REGULARITY OF HOLOMORPHIC CORRESPONDENCES AND APPLICATIONS TO THE MAPPING PROBLEMS

XIAOJUN HUANG

Abstract. We study the regularity results of holomorphic correspondences. As an application, we combine it with certain recently developed methods to obtain the extension theorem for proper holomorphic mappings between domains with real analytic boundaries in the complex 2-space.

Introduction

This paper is concerned with the boundary regularity problem for holomorphic mappings between domains in complex euclidean spaces. Our main purpose is to present certain results related to the following mapping conjecture in several complex variables:

Conjecture 0.1: Let M_{1} and M_{2} be two real analytic hypersurfaces of finite Dtype in C^{n} with n > 1. Suppose that D ⊂ C^{n} is a domain with M_{1} as part of its real analytic boundary. Also, suppose that f is a holomorphic mapping from D into C^{n}, that is continuous up to D ∪M_{1} and maps M_{1} into M_{2}. Then f admits a holomorphic extension across M_{1}.

Here, we recall that a real analytic hypersurface is of finite D-type [Da] if and only if it does not contain any non-trivial complex analytic variety.

The extensive study of such kinds of problems was initiated by the foundational work of Fefferman in the case of biholomorphic mappings between strongly pseudoconvex domains. Recent related papers include those by Lewy [Le], Pinchuk [Pi1], Webster [We], Diederich- Webster [DW], Bell [Be1] [Be3], Baouendi-Jacobowitz-Treves [BJT], Bedford-Bell [BB], Baouendi - Bell- Rothschild [BBR], Baouendi- Rothschild [BR1] [BR2] [BR3] [BR4], Bell-Catlin [BC1] [BC2], Diederich-Fornaess [DF1] [DF2]

[DF3] [DF4], Pinchuk-Tsyganov [PT], Diederich- Fornaess- Ye [DFY], Baouendi-Huang- Rothschild [BHR1] [BHR2], Huang - Pan [HP], Pan [Pan], and the references therein. For detailed surveys of this investigation, we refer the reader to the articles by Forstneric [For], Bedford [Bed], and Bell-Narashimhan [BN].

When M_{1} and M_{2} are pseudoconvex and when f is proper, a powerful method is to use Condition R introduced by Bell-Ligocka (see [Be2], for example) and the solutions to the \overline{∂}-Neumann problems to achieve the smooth extension. Then one applies various versions of the reflection principle to obtain the holomorphic extension ([BJT] [BBR] [BR1] [DF1]). However, because of an example by Barrett [Ba1], it is known that a different approach needs to be employed when the assumption of pseudoconvexity is dropped.

Supported in part by NSF DMS-9500881. Research at MSRI is supported in part by NSF DMS-9022140.

In [We], Webster used a class of invariant varieties, introduced by Segre [Se], to analyze biholomorphisms between non-degenerate algebraic hypersurfaces. Later, it became clear that these invariant objects called Segre surfaces may also be useful to study the above problem even in the non-pseudoconvex case (for example, see [DW], [DF2], [DFY]).

The plausible method based of the use of Segre surfaces might be carried out in two steps. First one proves the holomorphic extension under the assumption that the map extends as a holomorphic correspondence (see the following section for a precise definition of this terminology). Then, one reduces the general situation to the above case. A result along these lines was obtained in [DF2], where it was shown that a biholomorphism between two real analytic domains in C^{2}, that extends as a correspondence, admits a holomorphic extension up to the boundary. (See also related work in [BB], [Art], [BBR], [DF1] when the map is smooth). This, in particular, implies that any biholomorphic map between two bounded algebraic domains in C^{2} extends holomorphically across the boundary [DF2]. The idea used in [DF2] depends strongly on a theorem of Barrett [Ba2], and does not seem adaptable to the proper mapping case.

In this paper, we will establish the smoothness results for holomorphic correspondences coming from general holomorphic mappings (Theorem 1.1′). As an immediate application, we obtain a solution of Conjecture 0.1 in case n = 2, M_{1} and M_{2} are alegbraic, and f is proper. One of the other main results indicates that any continuous CR mapping between two variety-free real algebraic hypersurfaces in C^{2} is real analytic. This, in particular, gives a complete solution to Conjecture 0.1 in the 2dimensional algebraic case. Previously, Conjecture 0.1 was only known to hold in case hypersurfaces are strongly pseudoconvex. Our result seems to give the first Schwarz reflection principle for continuous CR mappings between a large class of hypersurfaces which may not be pseudoconvex, though much more has been well understood when the map is smooth and proper (see the papers mentioned above). Indeed, it is a wide open question to answer whether Conjecture 1.1 holds when the hypersurfaces are pseudoconvex of finite type ([BC]), or when f is smooth up to D ∪ M_{1} but the hypersurfaces may not be pseudoconvex ([BR2]).

Recently, based on the aforementioned result in [DF2] for biholomorphisms and the machinery developed in [We], Diederich -Pinchuk ( [DP]) announced a proof that any biholomorphic map between bounded real analytic domains in C^{2} extends as a holomorphic correspondence across the boundary (see also closely related work in [DFY]). This together with the result in [DF2] implies the holomorphic extension for biholomorphisms between real analytic domains in C^{2}. With the main result in the present paper as our major tool (Theorem 1.1 or Theorem 1.1′), and with the help of ideas the author learned from [DFY] and [DP], we will obtain, in the the last section of this paper, a solution of Conjecture 0.1 in case n = 2 and f is proper. Indeed, once Theorem 1.1 is known, the rest of the argument of the proof of Theorem 1.3 is a modification of the known methods and ideas for biholomorphic maps as appeared in [DFY], and, in particular, [DP]. However, since [DP] may not be presently available, we give a certain detailed discussion on this matter (§7) for completeness of the proof of Theorem 1.3.

Our method of the proof of Theorem 1.1, which is different from that in [DF2], can be described as follows: We first show that an attached reflection function (see, in particular, [BJT], [BBR]) extends holomorphically across the hypersurface. Here the main idea is to use the weak version of the edge of the wedge theorem (see Proposition 7.1 of [BHR1] or [Hu]). Using this reflection function, we will connect the branching locus of the map with the branching locus of the Segre varieties of the target hypersurface. This then gives us many restrictions on the branching behaviors of the map. Finally the above mentioned results and several technical lemmas (in particular, a modified version of the Baouendi-Rothschild Hopf lemma and a preservation principle) give the proof of Theorem 1.1. The proof of Theorem 1.2 is an easy corollary of Theorem 1.1 with the known results. Once one has Theorem 1.1 as the major tool, the proof of Theorem 1.3 can be achieved with the help of the existing ideas (see [DF1] [DFY] and, in particular, [DP]).

We would like to mention that certain notations in this paper (especially, those in §7) are adapted from [DFY] and, in particular, [DP].

Acknowledgment: The author is grateful to Professors S. Webster and R. Narashimhan for their many stimulating conversations (in particular, certain discussions related to Lemma 5.3), and constant encouragement. He would like to thank Professors S. Baouendi, S. Krantz, and L. Rothschild for their constant help and interest in this work. The author also thanks Professor S. Pinchuk for his explanation of his work with K. Diederich [DP] and for showing his lecture notes [DP] (on the construction of holomorphic correspondences for biholomorphic mappings).

Added in revision: This is the revised version of the author’s previous paper with the same title, in which Theorem 1.1 was proved only in case n = 1 or M_{1} and M_{2} are decoupled. Results related to Theorem 1.1′ were also announced by Diederich-Pinchuk in the workshop on Analytic and PDE methods in SCVs at MSRI (November,

1. Statement of main results and related observations

To state our main results, we need the following preliminary notation (see [BB], [DF2], and, in particular, [DFY], [DP]):

Let M_{1} and M_{2} be two real analytic hypersurfaces of finite D-type in C^{n+1} (n ≥ 1), and let p be a point in M_{1}. A CR mapping f from M_{1} into M_{2} is said to extend as a holomorphic correspondence to a neighborhood of p if (a) f is continuous near p; (b) for each component f_{j} of f, there is a polynomial P_{j}(z; X) in X with leading coefficient 1 and other coefficients holomorphic in a small neighborhood U_{1} of p, such that P_{j}(z; f_{j}(z)) = 0 for z ∈ U_{1} ∩ M_{1}. It can be shown that f extends as a holomorphic correspondence at p if and only if there are a small neighborhood U_{1} of p, a neighborhood U_{2} of M_{2}, and an analytic sub-variety V of U_{1} × U_{2} such that (a) V ⊃ Γ_{f} = {(z, w) : w = f(z), z ∈ M_{1} ∩ U_{1}} (b) the natural projection of V to U_{1} is locally proper.

We also recall that a real analytic hypersurface M ⊂ C^{n+1} is called rigid if, after a holomorphic change of coordinates, it can be defined by a real analytic function of the form ρ(z, \overline{z}) = z_{n+1} + \overline{z_{n+1}} + χ(z′, \overline{z′}), where z′ = (z_{1}, ⋯, z_{n}).

Theorem 1.1: Let M_{1} and M_{2} be two real analytic hypersurfaces of finite D-type in C^{n+1} (n ≥ 1). Let p ∈ M_{1}. Suppose that f is a CR mapping from M_{1} into M_{2} that extends as a holomorphic correspondence to a neighborhood of p. Then f admits a holomorphic extension near p, if either n = 1 or M_{1} and M_{2} are rigid.

See also Theorem 1.1′ of Section 5 for a more general version of this result and notice that in whole paper, only in §5, what we call Condition S is used.

Recall that a smooth hypersurface is called algebraic, if it can be defined by a real polynomial. Using certain known results in [We] and [DF2], one can apply Theorem 1.1 to obtain the following:

Theorem 1.2: Let M_{1} and M_{2} be two algebraic hypersurfaces of finite type in C^{2}. Let f be a continuous CR mapping from M_{1} to M_{2}. Then f is real analytic on M_{1}.

Corollary 1.2′: Let D and D′ be two smoothly bounded domains with real algebraic boundaries in C^{2}. Let f be a proper holomorphic mapping from D to D′. Then f admits a holomorphic extension to a neighborhood of \overline{D}.

We remark that Theorem 1.2 is sharp in a certain sense. In fact, using the inner functions defined on the ball in C^{2}, one can construct a bounded CR mapping between spheres in C^{2} which is even not continuous at any point. It is worthwhile to mention that the map in Theorem 1.2 is not assumed to be the boundary value of some globally defined proper holomorphic map. As is known, it is always a difficult problem to understand when a CR mapping comes from a globally defined proper holomorphic map. Also, we notice that when the map in Theorem 1.2 is assumed to be in the smoothness class C^{k} with k >> 1, then much less restriction to the hypersurfaces is required to obtain the real analyticity ( [BHR1]).

Using the two dimensional homeomorphic version of Theorem 1.1 (which has been established in [DF2]), Diederich-Pinchuk recently announced that any biholomorphic map between real analytic bounded domains in C^{2} admits a correspondence extension across the boundary [DP] (see also the related work in [DFY]). With Theorem 1.1 at our disposal and with the help of an idea of Diederich-Pinchuk [DP] on the application of the Bishop extension lemma, we will prove in §7 the following result (see [BBR] for the pseudoconvex case, and Theorem 1.3′′ in §7 for a local version of this result).

Theorem 1.3: Let D and D′ be two bounded domains in C^{2} with real analytic boundaries. Suppose that f is a proper holomorphic map from D to D′, which extends continuously across ∂D. Then f extends holomorphically across ∂D.

In higher dimensions, we will be content in this paper with the following application of Theorem 1.1 (see Theorem 1.1 ′ for a more general result on this):

Theorem 1.4: Let D and D′ be two smoothly bounded algebraic domains in C^{n} and let f be a proper holomorphic map from D to D′. If D and D′ are rigid then f admits a holomorphic extension to a neighborhood of \overline{D}.

The organization of the remainder of this paper is as follows: §2 through §5 is devoted to the proof of Theorem 1.1. In §2, we will study the holomorphic extendibility of a λ-function associated with the map and the target surface. This function was used in various different forms in [Le], [Pi], [BJT], [BBR], and [BR1] to study smooth CR-mappings (see, in particular, the work in [BBR] and [BR1]). In §3, we will present the connections between the branching properties of the map and the invariant varieties (Segre-surfaces). In §4, we will prove a localized Hopf lemma by modifying an argument that appeared in [BR2], which will then be used to prove a preservation principle for our (multiple-valued) map. Section 5 is devoted to the proof of Theorem 1.1, using results established in the previous sections. In §6, we will complete the proofs of Theorem 1.2 and Theorem 1.4. With Theorem 1.1 at our disposal and using some known ideas (in particular, those appeared in [DP]), in the last section (§7), we will present a proof of Theorem 1.3.

2. Basic set-ups and holomorphic extendibility of a reflection function

In this section, we first set up some notation which will be used throughout the paper. Then we study the holomorphic extendibility of a reflection function. We will use some notations, which the author adapts from [DP].

To start with, we let M_{1} ⊂ C^{n+1}, M_{2} ⊂ C^{n+1} be real analytic hypersurfaces of finite D-type, and let f be a non-constant CR mapping from M_{1} to M_{2}. Assume that f extends as a holomorphic correspondence to a neighborhood of p(∈ M_{1}). Our main objective from §2 to §5 is to understand under what circumstances f admits a holomorphic extension near p. Since this is purely a local problem, after holomorphic changes of coordinates and shrinking the size of the hypersurfaces, we can assume that p = f(p) = 0; M_{1} , M_{2} are connected and defined, respectively, by the functions of the following form (see [BJT], for example):

(2.1)  ρ_{1}(z, \overline{z}) = z_{n+1} + \overline{z_{n+1}} + ∑_{j=0}^{∞} ϕ_{j}(z′, \overline{z′}) (\frac{z_{n+1} − \overline{z_{n+1}}}{2i} )^{j} ,

(2.2)  ρ_{2}(z, \overline{z}) = z_{n+1} + \overline{z_{n+1}} + ∑_{j=0}^{∞} ψ_{j}(z′, \overline{z′}) (\frac{z_{n+1} − \overline{z_{n+1}}}{2i} )^{j} ,

with ϕ_{j}(z′, 0) = ϕ_{j}(0, \overline{z′}) = ψ_{j}(0, \overline{z′}) = ψ_{j}(z′, 0) ≡ 0 and ϕ_{j},ψ_{j} holomorphic in z′, \overline{z′} over certain fixed open neighborhood of (0, 0). Here we use z = (z′, z_{n+1}) with z′ = (z_{1}, ⋯, z_{n}) for the coordinates in C^{n+1}.

Applying a CR extension result in [BT] or [Tr] and making M_{1} sufficiently small, we can further assume that there is a domain D which contains M_{1} as part of its smooth boundary such that every CR function defined over M_{1} can be extended holomorphically to D. Hence, without loss of generality, we can assume, in all that follows, that f is holomorphic in D and continuous up to D ∪ M_{1}.

Applying the implicit function theorem to (2.1), (2.2) and shrinking the size of the hypersurfaces further (if necessary), we see that M_{1} and M_{2} can also be presented, respectively, by functions of the following form:

(2.1)′  ρ̃_{1}(z, \overline{z}) = z_{n+1} + \overline{z_{n+1}} + ∑_{j=0}^{∞} ϕ̃_{j}(z′, \overline{z′})z_{n+1}^{j} = 0,

(2.2)′  ρ̃_{2}(w, \overline{w}) = z_{n+1} + \overline{z_{n+1}} + ∑_{j=0}^{∞} ψ̃_{j}(z′, \overline{z′})z_{n+1}^{j} = 0,

where ϕ̃_{j} are determined by ϕ_{j}’s, ϕ̃_{j}(z′, 0) = ϕ̃_{j}(0, \overline{z}′) = ψ̃_{j}(z′, 0) = ψ̃_{j}(0, \overline{z′}) = 0; and

(2.3)  ρ̃_{j} = ρ_{j}h_{j}.

Here h_{j} is a real analytic function near M_{1} for j = 1, 2 and h_{j}(0, 0) = 1. Complexifying (2.3), we then obtain

(2.3)′  ρ̃_{j}(z, \overline{w}) = ρ_{j}(z, \overline{w})h_{j}(z, \overline{w}).

Therefore, for z, w close to 0, ρ_{j}(z, \overline{w}) = 0 if and only if ρ̃_{j}(z, \overline{w}) = 0.

We now recall the definition of Segre varieties and some related notions, which were first used in [We] (see also [DW] and [DF1]) for the study of the mapping problems.

Since M_{2} is completely symmetric to M_{1}, we will only introduce notations for M_{1} and add ′ for those corresponding to M_{2}.

We first choose a small neighborhood U_{3} of 0 such that ρ_{1}(z, w), ρ̃_{1}(z, w), h_{1}(z, w) are holomorphic over U_{3} × conj(U_{3}), and h_{1}(z, w) ≠ 0 in U_{3} × conj(U_{3}). Here and in what follows, for a subset A in the complex spaces we write conj(A) = \{\overline{z} : z ∈ A\}. For each point w ∈ U_{3}, the Segre variety associated to M_{1} and w is defined by Q_{w} = \{z ∈ U_{3} : ρ_{1}(z, \overline{w}) = 0\}. By making U_{3} small, it can be seen that Q_{w} is actually a complex submanifold of U_{3}. It is easy to show that there are two small neighborhoods U_{2}, U_{1} of 0 such that the following holds (see [DF1] for more details on this matter): (I.a) U_{3} ⊃⊃ U_{2} ⊃⊃ U_{1} ∋ 0. (I.b) U_{1} \ M_{1} has two connected components, which are in different sides of M_{1}. Moreover, they are homeomorphic to the ball. (I.c) For each w ∈ U_{1}, U_{2} ∩ Q_{w} is connected. (I.d) For a sequence {w_{j}} ⊂ U_{1} with w_{j} → w ∈ U_{1}, the limit set of {Q_{w_{j}}}, denoted by lim Q_{w_{j}}, is Q_{w}.

In what follows, we call {U_{3}, U_{2}, U_{1}} a Segre neighborhood system around 0. It is clear that for any open set containing 0, we can always construct inside this set a Segre neighborhood system around 0. Meanwhile, when U_{3} (U_{2}, respectively) is fixed, we can arrange U_{2} (U_{1}, respectively) as small as we need.

For convenience of the reader, we indicate here how (I.c) and (I.d) can be achieved (much more details for related discussion can be found in work of [DW] and [DF1]).

In the coordinates system where M_{1} is defined by (2.1), we can choose a small polydisc P = P (δ_{1}, ⋯, δ_{n+1}) = {z : |z_{j}| < δ_{j}( j = 1, ⋯, n + 1)}(⊂⊂ U_{3}). Then, applying the implicit function theorem, we can see that for each w with |w| small, Q_{w} ∩ P is contained in the graph of certain function , say z_{n+1} = g(z′, \overline{w}), over the polydisc P′ = P′(δ_{1}, ⋯, δ_{n}) = \{z′ : |z_{j}| < δ_{j}, j = 1, ⋯, n\}. Here g(z′, \overline{w}) is holomorphic for z, w ∈ U_{3} and g(z′, 0) = 0. Now, if we make w close enough to the origin, then Q_{w} ∩ P will be exactly the graph of g(z′, \overline{w}) over P′. Since it is biholomorphic to P′, it is of course connected. Hence, we can simply take U_{2} to be the polydisc P chosen in such a manner. Once U_{2} is chosen, it is then clear that we can simply take U_{1} to be a sufficiently small ball around 0. Next, if w_{j} → w ∈ U_{1}, then we clearly see that Q_{w_{j}} → Q_{w}; for g(z′, \overline{w_{j}}) → g(z′, \overline{w}). This gives (I.d).

We next set up the notation for a reflection mapping R.

By examining the proof of Lemma 1.1 of [BJT], it is clear that for each point p ∈ M_{1} close to 0, there exists a biholomorphic mapping Φ(·; p), which depends real analytically on p, such that the following holds:

(i) Φ(·; 0) = id and Φ(p; p) = 0. (ii) Φ(·; p) (and thus its inverse) is well-defined over some fixed open neighborhood of 0; (iii) Φ(M_{1}; p) is defined near 0 by the following equation:

ρ_{1}(z, \overline{z}; p) = z_{n+1} + \overline{z_{n+1}} + ∑_{j=0}^{∞} ϕ_{j}(z′, \overline{z′}; p) (\frac{z_{n+1} − \overline{z_{n+1}}}{2i} )^{j} ,

with ϕ_{j}(0, \overline{z′}; p) = ϕ_{j}(z′, 0; p) ≡ 0 and ϕ_{j} holomorphic in z′, \overline{z′} over certain fixed open neighborhood of (0, 0).

For a small positive ϵ_{0}, define Φ: (−ϵ_{0}, ϵ_{0})×M_{1} → C^{n+1} by Φ(t, p) = Φ^{−1}((0′, t); p). Using the facts that Φ(0, p) = p and Φ(t, 0) = (0′, t), it can be seen that Φ is an analytic diffeomorphism near (0, 0).

We now define the reflection mapping R as follows: For z close to 0, if Φ^{−1}(z) = (t, p), then R(z) = Φ(−t, p). Clearly, R is a diffeomorphism near the origin. By the way it was constructed, one can verify that R|_{M_{1}∩O(0)} = id, R^{2}(z) = z for z close to 0, R(D ∩ O(0)) ⊂ D^{c}, R(D^{c} ∩ O(0)) ⊂ D. Here D^{c} = U \ (D ∪ M_{1}) with U a neighborhood of M_{1}; and in all that follows we write O(a) for a small neighborhood of a, whose size may be different in different contexts.

We observe that for a real number t with |t| small, the Segre variety of Φ(M_{1}; p) at (0, t) is an open subset of the affine space {(z′, −t)} near (0, −t). Using the invariant property of Segre varieties under biholomorphic mappings ([We] [DW]) and by arranging the size of {U_{3}, U_{2}, U_{1}} suitably, one can see that R(z) ∈ Q_{z} for each z ∈ U_{2}.

Since we assumed that f extends as a correspondence to a neighborhood of 0, after shrinking M_{1} if necessary, we can assume that for each component f_{j} of f, there is an irreducible Weierstrass polynomial

(2.4)  P_{j}(z; X) = X^{N_{j}} + ∑_{0≤l<N_{j}} a_{j,l}(z)X^{l}

with a_{j,l} holomorphic over a neighborhood, say U, of M_{1} such that P_{j}(z; f_{j}(z)) = 0 for z ∈ M_{1}. Since f is known holomorphic on D, P_{j}(z; f_{j}(z)) = 0 for z ∈ D ∪ M_{1}. In all that follows, we always make (a) U ⊃⊃ U_{3} ∪ D; (b) D ⊃⊃ U_{3} ∩ D; (c) R is a diffeomorphism from U_{3}; and (d) R(D ∩ U_{1}) ⊂ D^{c} ∩ U_{2}, R(D^{c} ∩ U_{1}) ⊂ D ∩ U_{2}.

As usual, we define the branching locus E of f to be the union of the branching locus of f_{j} (j = 1, ⋯, n + 1). That is, a point z ∈ U is in E if and only if for some index j and number X, one has P_{j}(z; X) = \frac{∂P_{j}}{∂X} (z; X) = 0. Write E_{M_{1}} = E ∩ M_{1}. Then by the finite type assumption of M_{1}, one can see that E_{M_{1}} is a real analytic subset of M_{1} with codimension ≥ 2. Obviously, f admits a holomorphic extension across M_{1} \ E_{M_{1}}.

We now introduce the following λ-function associated to the map f and M_{2} (see [Le], [Pi1], [BJT], and, in particular, [BBR], for closely related notions):

(2.5)  G(f(z), λ) = −f_{n+1}(z) − ∑_{j=0}^{∞} ψ̃_{j}(f^{∗}(z), λ)f_{n+1}^{j}(z). z ∈ M_{1} ∪ D

Here ψ̃_{j} are the same as in (2.2)′ and we write f^{∗} = (f_{1}, ⋯, f_{n}). Then it is clear that G(f(z), λ) is holomorphic over D × O_{λ}(0) and continuous on (D ∪ M_{1}) × O_{λ}(0). Here and in all that follows, we put a subscript to O(a) to emphasize the coordinates used for the space where O(a) stays. For example, O_{b}(a) denotes a small open neighborhood of a in the complex b-space, whose size may be different in different contexts.

Our first lemma in this paper is to extend this function across M_{1} for small λ.

Lemma 2.1: Under the above notations and assumptions, G(f(z), λ) extends holomorphically to O_{z} × O_{λ}.

The proof we present here is to take the differentiation along the boundary, as did in [Le], [Pi], [BJT], [BBR], [BR1], [Hu], and [BHR1] (in particular, see [BJT], [BBR], and §2.4.2 of [Hu] or Proposition 7.1 of [BHR1]). However, there is an essential difference here. That is, our map is not assumed to be smooth. So, we can only do it almost everywhere. To reach the bad points, we jump into the domain and use the hypothesis to control the rate of blowing-up.

We first observe that when D ∪ M_{1} is pseudo-concave along M_{1}, then clearly f extends holomorphically across M_{1}. So, without loss of generality, we will always assume the existence of a non empty pseudoconvex piece in M_{1}. Furthermore, using the finite type assumption for M_{1}, we can then see the existence of strongly pseudoconvex points in M_{1} (see, for example, [BHR1]). Since it is now clear that strongly pseudoconvex points form a non empty open subset of M_{1} and since f extends holomorphically across M_{1} \ E_{M_{1}}, by our non constant assumption of f and by a result in [BR3], the Jacobian J_{f} of f is well-defined and non zero on an open dense subset in M_{1}.

Proof of Lemma 2.1: By the definition of G(f(z), λ) and using the assumption that f(M_{1}) ⊂ M_{2}, we have

(2.6)  \overline{f_{n+1}(z)} = G(f(z), \overline{f^{∗}(z)}) for z ∈ M_{1}.

Shrinking the size of M_{1} if necessary, we can choose a basis {L}_{j=1}^{n} of T^{(1,0)}M_{1}, whose coefficients are real analytic in z. As for the smooth mappings case (see [BJT], [BR1]), we apply \overline{L}_{j} to (2.6), to obtain

\overline{L}_{j}\overline{f}_{n+1}(z) = ∑_{l=1}^{n} \frac{∂G}{∂λ_{l}} \overline{L}_{j}\overline{f}_{l}, for z ∉ E_{M_{1}}.

Let J = det(\overline{L}_{j}\overline{f_{l}})_{1≤j,l≤n} and let J be the matrix (\overline{L}_{j}\overline{f_{l}})_{1≤j,l≤n} , which are welldefined on M_{1} \ E_{M_{1}}. We claim that J ≢ 0 on M_{1} \ E_{M_{1}}. To see this, by the above observation, we can assume that there is a strongly pseudoconvex point p_{0} ∈ M_{1} such that f extends biholomorphically to a neighborhood of p_{0} and maps p_{0} to a strongly pseudoconvex point q_{0} of M_{2}. Now, seeking a contradiction suppose J(z) ≡ 0 near p_{0}. Let J_{j}(z) = (L_{j}f_{1}, ⋯, L_{j}f_{n}). By passing to a nearby point, we can assume, without loss of generality, that the semi-continuous function d(z), which is defined as the complex dimension of the vector space V (z) spanned by {J_{j}(z)}, takes a local maximal value, say k (< n), at p_{0}. Here z is close to p_{0} but stays in M_{1}. Moreover, we can assume that {J_{1}(p_{0}), ⋯, J_{k}(p_{0})} forms a basis of V (p_{0}). Now, by the local maximality of d(z) at p_{0} and by the real analyticity of J_{j}(z)′s with respect to z, it is clear, from some simple arguments in linear algebra, that near p_{0} there exist real analytic functions a_{j}(z) (j = 1, ⋯, k) with z ∈ M_{1} sufficiently close to p_{0} such that J_{k+1}(z) = ∑_{j=1}^{k} a_{j}(z)J_{j}(z) for z(∈ M_{1}) ≈ p_{0}. Write T (z) = L_{k+1}(z) − ∑_{j=1}^{k} a_{j}(z)L_{j}(z). Then T (z) annihilates f^{∗}(z) for z(∈ M_{1}) ≈ p_{0}. Since T (p_{0}) ≠ 0 and M_{1} is strongly pseudoconvex at p_{0}, it follows that (dρ_{1}, [T, \overline{T}])(p_{0}) ≠ 0, and thus \{L_{1}, ⋯, L_{n}, [T, \overline{T}]\} forms a basis of T_{p_{0}}M_{1}. Write N(z) for the (1, 0)-component of [T, \overline{T}](z). Noticing that f is holomorphic near p_{0} and [T, \overline{T}]f^{∗} = T ((\overline{T} f^{∗}))−\overline{T} (T f^{∗}) = 0, it is clear that N(p_{0}) annihilates f^{∗}, too. Observe that {L_{1}, ⋯, L_{n}, N}(p_{0}) forms a basis of T_{p_{0}}^{(1,0)}C^{n+1}. Therefore, making use of the fact N(p_{0})(f^{∗}) = 0 and by a simple linear algebra argument, it follows that J_{f}(p_{0}) is given by some constant times \overline{J(p_{0})} and thus equals to 0. This contradicts our choice of p_{0}.

Now, we let Z^{∗} = {z ∈ M_{1} \ E_{M_{1}} : J(z) = 0}. Then, since M_{1} \ E_{M_{1}} has to be connected, we see that M_{1}\Z^{∗} is a dense open subset in M_{1}. In fact, with a little more effort, we will next show that Z^{∗} is contained in a real analytic subset of codimension at least 1. Indeed, notice that J(z) = J(z, \overline{z}, \overline{Df}) can be written as a polynomial in Df = (f_{1}′_{z_{1}}, ⋯, f_{n}′_{z_{n+1}}) with coefficients holomorphic in (z, \overline{z}) for z in a small neighborhood of M_{1}. Since (2.4) annihilates f_{j}, one can easily show that each component of Df can be presented as a rational function of f with coefficients holomorphic near M_{1}. Thus, by some standard algebra arguments involving symmetric functions, one can conclude that for some irreducible polynomial in X: P(z, w; X) = ∑_{j=0}^{N^{∗}} A_{j}(z, w)X^{j}, it holds that P(z, \overline{z}; J(z, \overline{z}, \overline{Df(z)})) = 0 for z ∈ M_{1} \ E_{M_{1}}. Here A_{j}(z, w) are holomorphic for z and \overline{w} in a small neighborhood of M_{1}. Since A_{j}(z, \overline{z}) = 0 on M_{1} if and only if A_{j}(z, \overline{z}) = ρ_{1}(z, \overline{z})h_{j}(z, \overline{z}) for some real analytic function h_{j} near M_{1}, after doing some cancellation, we may assume that not all A_{j}(z, \overline{z}) vanish on M_{1}. Now, since J(z) ≠ 0 on a dense open subset, the zero set of J(z) is obviously contained in the zero set of the coefficient A_{j^{∗}}(z, \overline{z}), where A_{j^{∗}}(z, \overline{z}) ≢ 0 but A_{j}(z, \overline{z}) ≡ 0 for z ∈ M_{1} and for j < j^{∗}. Hence Z^{∗} is contained in the proper real analytic subset of M_{1} consisting of E_{M_{1}} and the zeros of A_{j^{∗}}(z, \overline{z}) in M_{1}.

Away from Z^{∗}, we now have

(2.7)  (\frac{∂G}{∂λ_{1}} , ⋯, \frac{∂G}{∂λ_{n}} )^{t} = J^{−1}(z, \overline{z}, \overline{Df})(\overline{L}_{1}\overline{f}_{n+1}, ⋯, \overline{L}_{n}\overline{f}_{n+1})^{t}.

Writing (2.7) as n scalar equations, applying \overline{L}_{j} to each of them, and keeping doing in this manner, we see, by induction, that for each multi-index α = (α_{1}, ⋯, α_{n}); there are two holomorphic functions g_{α}^{(1)} and g_{α}^{(2)} in

(z, \overline{z}, \overline{f}, ⋯, \overline{D^{β}f}, ⋯)

( where β ≤ |α|) such that for each z ∈ M_{1} \ Z^{∗}, one has g_{α}^{(2)}(z, \overline{z}, ⋯, \overline{D^{β}f}, ⋯) ≠ 0 and

D_{λ}^{α}G(f, \overline{f^{∗}(z)}) = \frac{g_{α}^{(1)}(z, \overline{z}, ⋯, \overline{D^{β}f}, ⋯)}{g_{α}^{(2)}(z, \overline{z}, ⋯, \overline{D^{β}f}, ⋯)}.

Here D^{β} denotes the vector formed by all derivatives of f with order β. We remark that g_{α}^{(j)} is actually a polynomial in (\overline{Df}, ⋯, \overline{D^{β}f}, ⋯) (j = 1, 2) with coefficients real analytic in z for z close to M_{1}. By passing to the limit, we see that the function \frac{g_{α}^{(1)}}{g_{α}^{(2)}} has

a continuous extension to M_{1}, which we will denote by h_{α}(z, \overline{z}, ⋯, \overline{D^{β}f}, ⋯,). Notice also that for ∥w∥, ∥λ∥ ≪ 1, there exits a large constant R so that |D_{λ}^{α}G(w, λ)| <_{∼} α!R^{|α|}.

Denote all the solutions of (2.4) by η_{j,l}(z) (j = 1, ⋯, n + 1, l = 1, ⋯, N_{j}). We now can easily find a totally real submanifold M′ ⊂ M_{1} of dimension n + 1 such that 0 ∈ M′ and M′ ∩ Z^{∗} is contained in a real analytic subset of M′ with codimension at least 1 (see, for example, [Hu] or [BHR1]). Choose H: (C^{n+1}, 0) → (C^{n+1}, 0), a germ of biholomorphism which sends a small open neighborhood of 0 in R^{n+1} into M′ and maps the standard wedge W^{+} into D, W^{−} into D^{c}, respectively. Here the notation D^{c} is as before.

Consider the following equation in X:

∏_{l_{1},··· ,l_{2n+1}} (X − \frac{1}{α!}(D_{λ}^{α}G(η_{1,l_{1}} ∘ H(z), ⋯, η_{n+1,l_{n}} ∘ H(z), ⋯, \overline{η_{n,l_{2n+1}} ∘ H(\overline{z})}))

= X^{N} + ∑_{j<N} c_{j}(z)X^{j} = 0,

where l_{j} and l_{n+1+j} run from 1 to N_{j} for j ≤ n + 1, and N = N_{1}^{2} ⋯ N_{n}^{2}N_{n+1}. c_{j}(z)′s can be seen to be the symmetric functions of some Weierstrass polynomial equations and hence can be seen to be holomorphic near 0. Moreover, one can obtain the Cauchy estimates |c_{j}(z)| <_{∼} R^{N|α|}. Now, we notice that

h̃_{α}(z) = \frac{1}{α!}h_{α}(H(z), \overline{H(\overline{z})}, ⋯, \overline{D^{β}f(H(\overline{z}))}, ⋯)

is a solution of the above equation for almost all z ∈ R^{n+1} near 0. Meanwhile, it clearly extends to a meromorphic function to W^{−} ∩ O_{z}(0); for f is holomorphic over D. Thus, by the uniqueness of holomorphic functions [Pi2], it follows that

(h̃_{α}(z))^{N} + ∑_{j<N} c_{j}(z)(h̃_{α}(z))^{j} = 0,

for all

z ∈ W^{−} ∩ O_{z}(0) \ {the singular set of h̃_{α} in W^{−} ∩ O_{z}(0)}.

In particular, we see that h̃_{α}(z) is bounded. Using the Riemann extension theorem, we conclude that h̃_{α} extends holomorphically to W^{−} ∩ O_{z}(0). Moreover, we have |h̃_{α}(z)| <_{∼} R^{N|α|} for z ∈ W^{−} ∩ O_{z}(0) ; for its coefficients have the same sort of estimates.

Next, fix a small open subset U containing 0. Let

ϕ_{α}^{−}(z) = \frac{1}{α!} (D_{λ}^{α} ∑_{β} h̃_{β}(z) (λ − \overline{f^{∗} ∘ H(\overline{z})})^{β})_{λ=0}

for z ∈ W^{−} ∩ U and let ϕ_{α}^{+}(z) = \frac{1}{α!}D_{λ}^{α}G(f ∘ H(z), λ)|_{λ=0} for z ∈ W^{+} ∩ U. Then it can be seen that |ϕ_{α}^{+}(z)|, |ϕ_{α}^{−}(z)| <_{∼} R′^{|α|} for some large constant R′, where z stays in their defining regions, respectively. Notice that ϕ_{α}^{+} matches up with ϕ_{α}^{−} over U ∩R^{n+1} \{ a thin real analytic subset}. The classic edge of the wedge theorem then indicates that ϕ_{α}^{+} extends to a holomorphic function ϕ_{α} defined over some sufficiently small neighborhood U′ of 0, whose size depends only on the size of the wedges where ϕ_{α}^{±} are defined, and therefore is independent of α (see [Vad] or [BHR1], for example). Moreover, U′ can be filled in by analytic disks with boundary staying in the closure of (W^{+} ∪ W^{−}) ([Val]). So, the maximal principle tells that ϕ_{α} has the same kind of Cauchy estimates as ϕ_{α}^{+} and ϕ_{α}^{−} do. Now, (∑_{α} ϕ_{α}λ^{α}) ∘ H^{−1} clearly gives the holomorphic extension for G(f(z), λ) to O_{z}(0) × O_{λ}(0). The proof of Lemma 2.1 is now complete.

Remark 2.2: (a) As an immediate application of Lemma 2.1, by letting λ = 0 in (2.4), we conclude that f_{n+1}(z) admits a holomorphic extension near 0. Thus, P_{n+1}(z; X) = X − f_{n+1}(z).

(b) From the proof of Lemma 2.1, one can see that the same result holds whenever f extends to one side and M_{1} \ E contains a strongly pseudoconvex point where J_{f} is not 0. In particular, when f extends to one side, this can be applied to the case in which M_{1} and M_{2} are not Levi-flat in C^{2}.

(d) In the following sections, we always assume that G(f(z), λ) is holomorphic over the closure of U_{3} × O_{λ}(0)

3. Branches of F and Segre varieties

In this section, we start to study the branching property of the polynomial equations defining f. The argument will be based on the extensive use of Segre varieties and Lemma 2.1. Before proceeding, we set up the following notations:

Let E still be as before. For each z ∈ U_{3}, we write F(z) for the multiple-valued map formed by extending f from U_{3} ∩D. More precisely, when z ∉ E, f̃(z) ∈ F(z) if there exists a path γ ⊂ U_{3}\E with γ(0) ∈ D, γ(1) = z, and a continuous section G from [0, 1] to the sheaf of germs of holomorphic mappings to C^{n+1}, denoted by (O^{n+1}, π, C^{n+1}), such that G(0) coincides with the germ of f at γ(0) and g(γ(1)) = f̃(z), where g is a representation of G(γ(1)), i.e, the germ of g at γ(1) is G(γ(1)). When z ∈ U_{3} ∩ E, we let F(z) be the set of the limit points of all possible sequences {F(z_{j})} with z_{j} ∉ E and z_{j} → z. For each z ∈ U_{2}, we write F(Q_{z}) = ∪_{w∈Q_{z}∩U_{2}}F(w). We also notice that for each point in F(z), its m−th component is a solution of the equation P_{m}(z; X) = 0 ((2.4)). From this, it follows that when z is close to 0, then F(z) is close to the origin, too. Moreover, Lemma 2.1 now indicates that G(f̃(z), λ) is independent of the choices of f̃(z) ∈ F(z). So, in what follows, we use the notation G(F, λ) to replace G(f(z), λ).

For w ∈ U_{2}, we denote by A_{w} the set {ω ∈ U_{2} : Q_{w} ∩ U_{2} = Q_{ω} ∩ U_{2}}. We mention that A_{w} is invariantly attached to the hypersurafce M_{1}, and will play an important role in our proof of Theorem 1.1. By our finite D-type assumption for M_{1} and by shrinking U_{3}, we can make A_{0} = {0} and A_{z} finite for z ∈ U_{2} (see [DW], [BJT], and [DF1]). Similarly, by making U′_{3} small, we can assume that A′_{0} = {0} and #A′_{w} < ∞ for w ∈ U′_{2}. For more properties of A_{w}, we refer the reader to ([DF1], Lemma 1).

Before proceeding to the proof of the first lemma in this section, we mention that by shrinking U_{2} if necessary, we can assume that F(U_{2}) ⊂ U′_{1}. Meanwhile, it is clear that F(0) = 0. We also retain notations which have been set up so far.

Lemma 3.0 (i) #F(z) is constant for z ∈ U_{3} \ E. (ii) The limit set of F(z_{j}) is F(z_{0}) if z_{j} → z_{0}(∈ U_{3}). (iii) In case n = 1, F(z) = {(X, f_{2}(z)) : P_{1}(z; X) = 0}. (iv) For w_{1}, w_{2} ∈ U_{1}, if the variety Q_{w_{1}} ∩ Q_{w_{2}} ∩ U_{2} contains a point of complex dimension n, then Q_{w_{1}} ∩ U_{2} = Q_{w_{2}} ∩ U_{2}. Thus A_{w_{1}} = A_{w_{2}}.

Proof of Lemma 3.0: (i) is an immediate consequence of what is called the monodromy theorem. (iv) is a consequence of (I.c) and the fact that two connected closed complex submanifolds of a domain coincide if they have an open subset in common. (iii) follows easily from the irreducibility assumption of P_{1} and the following standard argument: For z ∉ E, write F(z) = {(g_{j}(z), f_{2}(z))}_{j=1}^{k}. Then (i) indicates that k is constant. Now, the Riemann extension theorem tells that P̃_{1} = ∏_{j=1}^{k}(X − g_{j}(z)) is a Weiestrass polynomial in X. It, of course, divides P_{1} by the previous observation. So, from the irreducibility of P_{1}, it follows that P_{1} = P̃_{1}, and this clearly completes the proof of (iii). So, it now suffices for us to give the proof of (ii) to complete the proof of Lemma 3.0.

When z_{0} ∉ E, then F(z) can be stratified into k holomorphic branches near z_{0} with k =the generic counting number of F(z). That is, we can write F(z) = {g_{1}(z), ⋯, g_{k}(z)} with g_{j}’s holomorphic near z_{0}. Now, when z_{j} is close enough to z_{0}, then F(z_{j}) = {g_{1}(z_{j}), ⋯, g_{k}(z_{j})}. From this, it is easy to see that lim F(z_{j}) = F(z_{0}).

When z_{0} ∈ E, by passing to a nearby point, we can assume, without loss of generality, that all z_{j}’s are not in E. Now, for another sequence {w_{j}}(⊂ U_{3} \ E) with w_{j} → z_{0}, we first find a family of curves {γ_{j}} such that ∥γ_{j} − z_{0}∥ → 0 and γ_{j}(0) = z_{j}, γ_{j}(1) = w_{j}. Still write F(z) = {g_{1}(z), ⋯, g_{k}(z)} for z ∉ E. Then for each l, there is a continuous map I_{l}(t) from [0, 1], which maps t to a point in F(γ_{j}(t)) such that I_{l}(0) = g_{l}(z_{j}) and I_{l}(1) = g_{σ_{z_{j}} (l)}(w_{j}), where σ_{z_{j}} is a permutation of {1, ⋯, k}. Moreover, it is clear that the m-th component of I_{l}(t) is a solution of P_{m}(γ_{j}(t); X) = 0, as observed above. Now, by Lemma 2.5 of [Mal] and our choices of γ_{j}, it follows that ∥g_{l}(z_{j}) − g_{σ_{z_{j}} (l)}(w_{j})∥ → 0. Therefore, we see that lim F(z_{j}) = lim F(w_{j}), which must be F(z_{0}) by the definition.

Lemma 3.1. After shrinking U_{1}, if necessary, then for each z ∈ U_{1}, it holds that

F(z) ⊂ A′ ; _{f(z)} and F(Q_{z}) ⊂ Q′_{_{f}(z)} for any f̃(z) ∈ F(z).

Proof of Lemma 3.1: We would like to mention that by a simple unique continuation argument and by using the invariant property of Segre varieties, one can easily show that for each nice branch f̃ with f̃|_{D} = f, it holds that f̃(Q_{z}) ⊂ Q_{f(z)} (see also [DF2], for example). The main idea of the proof of the lemma is to use the fact that ρ̃_{2}(F(z), f̃(ω)) is single-valued for each fixed ω, by the above established λ-function.

We let z ∈ U_{1} ∩ D, and assume that z, z^{∗} ∉ E. Here, we use z^{∗} to denote the reflection point of z, i.e, z^{∗} = R(z). We also choose a simply connected smooth curve γ : [0, 1] → U_{3} with γ ⊂ R(U_{1} \ D) \ E and γ(1) = z^{∗}, γ ∩ M_{1} = {γ(0)}. Here, when there is no confusion arising, we also use the letter γ to denote its image set. Moreover, we assume that γ intersects M_{1} transversally at γ(0). Let γ̃ = R(γ)∪γ. Then γ̃ is still a simply connected curve in U_{2} by the discussions in §2. Thickening γ̃ suitably, we can then obtain a simply connected domain, which we will denote by O(⊂ U_{2}). Now, we can well define a holomorphic map f̂ from O, which coincides with f on O ∩ D. Let M_{1c} = {(z, ω) ∈ U_{3}×conj(U_{3}) : ρ̃_{1}(z, ω) = 0} and consider M_{1c}^{∗} = M_{1c}∩{O×conj(O)}. Now, the following function is well-defined and holomorphic over M_{1c}^{∗}:

Ξ(z, ω) = \overline{f_{n+1}(\overline{ω})} − G(F(z), \overline{f̂(\overline{ω})}) (= ρ̃_{2}(F(z), \overline{f̂(\overline{ω})}))

which vanishes on M_{1}^{∗} = \{(a, \overline{a}) : a ∈ M_{1} ∩ U_{3}\}. Notice that M_{1}^{∗} is a totally real subset in M_{1c}^{∗} with maximal dimension, we conclude from the uniqueness property of holomorphic functions [Pi2] that Ξ(z, ω) ≡ 0 in the union, denoted by M_{1c}^{∗∗}, of the connected components of M_{1c}^{∗} which have non-empty intersections with M_{1}^{∗}. By our choice of γ̃, we see that (z, \overline{z^{∗}}), (z^{∗}, \overline{z}) ∈ M_{1c}^{∗∗}; for (γ(t), \overline{R(γ(t)})) ∈ M_{1c}^{∗} (R(γ(t)), \overline{γ(t)}) ∈ M_{1c}^{∗} (t ∈ [0, 1]) connect them to M_{1}^{∗}, respectively. Therefore, (z, Conj(Q_{z} ∩ O(z^{∗}))) ⊂ M_{1c}^{∗}, (z^{∗}, Conj(Q_{z^{∗}} ∩ O(z))) ⊂ M_{1c}^{∗} by the definition of Segre varieties. This implies that G(F(z), \overline{f̂(ω)}) = \overline{f_{n+1}(ω)} for ω ∈ Q_{z} ∩ O(z^{∗}); and G(F(z^{∗}), \overline{f̂(ω)}) = \overline{f_{n+1}(ω)} for ω ∈ Q_{z^{∗}} ∩O(z). Equivalently, we have ρ̃_{2}(F(z), \overline{f̂(ω)}) = 0 for ω ∈ Q_{z} ∩ O(z^{∗}); and ρ̃_{2}(F(z^{∗}), \overline{f̂(ω)}) = 0 for ω ∈ Q_{z^{∗}} ∩ O(z).

From a basic fact of Segre varieties: w ∈ Q_{z} if and only if z ∈ Q_{w}; it now follows

that f̂(Q_{z} ∩ O(z^{∗})) ⊂ ∩Q′ ; _{f(z)} with f̃(z) ∈ F(z); and f̂(Q_{z^{∗}} ∩ O(z)) ⊂ ∩Q′_{_{f}(z^{∗})} with

f̃(z^{∗}) ∈ F(z^{∗}).

By slightly perturbing z if necessary, we assume momentarily that Jf̂ ≠ 0 near z^{∗}. Now, since f̂(Q_{z} ∩ O(z^{∗})) ⊂ Q′_{_{f}(z)} for any f̃(z) ∈ F(z), and since each of them is

a connected complex submanifold of dimension n near f̂(z^{∗}), all these submanifolds therefore coincide near f̂(z^{∗}). Hence, it follows from Lemma 3.0 (iv) that all Q′_{_{f}(z)}∩U_{2}

are the same. So, F(z) ⊂ A′_{_{f}(z)} for any given f̃(z) ∈ F(z).

Assume also that J_{f}(z) ≠ 0. In a similar manner, we then also see that F(z^{∗}) ⊂ A′_{_{f}(z^{∗})} for any given f̃(z^{∗}) ∈ F(z^{∗}).

Next, we note that ρ̃_{2}(F(z), \overline{f̂}(ω)) = 0 if and only if ρ_{2}(F(z), \overline{f̂}(z)) = 0. Let q ∈ F(z). Then, by what we just obtained and by the reality of ρ_{2}, we have ρ_{2}(f̂(ω), \overline{q}) = 0 and therefore, ρ̃_{2}(f̂(ω), \overline{q}) = 0 for ω ∈ Q_{z} ∩ O(z^{∗}). Now, since Lemma 2.1 indicates that ρ̃_{2}(F(ω), \overline{q}) is well defined and holomorphic for ω ∈ U_{3}, we conclude that ρ̃_{2}(F(ω), \overline{q}) = 0 for ω ∈ U_{2} ∩ Q_{z}. This implies that F(Q_{z}) ⊂ Q′_{q}.

In a similar manner, we can also show that F(Q_{z^{∗}}) ⊂ Q′_{q^{∗}} for any q^{∗} ∈ F(z^{∗}). Now, we will complete the proof of Lemma 3.1 by passing to the limit. Let z close to 0 be such that either z ∈ E ∪ R(E) ∪ M_{1} or J_{f}(z) = 0, or J_{f}(z^{∗}) = 0.

By what we did above, we can find a sequence {z_{j}} with z_{j} → z so that F(z_{j}) ⊂ A′_{_{f}(z_{j})}

for any f̃(z_{j}) ∈ F(z_{j}). In the other words, Q′_{_{f}(z_{j})} = Q′_{f̂(z_{j})} for any f̃(z_{j}), f̂(z_{j}) ∈ F(z_{j}).

By Lemma (3.0) (ii) and (I.d), it then follows that Q′_{_{f}(z)} = Q′_{f̂(z)} for any f̃(z), f̂(z) ∈

F(z), i.e, F(z) ∈ Q′_{_{f}(z)} for any f̃(z) ∈ F(z). Similarly, we also have F(Q_{z}) ⊂ Q′_{_{f}(z)}

for any f̃(z) ∈ F(z).

Hence, after shrinking U_{1} one more time if necessary, we see the proof of Lemma 3.1.

Remark 3.2 (a) By Lemma 3.1, we can now well-define Q′_{F(z)} to be Q′_{q} and A′_{F(z)} = A′_{q} for some q ∈ F(z). Then Lemma 3.1 can be written as F(z) ⊂ A′_{F(z)} and F(Q_{z}) ⊂ Q′_{F(z)}.

(b) As an application of Lemma 3.1, we conclude that when A′_{w} is just a single point for w ≈ 0, then f extends holomorphically near 0. This is the case when the target point is a non-degenerate point or has some special bi-type property. Define A′ by sending each point w to A′_{w}. One can see that if f does not allow holomorphic extension, then A′ branches at 0 (we will make this more precise in §5). It is this fact that links the branching points of F with the singular points of A′, which will be the key observation for the proof of Theorem 1.1.

More specifically, let D_{1} = {(z_{1}, z_{2}) : |z_{1}|^{2} + |z_{2}|^{2} < 1} and D_{2} = {(z_{1}, z_{2}) : |z_{1}|^{4} + |z_{2}|^{2} < 1\}. Let G = (\sqrt{z_{1}}, z_{2}) be the multiple-valued map from D_{1} to D_{2}. Then the branching locus of G is given by Z = {(0, z_{2})}, and G(Z) = {(0, z_{2})}. We observe that G(Z) is exactly the branching locus of the A′-map of ∂D_{2}.

Lemma 3.3: (i) Let M_{2} be as given in (2.2). Let Ω^{±} be defined by ±ρ_{2} < 0 respectively. Write n̂^{±} = {b = (0′, b_{n+1}) : ±b_{n+1} < 0} with |b_{n+1}| small. Then A′_{b} ⊂ Ω^{±} when b ∈ n̂^{±}, respectively. (ii) If for some q ∈ F(z), it holds that q ∈ M_{2}, then F(z) ⊂ M_{2}. Hence, F(M_{1}∩U_{1}) ⊂ M_{2}. (iii) f_{n+1}(z) = z_{n+1}^{k}g(z) for some holomorphic function g(z) defined near 0. (iv) {F^{−1}(0)} ∩ U_{2} = {0} after shrinking U_{2}, i.e , f̃(z) ≠ 0 for any z(∈ U_{2}) ≠ 0 and f̃(z) ∈ F(z). Moreover F^{−1}(w) ∩ U_{2} is a finite set for any w ≈ 0; and for any analytic variety E passing through 0 ∈ U_{1}, F(E) also gives the germ of an analytic variety at 0 ∈ U′_{1} with dim_{0} E = dim_{0}(F(E)).

(v) After shrinking U_{1} if necessary, then for any z ∈ U_{1}, either F(z) ⊂ Ω^{+} or F(z) ⊂ Ω^{−}, or F(z) ⊂ M_{2}.

Proof of Lemma 3.3: (i): Let η = (η′, η_{n+1}) ∈ A′_{b}, i.e, Q′_{η} ∩ U′_{2} = Q′_{b} ∩ U′_{2}. We see that

\{(w′, w_{n+1}) ∈ U′_{2} : w_{n+1} + \overline{η_{n+1}} + ρ_{2}^{∗}(w′, \overline{η}′, \frac{w_{n+1} − \overline{η_{n+1}}}{2i}) = 0\}

= \{(w′, w_{n+1}) ∈ U′_{2} : w_{n+1} + \overline{b_{n+1}} + ρ_{2}^{∗}(w′, \overline{b}′, \frac{w_{n+1} − \overline{b_{n+1}}}{2i}) = 0\}.,

where ρ^{∗} = ∑_{j≥0} ψ_{j}(w′, \overline{w′})(Imw_{n+1})^{j}. Letting w′ = 0, we see that η_{n+1} = b_{n+1}. Meanwhile, since \{(w′, −\overline{η_{n+1}})\} ∩ U′_{2} = Q′_{η}, it follows that ρ_{2}^{∗}(w′, \overline{η′}, i\overline{η_{n+1}}) = 0 for all w′ with (w′, b_{n+1}) ∈ U′_{2}. We see, in particular, that ρ_{2}^{∗}(η′, \overline{η′}, i\overline{η_{n+1}}) = 0. Therefore, ψ_{0}(η′, \overline{η′}) = − ∑_{j>0}^{∞} ψ_{j}(η′, \overline{η′})(i\overline{η_{n+1}})^{j} = o(|η_{n+1}|). Now, ρ_{2}(η, \overline{η}) = η_{n+1} + \overline{η_{n+1}} + ρ_{2}^{∗}(η′, \overline{η′}, \frac{η_{n+1}−\overline{η_{n+1}}}{2i}) = η_{n+1} + \overline{η_{n+1}} + ψ_{0}(η′, \overline{η′}) + o(|η_{n+1}|) = 2η_{n+1} + o(η_{n+1}), which is positive when b_{n+1} = η_{n+1} > 0 and small; and negative when b_{n+1} < 0. This gives the proof of part (i).

(ii). Let q ∈ M_{2} ∩ F(z). Then q ∈ Q′_{q} = Q′_{_{f}(z)} for each f̃(z) ∈ F(z). Thus

f̃(z) ∈ Q′_{q} = Q′_{_{f}(z)}, by Lemma 3.1. So, f̃(z) ∈ M_{2} (see [DF1]).

(iii). By Lemma 3.1, it follows that F(Q_{0}) ⊂ Q′_{F(0)} = Q′_{0}. Thus, we see that f_{n+1}(z′, 0) ≡ 0. Since f_{n+1} is holomorphic near 0, we conclude that f_{n+1}(z′, z_{n+1}) = z_{n+1}^{k}g(z) for some holomorphic function g near 0 and some positive integer k.

(iv). The following argument will be based on some facts on local analytic covers and the elimination theorem (see [Chi], [Wh], or [GR]).

We start by setting up some notations:

Let Ṽ = {(z, w) ≈ (0, 0) : w_{n+1} = f_{n+1}(z), P_{j}(z, w_{j}) = 0, j = 1, ⋯, n} and let V be its irreducible component at (0, 0) which contains Γ_{f} = {(z, w) : w = f(z), z(≈ 0) ∈ D}. Let π and π′ be the natural projections from V to the first and second copies of C^{n+1}, respectively. Since P_{j}(0; X) = 0 has only zero solutions, i.e, π^{−1}(0) = 0, π is locally proper. Thus, we can choose a small neighborhood U of 0 such that (a) π^{−1}(U) is irreducible; (b) π is proper from π^{−1}(U); and (c) π is a sheeted covering map from π^{−1}(U) \ π^{−1}(E) to U \ E (see, for example , Theorem 1 of [Chi], pp 122). In the following discussion, we make U_{2} ⊃ U ⊃ U_{1} and restrict π to π^{−1}(U).

Now, by using some standard arguments as appeared in the proof of Lemma 3.0 (iii) and using the irreducibility of π^{−1}(U), one can see that F(z) = π′ ∘ π^{−1}(z), for z ∈ U_{2}. (Otherwise, we can form a proper analytic subvariety of V (of the same dimension), which is defined by {(z, w) : w ∈ F(z)}).

On the other hand, it obviously holds that F^{−1}(0) ∩ U = π(π′^{−1}(0) ∩ π^{−1}(U)). Thus, if 0 is an accumulation point of Y = F^{−1}(0), then Y ∩ U is an analytic variety of U (by the elimination theorem) and it must have positive dimension at 0. We will assume this and seek a contradiction.

Then, Y ∩ U contains some holomorphic curve Y^{∗} ⊂ U_{1} parametrized by z′ = ϕ(t), z_{n+1} = ψ(t), (t ∈ Δ, the unit disk in C^{1}) with ϕ(0) = ψ(0) = 0, ∥dϕ∥+∥dψ∥ ≠ 0 for t ≠ 0. For each z ∈ Y ∩ U_{1}, F(Q_{z}) ⊂ Q′_{0}, by Lemma 3.1. We claim that ∪Q_{z} with z ∈ Y^{∗} fills in an open subset in C^{n+1}. This then gives us a contradiction; for we assumed that J_{f} ≢ 0 and Q′_{0} is a complex hypersurface.

To see the size of U^{∗} = ∪Q_{z} with z ∈ Y^{∗}, we note that M_{1} is also defined by \overline{ρ̃_{1}(z, \overline{z})} = 0. Therefore, it can be seen that Q_{z} can be parametrized by

z_{n+1} = −\overline{ψ(t)} − ∑_{j=0}^{∞} \overline{ϕ̃_{j}}(\overline{ϕ(t)}, z′)\overline{ψ(t)}^{j} ≡ g(z′, \overline{t}).

Here and in what follows, we use the notation \overline{h}(z) for the function \overline{h(\overline{z})}. So, U^{∗} can be parametrized by the map

T (z′, \overline{t}) : C^{n} × Δ → C^{n+1}

T (z′, \overline{t}) = (z′, −\overline{ψ(t)}−∑_{j=0}^{∞} \overline{ϕ̃_{j}}(\overline{ϕ(t)}, z′)\overline{ψ(t)}^{j}). To see that T (z, t) is a biholomorphism at certain point (z′, \overline{t})(≈ (0, 0)), it suffices to show that \frac{∂g}{∂\overline{t}} ≠ 0 at (z′, \overline{t}). Indeed, this can be argued as follows: When ψ(t) ≢ 0, we can simply choose (0′, t) for some t with ψ′(t) ≠ 0; when ψ ≡ 0, if \frac{∂g}{∂\overline{t}} ≡ 0, then for any given z′ with |z′| small, \overline{ϕ̃_{0}}(\overline{ϕ(t)}, z′)

is independent of t. Hence it has to be 0; for ϕ(0) = 0 and thus \overline{ϕ̃_{0}}(ϕ(0), z′) = 0. Letting z′ = ϕ(t), it follows that \overline{ϕ̃_{0}}(\overline{ϕ(t)}, ϕ(t)) ≡ 0. This contradicts the finite type assumption on M_{1}; for it implies that Y^{∗} = (ϕ(t), 0) stays inside M_{1}. So, by making U_{2} small, it holds that F^{−1}(0) ∩ U_{2} = {0}.

Therefore, we conclude that 0 is an isolated point of π′^{−1}(0). Hence, π′ is locally finite near 0. Moreover, F^{−1}, when restricted as a map from O(0) to U_{2}, maps a small neighborhood 0 ∈ U′_{1} into a small neighborhood of 0 ∈ U_{1}.

Now, shrinking U_{2} if necessary, we assume that q ≈ 0, F^{−1}(q) ∩ U_{2} = π(π′^{−1}(q)) is finite. By the elimination theorem ([Chi], Theorem 1 of pp122), there exists an open neighborhood U′ of 0 such that π′ is proper from π′^{−1}(U′) to U′; and away from a proper analytic set, π′ gives a sheeted covering map. In particular, π′ is an open mapping (see [GR], Lemma 6 pp 102, and notice the openness follows from the onto part of that lemma).

In the following discussion, we make F(U) ⊂ U′, and we restrict π′ to π′^{−1}(U′). By the above discussion, we also observe the fact: π^{−1}(U) ⊂ π′^{−1}(U′).

Using again the elimination theorem and noting that π, π′ are local analytic covering maps, it follows that for any analytic variety E ⊂ U, F(E) ∩ π′(π^{−1}(U)) = π′(π^{−1}(E)∩π′(π^{−1}(U))) is an analytic variety in π′(π^{−1}(U)) with the same dimension at the origin (see for example, Theorem 11 E pp 68 of [Wh]; or Theorem 1, pp122, of [Chi]). This completes the proof of (iv).

(v) Since π is proper, it is also closed. Also, both are open mappings; for they are local analytic covering mappings, too. (See [GR], Lemma 6, pp102)

For any closed subset B of U′, we first notice that

F(B) ∩ U = π (π′^{−1}(B) ∩ π^{−1}(U)) .

Since π′^{−1}(B) is closed in π′^{−1}(B) ∩ π′^{−1}(U′), it is closed in π′^{−1}(B) ∩ π^{−1}(U) by the above arrangement. By using the closeness of π, it follows that F^{−1}(B) is closed in U. In particular, we see that M̃_{1} ∩ U = F^{−1}(M_{2} ∩ U′) is closed in U. Clearly, F(M̃_{1}) ⊂ M_{2} by part (ii) of this lemma.

Now, let U^{0} = U \M̃_{1}. Then it is open. We notice that ∂U^{0} is contained in ∂U ∪M̃_{1}. Since F = π′ ∘π^{−1}, the openness of π′ implies the openness of F from U as a multiplevalued map. Therefore it sends interior points to interior points. On the other hand, using the weak continuity of F (Lemma 3.0 (ii)), one sees that F maps the closure of U^{0} to the closure of F(U^{0}). Hence, a simple topological argument shows that the boundary of F(U^{0}) is contained in F(∂U) ∪ M_{2}. Since F^{−1}(0) ∩ U_{2} = {0}, it follows easily that 0 is not contained in the closure of F(∂U). Hence, we see that there is a small ball, denoted by B_{0}, centered at the origin such that B_{0} ∩ F(∂U) = ∅. Let U^{∗} be a small neighborhood of 0 such that F(U^{∗}) ⊂⊂ B_{0}. If U^{0} ∩ U^{∗} = ∅, then Part (ii) of this lemma shows that F(U^{∗}) ⊂ F(M̃_{1}) ⊂ M_{2} and thus we are done. So, without loss of generality, we assume that U^{∗} ∩ U^{0} ≠ ∅. Let D^{∗} be a connected component of U^{0} with U^{∗} ∩ U^{0} ≠ ∅. Note that ∂F(D^{∗}) ∩ B_{0} = ∅. Then a simple topological argument indicates that either F(D^{∗}) ⊃ B_{0} ∩ Ω^{+} or F(D^{∗}) ⊃ Ω^{−} ∩ B_{0} (see, for example, [BHR2]). Without loss of generality, we assume the first case. Then, there exists a point p_{0} ∈ D^{∗} and certain f̃(p_{0}) ∈ F(p_{0}) such that f̃(p_{0}) ∈ n̂^{+} and f̃(p_{0}) is close to 0. By the first part of this lemma and Lemma 3.1, we see that F(p_{0}) ⊂ Ω^{+}. Now, for any z ∈ D^{∗} \ E, let γ be a curve in D^{∗} with γ(0) = p_{0}, γ(1) = z, and γ((0, 1)) ∩ E = ∅. Then for any f̃(z) ∈ F(z), there is a branch f̂ of F, which is continuous on γ and f̂(z) coincides with f̃(z). Now, since f̂ ∘ γ does not meet M_{2} by our choice of U^{0}, and f̂ ∘ γ(t) ∈ Ω^{+} for t close enough to 0 (this can be seen by the fact that all limit points of f̂(γ(t)) (t → 0) are in F(p_{0})), we conclude that f̂(z) and thus f̃(z) have to be in Ω^{+}. For z ∈ E ∩ D^{∗}, by passing to a limit and noting F(z) ∩ M_{2} = ∅, we can also see that F(z) ⊂ Ω^{+}. Thus the proof of the last part of Lemma 3.3 is complete if we make U_{1} small.

Remark 3.4: The following observations will be used in the later discussions:

(a) From the proof of Lemma 3.3 (iv), we also notice the following property of F: There exists an open neighborhood U′_{0} of 0 such that U_{0} = F^{−1}(U′_{0}) ∩ U_{2} is an open subset of U_{1}. Moreover, F|_{U_{1}→F(U_{1})} and F^{−1}|_{U′_{0}→U_{0}} are multiple-valued open mappings.

(b) Let (V, π, π(V)) be the m-sheeted analytic covering space, as in Lemma 3.3 (iv). Let z_{0} ∈ E ∩ π(V) be close to the origin. Then, it is easy to see that in case #π^{−1}(z_{0}) < m, then there are two distinct sequences ξ_{j}, η_{j} ∈ π^{−1}(z_{j}) ⊂ V such that z_{j} → z_{0}, ξ_{j}, η_{j} → w_{0} ∈ π^{−1}(z_{0}).

(c) Still let V as above. Write (Y, σ, V) for the standard normalization of V (see [Wh], Chapter 8). We remark that after making V small, σ^{−1}(0) is a single point. Write E_{0} = π ∘ σ(S), where

S = {x ∈ Y : either x is singular or x is smooth but d_{x}σ is singular}.

Write S_{0} = (π∘σ)^{−1}(π∘σ)(S). Then π∘σ is a local biholomorphic mapping from Y \S_{0} to its image and gives a finitely sheeted covering mapping away from the singular set. (see [Wh], Chapter 8 and [GR], pp 108). Therefore, if V is singular at 0, i.e, f is not holomorphic at 0, then E_{0} = π ∘ σ(S) is an analytic variety of codimension 1 at 0. ( This follows from the fact that any finitely sheeted analytic covering space with ‘genuine’ singular set of codimension ≥ 2 is trivial) (see the following observation in Remark 3.4 (d)). Now, for each z ∈ E_{0} ∩ O(0)(⊂ E) and any small neighborhood U_{z} of z, the basic property for the normal space ([Wh], in particular Lemma 2F pp 254 ) indicates that V can not be smooth at π^{−1}(z). Therefore, #π^{−1}(z) < m.

(d) More generally, let P (z, X) = X^{N} + ∑_{j<N} a(z)X^{j} = 0 be an irreducible polynomial equation in X for z near 0. Let Z^{∗} be its branching locus. A point z_{0} is called a genuine branching point of P = 0 if P (z, X) can not be factorized into linear functions in X in the Noetherian ring O_{z_{0}}. Write Z for the collection of all genuine branching locus of P. The normalization argument as in (c) shows that Z is an analytic variety near the origin. Moreover, it can be seen that the following monodromy theorem holds: Let γ_{1}, γ_{2} be loops in O(0) \ Z, which are based at p_{0} ∈ O(0) \ Z^{∗} and are homotopic to each other in O(0) \ Z, relative to the base point. Then, any analytic continuation of the roots of P along γ_{1}, γ_{2}, starting with the same initial value, will end up with the same value when coming back to p_{0}. From this fact, if follows that in case Z ≠ ∅, Z must be of codimension 1 everywhere.

4. A local Hopf lemma and a preservation principle of F

Before proceeding further, we need to strengthen Lemma 3.3 (iii) to the following version:

Lemma 4.1: f_{n+1}(z) = z_{n+1}g(z) with g(0) ≠ 0.

This sort of the Hopf lemma was established in [BR2] in the case when the map is assumed to be smooth. Since we now have a nice control of the branches of F and we know that f_{n+1} is holomorphic, Lemma 4.1 can be proved by using the same approach and ideas as in [BR2]. Because our emphasis of the present paper is the two dimensional case, for completeness, we will present a detailed proof of Lemma 4.1 when n = 1 and leave the easy modification in general dimensions to Remark 4.1′.

We also mention that when f is a CR-homeomorphism, the proof of Lemma 4.1 readily follows from Lemma 3.3 (iii). To see this, write f^{−1} = (⋯, f_{n+1}^{−1}). Then the same argument as above shows that f_{n+1}^{−1}(w) = w^{m}g^{∗}(w) for some m ≥ 1. Thus z_{n+1} = z_{n+1}^{km}O(1). So k = m = 1.

Proof of Lemma 4.1: We will use the approach appeared in [BR2]. We assume n = 1. Seeking a contradiction, we suppose that f_{2}(z) = z_{2}^{k}g(z) with k ≥ 2.

We start with the function G(F, λ), which has the following property:

\overline{f_{2}(ω)} = G(F(z), \overline{f̃_{1}(ω)})

for any z ∈ Q_{ω} and (f̃_{1}, f_{2}) ∈ F. Write the defining equation of M_{1} in the following form (see (2.1)): t = ϕ(z_{1}, \overline{z_{1}}, s) with t = 2Rez_{2}, s = Imz_{2}, and ϕ = ρ_{1} − t. Clearly, ϕ(0, \overline{z_{1}}, s) = ϕ(z_{1}, 0, s) ≡ 0. Then z ∈ Q_{ω} reads as

z_{2} + \overline{ω_{2}} = ϕ(z_{1}, \overline{ω_{1}}, \frac{z_{2} − \overline{ω_{2}}}{2i}).

We let τ = −is + \frac{1}{2}ϕ(z_{1}, ξ, s) and R(z_{1}, ξ, τ) = is + \frac{1}{2}ϕ(z_{1}, ξ, s). As in [BR2], using the implicit function theorem, we can find a holomorphic function μ(z_{1}, ξ) near 0 such that R(z_{1}, ξ, μ(z_{1}, ξ)) ≡ 0 and μ(z_{1}, 0) = μ(0, ξ) ≡ 0. It is easy to verify that μ(z_{1}, ξ) ≢ 0 and thus \frac{∂μ}{∂z_{1}} (z_{1}, ξ) ≢ 0; for, otherwise, it implies that ϕ_{0} ≡ 0 and thus contradicts the finite type assumption of M_{1}. Now, one can directly verify that (z_{1}, R(z_{1}, ξ, τ)) ∈ Q\overline{(ξ,τ)}. So, we have \overline{(ξ, τ)} ∈ Q_{(z_{1},R(z_{1},ξ,τ))} and thus

\overline{f_{2}(z_{1}, R(z_{1}, ξ, τ))} = G(F(\overline{ξ}, \overline{τ}), \overline{f̃_{1}(z_{1}, R(z_{1}, ξ, τ)}).

As in the previous section, we write \overline{f̃_{1}}(z) = \overline{f̃_{1}(\overline{z})} and use the same notation for \overline{f_{2}}(z). Then

(4.1)  \overline{f_{2}}(\overline{z_{1}}, \overline{R(z_{1}, ξ, τ)}) = G(F(\overline{ξ}, \overline{τ}), \overline{f̃_{1}}(\overline{z_{1}}, \overline{R(z_{1}, ξ, τ)}).

We still let E be the branching locus of F, i.e, E = {z ∈ U_{3} : P_{1}(z, X) = \frac{∂P_{1}}{∂X} (z, X) = 0 for some X\}; and write F(z) = (F_{1}(z), f_{2}(z)).

We will assume momentarily the fact that 0 is an isolated point of E ∩Q_{0} = E ∩{z : z_{2} = 0}, which will be proved in Lemma 5.1 (a) of §5. (In n-dimensions, we observe the last statement of Lemma 5.1 (a) holds without Condition S).

Since {F^{−1}(w)} is finite for w ≈ 0 (Lemma 3.3 (iv)), we see that no branch of F is constant on an open subset of Q_{0}. We also notice, by Lemma 3.0 (iii), that F_{1}(z_{1}, 0) consists of exactly the solutions of P_{1}((z_{1}, 0); X) = 0. Using the Puiseaux expansion, we can write F_{1}(z_{1}, 0) = g(z_{1}^{1/k}), where g is a holomorphic function with g(z_{1}) ≠ 0 for each z_{1}(≠ 0) sufficiently close to 0.

Now, for each non zero z_{1} close to the origin, there is a small neighborhood O((z_{1}, 0)) near (z_{1}, 0) such that we can stratify F into several non-constant holomorphic branches. Next, we choose ξ sufficiently small and then let τ be sufficiently close to μ(z_{1}, ξ). Then R(z_{1}, ξ, τ) ≈ 0, too. After letting f̃ be a holomorphic branch of F over O((z_{1}, 0)), we then can take the derivative with respect to \overline{z}_{1} in (4.1), to obtain

(4.2)  \frac{∂\overline{f_{2}}}{∂z_{1}} + \frac{∂\overline{f_{2}}}{∂z_{2}} \overline{R′_{z_{1}}(z_{1}, ξ, τ)} = \frac{∂G}{∂λ}(\frac{∂\overline{f̃_{1}}}{∂z_{1}} + \frac{∂\overline{f̃_{1}}}{∂z_{2}} \overline{R′_{z_{1}}(z_{1}, ξ, τ)}).

Let τ = μ(z_{1}, ξ) in (4.2) and notice that f_{2} = z_{2}^{k}g with k ≥ 2. We obtain

(4.3)  \frac{∂G}{∂λ}(\frac{∂\overline{f̃_{1}}}{∂z_{1}} (\overline{z_{1}}, 0) + \frac{∂\overline{f̃_{1}}}{∂z_{2}} (z_{1}, 0)\overline{R′_{z_{1}}(z_{1}, ξ, μ(z_{1}, ξ))}) = 0.

By the above discussion, we see that \frac{∂\overline{_{f_{1}}}}{∂z_{1}} (\overline{z_{1}}, 0) ≠ 0 for z_{1} ≠ 0. So, for each z_{1} with |z_{1}| small, when ξ is chosen so that |ξ| is sufficiently smaller than |z_{1}|, using the fact that R′_{z_{1}}(z_{1}, 0, τ) = 0, we see that R′_{z_{1}}(z_{1}, ξ, μ(z_{1}, ξ)) ≈ 0. Moreover,

\frac{∂\overline{f̃_{1}}}{∂z_{1}} (\overline{z_{1}}, 0) + \frac{∂\overline{f̃_{1}}}{∂z_{2}} (z_{1}, 0)\overline{R′_{z_{1}}(z_{1}, ξ, μ(z_{1}, ξ))} ≠ 0.

Hence it follows from (4.3), that

\frac{∂G}{∂λ}(F(\overline{ξ}, \overline{μ(z_{1}, ξ)}), \overline{f̃_{1}}(\overline{z_{1}}, 0)) ≡ 0.

On the other hand, similar to an argument in [BR2], letting τ = μ(z_{1}, ξ) in (4.1), we have

G(F(\overline{ξ}, \overline{μ(z_{1}, ξ)}), \overline{f̃_{1}}(\overline{z_{1}}, 0)) ≡ 0.

Now, let q_{1} = G′_{λ}(w_{1}, 0, λ) and q_{0} = G(w_{1}, 0, λ). Notice that G(w_{1}, w_{2}, λ) = −w_{2} + q_{0} − ∑_{j>0} ψ̃_{j}(w_{1}, λ)w_{2}^{j} and q_{1} = (q_{0})′_{λ}. We see that the equations G(w_{1}, w_{2}, λ) = 0, G′_{λ}(w_{1}, w_{2}, λ) = 0 can be solved as w_{2} = q_{0}h^{∗}(q_{0}, w_{1}, λ)) and q_{1} = q_{0}ψ^{∗}(w_{1}, λ, q_{0}) for certain holomorphic functions h^{∗}, ψ^{∗}. In the following discussions, we always let τ = μ(z_{1}, ξ).

Let w_{1}(z_{1}, ξ) = f̂_{1}(\overline{ξ}, \overline{τ}) with f̂_{1}(\overline{ξ}, \overline{τ}) ∈ F(\overline{ξ}, \overline{τ}), and let λ(z_{1}) = \overline{f̃_{1}}(\overline{z_{1}}, 0) in the above formulas, we obtain the following equality

(4.4)  q_{1}(f̂_{1}(\overline{ξ}, \overline{τ}), \overline{f̃_{1}}(\overline{z_{1}}, 0)) = q_{0}(f̂_{1}(\overline{ξ}, \overline{τ}), \overline{f̃_{1}}(\overline{z_{1}}, 0)) × ψ^{∗}(f̂_{1}(\overline{ξ}, \overline{τ}), \overline{f̃_{1}}(\overline{z_{1}}, 0), q_{0}).

Here for clarity, we summarize the situations in which (4.4) makes sense and holds:

(a) z_{1} is an arbitrarily given non zero complex number near the origin. (b) f̃_{1} is some holomorphic branch of F over some small neighborhood O((z_{1}, 0)) of (z_{1}, 0), whose size depends on z_{1}. (c) ξ is taken in a small neighborhood O(z_{1}, f̃_{1}) of 0, whose size depends on the choice of O((z_{1}, 0)) and the choice of f̃_{1}. (d) f̂(\overline{ξ}, \overline{τ}) is any point in F_{1}(\overline{ξ}, \overline{τ}).

Since f_{2} ≢ 0 and \frac{∂μ}{∂z_{1}} ≢ 0, Z^{∗} = \{(z_{1}, ξ) : f_{2}(\overline{ξ}, \overline{μ(z_{1}, ξ)}) = 0\} is an analytic variety of dimension 1. So, we can assume that Z^{∗} has only one component of the form {(a, ξ)} near the origin, i.e, the one with a = 0. We therefore see that f_{2}(\overline{ξ}, \overline{μ(z_{1}, ξ)}), when regarded as a function in \overline{ξ}, is not constant for each fixed z_{1} ≠ 0; for if it were constant, then it would be identically 0. Meanwhile, noticing that

(f̂_{1}(\overline{ξ}, \overline{τ}), f_{2}(\overline{ξ}, \overline{τ})) ∈ Q′_{(_{f_{1}}(z_{1},0),0)}, we see that

(4.5)  f_{2}(\overline{ξ}, \overline{μ(z_{1}, ξ)}) = −ψ̃_{0}(f̂_{1}(\overline{ξ}, \overline{μ(z_{1}, ξ)}), \overline{f̃_{1}}(\overline{z_{1}}, 0)).

Let Z^{∗∗} ⊂ C^{2} be the zero set of q_{1}(w_{1}, λ) − q_{0}(w_{1}, λ)ψ^{∗}(w_{1}, λ, q_{0}(w_{1}, λ)). If it is not of dimension 2, then it has only finitely many irreducible one dimensional components near the origin. Thus, we can assume that for any λ(≠ 0) with |λ| small, Z^{∗∗} ∩ {(w_{1}, λ)} is of dimension 0. Returning to (4.4), one sees that for each z_{1} ≠ 0 close to the origin, f̂_{1}(\overline{ξ}, \overline{μ(z_{1}, ξ)}) takes only finitely many values for ξ ∈ O(z_{1}, f̃). This obviously contradicts (4.5); for f_{2}(\overline{ξ}, \overline{μ}(z_{1}, ξ)) can not take only finitely many values when ξ ∈ O(z_{1}, f̃_{1}) by the above argument.

Therefore, we showed that (q_{0})′_{λ} = q_{1}(w_{1}, λ) ≡ q_{0}(w_{1}, λ)ψ^{∗}(w_{1}, λ, q_{0}(w_{1}, λ)). Now, applying the same argument as in [BR2], we can see, by the basic theory of ODE and by the initial condition q_{0}(w_{1}, 0) = G(w_{1}, 0, 0) = 0, that q_{0} ≡ 0. This contradicts the finite type assumption of M_{1}.

Remark 4.1′: The above proof can be adapted to the general dimensions as in [BR2], almost without any change. However, we would like to include here some details just for convenience of the reader:

We first replace z_{1} by z′ ∈ C^{n} and define, in the same manner, the functions τ, μ, R. We then have, by Lemma 3.1, the following

\overline{f_{n+1}(z′, R(z′, ξ, τ))} = G(F(\overline{ξ}, \overline{τ}), \overline{f̃^{∗}(z′, R(z′, ξ, τ)});

for \overline{(ξ, τ)} ∈ Q_{(z′,R(z′,ξ,τ))} with ξ ∈ C^{n}.

Let E′ = E_{0} ∩ Q_{0}. By Lemma 5.1 (a), it is a proper analytic variety of Q_{0}. In the same manner, we let z′ ∈ Q_{0} \ E′ be close to 0 and choose a small neighborhood O((z′, 0)) of (z′, 0) such that F can be stratified into several (non-constant) holomorphic branches over O((z′, 0)). Arguing in the same way, we have for each l ≤ n,

\frac{∂\overline{f_{n+1}}}{∂z_{l}} + \frac{∂\overline{f_{n+1}}}{∂z_{n+1}} \overline{R′_{z_{l}}(z′, ξ, τ)} = ∑_{j=1,··· ,n} \frac{∂G}{∂λ_{j}} (\frac{∂\overline{f̃_{j}}}{∂z_{l}} + \frac{∂\overline{f̃_{j}}}{∂z_{n+1}} \overline{R′_{z_{l}}(z′, ξ, τ)}).

Let τ = μ(z′, ξ). Assume that f_{n+1} = z_{n+1}^{k}g with k > 1. One then has, for ξ with ∥ξ∥ sufficiently small, that

∑_{j=1,··· ,n} \frac{∂G}{∂λ_{j}} (\frac{∂\overline{f̃_{j}}}{∂z_{l}}(\overline{z′}, 0) + \frac{∂\overline{f̃_{j}}}{∂z_{n+1}} (z′, 0)\overline{R′_{z_{l}}(z′, ξ, τ)}) = 0.

Since F^{−1} is a finite-to-finite map, f^{∗}(z′, 0) is a finite-to-one map from O(z′, 0)∩Q_{0} too. Moving to a nearby point if necessary, we can assume that

det(\frac{∂\overline{f̃_{j}}}{∂z_{l}} (\overline{z′}, 0))_{1≤j,l≤n} ≠ 0.

Reasoning as in the proof of the two dimensional case and making |ξ| sufficiently small, we have for each l < n + 1

\frac{∂G}{∂λ_{l}}(F(\overline{ξ}, \overline{μ(z′, ξ)}), \overline{f̃^{∗}}(\overline{z′}, 0)) ≡ 0,

and

G(F(\overline{ξ}, \overline{μ(z′, ξ)}), \overline{f̃^{∗}}(\overline{z′}, 0)) ≡ 0.

Here, ∥ξ∥ is sufficiently small. Let q_{l} = G′_{λ_{l}}(w′, 0, λ) and q_{0} = G(w′, 0, λ). Similarly, we have

q_{l}(w′, λ) = q_{0}(w′, λ)h_{l}^{∗}(w′, λ, q_{0}(w′, λ)),

where w′ = f̂^{∗}(\overline{ξ}, \overline{τ}), τ = μ(z′, ξ), λ(z′) = \overline{f̃^{∗}}(\overline{z′}, 0), and h_{l}^{∗} is holomorphic in its variables. (See the notation introduced in the proof of Lemma 4.1).

Now, a similar argument as in the two dimensional case (involving the use of the n-dimensional version of (4.5)) indicates that

(4.6)  q_{l}(w′, λ) ≡ q_{0}(w′, λ)h_{l}^{∗}(w′, λ, q_{0})

for (w′, λ) ≈ 0 and each l < n + 1. Indeed, this can also be seen as follows: Let z′_{0} ∈ C^{n} be chosen as above such that λ(z′) = \overline{f̃^{∗}}(\overline{z′}, 0) is an open mapping from O((z′_{0}, 0)) ∩ Q_{0} to C^{n}. Fix each z′ ∈ O((z′_{0}, 0)) ∩ Q_{0} and consider the map H which sends each ξ(≈ 0) ∈ C^{n} to (f̂^{∗}(\overline{ξ}, \overline{μ(z′, ξ)}), f_{n+1}(\overline{ξ}, \overline{μ(z′, ξ)})). As argued before, we see that H(ξ) ∈ Q′_{(_{f^{∗}}(z′,0),0)} and is a finite to one map, too. Now, let π^{∗} be the natural projection from C^{n+1} to its first n-copies of C. Then, it follows easily that π^{∗} ∘ H is an open mapping. Hence, from the uniqueness theorem of holomorphic functions, we conclude that (4.6) holds identically. Finally, the same argument as in [BR2] gives a contradiction.

Lemma 4.2: After shrinking U_{1}, it holds that F(U_{1}∩D) ⊂ Ω and F(U_{1}∩D^{c}) ⊂ Ω^{c}. Here, Ω and Ω^{c} stay in different sides of M_{2}.

Proof of Lemma 4.2: We will still use the λ−function G(F, λ) introduced in Lemma 2.1. Let

ρ^{∗}(z, \overline{w}) = 1/NRe ( ∑_{{ \atop f(w)}∈F(z)} (\overline{f_{n+1}(w)} − G(F(z), \overline{{ \atop f(w)}}))) ,

where N is the generic counting number of F(z). Then by Lemma 2.1 and the Riemann extension theorem, we see that ρ^{∗}(z, \overline{w}) is holomorphic in (z, \overline{w}). In particular, ρ^{∗}(z, \overline{z}) is real analytic near 0. We claim that ρ^{∗}(z, \overline{z}) is < 0, when F(z) ⊂ Ω^{+}; > 0 when F(z) ⊂ Ω^{−}; and = 0 when F(z) ⊂ M_{2} (see Lemma 3.3 (a) for related notations). In fact, for each fixed f̃(z) ∈ F(z), we have

Re (\overline{f_{n+1}(z)} − G(f̃, \overline{f̃(z)})) = Reρ̃_{2} (f̃, \overline{f̃(z)})
= ρ_{2} (f̃, \overline{f̃(z)}) Reh_{2}(f̃(z), \overline{f̃(z)})) = ρ_{2}(f̃, \overline{f̃(z)}))(1 + o(∥f̃(z)∥).

Thus, the claim follows from Lemma 3.3 (v).

Next, applying Lemma 4.1, one sees that

ρ^{∗} = 2Re(f_{n+1}(z)) + o(|z|) = Re(g(0)z_{n+1}) + o(|z|)

with g(0) ≠ 0. Thus, d_{0}ρ^{∗} ≠ 0. Therefore, ρ^{∗} serves as a real analytic defining function of M_{1} near 0. Without loss of generality, let us assume that ρ^{∗}(z, \overline{z}) < 0 for z ∈ D. Then ρ(z, \overline{z}) > 0 for z ∈ D^{c}. Now, by making U_{1} small, the above argument shows that F(D ∩ U_{1}) ⊂ Ω^{+} and F(D^{c} ∩ U_{1}) ⊂ Ω^{−}. So, we can let the Ω in Lemma 4.2 be Ω^{+}. Of course, Ω^{c} is then Ω^{−}.

Remark 4.3 (a). In what follows, we will assume, without loss of generality, that Ω = Ω^{+}. Then we observe that the above results give the fact g(0) > 0. To see this, we write f_{n+1}(z) = αz_{n+1}+o(∥z∥) with α = α_{1}+\sqrt{−1}α_{2} ≠ 0. Letting t = x+\sqrt{−1}kx, then ρ^{∗}(t) = (α_{1} − α_{2}k)x + o(|x|). Notice that ρ^{∗}(x) < 0 for any given real number k and x(≈ 0) < 0. We conclude that (α_{1} − α_{2}k) > 0. Thus, we see that α_{2} = 0 and α_{1} > 0.

(b). Another immediate application of Lemma 4.2 is that for certain small neighborhoods U^{∗} and U^{∗}′ of 0, f is proper from U^{∗} ∩ D to U^{∗}′ ∩ Ω. In fact, let B be a sufficiently small ball centered at 0. By certain well-known construction (see [BC], for example) and Lemma 3.3 (iv), we can simply take U^{∗} to be the connected component of the set {z ∈ B ∩ D : f(z) ∉ ∂B ∩ D} such that f(U^{∗}) contains Ω ∩ O(0) with O(0) sufficiently small.

5. Completion of the proof of Theorem 1.1

We now proceed to the proof of our main technical theorem. For simplicity, we retain all the notation which we have set up in the previous sections.

As before, denote by A the map, which sends: w ∈ U_{1} to the finite set A_{w}. w_{0} is called a separable point of A if A_{w_{0}} = {w_{j}}_{j=1}^{N^{∗}} satisfies the following property: There exist open neighborhoods O(w_{j}) of w_{j} (j = 1, ⋯, N^{∗}) such that A(w) ∩ O_{w_{j}} = {w} for any w ∈ O_{w_{j}}. We write W = {w ∈ U_{1} : w is not a separable point of A}. We say that M_{1} satisfies Condition S at 0, if a holomorphic change of coordinates can be chosen so that M_{1} is still defined by an equation of the form as in (2.1) and (2.1)′; moreover, the following holds:

(a) W is contained in an analytic variety E spread over (z_{2}, ⋯, z_{n+1})-space, i.e, E is defined by an equation of the form: z_{1}^{N^{∗}} + ∑_{j=0}^{N^{∗}−1} a_{j}(z_{2}, ⋯, z_{n+1})z_{1}^{j} = 0 with a_{j}(z_{2}, ⋯, z_{n+1}) holomorphic near the origin.

(b) For each small t > 0, there is a sufficiently small positive number ϵ(t) such that for each b^{∗} = (b_{2}, ⋯, b_{n}) with |b^{∗}| < ϵ(t), the set E ∩ {(z_{1}, b^{∗}, −t) : z_{1} ∈ C^{1}} is contained in some connected component of D ∩ {(z_{1}, b^{∗}, −t) : z_{1} ∈ C^{1}}.

Similarly, we can define A′ and W′ and speak of Condition S for M_{2}. In this case, we use Ω to replace the role of D (see Remark 4.3(a)).

We will see in the following lemma that W and W′ can be used to control the branching locus of F.

Lemma 5.1: (a) Suppose that M_{1} and M_{2} satisfy Condition S at the origin. Then F(E_{0}∩O(0)) ⊂ W′, E_{0}∩O(0) ⊂ E, and E_{0}∩O(0)∩{(z_{1}, ⋯, z_{n}, 0)} is a proper variety near the origin of the {(z_{1}, ⋯, z_{n}, 0)}-space. Here E is chosen as in the definition of Condition S, and E_{0} is as defined in Remark 3.4 (c). (When n = 1, we always have F(E) ⊂ W′ and thus 0 is an isolated point of E ∩ Q_{0}). (b) M_{1} and M_{2} satisfy Condition S, when n = 1 or when M_{1} and M_{2} are rigid at the origin.

Proof of Lemma 5.1: We first prove Part (a) of the lemma:

Let z ∈ E_{0} be sufficiently close to 0. Then, by the definition of E_{0} and Remarks 3.4 (b) (c), it follows that there is a sequence z_{j} → z such that we can find two sequences {η_{j}} and {ξ_{j}} with ξ_{j}, η_{j} ∈ F(z_{j}) η_{j} ≠ ξ_{j}, but ξ_{j}, η_{j} → w_{0} for some w_{0}. By Lemma 3.1, it then follows that ξ_{j} ∈ A′_{η_{j}}. Hence, w_{0} is not a separable point of A′. Again by Lemma 3.1, we conclude that F(z) ⊂ W′. Notice that for this part, no Condition S is necessary.

We next prove that E_{0} ∩ O(0) ⊂ E. To this aim, we assume, without loss of generality, that E_{0} ≠ ∅. Let ∪_{j}E_{j} be the union of the irreducible components of E_{0} near 0. We recall by Remark 3.4 (d), that each E_{j} then must be of codimension 1 at 0. Now, since F(E_{j}) ⊂ W′ is also an analytic variety of codimension 1 at 0 (Lemma 3.3 (iv)), the assumption of Condition S for M_{2} indicates that F(E_{j}) ∩ Ω ≠ ∅ near 0. Therefore, it follows from Lemma 4.2, that E_{j} ∩ D is not empty in any neighborhood of 0. Now, pick any point z ∈ E_{j} ∩ D, which is sufficiently close to 0. Then, as above, we can find a sequence z_{j} → z and {η_{j}}, {ξ_{j}} with ξ_{j}, η_{j} ∈ F(z_{j}) η_{j} ≠ ξ_{j}, but ξ_{j}, η_{j} → w_{0} ∈ Ω. By making z sufficiently close to 0, we can assume by Remark (4.3)(b), that f(z_{0}) = w_{0} for some z_{0} close to 0. Moreover, f is proper from a small neighborhood U of z_{0} and \overline{U} ∩ f^{−1}(w_{0}) = \{z_{0}\}. Since f is single valued, there are two distinct points a_{j}, b_{j} ∈ U such that f(a_{j}) = ξ_{j} and f(b_{j}) = η_{j}. Now, the following Claim 5.2 indicates that a_{j} ∈ A_{b_{j}}. Since a_{j}, b_{j} clearly converge to z_{0} as j → ∞, we conclude that z_{0} ∈ W. Again by Lemma 3.1 and Claim 5.2, we see that z_{0}, z ∈ W. By the arbitrariness of the choice of z and the assumption of Condition S, this fact implies that E_{j} has an open subset contained in W and thus in E. From the uniqueness of analytic varieties, it follows that E_{j} ⊂ E for any j.

Claim 5.2: Let M_{1}, M_{2}, and f be as before. We have that b ∈ A_{a} if Q′_{F(a)} = Q′_{F(b)}, where a, b ∈ C^{n+1} are sufficiently close to 0. (For this claim, no Condition S is required).

Proof of Claim 5.2: Let a and b be as in the claim. We then need to show that Q_{a} ∩ U_{2} = Q_{b} ∩ U_{2}. To this aim, we consider the function ρ̃_{2}(F(z), \overline{f̃(ω)}), for any given f̃(ω) ∈ F(ω). By Lemma 2.1, ρ̃_{2}(F(z), \overline{f̃(ω)}) is well defined and holomorphic in (z, λ) with λ = \overline{f̃(ω)} for (z, λ) ∈ O_{z}(0) × O_{λ}(0). Meanwhile ρ̃_{2}(F(z), \overline{f̃(ω)}) = cz_{n+1} + c\overline{ω}_{n+1} + O(|z||λ| + o(ω)) for some c > 0, by Remark 4.3 (a). Thus, for each ω sufficiently close to 0, since |f̃(ω)| ≈ 0, the implicit function theorem indicates that ρ̃_{2}(F(z), \overline{f̃(ω)}) = 0 defines a connected complex hypersurface W_{ω} in some small neighborhood U^{#} of 0, whose size is independent of the parameter ω with |ω| << 1. Notice, by Lemma 3.1, that F(Q_{ω}) ⊂ Q′_{_{f}(ω)} near 0. We see that Q_{ω} ∩ U^{#} ⊂ W_{ω}. Thus it follows that W_{ω} = Q_{ω} in U^{#} when |ω| << 1; for Q_{ω} is also a connected complex submanifold of codimension 1 in U_{2}.

Now, if Q′_{F(b)} = Q′_{F(a)}, we then conclude that

ρ̃_{2}(F(z), \overline{F(b)}) = 0

and ρ̃_{2}(F(z), \overline{F(a)}) = 0 define the same variety. Hence, when a, b ≈ 0, we conclude, by Lemma 3.0 (iv), that Q_{a} = Q_{b}; for both of them have a small open subset of W_{a} ∩ U^{#} = W_{b} ∩ U^{#} in common. This gives the proof of Claim 5.2. (We mention that for the proof Claim 5.2, no assumption of Condition S is required).

The last statement in (a) follows clearly from the above facts and the assumption of Condition S for the hypersurfaces. More precisely, we notice that F(E_{0} ∩ O(0)) ⊂ W′ and F(Q_{0}) ⊂ Q′_{0}. From Lemma 3.3 (iv) and the fact that W′ ∩ {(z_{1}, ⋯, z_{n}, 0)} is contained in a proper subvariety (of codimension at least 1) in Q′_{0}, it follows that E_{0} ∩ Q_{0} is also of codimension at least 1 in Q_{0}. Actually, using the same proof as in Part (b) for the rigid case, and using a simple projection lemma, we notice that this part does not require Condition S neither. Also, we notice that the argument for this part does not use Claim 5.2, i.e, the use of Lemma 4.1, neither.

Now, we proceed to the proof of (b):

We first consider the case of n = 1. As before, we let M_{2} be defined by

z_{2} + \overline{z_{2}} + ∑_{j=0}^{∞} ψ̃_{j}(z_{1}, \overline{z_{1}})z_{2}^{j} = 0; or z_{2} + \overline{z_{2}} + ∑_{j=0}^{∞} \overline{ψ̃_{j}}(\overline{z_{1}}, z_{1})\overline{z_{2}^{j}} = 0.

For each b = (b_{1}, b_{2}) ≈ 0, A′_{b} = {(w_{1}, w_{2}) : Q′_{w} ∩ U′_{2} = Q′_{b} ∩ U′_{2}}. As did in Lemma 3.3, by letting z_{1} = 0 in the defining equations of Q′_{w} and Q′_{b}, one sees that w_{2} = b_{2}. Now, Q_{w} can also be defined by

z_{2} = −\overline{b_{2}} − ∑_{j=0}^{∞} \overline{ψ̃_{j}}(\overline{w}_{1}, z_{1})\overline{b_{2}}^{j},

where the notation \overline{ψ̃} is the same as explained before. Write

\overline{ψ̃_{j}}(\overline{w_{1}}, z_{1}) = ∑_{α>0} \overline{Ξ_{jα}}(\overline{w_{1}})z_{1}^{α}.

Then z_{2} = −\overline{b_{2}} − ∑_{α} ∑_{j} \overline{Ξ_{jα}}(\overline{w}_{1})\overline{b_{2}}^{j}z_{1}^{α}. So, the equation: Q′_{w} = Q′_{b} can be written as

∑_{j} \overline{Ξ_{jα}}(\overline{w}_{1})\overline{b_{2}}^{j} = ∑_{j} \overline{Ξ_{jα}}(\overline{b}_{1})\overline{b_{2}}^{j}

or

∑_{j} Ξ_{jα}(w_{1})b_{2}^{j} = ∑_{j} Ξ_{jα}(b_{1})b_{2}^{j},

for any α.

Choose k_{0} to be the smallest integer so that for certain α_{0}, Ξ_{0α_{0}}(w_{1}) = a^{∗}w_{1}^{k_{0}} + o(|w_{1}|^{k_{0}}) with a^{∗} ≠ 0.

Write P (w_{1}, b_{2}) = Ξ_{0,α_{0}}(w_{1}) + ∑_{j>0} Ξ_{j,α_{0}}(w_{1})b_{2}^{j}. For (b_{1}, b_{2}) close to the origin, we define

E = \{(b_{1}, b_{2}) : for some w_{1}^{0}, P (w_{1}^{0}, b_{2}) = P (b_{1}, b_{2}), \frac{∂P}{∂w_{1}} (w_{1}^{0}, b_{2}) = 0\}.

Then we first claim that near the origin, W ⊂ E. For this, let w = (ξ, b_{2}) ∉ E, and let C(w) = {(z_{1}, b_{2}) : P (z_{1}, b_{2}) = P (ξ, b_{2})}. Then by the definition of P, one has A′_{w} ⊂ C(w). Write A′_{w} = {(τ_{j}, b_{2})}_{j} and suppose that w ∈ W_{0}. Then for some j^{∗}, by the definition of W, there are two sequences {(η_{j}, c_{j})} and {(ξ_{j}, c_{j})} such that (a) η_{j} ≠ ξ_{j}; (b) η_{j}, ξ_{j} → τ_{j^{∗}} for some index j^{∗}, c_{j} → b_{2}; and (c) A′_{(η_{j},c_{j})} = A′_{(ξ_{j},c_{j})}.

From (c) and the definition of P, it follows that P (η_{j}, c_{j}) = P (ξ_{j}, c_{j}). Moreover, by the Taylor expansion: P (η_{j}, c_{j}) = P (ξ_{j}, c_{j}) + \frac{∂P}{∂w_{1}} (ξ_{j}, c_{j})(η_{j} − ξ_{j}) + o(|η_{j} − ξ_{j}|); we obtain \frac{∂P}{∂w_{1}} (ξ_{j}, c_{j}) = o(1). Letting j → 0, we therefore conclude that \frac{∂P}{∂w_{1}} (τ_{j^{∗}}, b_{2}) = 0, which together with the fact P (τ_{j^{∗}}, b_{2}) = P (ξ, b_{2}) implies that w ∈ E. This is a contradiction and thus proves our claim.

We next claim that E is an analytic variety spread over b_{2}-axis. To see that, we let

E^{∗} = \{(b_{1}, b_{2}, w_{1}) ∈ O(0) : P (w_{1}, b_{2}) = P (b_{1}, b_{2}), \frac{∂P}{∂w_{1}} (w_{1}, b_{2}) = 0\}.

Then E is the projection of E^{∗} to the first two copies of C^{1}. This projection map is obviously finite to one near the origin and thus is locally proper. So, E is an analytic variety near 0. Meanwhile, for each b_{2}, from \frac{∂P (w_{1},b_{2})}{∂w_{1}} = 0, we can solve out only finitely many w_{1}’s with |w_{1}| small. Moreover, for any solution w_{1}, we can easily see that |w_{1}| <_{∼} |b_{2}|^{\frac{1}{k_{0}−1}} . Thus it follows that there are also only finitely many b_{1}’s so

that P (w_{1}, b_{2}) = P (b_{1}, b_{2}) with \frac{∂P (w_{1},b_{2})}{∂w_{1}} = 0. Furthermore, one can easily see that |b_{1}| <_{∼} |b_{2}|^{1/k_{0}}. Therefore, we see that E is spread over b_{2}-axis. From the Weiestrass preparation theorem, it follows that E can be defined near 0 by an equation with the form as given in Part (a) of Condition S. Meanwhile, it is also easy to see that E ∩ (C^{1} × {b_{2}}) ⊂ Δ_{|b_{2}|^{α^{∗}}} × {b_{2}} for |b_{2}| sufficiently small, where Δ_{r} is used to stand for the disk in C^{1} with center at the origin and with radius r, α^{∗} is a constant between \frac{1}{k_{0}+1} and \frac{1}{k_{0}} . By our choice of k_{0}, it follows easily for sufficiently small |b_{2}|, that Δ_{|b_{2}|^{α^{∗}}} × {b_{2}} ⊂ D when b_{2} < 0 , and Δ_{|b_{2}|^{α^{∗}}} × {b_{2}} ⊂ D^{c} when b_{2} > 0. This completes the proof of Part (a) in case n = 1.

Finally, we show that M_{1} satisfies Condition S when M_{1} is rigid. For this purpose, we assume that M_{1} is defined by an equation of the form: z_{n+1} + \overline{z_{n+1}} + ∑_{|α|,|β|>0} a_{αβ}z′^{α}\overline{z′^{β}} = 0. Let P_{α}(z′) = ∑_{β} \overline{a_{αβ}}z′^{β}. Then for a = (a′, a_{n+1}) b = (b′, b_{n+1}) close to the origin, Q_{a} = Q_{b} if and only if a_{n+1} = b_{n+1} and P_{α}(a′) = P_{α}(b′). By the finite type assumption, it follows that the common zero of P′_{α}s is 0 near the origin. Using the Nullstellantz theorem, we see that for some finitely many indices {α_{j}}_{j=1}^{m}, the locus of {P_{α_{j}}}_{j=1}^{m} is also zero. Now, for a small neighborhood U of 0, let U′ = U ∩ {(z_{1}, ⋯, z_{n}, 0)}. Define Λ from a small neighborhood U′ of 0′ to C^{m} by Λ(z′) = (P_{α_{j}}(z′))_{j=1}^{m}. Then Λ is finite to 0 and proper from U′, after suitably shrinking of U. So, by the Remmert theorem, we conclude that the set E′ = Λ^{−1} (Λ({z ∈ U : dΛ does not have maximal rank at z})) is a proper variety of U′ (see [DF1] for a similar argument). It is easy to see that W ⊂ E, with E = (E′ × C^{1}) ∩ U_{1}. Now, after a linear change of coordinates in the (z_{1}, ⋯, z_{n})- space, we see that E is contained in an analytic variety defined by an equation of the form: z_{1}^{N^{∗}} + ∑_{j<N^{∗}} c_{j}(z_{2}, ⋯, z_{n})z_{1}^{j} = 0 with c_{j}(0) = 0. Since this equation does not contain z_{n+1}-variable, it follows easily that M_{1} satisfies Condition S at the origin.

We now give the following lemma, which indicates how the information on the branching locus can be used to obtain the holomorphic extendibility. We use Δ_{a} to denote the disc in C^{1} centered at 0 with radius a.

Lemma 5.3: Let M be a hypersurface near the origin of C^{n+1} defined by an equation of the form ρ(z, \overline{z}) = z_{n+1} + \overline{z_{n+1}} + ϕ(z, \overline{z}) = 0 with ϕ(z, z) = o(|z|). Let D be the side defined by ρ(z, \overline{z}) < 0. Assume that g is a holomorphic function over D that admits a holomorphic correspondence extension to Δ_{δ_{0}} × Δ_{δ}^{n} for some small δ_{0}, δ > 0, i.e, there exist holomorphic functions a_{j} ∈ Hol(Δ_{δ_{0}} × Δ_{δ}^{n}) such that P (z, g(z)) = g^{N}(z) + ∑_{j=0}^{N−1} a_{j}g^{j}(z) = 0 for z ∈ D. Here we assume that P (z, X) is irreducible in X, and Δ_{δ_{0}} ×Δ_{δ}^{n} ∩D ⊂⊂ D. Denote by Z^{∗} the branching locus of g and write Z for the genuine branching locus of g as defined in Remark 3.4 (d). Assume that Z is defined by P̃(z_{2}, ⋯, z_{n+1}; z_{1}) = z_{1}^{N_{1}} + ∑_{j=0}^{N_{1}−1} d_{j}(z_{2}, ⋯, z_{n+1})z_{1}^{j} = 0 for certain d_{j}(z_{2}, ⋯, z_{n+1}) ∈ Hol(Δ_{δ}^{n}) with d_{j}(0) = 0. Also, suppose that for each small positive number t, there is a positive number ϵ(t) such that when |b_{j}| < ϵ(t) with j = 2, ⋯, n, Z ∩ D^{∗}(b^{∗}, t) is contained in some connected open subset of D ∩ D^{∗}(b^{∗}, t), where D^{∗}(b^{∗}, t) = {z : z = (z_{1}, b_{2}, ⋯, b_{n}, −t) : z_{1} ∈ C^{1}}. Then g admits a holomorphic extension near 0.

Proof of Lemma 5.3: Without loss of generality, we assume that

P̃(z_{2}, ⋯, z_{n+1}; X) = 0

has no double roots, i.e, P̃(z_{2}, ⋯, z_{n+1}; X) and \frac{∂_{P}}{∂X} are relatively prime. Denote by Z̃ the branching locus of P̃, which is a subvariety of Δ_{δ}^{n}. Now, we choose a small t > 0 and b^{∗} = (b_{2}, ⋯, b_{n}) with |b_{j}| < ϵ(t) such that (b^{∗}, −t) ∉ Z̃ and Z^{∗}∩C^{1}×{(b^{∗}, −t)} is discrete. Write B_{η}(b^{∗}, −t) for the closed ball in C^{n} centered at (b^{∗}, −t) with radius η. Then, when η is sufficiently small, B_{η}(b^{∗}, −t) ∩ Z̃ = ∅ and 0 ∉ B_{η}(b^{∗}, −t). Construct a pseudoconvex domain B̃(t) ⊂⊂ Δ_{δ}^{n} such that the following holds: (i) 0 ∈ B̃(t), B̃(t) ⊃ B_{η}(b^{∗}, −t); and (ii) there is a contraction map T (·, τ) : B̃(t) → B̃(t) (τ ∈ [0, 1]) such that T (·, 0) = id, T (B̃(t), 1) = B_{η}(b^{∗}, −t), and for any τ, T (·, τ) is the identical map when restricted to B_{η}(b^{∗}, −t).

Now, using the hypothesis on Z and shrinking δ if necessary, we can find R > 0 with R < δ_{0} such that (Δ_{R} \ Δ_{\frac{1}{2} R}) × Δ_{δ}^{n} ∩ Z = ∅. Write Ω_{0} = (Δ_{R} × B̃(t)) \ \overline{(Δ_{\frac{1}{2} R} × (B̃(t) \ B_{η}(b^{∗}, −t)))}, which can also be written as

(Δ_{R} × int(B_{η}(b^{∗}, −t))) ∪ ((Δ_{R} \ \overline{Δ_{\frac{1}{2} R}}) × B̃(t)) .

Here, for a subset K, we write int(K) for the collection of the interior points of K. Then, it is easy to see that any (closed) loop in Ω_{0} \ Z based at certain z_{0} ∈ (Δ_{R} × int(B_{η}(b^{∗}, −t))) \ Z^{∗} can be deformed, relative to the base point and without cutting Z, to a loop in (Δ_{R} × B_{η}(b^{∗}, −t)) \ Z. Indeed, let γ = (γ_{1}, γ^{∗}) be a loop in Ω_{0} \ Z based at z_{0}. Then the above described deformation can be performed by (γ_{1}(t), T (γ^{∗}(t), τ)).

We next set up the following notation: For a given connected subset D′ (near the origin) with D′ \ Z pathwise-connected, a point z_{0} ∈ D′ \ Z^{∗}, and a complex number w_{0} with P (z_{0}, w_{0}) = 0, we define Val(z_{0}, w_{0}; D′) to be the collection of w′s, which is obtained as follows: There are a loop γ ⊂ D′ \ Z, based at z_{0}, and a continuous section G from [0, 1] to (O, π, C) with P (z; G) ≡ 0 such that each representation of G(0) takes value w_{0} and each representation of G(1) takes value w at z_{0}. We write P(z) = {w : P (z, w) = 0}. When Val(z_{0}, w_{0}; D′) is independent of the choice of w_{0}, we simply use the notation Val(z_{0}; D′). In particular, this is the case when #Val(z_{0}, w_{0}; D′) = N.

Claim 5.4: #Val(z_{0}; Δ_{R} × B_{η}(b^{∗}, −t)) = N, i.e,

Val(z_{0}; Δ_{R} × (B_{η}(b^{∗}, −t)) = {all solutions of P (z_{0}, X) = 0}.

Proof of Claim 5.4: Let Ω_{0} be as defined above. We then notice that the holomorphic hull of Ω_{0} is Δ_{R} × B̃(t). In fact, for any ϕ ∈ Hol(Ω_{0}), the following Cauchy integral gives the holomorphic extension of ϕ to Δ_{R} × B̃(t):

\frac{1}{2πi} ∫_{|ξ|=_{R}} \frac{ϕ(ξ, z_{2}, ⋯, z_{n+1})}{ξ − z_{1}}dξ, with R̃ → R.

Now, we fix a point z_{0} ∈ Ω_{0} \ Z^{∗} and a number w_{0} with P (z_{0}, w_{0}) = 0. For z ∈ Ω_{0} \Z^{∗}, let γ ⊂ Ω_{0} \Z be a curve connecting z_{0} to z. Denote by w be the value of the branch of P at z which is continuous along γ and takes the initial value w_{0}. Write Val(z, w; Ω_{0}) = {w_{1}, ⋯, w_{k(z,w)}}. Then, by the monodromy theorem (see Remark 3.4 (d)), one sees that k(z, w) is constant, say k′, for all z, w. Also, Val(z, w; Ω_{0}) is independent of the choice of γ, though w does. Now, P_{0}(z, X) = ∏_{j=1}^{k′}(X − w_{j}(z)) will give a well-defined polynomial in X with coefficients holomorphic and bounded in Ω_{0} \ Z^{∗}. Applying the Riemann extension theorem, we conclude that P_{0}(z, X) is a polynomial in X with coefficients holomorphic in Ω_{0}. The above observation then indicates that the coefficients of P_{0} are holomorphic in Δ_{R} × B̃(t), too. So, by the irreducibility of P, we conclude that k′ = N.

Now, from the argument preceding this claim and the monodromy theorem, the proof of Claim 5.4 follows.

Claim 5.5: Let b^{∗} and t be as chosen above and write p_{0} = (b^{∗}, −t). Let ϵ(t) be sufficiently small. Let z_{0} = (z_{1}, p_{0}) ∈ Δ_{R} × {p_{0}} \ Z^{∗} for a certain z_{1}. Then #Val(z_{0}; Δ_{R} × {p_{0}}) = N.

We first mention that the proof of Lemma 5.3 follows easily from Claim 5.5. Indeed, by the hypothesis, the finite set Z ∩{Δ_{R} ×{p_{0}}} stays in some connected open subset of D∩{C^{1}×{p_{0}}}, say D^{∗}. Since #(Z ∩{C^{1}×{p_{0}}}) is finite, we can choose a simply connected smooth domain D^{∗∗} ⊂⊂ D^{∗} with Z ∩ (C^{1} × {p_{0}}) ⊂ D^{∗∗} ∩ (C^{1} × {p_{0}}) (see Remark 5.6 (a) below). Therefore, by the Schoenflies theorem, there is a retraction from Δ_{R} × \{p_{0}\} to \overline{D^{∗∗}} that deforms Δ_{R} × \{p_{0}\} \ D^{∗∗} to the boundary of D^{∗∗}. Thus any loop in Δ_{R} that is based at z^{∗} ∈ D^{∗∗} \ Z^{∗} can be deformed, relative to the base point, to \overline{D^{∗∗}} without cutting Z ∩ Δ_{R} × \{p_{0}\}. Hence, by the monodromy theorem, it follows that #Val(z^{∗}; D^{∗}) = N for any given point z^{∗} ∈ D^{∗∗}. But this is impossible; for f is holomorphic in a small neighborhood of D^{∗}. So, to conclude the proof of Lemma 5.3, it suffices for us to prove Claim 5.5.

We now turn to the proof of Claim 5.5. By our choice of B_{η}(b^{∗}, −t), we know that

P̃(z_{2}, ⋯, z_{n+1}; z_{1}) = z_{1}^{N_{1}} + ∑_{j<N_{1}} d_{j}(z_{2}, ⋯, z_{n+1})z_{1}^{j} = 0

has roots

{ϕ_{1}(z_{2}, ⋯, z_{n+1}), ⋯, ϕ_{m}(z_{2}, ⋯, z_{n+1})},

which can be arranged such that ϕ_{j}(z^{∗}) ∈ Hol(B_{η}(b^{∗}, −t)) and ϕ_{j}(z^{∗}) ≠ ϕ_{l}(z^{∗}) for j ≠ l. So, for a given z^{∗} ∈ B_{η}(b^{∗}, −t), (Δ_{R} × {z^{∗}}) ∩ Z = {(ϕ_{j}(z^{∗}), z^{∗})}_{j=1}^{m}. Let p_{j}^{0} = ϕ_{j}(p_{0}). We will show that there exists a homeomorphism π_{z^{∗}}: Δ_{R} → Δ_{R}, depending continuously on z^{∗} ∈ B_{η}(b^{∗}, −t), so that π_{z^{∗}}(ϕ_{j}(z^{∗})) = p_{j}^{0}. If this can be done, then any loop γ(t) = (γ_{1}(t), γ^{∗}(t)), which is based at z_{0} and stays in (Δ_{R} × B_{η}(b^{∗}, −t)) \ Z can be deformed to a closed curve in Δ_{R} × {p_{0}} by the following map:

β(t, τ) = (π_{τp_{0}+(1−τ)γ^{∗}(t)}^{−1} ∘ π_{γ^{∗}(t)}(γ_{1}(t)), τp_{0} + (1 − τ)γ^{∗}(t))

with β(t, 0) = γ(t), β(t, 1) ∈ Δ_{R} × {p_{0}}, and β(0, τ) = β(1, τ) = z_{0}. Meanwhile, from the construction, one can verify that {β(t, τ)} ∩ Z = ∅. Thus, from the monodromy theorem stated in Remark 3.4 (d), it follows that #Val(z_{0}; Δ_{R} × {p_{0}}) = N.

To construct the above mentioned π_{z^{∗}} : Δ_{R} → Δ_{R}. We notice that

min_{j≠l,z^{∗}∈B_{η}(b^{∗},−t)}{|p_{j}(z^{∗}) − p_{l}(z^{∗})|} > 0.

By using a Möbius transform, we can assume that p_{1}(p_{0}) = p_{1}^{0} = 0. Suppose that for an index j < m, we can find a homeomorphic self map σ_{j}(·, z^{∗}) of Δ_{R}, which depends continuously on z^{∗} and maps p_{l}(z^{∗}) to p_{l}^{0} for l ≤ j. For the brevity of the notations, let us still write p_{j}(z^{∗}) for the transformed points: σ_{j}(p_{j}(z^{∗}), z^{∗}). We then wish to find a homeomorphic self map σ̃_{j+1}(·, z^{∗}) of Δ_{R}, depending continuously on z^{∗} ∈ B_{η}(b^{∗}, −t), such that σ̃_{j+1}(p_{l}^{0}, z^{∗}) = p_{l}^{0} for l ≤ j and σ̃_{j+1}(p_{j+1}(z^{∗}), z^{∗}) = p_{j+1}^{0}. If this can be done, letting σ_{j+1}(·, z^{∗}) = σ̃_{j+1}(·, z^{∗})∘σ_{j}(·, z^{∗}), and applying the induction argument, we see the existence of the aforementioned map π_{z^{∗}}. Indeed, the existence of σ̃_{j+1}(·, z^{∗}) clearly follows from the following Remark 5.6 (b) and by making ϵ(t) sufficiently small:

Remark 5.6: In this remark, we would like to say a few words about two simple facts in the set topology that were used in the proof of Lemma 5.2:

(a) Let D be a connected domain in C^{1}. Then, for any given finitely many points {p_{j}}_{j=1}^{k} ⊂ D, there is a smoothly bounded simply connected domain D^{∗} ⊂⊂ D such that {p_{j}}_{j=1}^{k} ⊂ D^{∗}.

Indeed, we can find a curve γ in D connecting all of the p_{j}’s, which, by the piecewise linear approximation, can be assumed to be piecewise linear. Making further division to the image of γ if necessary, we can assume that C = γ([0, 1]) is a linear graph (or, a finite CW-complex of dimension 1), which contains {p_{j}} in the set of its vertices. Now, a well-known result in Topology indicates that we can find a maximal tree C̃ inside C, which, by definition, is simply connected and contains all the vertices of C. Furthermore, the tubular neighborhood basis theorem shows the existence of a small neighborhood D̃(⊂ D^{∗}) of C̃, that can be retracted to C̃ and thus is simply connected, too. (See for example, J. F. Hudson: Piecewise Linear Topology, W. A. Benjamin, Inc, 1969; in particular, Theorem 2.11, pp57 ). Now, we have a Riemann mapping σ from D̃ to Δ. Then obviously, σ^{−1}(Δ_{r}) does our job when r < 1 is sufficiently close to 1. .

(b) Let {a, a_{1}, ⋯, a_{k}} be k + 1 distinct points in Δ. Suppose that p(z) is a continuous map from the ball the closed unit ball B_{n} ⊂ C^{n} to Δ such that ϵ = min_{j=1,··· ,k;z∈B_{n}} |p(z) − a_{j}| is a sufficiently small positive number, and |p(z) − p(z_{0})| < \frac{1}{4}ϵ. Then there exists a homeomorphic self map σ(·, z) of Δ_{R}, which depends continuously on z, such that σ(p(z), z) = a and σ(a_{j}, z) = a_{j} for j = 1, ⋯, k.

The proof of this fact can be seen as follows: Pick a continuous retraction π from B_{n} to z_{0}. We will then construct a homeomorphic self map σ(·, z) of Δ, that depends continuously on z ∈ B_{n}, such that the following holds: (a) σ(a_{j}, z) = a_{j} (j = 1, ⋯, k);

(b) σ(·, z_{0}) = id; (c) σ(p(z), z) = p(z_{0}) for z ∈ B_{n}. To this aim, we assume that p(z_{0}) = 0 to simplify the notation. Notice that |p(z) − p(z_{0})| < ϵ/4 and thus p(z) is contained in the disk Δ_{ϵ/4}. We also observe, by the hypothesis, that a_{j} ∉ Δ_{ϵ}. Now, we let τ(·, z) be the Möbius transform from Δ_{ϵ/4} to itself that maps p(z) for z to p(z_{0}). Meanwhile, we impose the condition that τ′(p(z), z) > 0 to make it unique. From the explicit formula of τ, it is clear that τ(·, z) depends continuously on z ∈ B_{n}.

For w = ϵ/4e^{\sqrt{−1}θ} with 0 ≤ θ < 2π, write τ(w, z) = \frac{ϵ}{4}e^{\sqrt{−1}r(θ,z)}, where r(0, z) ∈ [0, 2π), r(2π, z) − r(0, z) = 2π, and r(θ, z) is a strictly increasing function of θ for fixed z. Moreover, r(θ, z) depend continuously on (θ, z). The homeomorphism σ(·, z) from Δ to itself can be defined as follows:

(i) σ(·, z_{0}) = id. For z ∈ B_{n}, we let σ(w, z) = w, for w ∉ Δ_{ϵ}; σ(w, z) = τ(w, z), for w ∈ Δ_{ϵ/4}; and otherwise

σ(w, z) = |w|exp (\sqrt{−1}\frac{4}{3ϵ} ((|w| − ϵ/4)θ + (ϵ − |w|)h(θ, z))) ,

where w = |w|e^{\sqrt{−1}θ} with |w| ∈ [ϵ/4, ϵ].

It can be easily verified that σ possesses all the properties we imposed.

Completion of the proof of Theorem 1.1. We now are ready to apply the previously established results to present a proof of Theorem 1.1. Indeed Theorem 1.1 follows from the following Theorem 1.1′ and Lemma 5.1 (b).

Theorem 1.1′: Let M_{1} ⊂ C^{n+1} and M_{2} ⊂ C^{n+1} be real analytic hypersurfaces of finite D-type. Suppose that f is a CR mapping from M_{1} to M_{2}. Let p ∈ M_{2}. Assume that f extends as a holomorphic correspondence near p and assume furthermore that M_{1}, M_{2} satisfy Condition S at p, and f(p), with respect to D and Ω as introduced before, respectively. Then f admits a holomorphic extension near p.

Proof of Theorem 1.1′: For each component f_{j} of f, we see that the genuine branching locus of the polynomial equation defining f_{j}, denoted by E_{0}^{(j)}, is contained in E_{0}. By Lemma 5.1, it then follows that for sufficiently small t > 0, one can always find ϵ(t) > 0 such that when |b_{j}| < ϵ(t) (j = 2, ⋯, n), E_{0}^{(j)} ∩ {(z_{1}, b_{2}, ⋯, b_{n}, −t)} is contained in some connected component of D ∩ {(z_{1}, b_{2}, ⋯, b_{n}, −t)}. Now, applying Lemma 5.3, we conclude that f_{j} extends holomorphically across 0 for each j.

Remark 5.7: By examing the proof, it can be seen that Theorem 1.1′ also holds when M_{1} and M_{2} are merely assumed to be essentially finite at the origin in the sense that A_{0} = A′_{0} = {0} (see [BJT] or [DW]).

6. Proofs of Theorem 1.2 and Theorem 1.4

Now, we pass to the proofs of Theorem 1.2 and Theorem 1.4. We first let M_{1}, M_{2}, and f be as in Theorem 1.2. We also assume that f is not constant. Then we claim that f must be an algebraic map. To see this, we first stratify M_{2} into the disjoint union of the sets M_{2}^{±}, S_{2}, S_{1}, and S_{0}. Here M_{2}^{±} is the set of points in M_{2} where the Levi form is non-zero; S_{2} is a locally finite union of 2−dimensional real analytic totally real submanifolds in M_{2} \ M_{2}^{±}; S_{1} is a locally finite union of 1-dimensional real analytic submanifolds in M_{2} \ (M^{±} ∩ S_{2}); and S_{0} is a locally finite subset. For the existence of such a semi-analytic stratification, we refer the reader to a related explanation appeared in [DFY]. Assume that 0 ∈ M_{1} and f(0) = 0. Since the algebraicity is a global property, we need only to show the map is algebraic at some small piece of M_{1}. Using the finite type assumption of M_{1}, we can assume the existence of some non empty open piece U_{0} in M_{1}^{±}, where M_{1}^{±} denotes the set of points in M_{1} where the Levi form do not vanish. When f(U_{0}) ∩ M_{2}^{±} ≠ ∅, by results of Pinchuk-Tsyganov [PS], Lwey [Le], and Pinchuk [Pi], we see that f is actually real analytic at some point p in U_{0}. Thus applying a result of Baouendi-Rothschild [BR3], we see J_{f} ≢ 0 near p. Now, the algebraicity follows from a theorem of Webster [We].

When f(U_{0}) ⊂ S_{2}, by shrinking U_{0} if necessary, we assume that f extends to a certain side, say D, of M_{1}. As in the proof of Lemma 2.1, we can find a totally real submanifold E ⊂ U_{0} and a wedge W^{+} ⊂ D with edge E. Moreover, we can assume that f(E) is contained in a connected piece of S_{2}. Now, by the reflection principle, we conclude that f admits a holomorphic extension near E. Since f maps a neighborhood of E in U_{0} to a two dimensional manifold, we easily see that J_{f} ≡ 0 near E. This contradicts the non constant assumption for f, by [BR3]. Similarly, we can exclude the case: f(U_{0}) ⊂ S_{1} ∪ S_{0}. Summarizing the above arguments, we can conclude the algebraicity of f.

We also recall a result in [DF2], which states that any proper holomorphic map between two bounded algebraic domains in C^{n} (n > 1) is continuous up to the boundary. Meanwhile, it is also algebraic by [We] (see also [DF2]). Thus, the proofs of Theorem 1.2, Corollary 1.2′, and Theorem 1.4 now clearly follow from Theorem 1.1 and the following general result:

Lemma 6.1: Let M_{1} and M_{2} be two connected algebraic hypersurfaces of finite D-type in C^{n} (n > 1) and assume that D is bounded domain with M_{1} as part of its boundary. Suppose that f is an algebraic holomorphic map from D to C^{n}, that is continuous up to D ∪ M_{1}, and maps M_{1} into M_{2}. Then for each point p ∈ M_{1}, f extends as a holomorphic correspondence to a neighborhood of p.

Proof of Lemma 6.1: Let M_{1} and M_{2} be defined by two real polynomials ρ_{1}(z, \overline{z}) and ρ_{2}(z, \overline{z}), respectively. Without loss of generality, we assume that p = 0 and f(0) = 0. By the hypothesis, we suppose that the j^{th} component f_{j} of f satisfies the irreducible polynomial equation P_{j}(z, X) = ∑_{l=0}^{N_{j}} a_{jl}(z)X^{l} = 0. As mentioned before, by a result of Baouendi-Treves [BT] (see also [Tr]), we can assume, without loss of generality, that any CR function defined over M_{1} can be extended to D.

Write V for the irreducible variety in C^{n} × C^{n}, defined by P_{j}(z, w) = 0 for j = 1, ⋯, n, which contains the graph Γ_{f} of f over D. Write π, π′ for the nature projections from V to the first and the second copies of C^{n}, respectively. Define E to be the collection of points which are either the zeros of a_{jN_{j}}(z) for some j, or are the branching points of of P_{j}(z, X) for certain j. Notice that E is an analytic variety of codimension ≥ 1.

Claim 6.2: Fix a point z_{0} ∈ (M_{1} \ E) ∩ U_{1} and choose a sufficiently small ball B centered at z_{0}. Consider the multiple-valued extension F_{j} of f_{j} from B ∩ D to B ∪ (D^{c} ∩ U_{1}) \ E with U_{1} as chosen before (see §3 for a precise definition of F_{j}) . If there is a constant C such that |f̃(z)| ≤ C for any z ∈ B ∪(D^{c}∩U_{1}) and f̃(z) ∈ F_{j}(z), then f_{j}, near 0, satisfies a polynomial equation with leading coefficient 1.

Proof of Claim 6.2: For each z ∈ B ∪ (D^{c} ∩ U_{1}) \ E, let

F_{j}(z) = {f_{j}^{(1)}(z), ⋯, f_{j}^{(k(z))}(z)}

. First, by the monodromy theorem, k(z) is constant. Now, suppose that all elements in F_{j}(z) is bounded by a constant independent of z. As we did before, using the Riemann extension theorem, we conclude that P̃_{j}(z, X) = ∏_{l}(X − f_{j}^{(l)}(z)) is a polynomial in X with coefficients holomorphic and bounded in D^{c} ∩ U_{1}. Now, by the aforementioned result of Baouendi-Treves and Trepreau, one sees that the coefficients of P̃_{j} can be extended holomorphically to a neighborhood of 0. P̃_{j} has leading coefficient 1 and annihilates f_{j}(z) when z ∈ U_{1} .

Now, seeking a contradiction, suppose that Lemma 6.1 is false.

Then Claim 6.2 indicates that no matter how we shrink U_{1} and make z_{0} close to 0, there exists a fixed index j_{0} such that one can always find some point z^{1} ∈ B ∪ (D^{c} ∩ U_{1}) \ E with |f̂_{j_{0}}(z^{1})| > 1 for certain f̂_{j_{0}}(z) ∈ F_{j_{0}}(z). Observe that for a given curve γ ⊂ B ∪ (D^{c} ∩ U_{1}) \ E with γ(0) = z_{0} and γ(1) = z^{1}, there is a certain holomorphic branch f̃_{j_{0}} of F_{j_{0}} along γ, which takes value f_{j_{0}}(z_{0}) at z_{0} and takes value f̂_{j_{0}}(z^{1}) at z^{1}.

On the other hand, by the Whitney approximation theorem (see J. Milnor: Differential Topology, Lecture Notes by Munkress; in particular, Theorem 1.28 and Lemma 1.29) and by noting that B∪(D^{c}∩U_{1})\E can be retracted to D^{c}∩U_{1}\E, we notice the existence of a smooth simple curve γ^{∗} connecting z_{0} to z with γ((0, 1]) ⊂ (D^{c} ∩U_{1})\E and γ transversal to M_{1} at z_{0}. Moreover γ^{∗} is homorpotic to γ (relative to the base point) in B ∪ (D^{c} ∩ U_{1}) \ E.

So, using the Monodromy theorem we can simply assume that γ = γ^{∗}. As what we did in Lemma 3.1, we can thicken g̃ = γ ∪ R(γ) and then apply the uniqueness theorem of holomorphic functions and the invariant property of Segre varieties to conclude that f(Q_{z} ∩ O(z^{∗})) ⊂ Q′_{_{f}(z)}, for any z ∈ γ, where the branch f̃ of F is

determined by the initial value condition f̃(z_{0}) = f(z_{0}).

Since f(0) = 0, for any small ϵ > 0, by changing the size of U_{1}, we see the existence of a sequence p̃_{j} → 0 such that |f̃(p̃_{j})| = ϵ. In particular, we see that the cluster set Cl_{f}(0) of f̃ at 0 contains points with norm ϵ. Here, the branch f̃ is as explained above. We now are ready to use the following argument to show that this is impossible, which was first used in [DF2] for a different purpose.

Let V̂ be the algebraic closure of V in P^{n} × P^{n} (see [Wh]). Without loss of generality, we can further assume that V̂ is irreducible and contains Γ_{f}. Let π̂ and π̂′ be the nature projection from V̂ to the first and the second copies of P^{n}, respectively. Since V̂ is of dimension n, away from a proper subvariety, we see that π̂ and π̂′ are local biholomorphisms. For each z, let Q_{z} = \{w ∈ C^{n} ∩ U_{1} : ρ_{1}(w, \overline{z}) = 0\} as before, and let Q̂_{z} be the compactification of Q_{z} in P^{n}. Now, for each w ∈ Cl_{f}(0), the cluster set of f̃ at 0, let z_{j} → 0 with f̃(z_{j}) = w_{j} → w_{0}. Then z_{j} ∈ D^{c} and f(Q_{z_{j}} ∩ O(z_{j}^{∗})) ⊂ Q′_{w_{j}}. This gives that π̂′ ∘ π̂^{−1}(Q̂_{z_{j}}) ⊃ Q′_{w_{j}} ∩ U′_{2}, where U′_{2} is as defined in §2. Now, letting j → ∞, we see that π̂′ ∘ π̂^{−1}(Q̂_{0}) ⊃ Q_{w′} ∩ U′_{1}. Notice that π′ ∘ π̂^{−1}(Q̂_{0}) is an analytic variety of P^{n} of codimension 1, by the Remmert mapping theorem. Hence, it follows that Q̂_{w′} ∩ U′_{2} has to be contained in one of the finite irreducible components of π′ ∘ π^{−1}(Q̂_{0}). Since #A′_{w} < ∞ (w ∈ U′_{2}), for each connected component of π′ ∘ π^{−1}(Q̂_{0}), there are only finitely many w’s with |w| small such that the above property holds. This implies that #(Cl_{f}(0) ∩ O(0)) < ∞ and contradicts our assumption. The proof of Lemma 6.2 is complete.

7. Extending proper holomorphic mappings in C^{2}- Proof of Theorem 1.3

With Theorem 1.1 at our disposal, we now use the ideas appeared in [DFY] and, in particular, [DP] to present a proof of the following Theorem 1.3′, which together with Theorem 1.1 gives Theorem 1.3. We would like to mention again the whole strategy is similar to that in [DP] (see also closely related work in [DFY]) for the study of biholomorphic maps.

Theorem 1.3′: Let D and D′ be two smoothly bounded domains with real analytic boundaries in C^{2}. Let f be a proper holomorphic map from D to D′. Suppose that f extends continuously up to ∂D. Then f extends as a holomorphic correspondence across ∂D. By Theorem 1.1, it thus follows that f admits a holomorphic extension to \overline{D}.

To start with, we let M be a real analytic hypersurface of finite type in C^{2}, and let Ω be a domain with M as part of its smooth boundary. Then one has the following semi-analytic stratification as introduced in [DFY]:

M = M^{+} ∪ M^{−} ∪ T_{2}^{+} ∪ T_{2}^{−} ∪ T_{2}^{±} ∪ T_{1} ∪ T_{0}.

Here

(a) M^{+} (M^{−}) is the set of strongly pseudoconvex (strongly pseudoconcave, respectively) boundary points of Ω ∪ M;

(b) T_{2}^{+} (T_{2}^{−}) is a locally finite union of 2-dimensional totally real analytic submanifolds, where the vanishing order of the Levi form of M, as the boundary of Ω ∪ M, is even and positive (negative, respectively). Therefore, Ω is pseudoconvex near T_{2}^{+} and pseudoconcave near T_{2}^{−}; (c) T_{2}^{±} is a locally finite union of 2-dimensional totally real submanifolds, where the vanishing order of the Levi form is odd. It is known (see [DF3] or [BCT]) that any CR functions defined near T_{2}^{±} can be holomorphically extended to both sides of M near T_{2}^{±};

(d) T_{1} is a locally finite union of one dimensional real analytic curves; and T_{0} is a locally finite subset of M.

Now, let D and D′ be as in Theorem 1.3′. After making the above type of the stratifications, we have the following disjoint unions:

D = ∂D^{+} ∪ ∂D^{−} ∪ T_{2}^{+} ∪ T_{2}^{−} ∪ T_{2}^{±} ∪ T_{1} ∪ T_{0},
D′ = ∂D′^{+} ∪ ∂D′^{−} ∪ T′_{2}^{+} ∪ T′_{2}^{−} ∪ T′_{2}^{±} ∪ T′_{1} ∪ T′_{0}.

In what follows, we write Σ′ = ∂D′^{+} ∪ ∂D′^{−} ∪ T′_{2}^{+} ∪ T′_{2}^{−} ∪ T′_{2}^{±}, and write Σ for the boundary points in ∂D where f extends holomorphically.

Lemma 7.1: (a) For any boundary point p ∈ ∂D, if f(p) ∈ Σ′, then f extends holomorphically across p. (b) f extends almost everywhere in ∂D and Σ∩T_{2}^{+} is dense in T_{2}^{+}.

Proof of Lemma 7.1: We first note that in case q = f(p) ∈ ∂D′^{−} ∪ T′_{2}^{±} ∪ T′_{2}^{−}, then p stays inside the holomorphic hull of D (see [BHR2]). Thus f extends automatically at p. We now assume that q = f(p) ∈ ∂D′^{+} ∪ T′_{2}^{+}. Pick a small pseudoconvex piece M′ ⊂ ∂D of q, and let M be the connected component of f^{−1}(M′) which contains p. Then since f(∂D) ⊂ ∂D′, it is clear that f^{−1}(q) ⊂⊂ M. We notice that M has to be pseudoconvex, by the properness and boundary continuity of f. Hence, using a result of Bell-Catlin [BC], it follows that f is smooth at p. By [BBR], we therefore conclude that f extends holomorphically across p.

Next, if for some point p and a small open subset U_{p} ⊂ ∂D near p, f(T_{2}^{+} ∩ U_{p}) ⊂ T′_{1} ∪ T′_{0}, then it is easy to see that there is a wedge W^{+} with edge U_{p} ∩ T_{2}^{+} such that W^{+} ⊂ D. Indeed, after a local change of coordinates, one can assume that p = 0, T_{2}^{+} = iR^{2}, and T_{0}^{(1,0)} is given by z_{1}-axis. Then, one can simply take W^{+} = {(z_{1} = x_{1} + iy_{1}, z_{2} = x_{2} + iy_{2}) ≈ (0, 0) : x_{1} < 0, |x_{j}| > |y_{j}|}.

Notice that T′_{1} ∪ T′_{0} is contained in the union of finitely many analytic arcs and finitely many points. Using the continuity of f and applying the reflection principle, one sees that f extends holomorphically in a small neighborhood of some point z ∈ U_{p} ∩ T_{2}^{+}. Now, notice that J_{f} ≢ 0 for w ∈ U_{p} ∩ T_{2}^{+} (≈ z). This gives a contradiction; for f maps the two dimensional manifold U_{p}∩T_{2}^{+} into a one dimensional semi-analytic subset. Now, from the above argument, it follows that Σ ∩ T_{2}^{+} is dense in T_{2}^{+}.

Similarly, one can see that f extends holomorphically across almost every point in ∂D, as did in the beginning of §6.

Now, we fix p_{0} ∈ ∂D and q_{0} = f(p_{0}). We will show that f extends as a holomorphic correspondence near p_{0}. We can also assume, without loss of generality [BT] [Tr], that every CR function defined on ∂D near p_{0} can be extended holomorphically to the side in D.

For each p ≈ p_{0} and q ≈ q_{0} with q_{0} = f(p_{0}), we introduce Segre neighborhood systems near p and q, as did in §2: U_{3}(p) ⊃⊃ U_{2}(p) ⊃⊃ U_{1}(p) ∋ p; U′_{3}(q) ⊃⊃ U′_{2}(q) ⊃⊃ U′_{1}(q) ∋ q. As in [DFY] and [DP] (in particular, see [DP]), we define the “potential” correspondence of f near p_{0} as follows:

V(p_{0}, q_{0}) = {(w, w′) ∈ U_{1}(p_{0}) × U′_{1}(q_{0}) : f(Q_{w}(p_{0}) ∩ U_{2}(p_{0}) ∩ D) ⊃ _{w^{′∗}}Q′_{w′}(q_{0})}.

Here, as in §2, we write w′^{∗} for the reflection point R′(w′) of w′; and we use _{w′∗}Q′_{w′} to denote the germ of the variety Q′_{w′} at w^{′∗}. Also, the Segre varieties are chosen in the indicated Segre systems.

In what follows, we will drop the reference points for the Segre systems and Segre varieties to simplify the notation, when there is no confusion arising.

By the invariant property of Segre surfaces, it can be seen that if f extends holomorphically in a small neighborhood G of some point p ∈ U_{1}(p_{0}) ∩ ∂D, then V(p_{0}, q_{0}) ⊃ \{(w, w′) ≈ (p, f(p)) : w′ = f(w), w ∈ G \ \overline{D}\}.

The key point in proving Theorem 1.3′ is to show that V(p_{0}, q_{0}) extends as an analytic variety in U_{1}(p_{0})×U′_{1}(q_{0}) with the first projection map π : V(p_{0}, q_{0}) → U_{1}(p_{0}) locally proper. For this purpose, we will use the strategy of Diederich-Pinchuk [DP] of applying the extension lemma of Bishop ( [Chi]).

Before arguing further, we need recall some useful facts: (7.a)

∂D \ (U_{2}(p_{0}) ∩ ∂D ∩ Q_{T})

is dense in ∂D ∩ U_{1}(p_{0}), where Q_{T} = ∪_{z∈(T_{1}∪T_{0})∩U_{1}(p_{0})}Q_{z}(p_{0}). (7.b) ([DP]): Making U′_{1} small, then (T′_{1} ∪ T′_{0}) ∩ U′_{1}(q_{0}) is contained in a pluripolar set T̃ in U′_{1}. That is, there is a plurisubharmonic function h (≢ −∞) such that T′_{0} ∪ T′_{0} ⊂ T̃ = {z ∈ U′_{1} : h(z) = −∞}. (This follows from the fact that the semianalytic subset (T′_{1} ∪ T′_{0}) ∩ U′_{1} is contained in a one dimensional real analytic subset [BM], which can be complexified.) (7.c) Fix U′_{1} . Using the properness and the boundary continuity of f, one can always choose U_{2} small enough so that V has no limit points in (U_{1} \ \overline{D}) × ∂U′_{1}. (7.d) ([DP]) For each p ∈ T_{2}^{+}, after making U_{2}(p) small, it holds that Q_{p}∩U_{2}(p)∩∂D ⊂ ∂D^{+} ∪ {p}. (7.e) For each p ∈ ∂D^{+}, after making U_{2}(p) small and then making U_{1}(p) sufficiently small, it then holds that Q_{p} ∩ U_{2}(p) ∩ ∂D = {p̃} for any p̃ ∈ U_{1}(p). (7.f) ([BR2]) Let M_{1} and M_{2} be two connected oriented real analytic hypersurfaces in C^{n} of finite D-type. Let f be a non constant holomorphic map from M_{1} into M_{2} with f(M_{1}) ⊂ M_{2}. Then f is locally finite and the normal component of f has non vanishing normal derivative at each point of M_{1}. So, by the standard result, it follows that f is locally proper at each point in M_{1} and preserves the sides. (7.h) (Bishop’s Lemma [Chi]): Let E be a (complete) pluripolar set in the domain U ×U′ ⊂ C^{4}, where U and U′ are domains in C^{2}. Let A ⊂ (U ×U′)\E be an analytic set of pure dimension 2 with no limit points in U × ∂U′. Suppose that there exists an open subset V ⊂ U such that \overline{A} ∩ ((V × U′)) is an analytic variety in V × U′ of pure dimension 2. Then \overline{A} is an analytic set in U × U′.

Remark (7.1)′(a): We observe that (7.d) can be strengthened as follows: For any point p̃(∈ T_{2}^{+}) ≈ p, it also holds that Q_{p} ∩ U_{2}(p) ∩ ∂D ⊂ ∂D^{+} ∪ p̃. The proof of this fact is the same as that for (7.d) (see [DP]). However, since this fact will be important in our later discussion, for the convenience of a reader, we give the following details (see a similar argument in [DP]):

After an appropriate holomorphic change of coordinates, we can assume that p = 0 and T_{2}^{+} near p is iR^{2} = {(iy_{1}, iy_{2})}. We also choose z_{1}-axis for the complex tangent space of ∂D at 0. By the way we chose T_{2}^{+}, we can assume that D is defined near 0 by ρ = 2x_{2} + (2x_{1})^{k}a(z_{1}, \overline{z_{1}}, y_{2}), where m > 1 and a|_{iR^{2}} ≢ 0. By the way we chose T_{2}^{+}, we can see that k is even and a(0) > 0 (see [DP]). Now, for w = (ib_{1}, ib_{2}) ∈ iR^{2}, Q_{w} is defined by the equation: z_{2} − ib_{2} + (z_{1} − ib_{1})^{2m}a(z_{,}\overline{w}) = 0. So, it can be seen that near the origin, Q_{w} ∩ iR^{2} = {w}. This completes the proof of the above fact.

(b) To see (7.a), we notice the facts that Q_{z} ∩ ∂D is at most a real analytic subset of dimension 1 and T_{1} ∪ T_{0} is contained in a real analytic set of at most dimension 1 [BM]. Therefore, U_{2}(p_{0}) ∩ ∂D ∩ Q_{T} is contained in a real analytic space which can be fibered along a one dimensional real analytic curve and each fiber has at most dimension 1. Thus, U_{2}(p_{0}) ∩ ∂D ∩ Q_{T} has at most dimension 2.

(c) (7.e) can be seen as follows: For each p̃ close to p, after a normal holomorphic change of coordinates (see related discussions in §2) and using the strong pseudoconvexity of ∂D at p̃, one sees clearly that Q_{p} cuts ∂D only at p̃. Since this process can be done uniformly, we can achieve (7.e) after shrinking the size of U_{2}.

Lemma 7.2: Shrinking U_{1}(p) if necessary, one can make V(p, q) having no limit points in (U_{1}(p) \ \overline{D}) × Σ′.

Proof of Lemma 7.2: Suppose that (w_{v}, w′_{v}) ∈ V with (w_{v}, w′_{v}) → (w_{0}, w′_{0}) ∈ (U_{1} \ \overline{D}) × Σ′. Then f(Q_{w_{v}} ∩ U_{2} ∩ D) ⊃ _{w′_{v}^{∗}}Q′_{w′_{v}}. In particular, from the properness of f, it can be seen that there exists a point ξ_{v}^{∗} ∈ Q_{w_{v}} ∩\overline{U}_{2} ∩D such that f(ξ_{v}^{∗}) = w_{v}^{′∗} and f(Q_{w_{v}} ∩ O(ξ_{v}^{∗})) ⊃_{w′_{v}^{∗}} Q′_{w′_{v}} Assume, without loss of generality, that ξ_{v}^{∗} → ξ_{0} ∈ \overline{U}_{2} \ D and w_{v}^{∗} → w′_{0}. Then f(ξ_{0}) = w′_{0} and ξ_{0} ∈ ∂D. By Lemma 7.1, it follows that f extends holomorphically to a neighborhood of ξ_{0}. Therefore, there is a small open subset Ω near ξ_{0} such that f is a proper holomorphic map from Ω to f(Ω) and f|\overline{Ω}^{−1}(f(ξ_{0})) = ξ_{0} (see (7.f)). Meanwhile, f(Ω∩D) ⊂ D′, f(Ω∩D^{c}) ⊂ D′^{c} ∩U′_{2}. Now, let η_{v} ∈ Ω be such that f(η_{v}) = w′_{v}. Then, for v ≫ 1, it holds that f(Q_{η_{v}} ∩ O(η_{v}^{∗})) ⊂ Q′_{w′_{v}}. Obviously, η_{v} → ξ_{0} and f(Q_{η_{v}} ∩ Ω) ⊃ f(Q_{w_{v}} ∩ O(ξ_{0})). Write f|_{Ω}^{−1} = {σ_{1}, ⋯, σ_{k}}. Hence, we see that Q_{w_{v}} ∩ O(ξ_{0}) ⊂ ∪_{j=1}^{k}σ_{j}(f(Q_{η_{v}} ∩ Ω)).

Similar to Claim 5.2, we have the following

Claim 7.2′: For v sufficiently large, Q_{η_{v}} ∩ U_{2} = Q_{w_{v}} ∩ U_{2}. Thus, by passing to the limit, it holds that w_{0} ∈ A_{ξ_{0}}.

Proof of Claim 7.2′: Let ρ and ρ′ be the defining functions of D and D′, respectively. Then we notice that ρ′(f(z), \overline{f}(w)) = ρ(z, \overline{w})h(z, \overline{w}) for z, w ∈ Ω (see (7.f)) and h ≠ 0 (we might need to shrink Ω here).

For z ∈ ∪_{j=1}^{k}σ_{j}(f(Q_{η_{v}} ∩ Ω)), there is a point z̃ ∈ Q_{η_{v}} ∩ Ω such that f(z) = f(z̃). Now, from

ρ′(f(z), \overline{f(η_{v})})) = ρ(z, \overline{η_{v}})h(z, \overline{η_{v}})

and ρ′(f(z̃), \overline{f(η_{v})}) = ρ(z̃, \overline{η_{v}})h(z̃, \overline{η_{v}}) = 0, it follows that ρ(z, \overline{η_{v}}) = 0. Thus, we conclude that z ∈ Q_{η_{v}}. Since both Q_{w_{v}} and Q_{η_{v}} are connected complex curves in U_{2}, we see the Q_{η_{v}} ∩ O(ξ_{0}) = Q_{w_{v}} ∩ O(ξ_{0})

Passing to a limit, we now conclude that Q_{ξ_{0}} ∩ U_{2} = Q_{w_{0}} ∩ U_{2}.

Next, w_{0} ∈ A_{ξ_{0}}. Since ξ_{0} ∈ ∂D, w_{0} ∈ ∂D. This is a contradiction and thus completes the proof of Lemma 7.2.

In what follows, we always make U_{1} small such that Lemma 7.2 holds. Following the strategy of Diederich-Pinchuk, we now prove the following lemma.

Lemma 7.3: Suppose that for certain U_{2}(p) small enough, it holds that (Q_{p}∩∂D∩ U_{2}(p)) \ {p} ⊂ Σ. Then after shrinking U_{1} and U′_{1} if necessary, V(p, q) is an analytic sub-variety of (U_{1}(p)\\overline{D})×(U′_{1}(p)\\overline{D′}) which has no limit points in ((U_{1} \ \overline{D}) × ∂U′_{1})∪

((U_{1} \ \overline{D}) × Σ′). We mention that in this lemma, (Q_{p} ∩ ∂D ∩ U_{2}(p)) \ \{p\} can be empty, and all Segre surfaces are defined in the Segre system as chosen here.

Proof of Lemma 7.3: Part of the proof is greatly motivated by an argument in [DP] for the study of biholomorphic mappings.

Let U_{2} be sufficiently small so that (Q_{p} ∩ \overline{D} ∩ \overline{U}_{2}) \ \{p\} ⊂ Σ. We also assume the property in (7.c). When (Q_{p} ∩ \overline{D} ∩ ∂U_{2}) \ \{p\} ≠ ∅, then by the hypothesis, there is a small ϵ > 0 such that f extends holomorphically to \overline{B_{3ϵ}(E_{p})}, where E_{p} = Q_{p} ∩\overline{D}∩∂U_{2} and B_{δ}(E_{p}) = {z : dist(z, E_{p}) < δ} for δ > 0. Moreover, by (7.f), we can assume that f is finite to one from B_{3ϵ}(E_{p}) and p ∉ B_{4ϵ}(E_{p}). Now, we let R′(p) = {w′ ∈ U′_{1} \ D′ : f(Q_{p} ∩ B_{2ϵ}(E_{p})) ⊃ _{w′^{∗}}Q′_{w′}}. Then we claim that R′(p) is a finite set. Indeed, this can be seen as follows: Let {w′_{v}(∈ R′(p))} with w′_{v}^{∗} → w′^{∗}. Choose w_{v} ∈ Q_{p} ∩B_{2ϵ}(E_{p}) such that f(w_{v}) = w′_{v}^{∗}. Assume also that w_{v} → w ∈ Q_{p} ∩ \overline{B_{2ϵ}(E_{p})}. Then as above, f is a proper holomorphic map from a small neighborhood of w. This implies that f(Q_{p} ∩ O(w)) is an analytic curve near w′^{∗}. Thus, it follows that any Q′_{w′_{v}} coincides with Q′_{w′} when v ≫ 1. So, by the finiteness of A′_{w′}, it follows that {w′_{v}} is a finite sequence.

Let R(p) = {w ∈ Q_{p} ∩ B_{2ϵ}(E_{p}) : f(w) = w′^{∗} for some w′ ∈ R′(p)}. Then R(p) is also finite. Slightly shrinking U_{2} and ϵ if necessary, we can assume that R(p) ∩ ∂U_{2} ∩ \overline{D} = ∅.

To complete the proof of Lemma 7.3, we need to show that after shrinking U_{1}, V is an analytic sub-variety in (U_{1} \ \overline{D}) × (U′_{1} \ \overline{D′}), which has no limit points in ((U_{1} \ \overline{D}) × ∂U′_{1}) ∪ ((U_{1} \ \overline{D}) × Σ′).

We first claim that V is a local variety. To this aim, we will prove that after shrinking the size of U_{1}, for any (w, w′) ∈ V, then E^{∗}(w, w′)∩∂U_{2}∩Q_{w} ∩\overline{D} = ∅, where E^{∗}(w, w′) = \{ξ ∈ (U_{2}∩D)∪B_{ϵ}(E_{w})∩Q_{w} : f(_{ξ}Q_{w}) ⊃_{w^{′∗}} Q′_{w′}\}. Here E_{w} = Q_{w}∩\overline{D}∩∂U_{2} and B_{ϵ}(E_{w}) is defined in a similar way. Also, by f(_{ξ}Q_{w}) ⊃_{w^{′∗}} Q′_{w′}, we mean that for each small neighborhood O(ξ) of ξ, it holds that f(Q_{w} ∩ O(ξ)) ⊃_{w^{′∗}} Q′_{w′}.

We also make \overline{B_{ϵ}(E_{w})} ⊂ B_{2ϵ}(E_{p}) by shrinking U_{1}.

Indeed, when E_{p} = (Q_{p} ∩ \overline{D} ∩ ∂U_{2}) = ∅, a continuity argument then shows that E_{w} = (Q_{w} ∩ \overline{D} ∩ ∂U_{2}) = ∅ for w ≈ p.

Now, if the above assertion does not hold in case E_{p} ≠ ∅, after shrinking U_{1} again, we can then find several sequences \{z_{j}\} ⊂ ∂U_{2} ∩ \overline{D} ∩ E^{∗}(w_{j}, w′_{j}) with z_{j} → z_{0}, \{w_{j}\} ⊂ U_{1} \ \overline{D} with w_{j} → p, and \{w′_{j}\} ⊂ U′_{1} \ \overline{D}′ with w′_{j} → w′_{0} ∈ U′_{1} such that the following holds:

(a) z_{j} ∈ Q_{w_{j}}, (b) f(z_{j}) = w′_{j}, and (c) f(_{z_{j}}Q_{w_{j}} ∩ B_{ϵ}(E_{w_{j}})) ⊃_{w_{j}^{′∗}} Q_{w′_{j}} when j >> 1. We observe that f extends to a proper holomorphic map from O(z_{0})∩B_{ϵ}(E_{p}), Hence, by passing to the limit, we can see that f(_{z_{0}}Q_{p} ∩ B_{ϵ}(E_{p})) ⊃_{w_{0}^{′∗}} Q′_{w′_{0}}, f(z_{0}) = w′_{0}, and z_{0} ∈ Q_{p} ∩∂U_{2} ∩\overline{D}. Hence, we have a contradiction; for it implies that z_{0} ∈ R(p)∩∂U_{2}.

Now, by making the size of U_{1} sufficiently small, we can assume that for each (w, w′) ∈ V, E^{∗}(w, w′)∩D∩\overline{U}_{2} ⊂ U_{2} ∩D. Write E^{∗}(w, w′)∩D∩\overline{U}_{2} = \{a_{1}, ⋯, a_{m}\} ⊂ D ∩ U_{2}. Let (w̃, w̃′)(∈ V) ≈ (w, w′). Then, since E(w̃, w̃′) ∩ D ∩ ∂U_{2} = ∅, it can be seen that there exists a point ξ̃ so that ξ̃ ∈ Q_{w} ∩ U_{2} ∩ D, f(_{ξ}Q_{w}) ⊃_{w′}^{∗} Q′_{w′}, and

f(ξ̃) = w̃′^{∗}. Indeed, let \{ξ̃_{j}\} = f^{−1}(w̃^{′∗}) ∩ \overline{U}_{2} ∩ D. Then since f is locally proper near each ξ̃_{j}, the choice of our w′ indicates that f(O(ξ̃_{l}) ∩ Q_{w}) ⊃_{w′}^{∗} Q′_{w′} for a certain l. So, we can take ξ̃ to be this ξ̃_{l}, which has to stay in U_{2} ∩ D.

By the properness of f and by the continuous dependence of Segre surfaces on the base points, it follows that ξ̃ has to be close to one of the a′_{j}s when w̃ is close to w; for otherwise we would have E^{∗}(w, w′)∩\overline{D}∩∂U_{2} ≠ ∅. We let ρ_{2} be a real analytic defining function of ∂D′ near q = f(p), and we choose a coordinates system near p so that the tangent space T_{p}^{(1,0)}∂D is the z_{1}-axis. Then Q_{w} is given by z_{2} = h(z_{1}, \overline{w̃}) for some holomorphic function h. We claim that the condition that f(Q_{w} ∩ D ∩ U_{2}) ⊃ _{w′}^{∗}Q′_{w′} for (w̃, w̃′) ≈ (w, w′), can be expressed by the equation ρ_{2}(f(z), \overline{w̃}′) = 0 with z ∈ Q_{w}∩ O(a_{j}) for some j. Indeed, ρ_{2}(f(z), \overline{w̃}′) = 0 implies that f(Q_{w} ∩ O(a_{j})) ⊂ Q′_{w′}. Since we can make O(a_{j}) sufficiently small so that f is proper from O(a_{j}) to a neighborhood, say O′_{j}, of w^{′∗}, f(Q_{z} ∩ O(a_{j})) is also an analytic variety of O′_{j} of codimension 1. Since we can choose O_{j} and O′_{j} suitably so that O′_{j} ∩ Q′_{w′} is connected for w̃′ ≈ w′, thus f(Q_{w} ∩ O(a_{j})) = Q_{w′} ∩ O′_{j}. Now, if we make w̃′ close enough to w′, then by the continuity of the R-operator, w̃^{′∗} ∈ f(Q_{w} ∩ O(a_{j})). This verifies our claim.

On the other hand, we have the following Taylor expansion near a_{l}^{(1)} for each l, where a_{l}^{(1)} is the first coordinate of a_{l}:

ρ_{2}(f(z), \overline{w̃}′) = ∑_{j} b_{j}^{l}(\overline{w̃}, \overline{w̃′})(z_{1} − a_{l}^{(1)})^{j}

Let V_{l} be the common zeros of b_{j}^{l}(\overline{w̃}, \overline{w̃′}) for j = 0, 1, · · ·. Then clearly near (w, w′), V is the finite union of V_{l}’s. This shows that V is a local variety.

We next let (w_{v}, w′_{v})(∈ V) → (w_{0}, w′_{0}) ∈ (U_{1} \ \overline{D}) × (U′_{1} \ \overline{D}′). We then show that (w_{0}, w′_{0}) ∈ V, too. Notice that f(Q_{w_{v}} ∩ U_{2}) ⊃ _{w′}_{v}^{∗}Q′_{w′_{v}}. So, as argued before, there is a point ξ_{v} ∈ Q_{w_{v}} ∩ U_{2} ∩ D such that f(_{ξ_{v}}Q_{w_{v}}) ⊃_{w_{v}^{′∗}} Q′_{w′_{v}}. Assume, without loss of generality, that ξ_{0} = lim ξ_{v}. Letting v → ∞, since ξ_{0} ∈ Q_{w_{0}} ∩ \overline{U}_{2} ∩ D and since f is locally proper near ξ_{0}, it follows that f(O(ξ_{0}) ∩ Q_{w_{0}} ∩ (B_{ϵ}(E_{w_{0}}) ∪ (U_{2} ∩ D)) ⊃_{w_{0}^{′∗}} Q′_{w′_{0}}. Thus, by the above choice of U_{2}, it follows that ξ_{0} ∈ U_{2}. We therefore conclude that f(Q_{w_{0}} ∩ U_{2} ∩ D) ⊃ _{w′}_{0}^{∗}Q′_{w′_{0}}. This shows that (w_{0}, w′_{0}) ∈ (U_{1} \ \overline{D}) × (U′_{1} \ \overline{D′}).

Next, applying Lemma 7.2 and (7.b), we conclude that V has no limit points in (U_{1} \ \overline{D}) × ∂U′_{1} ∪ (U_{1} \ \overline{D}) × Σ′.

We still keep the above notation and assume the hypothesis in Lemma 7.3. Notice that V contains points of dimension 2 (see the observation made after Lemma 7.1). We see that V is an analytic variety of dimension 2 in ((U_{1} \ \overline{D}) × U′_{1}) \ A). Here A = (U_{1} \ \overline{D}) × (T′_{1} ∪ T′_{0}).

Decompose V into a locally finite union of its irreducible components. In what follows, we will be only interested in the locally finite union of those two dimensional components, which we will still denote by V, for the brevity of notations. (The new V can be formed by considering all smooth points of dimension 2 in the old V and then adding new points by taking the closure). Now, we can say that V is an analytic variety in ((U_{1} \ \overline{D}) × U′_{1})) \ A, which has pure dimension 2.

Next, we let Ã = (U_{1} \ \overline{D}) × T̃(⊃ A), which is a pluripolar set in (U_{1} \ \overline{D}) × U′_{1} by (7.b). We also write Ṽ = V \ ((U_{1} \ \overline{D}) × T̃)). We now think of V and Ṽ as subsets in (U_{1} \ \overline{D}) × U′_{1}. Then, we see that Ṽ is an analytic variety of pure dimension 2 in ((U_{1} \ \overline{D}) × U′_{1}) \ Ã.

Lemma 7.4: As before, let {U_{j}}_{j=1}^{3} be a Segre neighborhood system at p and assume the hypothesis in Lemma 7.3. If there exists a sequence {p_{j}} ⊂ Σ with p_{j} → p, such that Q_{p_{j}} ∩ ∂D ∩ U_{2} ⊂ Σ, then V extends as an analytic variety of pure dimension 2 in U_{1} × U′_{1}, after shrinking U_{1} and U′_{1}. Thus, by Theorem 1.1, f admits a holomorphic extension at p.

Proof of Lemma 7.4: The main idea of the proof is to apply the Bishop lemma, as first used in the work of Diederich-Pinchuk [DP]. Without loss of generality, we also assume that any CR function defined near p ∈ ∂D can be holomorphically extended to D (near p) (see [BT] and [Tr]).

We let z be a certain p_{j} with j ≫ 1. Slightly shrinking U_{2}, we observe that Q_{z} ∩ ∂D ∩ \overline{U}_{2} ⊂ Σ. It thus follows that for a small neighborhood G of z, one also has Q_{z} ∩ ∂D ∩ \overline{U}_{2} ⊂ Σ for any z̃ ∈ \overline{G}. Now, we consider V^{∗} = (V ∩ ((G \ \overline{D}) × U′_{1})). Then as did in Lemma 7.2, one sees that V^{∗} has no limit points in (G \ \overline{D}) × Σ′. We claim that V^{∗} has no limit points in (G \ \overline{D}) × (T′_{1} ∪ T′_{0}), neither. To see this, we let (w_{v}, w′_{v})(∈ V^{∗}) → (w_{0}, w′_{0}) ∈ (G \ \overline{D}) × (T′_{1} ∪ T′_{0}). Then, as in Claim 7.2 ′, one sees that for certain ξ_{0} ∈ Q_{w_{0}} ∩ ∂D ∩ \overline{U}_{2}, it holds that f(ξ_{0}) = w′_{0}. However, by our choice of G, we know that ξ_{0} ∈ Σ. Thus, a similar argument as in Claim 7.2′ indicates that w_{0} ∈ A_{ξ_{0}} ∈ ∂D. This is a contradiction.

Hence, from the above argument, it follows that \overline{V^{∗}} ∩ ((G \ \overline{D}) × U′_{1}) = V^{∗} is an analytic variety of pure dimension 2 in (G \ \overline{D}) × U′_{1} without limit points in U_{1} × ∂U′_{1}. Now, we let Ṽ^{∗} = V^{∗} \ Ã. Then \overline{Ṽ^{∗}} ∩ ((G \ \overline{D}) × U′_{1}) = \overline{Ṽ} ∩ ((G \ \overline{D}) × U′_{1}).

Claim 7.5: \overline{Ṽ^{∗}} ∩ ((G \ \overline{D}) × U′_{1}) = V^{∗}. Thus, \overline{Ṽ} ∩ ((G \ \overline{D}) × U′_{1}) is an analytic variety of pure dimension 2 in (G \ \overline{D}) × U′_{1}.

Proof of Claim 7.5: Let π_{G} : V^{∗} → G and π′_{G} : V^{∗} → U′_{1}. We first show that π′_{G} is finite to one, after shrinking G if necessary. To this aim, assume that w′ ∈ U′_{1} and let \{z_{j}\} be a sequence in π_{G}(π′_{G}^{−1}(w′)) with z_{j} → z_{0}(∈ \overline{G}). As in Lemma 7.3, we can then assume that for a certain sequence \{ξ_{j}(∈ Q_{z_{j}})\}, it holds that ξ_{j}(∈ Q_{z_{j}} ∩ \overline{U}_{2}) → ξ_{0} ∈ Q_{z_{0}} ∩ \overline{U}_{2} ∩ \overline{D}, f(ξ_{j}) → w′^{∗}, and f(_{ξ_{j}}Q_{z_{j}}) ⊃ _{w′}^{∗}Q′_{w′}. As before, by our choice of G, if follows that there is a small neighborhood of ξ_{0} which is mapped properly by f to a neighborhood of w′^{∗}. Thus, f^{−1}( _{w′^{∗}}Q_{w′}) has only finitely many components near ξ_{0}. Therefore, we conclude that {Q_{z_{j}}} is a finite sequence. Making use of the fact that A_{z} is finite for each z ∈ G, we see that π_{G}^{−1}(w′) is a finite set and thus π_{G} is a local analytic covering map. In particular, π′_{G} is an open mapping.

Next, we consider V^{∗} ∩ ((G \ \overline{D}) × T̃). Then the above fact indicates that it does not contain any non-empty open subset of V^{∗}. In fact, if this is not the case, π′_{G}(V^{∗} ∩ ((G \ \overline{D}) × T̃) ⊂ T̃ would contain open subsets of U′, which contradicts the property of T̃ (see (7.b)). Hence, we see that \overline{Ṽ^{∗}} ∩ ((G \ \overline{D}) × U′_{1}) = V^{∗}.

Now, we can apply the Bishop extension lemma (7.h) to conclude that Ṽ extends as an analytic variety of dimension 2 in (U_{1} \ \overline{D}) × U′_{1}, which has no limit points in U_{1} × ∂U′_{1}. Let us still write the extended variety as V. Since any compact analytic variety in C^{2} has to be a finite set, we can see, by using the above mentioned property of V, that π^{−1}(z) is finite for each z ∈ U_{1} \\overline{D}, and π is proper. Hence, V can have only finitely many irreducible components, {V}_{j=1}^{m}, (of dimension 2); and each of them has to be a sheeted analytic covering space over U_{1} \ \overline{D} ([Wh]).

So, each V_{i} can be defined by equations of the form ([Wh]):

P_{ik}(w, w′_{k}) = w′_{k}^{N_{ik}} + ∑_{0≤j<N_{ik}} a_{ikj}(w)w′_{k}^{j}

with a_{ikj}(w) holomorphic and bounded in U_{1} \ \overline{D}. By the result of Baouendi-Treves [BT] and Trepreau [Tr], we see that a_{ikj} can be holomorphically extended to a neighborhood of p. So each V_{i} extends to a neighborhood of p. Notice that there are points arbitrarily close to p where f extends, we conclude that Γ_{f} ⊂ ∪V_{i}. This tells that f extends as a holomorphic correspondence near p. Applying Theorem 1.1, we see that f extends holomorphically across p.

We are now ready to complete the proof of Theorem 1.3′ by the argument appeared in [DFY]: First, Σ ⊃ ∂D^{−} ∪ T_{2}^{−} ∪ T_{2}^{±}. Since f extends almost everywhere, from (7.e), Lemma 7.3, and Lemma 7.4, it follows that ∂D^{+} ⊂ Σ. Applying Lemma 7.1, Lemma 7.3 and Lemma 7.4 with Remark 7.1′(a), we conclude, from the fact ∂D^{+} ⊂ Σ, that T_{2}^{+} ⊂ Σ. Now, we can use (7.a) to show that T_{1} ∪ T_{0} ⊂ Σ:

Let z(≈ p_{0}) ∈ T_{1}. Since A_{p_{0}} is finite, it follows that only finitely many z’s can make Q_{z} ∩ ∂D to be a real analytic set of dimension 1 in T_{1}. Thus, by Lemma 7.3, Lemma 7.4, and (7.a); it follows that f extends at most except at finitely many points near p_{0}. Applying again Lemma 7.3 and Lemma 7.4, we see that f extends also as a holomorphic correspondence at those isolated points, too. The proof of Theorem 1.3′ is now complete.

Remark 7.6: Except the first part of Lemma 7.1, the entire proof of Theorem 1.3′ is purely a local argument. So, using the following Lemma 7.1′′ to replace part of Lemma 7.1, we also see the proof of the following local version of Theorem 1.3

Theorem 1.3′′: Let D_{1} and D_{2} be two bounded domains in C^{2}. Assume that M_{1} and M_{2} are contained in the real analytic boundaries of D_{1} and D_{2}, respectively. Suppose that f is a proper holomorphic mapping from D_{1} to D_{2}, that is continuous on D_{1} ∪ M_{1} and sends M_{1} into M_{2}. Also assume that M_{1} and M_{2} are of finite type. Then f admits a holomorphic extension across M_{1}.

Lemma 7.1′′: Assume the notation and hypothesis in Theorem 1.3′′. Let p_{0} ∈ M_{1} be such that D_{2} is pseudoconvex near q_{0} = f(p_{0}). Then f admits a holomorphic extension at p_{0}.

Proof of Lemma 7.1′′: Construct a pseudoconvex domain of finite type Ω_{2} ⊂ D_{2} with ∂Ω_{2} containing a small piece of M_{2} near q_{0}. Let Ω_{1} be a connected component of f^{−1}(Ω_{2}) that contains p_{0} in part of its smooth boundary. Then f is proper from Ω_{1} to Ω_{2} and maps M_{1} near p_{0} to M_{2} near q_{0}. Meanwhile, it also follows that Ω_{1} is pseudoconvex. Now, by examining the proof of Theorem 2 of [BC] and by [BBR], to see the extension of f at p_{0}, it suffices for us to show the for z(≈ z_{0}) ∈ Ω_{1}, it holds that dist(f(z), M_{2})^{γ} <_{∼} dist(z, M_{1}), for some positive integer γ. But this can be seen by applying the Hopf lemma, a classical argument of Henkin, and using the existence of family of plurisubharmonic peaking functions near p_{0}; as did in [Ber] (pp 629, line -15, pp 629, line 11) and [BC] (pp359 line -6, pp 360 line 7).

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Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA Current address: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley,

CA 94720, USA

  E-mail address: xhuang@math.uchicago.edu; xhuang@msri.org