂(Img)}{∂(Rew)}|_{0} = 0, \frac{∂g}{∂w}|_{0} = λ.

Next, we notice that the assumption F (M_{1}) ⊂ M_{2} gives the following non-linear functional equation with respect to F:

(2.2)  \frac{g − \overline{g}}{2i} = ∑_{j=1}^{n−1} |f_{j}|^{2} + ∑_{j=1}^{N−n} |ϕ_{j}|^{2}, (z, w) ∈ M_{1}.

Applying L_{l}, L_{k}L_{l} and \overline{L_{k}}L_{l} to (2.2), we have

\frac{L_{l}g}{2i} = ∑_{j} L_{l}f_{j}\overline{f_{j}} + ∑_{j} L_{l}ϕ_{j}\overline{ϕ_{j}},

\begin{aligned}\frac{L_{k}L_{l}g}{2i} = & ∑_{j} L_{k}(L_{l}f_{j})\overline{f_{j}} + ∑_{j} L_{k}(L_{l}ϕ_{j})\overline{ϕ_{j}}, \\ \frac{\overline{L_{k}}L_{lg}}{2i} = & ∑_{j} (\overline{L_{k}}L_{l}f_{j} · \overline{f_{j}} + L_{l}f_{j} · \overline{L_{k}f_{j}}) \\ & + ∑_{j} (\overline{L_{k}}L_{l}ϕ_{j} · ϕ_{j} + L_{l}ϕ_{j} · \overline{L_{k}ϕ_{j}}) . \end{aligned} (2.2^{∗})

Letting (z, w) = (0, 0) in the above first two equations, we obtain

\frac{∂g}{∂z_{l}} |_{0} = \frac{∂^{2}g}{∂z_{k}∂z_{l}} |_{0} = 0.

On the other hand, write f̃ = (f, ϕ),

E_{l} = (\frac{∂f̃}{∂z_{l}})|_{0} = (\frac{∂f_{1}}{∂z_{l}} , ⋯ \frac{∂f_{n−1}}{∂z_{l}} , \frac{∂ϕ_{1}}{∂z_{l}} , ⋯ \frac{∂ϕ_{N−n}}{∂z_{l}})|_{0},

and

E_{w} = (\frac{∂f̃}{∂w})|_{0} = (\frac{∂f_{1}}{∂w} , ⋯ \frac{∂f_{n−1}}{∂w} , \frac{∂ϕ_{1}}{∂w} , ⋯ \frac{∂ϕ_{N−n}}{∂w})|_{0}.

Notice that \frac{\overline{L_{k}}L_{l}g}{2i} |_{0} = δ_{l}^{k} \frac{∂g}{∂w}|_{0} = δ_{l}^{k}λ with δ_{k}^{l} = \begin{cases}0, & if k ≠ l \\ 1, & if k = l \end{cases}. Use of

(2.2^{∗}) leads to that E_{l}\overline{E_{k}}^{t} = λδ_{k}^{l}. Extend \{E_{1}/\sqrt{λ}, ⋯, E_{n−1}/\sqrt{λ}\} to a certain orthonormal basis \{E_{1}/\sqrt{λ}, ⋯, E_{n−1}/\sqrt{λ}, C_{1}, ⋯, C_{N−n}\} of C^{N−1}. Denote by A the (N − 1) × (N − 1) (constant unitary) matrix

A := (\begin{matrix}E_{1}/\sqrt{λ} \\ · \\ · \\ · \\ E_{n−1}/\sqrt{λ} \\ C_{1} \\ · \\ · \\ C_{N−n} \end{matrix}) .

Let

\begin{matrix}F^{∗} = (f̃^{∗}, g^{∗}) := & (f_{1}^{∗}, ⋯, f_{n−1}^{∗}, ϕ_{1}^{∗}, ⋯, ϕ_{N−n}^{∗}, g^{∗}) \\ = & \frac{1}{\sqrt{λ}}F · (\begin{matrix}\overline{A^{t}} & 0 \\ 0 & 1/\sqrt{λ} \end{matrix}) . \end{matrix} (2.3)

Then F^{∗}(M_{1}) ⊂ ∂H_{N} and

F^{∗}′(0) = (\begin{matrix}(id)_{(n−1)×(n−1)} & (0)_{(n−1)×(N−n)} & 0 \\ ∗ & ∗ & 1 \end{matrix}) .

Notice that g = λg^{∗}. One can thus write

f_{j}^{∗} = z_{j} + a_{j}w + ∑_{k,l=1}^{n−1} e_{kl}^{j}z_{k}z_{l} + O(|w|^{2}) + o(|(z, w)|^{2}),

(2.3′)  ϕ_{j}^{∗} = b_{j}w + ∑_{k,l=1}^{n−1} q_{kl}^{j}z_{k}z_{l} + O(|w^{2}| + |zw|) + o(|(z, w)|^{2}),
g^{∗} = w + dw^{2} + O(|zw|) + o(|(z, w)|^{2}),

where a_{j}, b_{j}, e_{kl}^{j}, q_{kl}^{j} and d are certain constants with e_{kl}^{j} = e_{lk}^{j}, q_{kl}^{j} = q_{lk}^{j}. Write

a = (a_{1}, ⋯, a_{n−1}, b_{1}, ⋯, b_{N−n}), r = \frac{1}{2}Re\{\frac{∂^{2}g^{∗}}{∂w^{2}} |_{0}\}, (2.3′′)

and define G ∈ Aut_{0}(H_{N}) ([29, 1.2.1, p.164]) by

\begin{aligned}G = & (\frac{z^{∗} − aw^{∗}}{1 + 2i < z^{∗}, \overline{a} > −(−r + i < a, \overline{a} >)w^{∗}} , \\ & \frac{w^{∗}}{1 + 2i < z^{∗}, \overline{a} > −(−r + i < a, \overline{a} >)w^{∗}} ) , \end{aligned} (2.4)

where we have used the standard notation < α, β >= ∑_{j=1}^{k} α_{j}β_{j} for any vectors α, β ∈ C^{k}. Thus by a direct computation, we see that F^{∗∗} = (f̃^{∗∗}, g^{∗∗}) := G ∘ F^{∗} still maps M_{1} into ∂H_{N}. Moreover, F^{∗∗} satisfies the following normalization condition (compare (2.5) with [8, (2.9), p. 231]):

\begin{matrix}F^{∗∗}, \frac{∂(f^{∗∗} − z)}{∂z_{l}}, \frac{∂f^{∗∗}}{∂w} , \frac{∂ϕ^{∗∗}}{∂z_{l}} , \frac{∂ϕ^{∗∗}}{∂w} , \\ \frac{∂g^{∗∗}}{∂z_{l}} , \frac{∂(g^{∗∗} − w)}{∂w}, \frac{∂^{2}g^{∗∗}}{∂z_{l}∂z_{k}} , Re\frac{∂^{2}g^{∗∗}}{∂w^{2}} = 0, at (z, w) = 0. \end{matrix} (2.5)

Notice

\begin{matrix}a_{l} = \frac{1}{λ}E_{w} · \overline{E_{l}}^{t}, b_{l} = \frac{1}{\sqrt{λ}}E_{w} · \overline{C_{l}}^{t}, \\ e_{kl}^{j} = \frac{1}{2λ}f̃′′_{z_{k}z_{l}}(0) · \overline{E_{j}}^{t}, q_{kl}^{j} = \frac{1}{2\sqrt{λ}}f̃′′_{z_{k}z_{l}}(0) · \overline{C_{j}}^{t}. \end{matrix} (2.6)

We have the following, which will be used for our proof of Theorem A.

Lemma 2.1. Assume the notation which we have set up so far. Suppose that

f^{∗∗} = z + O(|w|^{2} + |zw|) + o(|(z, w)|^{2})

and

ϕ^{∗∗} = O(|w|^{2} + |zw|) + o(|(z, w)|^{2}).

Namely, suppose that \frac{∂^{2}f^{∗∗}}{∂z_{k}z_{j}} |_{0} = \frac{∂^{2}ϕ^{∗∗}}{∂z_{k}z_{j}} |_{0} = 0 for any k, j. Then one has

the following feedback for f̃:

f̃′′_{z_{k}z_{l}}(0) = \frac{2\sqrt{−1}}{λ}(\overline{E_{w}} · E_{l}^{t})E_{k} + \frac{2\sqrt{−1}}{λ}(\overline{E_{w}} · E_{k}^{t})E_{l}.

Proof of Lemma 2.1. Notice that

ϕ_{j}^{∗∗} = \frac{ϕ_{j}^{∗} − b_{j}g^{∗}}{1 + 2i < f̃^{∗}, \overline{a} > −(−r + i < a, \overline{a} >)g^{∗}} .

Comparing the coefficients of terms of the form z_{l}z_{k} in the Taylor expansion of the above expression and using the given hypothesis, we get easily that q_{kl}^{j} = 0, for any j, k, l. Since q_{kl}^{j} = \frac{1}{2\sqrt{λ}} f̃′′_{z_{l}z_{k}}(0) · \overline{C_{j}}^{t}, it

follows that the (N − 1)-tuple f̃′′_{z_{l}z_{k}}(0) stays in the space spanned by {E_{1}, ⋯, E_{n−1}}, so that there exist constants {λ_{kl}^{j}} such that

(2.7)  f̃′′_{z_{k}z_{l}}(0) = ∑_{j=1}^{n−1} λ_{kl}^{j}E_{j}.

Next, considering the Taylor expansion of f_{j}^{∗∗} in the following

f_{j}^{∗∗} = \frac{f_{j}^{∗} − a_{j}g^{∗}}{1 + 2i < f̃^{∗}, \overline{a} > −(−r + i < a, \overline{a} >)g^{∗}} (2.7′)

and using the hypothesis, we obtain ∑ e_{kl}^{j}z_{k}z_{l}−2i ∑_{l} \overline{a_{l}}z_{j}z_{l} ≡ 0. Hence, it follows that e_{kl}^{j} = \sqrt{−1}(δ_{k}^{j}\overline{a_{l}} + δ_{l}^{j}\overline{a_{k}}), so that, by (2.6),

\frac{1}{2λ}f̃′′_{z_{k}z_{l}}(0)\overline{E_{j}}^{t} = \frac{\sqrt{−1}δ_{k}^{j}}{λ} \overline{E_{w}} · E_{l}^{t} + \frac{\sqrt{−1}δ_{l}^{j}}{λ} \overline{E_{w}} · E_{k}^{t}.

Combing this with (2.7) and making use of the orthogonality: E_{l} ·\overline{E_{j}}^{t} = λδ_{l}^{j} gives

λ_{kl}^{j} = \frac{2\sqrt{−1}}{λ}(δ_{k}^{j}\overline{E_{w}} · E_{l}^{t} + δ_{l}^{j}\overline{E_{w}} · E_{k}^{t}).

Returning to (2,7) completes the proof of Lemma 2.1. q.e.d.

3. Chern-Moser Lie-derivate and a semi-linear equation

In this and the following sections, we will present the proof of Theorem A. After the discussion in §2, the C^{2}-smooth CR map F = (f, ϕ, g) sending M_{1} into M_{2} can, now and in what follows, be assumed to take the following form:

\begin{matrix}f = z + f̂, g = w + ĝ with f̂, ĝ, ϕ = O(|(z, w)|^{2}), \\ \frac{∂^{2}ĝ}{∂z_{l}∂z_{k}} |_{0}, Re\frac{∂^{2}ĝ}{∂w^{2}} |_{0} = 0. \end{matrix} (3.1)

Our idea for the proof of Theorem A can be briefly described as follows: We first use the Moser formal theory to verify the hypothesis in Lemma 2.1. Then, we compose f with the automorphisms of the Heisenberg hypersurfaces to produce a continuous family of mappings, from which we will derive a differential equation (See (4.1′) in §4). This equation will be good enough to tell that F = (z, 0, w). In this section, we will extend the lower order case of a fundamental lemma of Chern-Moser to the case applicable in our setting.

To proceed, we first recall some notation and definitions used in [8] and [25], and assign the weights 1 and 2 to z and w, respectively. Then a (possibly real) polynomial h(z, w, \overline{z}, \overline{w}) is a weighted homogeneous polynomial of degree s if for each t > 0, h(tz, t^{2}w, t\overline{z}, t^{2}\overline{w}) = t^{s}h(z, w, \overline{z}, \overline{w}). We say that a function h defined over M_{1} is in the class P + o_{wt}(s) if there is a polynomial h_{1} and a function h_{2} such that h = h_{1} + h_{2} with lim_{t→0^{+}} \frac{h_{2}(tz,t^{2}w,t\overline{z},t^{2}\overline{w})}{t^{s}} → 0 uniformly with respect to (z, w) ≈ (0′, 0). When the above h_{1} ≡ 0, we say h ∈ o_{wt}(s).

Now, substituting (3.1) into (2.2), we obtain

(3.2)  Im(w + ĝ) = ∑_{j=1}^{n−1} |z_{j} + f̂_{j}|^{2} + ∑_{j=1}^{N−n} |ϕ_{j}|^{2}, (z, w) ∈ M_{1}.

Suppose that we have the decompositions: f̂_{j} = ∑_{s=2}^{m−1} f̂_{j}^{(s)}+o_{wt}(m−1), ĝ = ∑_{s=2}^{m} ĝ^{(s)} + o_{wt}(m), and ϕ_{j} = ∑_{s=l}^{m−l} ϕ_{j}^{(s)} + o_{wt}(m − l) (with l ≥ 2).

Here, as in [25], we use the notation (·)^{(σ)} to denote a certain weighted holomorphic homogeneous polynomial of weighted degree σ. Then, after collecting terms in (3.2) of weighted degree k, we have the following k-th semi-linearization of (3.2):

\begin{aligned}Im & (ĝ^{(k)}(z, w) − 2i < \overline{z}, f̂^{(k−1)}(z, w) >) \\ & = ∑_{j=1}^{N−n} ∑_{p=l}^{k−l} ϕ_{j}^{(k−p)}\overline{ϕ_{j}}^{(p)} + N^{(k)}, (z, w) ∈ M_{1}, \end{aligned} (3.3)

where m ≥ k ≥ l, and N^{(k)} is a weighted homogeneous polynomial of weighted degree k, contributed by f̂^{(σ−1)} with σ ≤ k − 1. Moreover, N^{(k)} ≡ 0 if f̂^{(σ−1)} ≡ 0 for σ ≤ k − 1. Therefore, when ĝ^{(σ_{1})}, f̂^{(σ_{1}−1)}, ϕ^{(σ_{2})} ≡ 0 for σ_{1} < 2k and σ_{2} < k (2k ≤ m), the 2k-th linearization of (3.2) can be simplified as

\begin{aligned}Im & (ĝ^{(2k)}(z, w) − 2i < \overline{z}, f̂^{(2k−1)}(z, w) >) \\ & = ∑_{j=1}^{N−n} ϕ_{j}^{(k)}\overline{ϕ_{j}}^{(k)}, (z, w) ∈ M_{1}. \end{aligned} (3.3′)

Similarly, when ĝ^{(σ_{1})}, f̂^{(σ_{1}−1)}, ϕ^{(σ_{2})} ≡ 0 for σ_{1} ≤ 2k and σ_{2} ≤ k (2k+1 ≤ m), the (2k + 1)-th linearization of (3.2) reduces to

\begin{matrix}Im (ĝ^{(2k+1)}(z, w) − 2i < \overline{z}, f̂^{(2k)}(z, w) >) = 0, \\ (z, w) ∈ M_{1}. \end{matrix} (3.3′)

Write L(f̂^{(σ−1)}, ĝ^{(σ)}) = Im(ĝ^{(σ)} − 2i < \overline{z}, f̂^{(σ−1)} >)|_{M_{1}}, which is the Lie derivative of ρ_{1} = Imw − ∑_{j=1}^{n−1} |z_{j}|^{2} along the vector field

X = 2Re(ĝ^{(σ)} \frac{∂}{∂w} + ∑ f̂_{j}^{(σ−1)} \frac{∂}{∂z_{j}}) restricted to M_{1}. The operator L is the basic tool for the construction of the local normal form of strongly pseudoconvex hypersurfaces, and for many other studies (see, for instance, the work [8], [29], etc). In particular, we mention the following lemma of Chern-Moser, which partially motivated our present study:

Lemma 3.0 ([8, Lemma 2.1, p.233]). Consider the linear equation L(f, g) = Im(g − 2i < \overline{z}, f >) = 0 with f, g holomorphic near the origin. Then it has a unique solution (f, g) ≡ (0, 0) if the following normalization condition holds:

f|_{0}, g|_{0}, \frac{∂f}{∂w} |_{0}, \frac{∂f}{∂z_{l}} |_{0}, \frac{∂g}{∂z_{l}} |_{0}, \frac{∂g}{∂w} |_{0}, \frac{∂^{2}g}{∂z_{l}z_{k}|_{0}} , Re(\frac{∂^{2}g}{∂w^{2}} )|_{0} = 0.

For the proof of Theorem A, the following lemma involving L is of fundamental importance:

Proposition 3.1. Consider the following semi-linear equation of weighted degree 4 with respect to the holomorphic polynomials P^{(4)}, Q^{(3)} = (Q_{1}^{(3)}, ⋯, Q_{n−1}^{(3)}), and Φ_{j}^{(2)} (j = 1, ⋯, k) of weighted degree 4, 3, and 2, respectively:

(3.4)  L(Q^{(3)}, P^{(4)}) = ∑_{j=1}^{k} |Φ_{j}^{(2)}(z, w)|^{2}, (z, w) ∈ M_{1}.

Assume the following normalization condition holds:

\frac{∂Φ_{j}^{(2)}}{∂w} = Re\frac{∂^{2}P^{(4)}}{∂w^{2}} = 0. (3.4′)

Then P^{(4)} ≡ 0, Q^{(3)} = a^{(1)}(z)w with

−2i < a^{(1)}(z), \overline{z} > |z|^{2} = ∑_{j=1}^{k} |Φ_{j}^{(2)}(z)|^{2},

where a^{(1)}(z) is a vector-valued holomorphic polynomial of degree one. Moreover, if k ≤ n−2, then (3.4) has only trivial solution Q_{j}^{(3)} = P^{(4)} =

Φ_{j}^{(2)} ≡ 0.

We should mention that when all Φ_{j}^{(2)} are assumed to be 0, then Proposition 3.1 reduces to 4^{th}-order case of Lemma 3.0. Also, we point out that when k ≥ n − 1, The last statement in Proposition 3.1 fails. Indeed, this is the only place in the course of the proof of Theorem A, where the codimension restriction is used. The geometric interpretation of Proposition 3.1 is that the expression ∑_{j=1}^{k} |Φ_{j}^{(2)}|^{2} stays in the Moser normal space N^{4} ([8]) when k ≤ n − 2. We remark that the last statement in Proposition 3.1 may hold also in any higher degree case. However, we will not pursue this because the present version of Proposition 3.1 is good enough for our purpose later.

Proof of Proposition 3.1. By (3.4′), we can write

Q^{(3)}(z, w) = a^{(1)}(z)w + b^{(3)}(z),

P^{(4)}(z, w) = A^{(4)}(z) + B^{(2)}(z)w + D_{0}w^{2}

with Re(D_{0}) = 0, and Φ_{j}^{(2)}(z, w) = h_{j}^{(2)}(z). Here a^{(1)} and b^{(3)} are vectorvalued holomorphic polynomials of degrees 1 and 3, respectively. Then (3.4) reads as

\begin{aligned}Im & (A^{(4)} + B^{(2)}w + D_{0}w^{2} − 2i < \overline{z}, a^{(1)} > w − 2i < \overline{z}, b^{(3)} >) \\ & = ∑_{j=1}^{k} |h_{j}^{(2)}(z)|^{2}, (z, w) ∈ M_{1}, \end{aligned} (3.5′)

where w = u + i|z|^{2} with u = Re(w). Noticing that {z, u} are independent variables, (3.5′) can be split into the following equations:

Im (A^{(4)}(z) + iB^{(2)}(z)|z|^{2} − D_{0}|z|^{4} + 2 < \overline{z}, a^{(1)}(z)|z|^{2} − ib^{(3)}(z) >)

(3.5′′)  = ∑_{j=1}^{k} |h_{j}^{(2)}(z)|^{2},

\begin{matrix}Im (B^{(2)}(z) + 2iD_{0}|z|^{2} − 2i < \overline{z}, a^{(1)}(z) >) u = 0, \\ Im(D_{0}u^{2}) = 0. \end{matrix} (3.5′′′)

Now, (3.5′′′) together with the normalization condition gives that D_{0} = 0. Moreover, noting that u is real and collecting terms of the form z_{l}z_{k}, we can also conclude from (3.5′′′) that B^{(2)} ≡ 0. Hence

Im(i < \overline{z}, a^{(1)}(z) >) ≡ 0.

Returning to (3.5′′), we similarly obtain that A^{(4)} ≡ 0 and b^{(3)} ≡ 0. Therefore, P^{(4)} ≡ 0 and Q^{(3)} = a^{(1)}(z)w. Furthermore, combining these with (3.5′′), we get

−2i < \overline{z}, a^{(1)}(z) > |z|^{2} = ∑_{j=1}^{k} |h_{j}^{(2)}(z)|^{2}.

To complete the proof of the second part of Proposition 3.1, it now suffices for us to apply the following Lemma 3.2 to the equation −2i < \overline{z}, a^{(1)}(z) > |z|^{2} = ∑_{j=1}^{k} |h_{j}^{(2)}(z)|^{2}:

Lemma 3.2. Let {ψ_{j}}_{j=1}^{k} and {χ_{j}}_{j=1}^{k} be holomorphic functions in z ∈ C^{n} near the origin. Assume that ψ_{j}(0) = χ_{j}(0) = 0. Let H(z, \overline{z}) be a real analytic function for z ≈ 0 such that

(3.6)  ∑_{j=1}^{k} ψ_{j}(z)\overline{χ_{j}(z)} = |z|^{2}H(z, \overline{z}).

Then when k ≤ n − 1, H(z, \overline{z}) ≡ 0.

Proof of Lemma 3.2. Complexifying (3.6), we have

∑_{j=1}^{k} ψ_{j}(z)\overline{χ_{j}(ξ)} =< z, \overline{ξ} > H(z, \overline{ξ}) (3.6′)

with z, ξ independent variables. Without loss of generality, we can further assume that ψ_{j} ≢ 0 for each j. Therefore, we can find a point z_{0} sufficiently close to the origin such that ψ_{j}(z_{0}) = ϵ_{j} ≠ 0 for each j.

By the assumption: k ≤ n − 1, we can easily see that

V_{z_{0}} = {z : ψ_{j}(z) = ψ_{j}(z_{0}), j = 1, ⋯, k}

defines a complex analytic variety of dimension at least 1 near z_{0}. By our choice of z_{0} and by ψ_{j}(0) = 0, it is clear that V_{z_{0}} can not contain a complex line passing through the origin. Hence, there exists a point z^{∗} ∈ V_{z_{0}} such that V_{z_{0}} contains a complex curve C^{∗} near z^{∗} parametrized by an equation of the form

(3.7)  z(t) = z^{∗} + vt + o(t),

where {z^{∗}, v} are independent vectors and |t| < 1. Notice that for each z ∈ C^{∗} and ξ with < z, \overline{ξ} >= 0 and ξ close to 0, we have ∑ \overline{ϵ_{j}}χ_{j}(ξ) = 0. Also notice that a direction computation based on (3.7) shows that all such ξ fill in an open subset of C^{n}. We conclude that ∑_{j} \overline{ϵ_{j}}χ_{j}(z) ≡ 0. Hence, (3.6) can be reduced to

∑_{j=1}^{k−1}(ψ_{j}(z) − \frac{ϵ_{j}}{ϵ_{k}} ψ_{k}(z))\overline{(χ_{j}(z))} =< z, \overline{z} > H(z, \overline{z}).

Now, applying an induction argument, it follows easily that ∑ ψ_{j}χ_{j} ≡ 0 and H ≡ 0. q.e.d.

4. A criterion for linearity

We now give a criterion on linearity for C^{2}-smooth CR mappings between spheres. Notice that the results of the present section applies to any codimensional case. Namely, N can be any positive integer with N ≥ n.

To start, we let T = \frac{∂}{∂u} be the real tangent vector field along ∂H_{n} transversal to T^{(1,0)}∂H_{n} + T^{(0,1)}∂H_{n}. We notice that

(4.0) \{L_{1}, ⋯, L_{n−1}, \overline{L_{1}}, ⋯, \overline{L_{n−1}}, T \} forms a basis of CT∂H_{n}.

Parametrize M_{1} by R^{2n−1} through (z, u) → (z, w) = (z, u + i|z|^{2}), and introduce the standard pairing (·, ·) over M_{1}. For any differential operator Y with smooth coefficients along M_{1}, we can find its formal adjoint differential operator Y^{∗} with smooth coefficients such that (Y (χ), ρ) = (χ, Y^{∗}(ρ)). Here χ ∈ C^{∞}(M_{1}), ρ ∈ C_{0}^{∞}(M_{1}). Then for each continuous function h and a differential operator with smooth coefficients Y along M_{1}, Y (h) can be identified as a distribution acting on the testing function space C_{0}^{∞}(M_{1}) in the canonical way: (Y (h), ρ) = (h, Y^{∗}(ρ)). A continuous function h is called a CR function if \overline{L}_{j}(h) ≡ 0 in the sense of distribution for each j.

With the above notion, we can state the following useful fact, whose proof is quite straightforward:

Lemma 4.1. (a) T commutes with L_{j} for each j. Hence, for any continuous CR function h over M_{1}, T h is a CR distribution over M_{1}, too. Moreover, T h = −\frac{1}{2i}[L_{j}, \overline{L_{j}}]h = \frac{1}{2i} \overline{L_{j}}(L_{j}h).

(b) Let h be a C^{2} CR function over M_{1} and let χ be a C^{1} function over M_{1}. Then for any k, it holds, in the sense of distribution that

\begin{matrix}\overline{L_{k}}(L_{k}^{2}(h)χ) = 4iL_{k}(T (h))χ + L_{k}^{2}(h)\overline{L_{k}}(χ); \\ \overline{L_{k}}(L_{k}(T (h))χ) = 2iT^{2}(h)χ + L_{k}(T (h))\overline{L_{k}}(χ). \end{matrix} (4.1)

Next, let F be a twice continuously differentiable CR mapping from M_{1} into M_{2} such that F (0) = 0, which satisfies the normalization in (3.1). For each (z_{0}, w_{0}) ∈ M_{1}, write σ_{(z_{0},w_{0})} ∈ Aut(H_{n}) for the map sending (z, w) to (z + z_{0}, w + w_{0} + 2i < z, \overline{z_{0}} >). Define τ_{(z_{0},w_{0})} ∈ Aut(H_{N}) by

τ(z^{∗}, w^{∗}) = (z^{∗} − f̃(z_{0}, w_{0}), w^{∗} − \overline{g(z_{0}, w_{0})} − 2i < z^{∗}, \overline{f̃(z_{0}, w_{0})} >), where f̃ = (f, ϕ) is as before. Then

F_{(z_{0},w_{0})}(z, w) = (f_{(z_{0},w_{0})}, ϕ_{(z_{0},w_{0})}, g_{(z_{0},w_{0})}) = τ_{(z_{0},w_{0})} ∘ F ∘ σ_{(z_{0},w_{0})}

is still a twice continuously differentiable CR map from a small open piece of ∂H_{n} containing 0 into ∂H_{N} with F_{(z_{0},w_{0})}(0) = 0. As in §2, we can similarly define the data λ_{(z_{0},w_{0})}, (E_{l})_{(z_{0},w_{0})}, (E_{w})_{(z_{0},w_{0})}, (C_{l})_{(z_{0},w_{0})}, A_{(z_{0},w_{0})}, F_{(z_{0},w_{0})}^{∗}, a_{(z_{0},w_{0})}, r_{(z_{0},w_{0})}, (e_{kl}^{j})_{(z_{0},w_{0})}, (q_{kl}^{j})_{(z_{0},w_{0})}, and G_{(z_{0},w_{0})}, etc, which are now subscripted by (z_{0}, w_{0}) for they depend on the choice of (z_{0}, w_{0}). As in §2, we can normalize F_{(z_{0},w_{0})}^{∗} by composing it with G_{(z_{0},w_{0})} to get the map F_{(z_{0},w_{0})}^{∗∗} = (f_{(z_{0},w_{0})}^{∗∗}, ϕ_{(z_{0},w_{0})}^{∗∗}, g_{(z_{0},w_{0})}^{∗∗}) so that the normalization condition as in (2.5) holds for F_{(z_{0},w_{0})}^{∗∗}. Notice that F_{(z_{0},w_{0})}^{∗∗} also depends on the choice of (C_{l})_{(z_{0},w_{0})}. The purpose of this section is to prove the following linearity criterion, which should be also useful for other related investigations.

Theorem 4.2. Let F be a twice continuously differentiable CR mapping from M_{1} into M_{2} such that F (0) = 0, which satisfies the normalization in (3.1). Assume the new map F_{(z_{0},w_{0})}^{∗∗} is as defined in the above paragraph. Suppose that for each (z_{0}, w_{0})(≈ 0) ∈ M_{1}, it always holds that

f_{(z_{0},w_{0})}^{∗∗} = z + O(|w|^{2} + |z||w|) + o(|(z, w)|^{2})

and

ϕ_{(z_{0},w_{0})}^{∗∗} = O(|w|^{2} + |z||w|) + o(|(z, w)|^{2}).

Then F (z, w) ≡ (z, 0, w).

We remark that in terms of Lemma 5.3 to be presented in the next

section, the assumption in Theorem 4.2 is equivalent to that \frac{∂^{2}(ϕ_{(z_{0},w_{0})}^{∗∗})}{∂z_{k}∂z_{l}}|_{0}

= 0, or that \frac{∂^{2}(f_{(z_{0},w_{0})}^{∗∗})}{∂z_{k}∂w}|_{0} = 0 for each k, l and (z_{0}, w_{0}) ≈ 0.

Before proceeding to the proof of Theorem 4.2, we need to put Lemma 2.1 in a more applicable form. To this aim, we first write z_{0} = (z_{1,0}, ⋯, z_{n−1,0}), and

f̃_{(z_{0},w_{0})} = ((f_{1})_{(z_{0},w_{0})}, ⋯, (f_{n−1})_{(z_{0},w_{0})}, (ϕ_{1})_{(z_{0},w_{0})}, ⋯, (ϕ_{N−n})_{(z_{0},w_{0})}) .

In the following, we first collect certain expressions whose proofs follow from a direct computation:

(E_{w})_{(z_{0},w_{0})} =(f̃_{(z_{0},w_{0})})′_{w}(0) = f̃′_{w}(z_{0}, w_{0});
(E_{l})_{(z_{0},w_{0})} =(f̃_{(z_{0},w_{0})})′_{z_{l}}(0) = f̃′_{z_{l}}(z_{0}, w_{0}) + 2i\overline{z_{l,0}}f̃′_{w}(z_{0}, w_{0})
=L_{l}(f̃)(z_{0}, w_{0}),
(g_{(z_{0},w_{0})})′_{z_{l}} (0) =L_{l}(g − 2i < f̃, \overline{f̃} >)(z_{0}, w_{0});
(g_{(z_{0},w_{0})})w^{′}(0) =gw^{′}(z_{0}, w_{0}) − 2i < f̃′_{w}(z_{0}, w_{0}), \overline{f̃(z_{0}, w_{0})} >,
λ(z_{0}, w_{0}) :=λ_{(z_{0},w_{0})} = g′_{w}(z_{0}, w_{0}) − 2i < f̃′_{w}(z_{0}, w_{0}), \overline{f̃(z_{0}, w_{0})} >;
\binom{}{f_{(z_{0},w_{0})}}z_{l}z_{k}^{′′} (0) =(f̃)z_{k}z_{l}^{′′}(z_{0}, w_{0}) + 2i\overline{z_{0,l}}(f̃)z_{k}w^{′′}(z_{0}, w_{0})

+ 2i\overline{z_{0,k}}(f̃)z_{l}w^{′′}(z_{0}, w_{0}) − 4\overline{z_{0,l}z_{0,k}}(f̃)ww^{′′}(z_{0}, w_{0})

=L_{l}(L_{k}(f̃))(z_{0}, w_{0}).

Noticing the arbitrariness of (z_{0}, w_{0}) and applying Lemma 2.1 to F_{(z_{0},w_{0})}^{∗∗}, the above formulas then allow Lemma 2.1 to be restated as follows:

Lemma 4.3. Assume the above notation and assume the hypothesis in Theorem 4.2. For each j, k and at each point (z, w)(∈ M_{1}) ≈ 0, we then have the following:

\begin{aligned}L_{j}(L_{k}(f̃)) = & \frac{2i}{λ(z, w)}(\overline{f̃′}_{w} · (L_{j}f̃)^{t})L_{k}(f̃) \\ & + \frac{2i}{λ(z, w)}(\overline{f̃′}_{w} · (L_{k}f̃)^{t})L_{j}(f̃). \end{aligned} (4.1′)

Proof of Theorem 4.2. Write A_{k}(z, w) = \frac{2i\overline{f̃′_{w}}·L_{k}(f̃)^{t}}{λ(z,w)} , which is clearly C^{1} over M_{1}. Assume, without loss of generality that L_{k}(f_{k}) ≠ 0 over M_{1}. Then we first notice that Lemma 4.3 gives immediately that

\begin{matrix}A_{k}(z, w) = & \frac{L_{k}^{2}(f_{k})}{2L_{k}(f_{k})}, \\ L_{j}(L_{k}(ϕ)) = & A_{j}(z, w)L_{k}(ϕ) + A_{k}(z, w)L_{j}(ϕ). \end{matrix} (4.2)

In particular, we obtain

(4.3)  L_{k}^{2}(ϕ) = 2A_{k}(z, w)L_{k}(ϕ).

In the following, we will often use the simple fact that two continuous functions are the same in the usual sense if and only if they are the same in the distribution sense. Now, Applying \overline{L_{k}} to (4.3) and using Lemma 4.1(b), we can easily get

(4.4)  4iL_{k}(T (ϕ)) = B_{k,1}(z, w)L_{k}(ϕ) + B_{k,2}T (ϕ),

where B_{k,2} = 4iA_{k}(z, w) ∈ C^{1}(M_{1}) and B_{k,1} = \overline{L_{k}}(2A_{k}(z, w)) ∈ C^{0}(M_{1}). Notice that there are certain functions b_{k,j} such that

B_{k,1} = b_{k,1}(z, w)L_{k}(T (f_{k})) + b_{k,2}(z, w)L_{k}^{2}(f_{k}), (4.4′)

where b_{k,j}(z, w) ∈ C^{1}(M_{1}) for j = 1, 2. Indeed, by Lemma (4.2)(b) and (4.2), we have

\overline{L_{k}}(A_{k}) = \overline{L_{k}}(\frac{1}{2L_{k}(f_{k})})L_{k}^{2}(f_{k}) + \frac{2i}{L_{k}(f_{k})}L_{k}(T (f_{k})).

Hence, to see (4.4′), it suffices to take b_{k,2}(z, w) = \overline{L_{k}}(\frac{1}{L_{k}(f_{k})}) and b_{k,1} = \frac{2i}{L_{k}(f_{k})}, which are apparently C^{1} over M_{1}.

Applying \overline{L_{k}} to (4.4), we obtain in the sense of distribution

(4.5)  −8T^{2}(ϕ) = h + \overline{L_{k}}(B_{k,2})(T (ϕ)),

where h = \overline{L_{k}}((b_{k,1}L_{k}(T (f_{k})) + b_{k,2}L_{k}^{2}(f_{k}))L_{k}(ϕ)). By Lemma 4.1 (b), h can be identified with

(c_{k,1}L_{k}(T (f_{k})) + c_{k,2}T^{2}(f_{k}) + c_{k,3}L_{k}^{2}(f_{k}))) L_{k}(ϕ)
+ (c_{k,4}L_{k}(T (f_{k})) + c_{k,5}L_{k}^{2}(f_{k})) T (ϕ)

for certain c′_{k,j}s, which are continuous over M_{1}. Hence, h is equivalent to a continuous function and we thus obtain point-wisely that

(4.6)  T^{2}(ϕ) = C_{k,1}(z, w)L_{k}(ϕ) + C_{k,2}(z, w)T (ϕ),

where C_{k,j} are certain continuous functions over M_{1}.

Now, since ϕ are CR over M_{1}, \overline{L_{k}}T (ϕ) and \overline{L_{j}}(\overline{L_{k}}(ϕ)) ≡ 0. Identify M_{1} with a neighborhood of 0 in R^{2n−1} through the map (x, y, u) → (z = (x+iy), w = u+i|z|^{2}) and write X for the vector (\frac{∂ϕ}{∂x}, \frac{∂ϕ}{∂y}, \frac{∂ϕ}{∂u}). We then see from (4.0), (4.2), (4.4) and (4.6) that there is a certain matrix Λ(x, y, u) whose entries are continuous functions over M_{1} such that X satisfies the following

(4.7)  DX = Λ(x, y, u)X^{t}, X(0) = 0.

Regarding (4.7) as a differential equation in X and applying the uniqueness theory, it follows that X ≡ 0. Indeed, for any p = (x_{0}, y_{0}, u_{0}) ≈ 0, let X_{p}(t) = X(tp) for 0 ≤ t ≤ 1. Then \frac{dX_{p}}{dt} = Λ_{p}(t)X_{p}(t)^{t} by (4.7) for a certain continuous matrix function (in t ∈ [0, 1]) Λ_{p}(t) with ∥Λ_{p}(t)∥ <_{∼} ∥p∥, where ∥ · ∥ stands for the super-norm. Since X_{p}(0) = 0,

X_{p}(t) = ∫_{0}^{t} Λ_{p}(pτ)X_{p}(τ)^{t}dτ. Hence ∥X_{p}∥ ≤ C∥p∥∥X_{p}∥ for some constant C > 0 independent of p. It thus follows that X_{p} ≡ 0 once ∥p∥ < \frac{1}{C}. Since ϕ(0) = 0, this also gives that ϕ ≡ 0 over M_{1}.

Once we showed that ϕ ≡ 0, then (f, g) is a C^{2} CR diffeomorphism between two small open pieces (near the origin) of ∂H_{n}. Applying a classical theorem of Alexander, we see that (f, g) extends to an automorphism of H_{n}. Since (f, g) satisfies the normalization condition in (3.1), we get that (f, g) = (z, w), say, by Lemma 3.0. (Here, after knowing ϕ ≡ 0, we remark that we can also directly get (f, g) = (z, w) by studying the uniqueness of the first n−1 components of Equation (4.1′), coupled with L_{j}(L_{k}(g)) = 2iL_{j}(L_{k}(f)) · \overline{f}.) The proof of Theorem 4.2 is complete. q.e.d.

5. Completion of the proof of Theorem A

We now proceed to the proof of Theorem A. We start with the following:

Proposition 5.1. Let F = (f, ϕ, g) be any twice differentiable CR mapping from M_{1} into M_{2} satisfying the normalization condition (3.1). Assume that N ≤ 2n − 2. Then f = z + o_{wt}(3), g = w + o_{wt}(4) and ϕ = o_{wt}(2).

In what follows, to simplify the notation, we use P_{1} and P_{2} for polynomials which may be different in different contexts. We say that wt(P_{1}) ≥ k if |P_{1}(tz, t^{2}w, t\overline{z}, t^{2}\overline{w})| ≤ Ct^{k}.

We proceed by presenting the following Lemma 5.2 and Lemma 5.3. Since Lemma 5.2 is elementary, we leave its proof to the reader. (Different from Proposition 5.1, the following two lemmas hold in any codimensional case.)

Lemma 5.2. (a) For any function χ ∈ C^{k}(M_{1})∩o_{wt}(k) with k ≥ 1, L_{j}χ, \overline{L_{j}}χ, ∈ o_{wt}(k − 1), and T χ ∈ o_{wt}(k − 2).

(b) If χ ∈ C^{1}(M_{1}) is such that \overline{L_{j}}(χ), L_{j}(χ) ∈ P + o_{wt}(k − 1) for each j and T (χ) ∈ P + o_{wt}(k − 2), then χ ∈ P + o_{wt}(k). Moreover, write χ = h_{1} + h_{2}, where h_{1} is a polynomial with weighted degree not larger than k and h_{2} ∈ o_{wt}(k). Then in case χ is CR, h_{1} must be a holomorphic polynomial in (z, w).

Lemma 5.3. Let F = (f, ϕ, g) = (f_{1}, ⋯, f_{n−1}, ϕ_{1}, ⋯, ϕ_{k}, g) be a twice differentiable CR map from M_{1} into M_{2} satisfying the normalization condition (3.1). Then, f = Q + o_{wt}(3) and g = P + o_{wt}(4) with P, Q certain holomorphic polynomials. In fact, we can choose P, Q such that P = w and Q = z + \frac{iw}{2}a^{(1)}(z), where a^{(1)}(z) is a certain (vector-valued) holomorphic polynomial of degree one with < a^{(1)}(z), \overline{z} >= ∑_{j=1}^{k} |ϕ_{j}^{(2)}(z)|^{2}.

Proof of Lemma 5.3. We first recall that

(5.0)  \frac{1}{2i}L_{l}g = ∑ L_{l}(f_{j})\overline{f_{j}} + ∑ L_{l}(ϕ_{j})\overline{ϕ_{j}}.

Also, it is easy to verify the following:

(5.a1) L_{l}f_{j}\overline{f_{j}} = (P_{1} + o_{wt}(1))(P_{2} + o_{wt}(2)) ∈ P + o_{wt}(2),

where wt(P_{2}) ≥ 1; and

(5.a2) L_{l}ϕ_{j}\overline{ϕ_{j}} = (P_{1} + o_{wt}(1))(P_{2} + o_{wt}(2)) ∈ P + o_{wt}(3),

where wt(P_{1}) ≥ 1, wt(P_{2}) ≥ 2. Hence, L_{l}g ∈ P + o_{wt}(2).

On the other hand, since g ∈ C^{2}(M_{1}), T (g) ∈ P +o_{wt}(1). By Lemma 5.1 (b), we see that g ∈ P + o_{wt}(3).

Notice g = w + o_{wt}(2). Write ĝ = ĝ_{3} + o_{wt}(3) with ĝ_{3} a weighted homogeneous polynomial of degree 3. By Lemma 5.2(b), ĝ_{3} is holomorphic. To see that ĝ^{(3)} = ĝ_{3} = 0 and f̂^{(2)} = 0, it now suffices for us to apply Lemma 3.0 to (3.3′′) with k = 1 (this can also be quite easily seen by a straightforward computation).

Next, we apply L_{j} to (5.0) to yield \frac{1}{2i}L_{j}(L_{l}g) = L_{j}(L_{l}(f̃)) · \overline{f̃}. Applying the just obtained equation by \overline{L_{k}} and making use of Lemma 4.1, one gets L_{j}(L_{l}(f̃)) · \overline{L_{k}f̃} = 0, for j, l different from k, and

L_{j}(L_{l}(f̃)) · \overline{L_{k}f̃}+ cT (L_{v}(f̃)) · \overline{f̃} = \frac{c}{2i}T L_{v}(g),

for k = j or k = l. Here when l ≠ j, we have c = 2i, v = {l, j} \ {k}; and when l = j, we have c = 4i, v = k. Since L_{k}(f_{μ}) = δ_{μ}^{k} + o_{wt}(1), we can easily solve the above equations L_{j}L_{l}(f_{k}) and then verify that

L_{j}L_{l}(f_{k}) − \frac{cT L_{l}(g)}{2iL_{k}(f_{k})} ∈ P + o_{wt}(1).

Since

2iT (L_{l}(g)) = L_{l}(T (f̃)) · \overline{f̃}+ (L_{l}(f̃)) · T (f̃),

it follows readily that T (L_{l}(g)) ∈ P + o_{wt}(1). Therefore, we prove that L_{j}(L_{l}(f_{k})) ∈ P + o_{wt}(1) for any j, l, k.

Since F is assumed to be twice differentiable, \overline{L_{j}}(L_{l}f_{k}) = δ_{l}^{j}T (f_{k}) ∈ P+o_{wt}(1) and T (L_{l}(f_{k})) ∈ P+o_{wt}(0). Making use of Lemma 5.2, we see that L_{l}(f_{k}) ∈ P + o_{wt}(2) for any l, k. Again since T (f_{k}) ∈ P + o_{wt}(1), f_{k} ∈ P + o_{wt}(3) for any k. Also, by Lemma 5.2 (b), we can write f = Q + o_{wt}(3) for a certain holomorphic polynomial Q. Making use of what we obtained above, it is now easy to verify that the expressions L_{l}f_{j}\overline{f_{j}} and ∥L_{l}f_{j}∥^{2} stay in P+o_{wt}(3) and P+o_{wt}(2), respectively. By the identity (5.0), 2iT (g) =< 2iT f, \overline{f} > +∥L_{l}f∥^{2}+ < 2iT ϕ, \overline{ϕ} > +∥L_{l}ϕ∥^{2}, and the same argument as before, we can see that L_{j}(g) ∈ P+o_{wt}(3) and T (g) ∈ P + o_{wt}(2). Similarly, g = P + o_{wt}(4) for a certain holomorphic polynomial P. Finally, applying the first part of Proposition 3.1 to (3.3′), we complete the proof of Lemma 5.3. q.e.d.

Proof of Proposition 5.1. By Lemma 5.3, we can write f = z+f̂^{(3)}+ o_{wt}(3) and g = w + o_{wt}(4), where f̂^{(3)} is a holomorphic polynomials of weighted degree 3. To finish the proof of Proposition 5.1, it suffices to apply the last part of Proposition 3.1 to (3.3′). q.e.d.

Finally, we are ready to finish the proof of Theorem A:

Proof of Theorem A. Let F be as in Theorem A and assume that N ≤ 2n − 2. After the normalization in §2, we may assume that F satisfies the normalization condition in (3.1). Now, as did in §4, using the Heisenberg group structures of ∂H_{n} and ∂H_{N}, for any (z_{0}, w_{0}) ≈ 0 and (C_{l})_{(z_{0},w_{0})} as chosen before, we have the new twice differentiable CR map F_{(z_{0},w_{0})}^{∗∗} from M_{1} into M_{2}, which still satisfies the normalization condition (3.1). Applying Proposition 5.1, we see that f_{(z_{0},w_{0})}^{∗∗} = z + o_{wt}(3), g_{(z_{0},w_{0})}^{∗∗} = w + o_{wt}(4) and ϕ_{(z_{0},w_{0})}^{∗∗} = o_{wt}(2) always hold. Hence, from Theorem 4.2 it follows that F = (z, 0, w). This completes the proof of Theorem A. q.e.d.

References

[1] H. Alexander, Holomorphic mappings from ball and polydisc, Math. Ann. 209 (1974) 245-256.

[2] , Proper holomorphic maps in C^{n}, Indiana Univ. Math. J. 26 (1977) 137-146.

[3] M. S. Baouendi, H. Jacobowitz & F. Treves, On the real analyticity of CR mappings, Ann. of Math. 122 (1985) 365-400.

[4] M. S. Baouendi & L. P. Rothschild, Geometric properties of smooth and holomorphic mappings between surfaces in complex spaces, J. Differential Geom. 31 (1990) 473-499.

[5] S. Baouendi, X. Huang & L. Rothschild, Regularity of CR mappings between algebraic hypersurfaces, Invent. Math. 125 (1996) 13-36.

[6] S. Bell & R. Narasimhan, Proper holomorphic mappings of complex spaces, Encyclopedia Math. Sci. 69, Several Complex Variables VI (eds. W. Barth and R. Narasimhan), Springer, Berlin, 1990.

[7] D. Catlin & J.P. D’Angelo, A stabilization theorem for Hermitian forms and applications to holomorphic mappings, Math. Res. Lett. 3 (1996) 149–166.

[8] S. S. Chern & J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974) 219-271.

[9] J. Cima & T. J. Suffridge, A reflection principle with applications to proper holomorphic mappings, Math. Ann. 265 (1983) 489-500.

[10] , Boundary behavior of rational proper maps, Duke Math. J. 60 (1990) 135-138.

[11] J. Cima, S. Krantz & T. J. Suffridge, A reflection principle for proper holomorphic mappings of strictly pseudoconvex domains and applications, Math. Z. 186 (1984) 1-8.

[12] H. Cao & N. Mok, Holomorphic immersions between compact hyperbolic space forms, Invent. Math. 100 (1990) 49–61.

[13] J. P. D’Angelo, Several complex variables and the geometry of real hypersurfaces, CRC Press, Boca Raton, 1993.

[14] J. Faran, Maps from the two ball to the three ball, Invent. Math. 68 (1982) 441-475.

[15] , On the linearity of proper maps between balls in the lower dimensional case, J. Differential Geom. 24 (1986) 15-17.

[16] , A reflection principle for proper holomorphic mappings and geometric invariants, Math. Z. 203 (1990) 363-377.

[17] F. Forstneric, Proper holomorphic maps between balls, Duke Math. J. 53 (1986) 427-440.

[18] , Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989) 31-62 .

[19] , A survey on proper holomorphic mappings, Proc. Year in Several Complex Variables at Mittag-Leffler Institute, Math. Notes 38 (1992), Princeton University Press, Princeton, NJ.

[20] M. Hakim & N. Sibony, Fonctions holomorphes bornes sur la boule unite de C^{n}, Invent. Math. 67 (1982) 213–222.

[21] X. Huang, On the mapping problem for algebraic hypersurfaces in complex spaces of different dimensions, Ann. Inst. Fourier (Grenoble) 44 (1994) 433-463.

[22] L. Lempert, Embedding strictly pseudoconvex domains into the ball, Amer. J. Math. 104 (1982) 901-904.

[23] , Embedding Cauchy-Riemann manifolds into a sphere, Internat. J. Math. (1990) 91-100.

[24] E. Low, Embeddings and proper holomorphic mappings of strictly pseudoconvex domains into polydiscs and balls, Math. Z. 190 (1985) 401-410.

[25] J. Moser, Analytic surfaces in C^{2} and their local hull of holomorphy, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985) 397–410.

[26] H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo, (2) 23 (1907) 185-220.

[27] B. Stensones, Proper holomorphic mappings between the balls, J. Geom. Anal., to appear, 1994.

[28] N. Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan 14 (1962) 397-429.

[29] V. Vitushikin, Holomorphic mappings and the geometry of hypersurfaces, several complex variables. I, Encyclopedia Math. Sci., 7. Springer, Berlin, (1990) 159-215.

[30] S. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977) 53-68.

[31] , On mappings an (n+1)-ball in the complex space, Pacific J. Math. 81 (1979) 267-272.

[32] C. Yue, Dimension and rigidity of quasi-Fuchsian representations, Ann. of Math. (2) 143 (1996) 331–355.

Rutgers University