\documentstyle[12pt]{amsart} \textheight=574pt \textwidth=432pt \oddsidemargin=18.88pt \evensidemargin=18.88pt \topmargin=14.21pt \begin{document} \def\CC{\bf C} \def\RR{\bf R} \def\C{\hbox{C}} \def\g{\gamma} \def\rh{\rho} \def\DD{\D_{\a^*}} \def\ale{\mathrel{\mathop{<}\limits_{\sim}}} \def\ld{\lambda} \def\sm {\setminus} \def\O{\Omega} \def\b{\beta} \def\o{\omega} \def\D{\Delta} \def\ss{\subset} \def\1{{(0,\gamma)\OO \|\gamma\|}} \def\L{\Lambda} \def\ic{e^{i\theta}} \def\pd{\phi(\Delta)} \def\h{\hbox} \def\df{{d\Phi^*\OO d\xi}} \def\q{Q(X,\xi)} \def\wt{\widetilde} \def\ra{\rightarrow} \def\ih{I_{2n+1}+i{\partial H(X)\over\partial X }} \def\PP{\partial} \def\OO{\over} \def\d{\delta} \def\l{\lambda} \def\Sqt{\sqrt} %\def\endpf{\hbox{\vrule height1.5ex width.5em}} \def\endpf${$\qed} \def\H{\hat} \def\a{\alpha} \def\d{\delta} \def\V{\h{Val}} \def\Ol{\overline} \def\OO{\over} \def\R{\h{Re}} \def\I{\h{Im}} \def\C{\hbox{C}} %\hoffset=0.5in %\voffset=1.0in \hsize=6.0truein \vsize=8.0truein \def\D{\Delta} \def\d{\delta} \def\a{\alpha} \def\zn{z_{n+1}} \def\Sqt{\sqrt} \title[Regularity of Holomorphic Correspondences] {Regularity of Holomorphic Correspondences and Applications to the Mapping Problems} \author{Xiaojun Huang} \address{ Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA} \curraddr{ Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720, USA} \email{xhuang@@math.uchicago.edu; xhuang@@msri.org} \thanks{Supported in part by NSF DMS-9500881. Research at MSRI is supported in part by NSF DMS-9022140.} \begin{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. \end{abstract} \maketitle \section*{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: \bigbreak {\bf Conjecture 0.1}: Let $M_1$ and $M_2$ be two real analytic hypersurfaces of finite D-type in $\bf C^n$ with $n>1$. Suppose that $D\subset {\bf C^n}$ is a domain with $M_1$ as part of its real analytic boundary. Also, suppose that $f$ is a holomorphic mapping from $D$ into $\bf C^n$, that is continuous up to $D\cup M_1$ and maps $M_1$ into $M_2$. Then $f$ admits a holomorphic extension across $M_1$. \bigbreak 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. % (see [Da], for example). 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 $\overline {\PP}$-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. 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 $\bf C^2$, 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 $\bf C^2$ 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 $\bf C^2$ is real analytic. This, in particular, gives a complete solution to Conjecture 0.1 in the 2-dimensional 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 {\it 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 $D\cup M_1$ 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 $\bf C^2$ 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 $\bf C^2$. 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 ($\S 7$) for completeness of the proof of Theorem 1.3. \bigbreak 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 $\S 7$) are adapted from [DFY] and, in particular, [DP]. \bigbreak {\bf Acknowledgment}: The author is grateful to Professors S. Webster and R. Nara\-shim\-han for their many stimulating conversations %encouragement during the preparation of this paper (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). \bigbreak %Unpon the completion of the present paper, %we learnt that %After the compleletion of this paper, {\it 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, 1995). \section{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 $\bf C^{n+1}$ ($n\ge 1$), 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 {\it 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 ${\cal P}_j(z;X)$ in $X$ with leading coefficient 1 and other coefficients holomorphic in a small neighborhood $U_1$ of $p$, such that ${\cal P}_j(z;f_j(z)) = 0$ for $z\in U_1\cap M_1$. 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 $U_1\times U_2$ such that (a) $ V\supset \Gamma_f=\{(z,w): w=f(z),\ z\in M_1\cap U_1\}$ (b) the natural projection of $V$ to $U_1$ is locally proper. We also recall that a real analytic hypersurface $M\subset {\bf C^{n+1}}$ is called {\it rigid} if, after a holomorphic change of coordinates, it can be defined by a real analytic function of the form $\rho(z,\overline {z})=\zn+\overline {\zn}+\chi(z',\overline {z'})$, where $z'=(z_1,\cdots,z_n)$. \bigbreak {\bf Theorem 1.1}: Let $M_1$ and $M_2$ be two real analytic hypersurfaces of finite D-type in $\bf C^{n+1}$ ($n\ge 1$). Let $p\in M_1$. 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. % if either $n=1$ or $M_2$ is decoupled. \bigbreak See also Theorem 1.1$'$ of Section 5 for a more general version of this result and notice that in whole paper, only in $\S 5$, 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: \bigbreak {\bf Theorem 1.2}: Let $M_1$ and $M_2$ be two algebraic hypersurfaces of finite type in $\bf C^2$. Let $f$ be a continuous CR mapping from $M_1$ to $M_2$. Then $f$ is real analytic on $M_1$. \bigbreak {\bf Corollary 1.2$'$}: Let $D$ and $D'$ be two smoothly bounded domains with real algebraic boundaries in $\bf C^2$. Let $f$ be a proper holomorphic mapping from $D$ to $D'$. Then $f$ admits a holomorphic extension to a neighborhood of $\overline {D}$. \bigbreak We remark that Theorem 1.2 is sharp in a certain sense. In fact, using the inner functions defined on the ball in $\bf C^2$, one can construct a bounded CR mapping between spheres in $\bf C^2$ which is even not continuous at any point. %Another observation we would like to make is %that in the first part of Theorem 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 $\bf C^2$ 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 $\S 7$ the following result (see [BBR] for the pseudoconvex case, and Theorem 1.3$''$ in $\S 7$ for a local version of this result). \bigbreak {\bf Theorem 1.3}: Let $D$ and $D'$ be two bounded domains in $\bf C^2$ with real analytic boundaries. Suppose that $f$ is a proper holomorphic map from $D$ to $D'$, which extends continuously across $\PP D$. Then $f$ extends holomorphically across $\PP D$. % In addition, when $D$ and $D'$ have algebraic boundaries, %the assumption of continuity of $f$ up to $\PP D$ can be dropped. \bigbreak 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): \bigbreak {\bf Theorem 1.4}: Let $D$ and $D'$ be two smoothly bounded algebraic domains in $\bf C^n$ 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 $\overline {D}$. \bigbreak The organization of the remainder of this paper is as follows: $\S 2$ through $\S 5$ is devoted to the proof of Theorem 1.1. In $\S 2$, we will study the holomorphic extendibility of a $\l$-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 $\S 3$, we will present the connections between the branching properties of the map and the invariant varieties (Segre-surfaces). In $\S 4$, 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 $\S 6$, 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 ($\S 7$), we will present a proof of Theorem 1.3. \section{Basic set-ups and holomorphic extendibility of a reflection function} \bigbreak 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 $M_1\subset {\bf C^{n+1}}$, $M_2\subset {\bf C^{n+1}}$ be real analytic hypersurfaces of finite D-type, and let $f$ be a {\it non-constant} CR mapping from $M_1$ to $M_2$. Assume that $f$ extends as a holomorphic correspondence to a neighborhood of $p(\in M_1)$. Our main objective from $\S 2$ to $\S 5$ 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): $$\rho_1(z,\overline z)=z_{n+1}+\overline {z_{n+1}}+\sum_{j=0}^{\infty}\phi_j(z',\overline {z'}) \left({z_{n+1}-\overline {\zn}\OO 2i}\right)^j,\leqno (2.1)$$ %with $\phi_j(0,\overline {z'};)=\phi_j(z',0)\equiv 0$ and $\phi_j$ holomorphic %in $z',\overline {z'}$ over certain fixed open neighborhood of $0$; $$\rho_2(z,\overline z)=z_{n+1}+\overline {z_{n+1}}+\sum_{j=0}^{\infty}\psi_j(z',\overline {z'}) \left({z_{n+1}-\overline {\zn}\OO 2i}\right)^j,\leqno (2.2)$$ with $\phi_j(z',0)=\phi_j(0,\overline {z'})=\psi_j(0,\overline {z'})=\psi_j(z',0)\equiv 0$ and $\phi_j$,$\psi_j$ holomorphic in $z',\overline {z'}$ over certain fixed open neighborhood of $(0,0)$. Here we use $z=(z',\zn)$ with $z'=(z_1,\cdots,z_n)$ for the coordinates in $\bf C^{n+1}$. 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 $D\cup M_1$. 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: $$\wt{\rho_1 }(z,\overline {z})=\zn+\overline {\zn}+\sum_{j=0}^{\infty}\wt{\phi_j}(z',\overline {z'}) \zn ^j=0,\leqno (2.1)'$$ $$\wt{\rho_2 }(w,\overline {w})=z_{n+1}+\overline {z_{n+1}}+\sum_{j=0}^{\infty}\wt{\psi_j} (z',\overline {z'}) z_{n+1}^j=0,\leqno(2.2)'$$ where $\wt{\phi_j}$ are determined by $\phi_j$'s, $\wt{\phi_j}(z',0)=\wt{\phi_j}(0,\overline z')=\wt{\psi_j}(z',0) =\wt{\psi_j}(0,\overline {z'})=0$; and $$\wt{\rho_j}=\rho_j h_j.\leqno(2.3)$$ %%%%%In particular, it follows that %${\rho_j}(w,\overline \theta)=0$ if and only %if %$\wt{\rho_2}(w,\overline \theta) %=0$. 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 $$\wt{\rho_j}(z,\overline w)=\rho_j(z, \overline w) h_j(z, \overline w).\leqno(2.3)'$$ Therefore, for $z, w$ close to $0$, ${\rho_j}(z,\overline w)=0$ if and only if $\wt{\rho_j}(z, \overline w)=0$. 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 ${\cal U}_3$ of $0$ such that $\rho_1(z,w)$, $\wt{\rho_1}(z,w)$, $h_1(z,w)$ are holomorphic over ${\cal U}_3\times \h{conj}({\cal U}_3)$, and $h_1(z,w)\not =0$ in ${\cal U}_3\times \h{conj}({\cal U}_3)$. Here and in what follows, for a subset $A$ in the complex spaces we write $\h{conj}(A)=\{\overline {z}: z\in A\}$. For each point $w\in {\cal U}_3$, the Segre variety associated to $M_1$ and $w$ is defined by $Q_{w}=\{z\in {\cal U}_3: \rho_1(z,\overline {w})=0\}$. By making ${\cal U}_3$ small, it can be seen that $Q_{w}$ is actually a complex submanifold of ${\cal U}_3$. It is easy to show that there are two small neighborhoods $ {\cal U}_2,\ {\cal U}_1 $ of $0$ such that the following holds (see [DF1] for more details on this matter): \noindent (I.a) ${\cal U}_3\supset \supset {\cal U}_2\supset\supset {\cal U}_1\ni 0$. \noindent (I.b) ${\cal U}_1\sm M_1$ has two connected components, which are in different sides of $M_1$. Moreover, they are homeomorphic to the ball. \noindent (I.c) For each $w\in {\cal U}_1$, ${\cal U}_2\cap Q_w$ is connected. \noindent (I.d) For a sequence $\{w_j\}\subset {\cal U}_1$ with $w_j\ra w\in {\cal U}_1$, the limit set of $\{Q_{w_j}\}$, denoted by $\lim Q_{w_j}$, is $Q_{w}$. In what follows, we call $\{{\cal U}_3, {\cal U}_2, {\cal U}_1\}$ 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 ${\cal U}_3$ (${\cal U}_2$, respectively) is fixed, we can arrange ${\cal U}_2$ (${\cal U}_1$, 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 $P=P(\delta_1,\cdots,\delta_{n+1})= \{z:\ |z_j|<\delta_j (\ j=1,\cdots, n+1)\}(\subset\subset {\cal U}_3)$. Then, applying the implicit function theorem, we can see that for each $w$ with $|w|$ small, $Q_{w}\cap P$ is contained in the graph of certain function , say $z_{n+1}=g(z',\overline w)$, over the polydisc $P'=P'(\delta_1,\cdots,\delta_n)= \{z':\ |z_j|<\delta_j,\ j=1,\cdots, n\}$. Here $g(z',\overline w)$ is holomorphic for $z, w\in {\cal U}_3$ and $g(z',0)=0$. Now, if we make %$ $\delta_{j}$ ($jk_0}a_{ij}f^i_1(z) %+\sum_{ik;k>0}a_{ijk}f^{i}_1(z)f_2^{k}(z)=-{1\OO j!}{\PP G^{j}\OO\PP \l^{j}}(f%,0),$$ %where $\wt{\psi}_k(z_1,\overline {z_1})=\sum a_{ijk}z_1^i\overline {z_1}^j$. %Applying the Weiestrass preparation theorem, we see that %$f_1$ satisfies a Weiestrass polynomial % equation of degree $k_0$ with coefficients %holomorphic near $0$. %Since $f$ is continuous on $D\cup M_1$, %we conclude that $f_1$ is H\"older-$1/k_0$ continuous over %$(D\cup M_1)\cap O(0)$ [Mal]. In particular, %for any point $b=(0,b_2)$ with $b_2<0$ and $|b_2|$ sufficiently small, % we have %$|f_1(b)-f_1(0)|\ale |b_2|^{1/k_0}$ (see pp 55 of [Mal], for example). % (c) Let $(b_1, b_2)\in \bf C^2$ with $b_2$ real, %and fix $\beta\in (1/(k_0+1),1/k_0)$. Then when $|b_1|\ale |b_2|^{\b}$ %with $|b_2|$ sufficiently small, %we have $w=(b_1,b_2)\in \O^{\pm}$ %depending on whether $-b_2$ is positive or negative. Here and in %what follows, $\O^{\pm}$ is defined by $\pm\rho_2(z,\overline {z})<0$, respectively. %To see this, by our choice of $k_0$ and by noting %that $\wt{\psi_0}$ has no harmonic terms, %we have %that ${\wt \rho}_2(w,\overline {w}))=2b_2+ o(|b_2|)$ and thus %${ \rho}_2(w,\overline {w}))=b_2\h{Re}(2+ o(1))$ for small $b_2$. %Hence $\rho_2(w,\overline w)$ has the same sign as $b_2$ does. (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\sm {\cal 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 $\bf C^2$. (d) In the following sections, we always assume that $G(f(z),\l)$ is holomorphic over the closure of ${\cal U}_3\times { O}_{\l}(0)$ %Observations in (b) and (c) will be used in %$\S 5$. \section{Branches of $\cal 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 $\cal E$ still be as before. For each $z\in {\cal U}_3$, we write ${\cal F}(z)$ for the multiple-valued map formed by extending $f$ from ${\cal U}_3\cap D$. More precisely, when $z\not\in {\cal E}$, $\wt{f}(z)\in {\cal F}(z)$ if there exists a path $\gamma\subset {\cal U}_3\setminus {\cal E}$ with $\gamma (0)\in D$, $\g(1)=z$, and a continuous section ${\cal G}$ from $[0,1]$ to the sheaf of germs of holomorphic mappings to $\bf C^{n+1}$, denoted by $({\cal O}^{n+1},\pi, {\bf C^{n+1}})$, such that ${\cal G}(0)$ coincides with the germ of $f$ at $\g(0)$ and $g(\g(1))=\wt{f}(z)$, where $g$ is a representation of ${\cal G}(\g(1))$, i.e, the germ of $g$ at $\g(1)$ is ${\cal G}(\g(1))$. When $z\in {\cal U}_3\cap {\cal E}$, we let ${\cal F}(z)$ be the set of the limit points of all possible sequences $\{{\cal F}(z_j)\}$ with $z_j\not \in {\cal E}$ and $z_j\rightarrow z$. For each $z\in {\cal U}_2$, we write ${\cal F}(Q_z)=\cup_{w\in Q_z\cap {\cal U}_2}{\cal F}(w).$ We also notice that for each point in ${\cal F}(z)$, its $m-$th component is a solution of the equation ${\cal P}_m(z;X)=0$ ((2.4)). From this, it follows that when $z$ is close to $0$, then ${\cal F}(z)$ is close to the origin, too. Moreover, Lemma 2.1 now indicates that $G({\wt{f}}(z),\l)$ is independent of the choices of $\wt{f}(z)\in {\cal F}(z)$. So, in what follows, we use the notation $G({\cal F},\l)$ to replace $G(f(z),\l)$. For $w\in {\cal U}_2$, we denote by $A_w$ the set $\{\o\in {\cal U}_2: Q_w\cap {\cal U}_2=Q_{\o}\cap {\cal 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 ${\cal U}_3$, we can make $A_0=\{0\}$ and $A_z$ finite for $z\in {\cal U}_2$ (see [DW], [BJT], and [DF1]). Similarly, by making ${\cal U}'_3$ small, we can assume that $A_0'=\{0\}$ and $\#A'_{w}<\infty$ for $w\in{\cal 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 ${\cal U}_2$ if necessary, we can assume that ${\cal F}({\cal U}_2)\subset {\cal U}_1'$. Meanwhile, it is clear that ${\cal F}(0)=0$. We also retain notations which have been set up so far. \bigbreak {\bf Lemma 3.0} % (i) If $z_j\ra z$ with $\{z,z_j\}_j\subset {\cal U}_1$, then the limit sets %of $A_{z_j}$ is contained in $A_{z}$. (i) $\# {\cal F}(z)$ is constant for $z\in {\cal U}_3\sm {\cal E}$. \noindent (ii) The limit set of ${\cal F}(z_j)$ is ${\cal F}(z_0)$ if $z_j\ra z_0 (\in {\cal U}_3)$. \noindent (iii) In case $n=1$, ${\cal F}(z)=\{(X,f_2(z)):\ {\cal P}_1(z;X)=0\}$. \noindent (iv) For $w_1, w_2\in {\cal U}_1$, if the variety $Q_{w_1}\cap Q_{w_2}\cap {\cal U}_2$ contains a point of complex dimension $n$, then $Q_{w_1}\cap {\cal U}_2= Q_{w_2}\cap {\cal U}_2$. Thus $A_{w_1}=A_{w_2}$. \bigbreak {\it 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 ${\cal P}_1$ and the following standard argument: For $z\not \in {\cal E}$, write ${\cal 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 $\wt{{\cal P}_1}=\prod_{j=1}^{k} (X-g_j(z))$ is a Weiestrass polynomial in $X$. It, of course, divides ${\cal P}_1$ by the previous observation. So, from the irreducibility of ${\cal P}_1$, it follows that ${\cal P}_1=\wt {\cal P}_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\not \in {\cal E}$, then ${\cal F}(z)$ can be stratified into $k$ holomorphic branches near $z_0$ with $k=$the generic counting number of ${\cal F}(z)$. That is, we can write ${\cal F}(z)=\{g_1(z),\cdots, g_{k}(z)\}$ with $g_j$'s holomorphic near $z_0$. Now, when $z_j$ is close enough to $z_0$, then ${\cal F}(z_j)=\{g_1(z_j),\cdots,g_k(z_j)\}$. From this, it is easy to see that $\lim {\cal F}(z_j)= {\cal F}(z_0)$. When $z_0\in {\cal E}$, by passing to a nearby point, we can assume, without loss of generality, that all $z_j$'s are not in ${\cal E}$. Now, for another sequence $\{w_j\}(\subset {\cal U}_3\sm {\cal E})$ with $w_j\ra z_0$, we first find a family of curves $\{\g_j\}$ such that $\|\g_j-z_0\|\ra 0$ and $\g_j(0)=z_j$, $\g_j(1)=w_j$. Still write ${\cal F}(z)=\{g_1(z),\cdots, g_k(z)\}$ for $z\not\in {\cal E}$. Then for each $l$, there is a continuous map $I_l(t)$ from $[0,1]$, which maps $t$ to a point in ${\cal F}(\g_j(t))$ such that $I_l(0)=g_{l}(z_j)$ and $I_l(1)=g_{\sigma_{z_j} (l)}(w_j)$, where $\sigma_{z_j}$ is a permutation of $\{1,\cdots,k\}$. Moreover, it is clear that the m-th component of $I_l(t)$ is a solution of ${\cal P}_m(\g _j(t);X)=0$, as observed above. Now, by Lemma 2.5 of [Mal] and our choices of $\g_j$, it follows that $\|g_l(z_j)-g_{ \sigma_{z_j}(l)}(w_j)\|\ra 0$. Therefore, we see that $\lim {\cal F}(z_j)=\lim {\cal F}(w_j)$, which must be ${\cal F}(z_0)$ by the definition. $\endpf$ \bigbreak {\bf Lemma 3.1}. After shrinking $ {\cal U}_1$, if necessary, then for each $z\in {\cal U}_1$, it holds that ${\cal F}(z)\subset A'_{\wt{f(z)}}$ and ${\cal F}(Q_z)\subset Q'_{\wt f(z)}$ for any $\wt{f}(z)\in {\cal F}(z)$. \bigbreak {\it 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 $\wt{f}$ with $\wt{f}|_{D}=f$, it holds that $\wt{f}(Q_z)\subset Q_{\wt {f(z)}}$ (see also [DF2], for example). The main idea of the proof of the lemma is to use the fact that $\wt{\rho}_2({\cal F}(z),\wt{f}(\o))$ is single-valued for each fixed $\o$, by the above established $\l$-function. We let $z\in {\cal U}_1\cap D$, and assume that $z,\ z^*\not\in {\cal E}$. Here, we use $z^*$ to denote the reflection point of $z$, i.e, $z^*={\cal R}(z)$. We also choose a simply connected smooth curve $\gamma : [0,1]\ra {\cal U}_3$ with $\gamma \subset {\cal R}({\cal U}_1\sm D)\sm {\cal E}$ and $\g(1)=z^*$, $\g\cap M_1=\{\g(0)\}$. Here, when there is no confusion arising, we also use the letter $\gamma$ to denote its image set. Moreover, we assume that $\g$ intersects $M_1$ transversally at $\g(0)$. Let $\wt{\g}={\cal R} (\g) \cup \g$. Then $\wt{\g}$ is still a simply connected curve in ${\cal U}_2$ by the discussions in $\S 2$. Thickening $\wt{\g}$ suitably, we can then obtain a simply connected domain, which we will denote by $O (\subset {\cal U}_2)$. Now, we can well define a holomorphic map $\H{f}$ from $O$, which coincides with $f$ on $O\cap D$. Let $M_{1c}=\{(z,\o)\in {\cal U}_3 \times \h{conj}({\cal U}_3): \wt{\rho_1}(z,\o)=0\}$ and consider $M^*_{1c}=M_{1c}\cap \{O\times\h{conj}(O)\}$. Now, the following function is well-defined and holomorphic over $M^*_{1c}$: $$\Xi(z,\o)=\overline {f_{n+1}(\overline \o)}-G({\cal F}(z),\overline {\H{f}(\overline \o)}) \left (=\wt{\rho_2}({\cal F}(z),\overline {\H f (\overline {\o})})\right )$$ which vanishes on $M_1^*=\{(a,\overline a):a\in M_1\cap {\cal 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 $\Xi(z,\o)\equiv 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 $\wt{\g}$, we see that $(z,\overline {z^*}),\ (z^*,\overline {z})\in M^{**}_{1c}$; for $(\g(t),\overline {{\cal R}(\g(t)}))\in M^*_{1c}$ $({{\cal R}(\g(t)}),\overline {\g(t)})\in M^*_{1c}$ ($t\in [0,1]$) connect them to $M^*_1$, respectively. Therefore, %$(Q_z\cap { O}, z^*)\subset M^*_{1c}$, $(z,\h{Conj}(Q_{z}\cap O(z^*)))\subset M_{1c}^*$, $(z^*,\h{Conj}(Q_{z^*}\cap O(z)))\subset M_{1c}^*$ by the definition of Segre varieties. This implies that $G({\cal F}(z),\overline {\H{f}(\o)})=\overline {f_{n+1}(\o)}$ for $\o\in Q_{z}\cap O(z^*)$; and $G({\cal F}(z^*),\overline {\H{f}(\o)})=\overline {f_{n+1}(\o)}$ for $\o\in Q_{z^*}\cap O(z)$. Equivalently, we have $\wt{\rho}_2({\cal F}(z),\overline {\H{f}(\o)})=0$ for $\o\in Q_{z}\cap O(z^*)$; and $\wt{\rho}_2({\cal F}(z^*),\overline {\H{f}(\o)})=0$ for $\o \in Q_{z^*}\cap O(z)$. From a basic fact of Segre varieties: $w\in Q_z$ if and only if $z\in Q_w$; it now follows that $\H{f}(Q_{z}\cap O(z^*))\subset \cap Q'_{\wt{f(z)}} $ with $\wt{f}(z)\in {\cal F}(z)$; and ${\H f}(Q_{z^*}\cap O(z))\subset \cap Q'_{\wt{f}(z^*)}$ with ${\wt f}(z^*)\in {\cal F}(z^*)$. By slightly perturbing $z$ if necessary, we assume momentarily that $J_{\H{f}}\not = 0$ near $z^*$. Now, since $\H{f}(Q_z\cap O(z^*)) \subset Q'_{\wt{f}(z)}$ for any $\wt{f}(z)\in {\cal F}(z)$, and since each of them is a connected complex submanifold of dimension $n$ near $\H{f}(z^*)$, all these submanifolds therefore coincide near $\H{f}(z^*)$. Hence, it follows from Lemma 3.0 (iv) that all $Q'_{\wt{f}(z)}\cap {\cal U}_2$ are the same. So, ${\cal F}(z)\subset A'_{\wt f(z)}$ for any given $\wt{f}(z)\in {\cal F}(z)$. Assume also that $J_{f}(z)\not =0$. In a similar manner, we then also see that ${\cal F}(z^*)\subset A'_{\wt{f}(z^*)}$ for any given $\wt{f}(z^*)\in {\cal F}(z^*)$. Next, we note that ${\wt \rho_2}({\cal F}(z),\overline {\H f}(\o))=0$ if and only if $\rho_2({\cal F}(z),\overline {\H f}(z))=0$. Let $q\in {\cal F}(z)$. Then, by what we just obtained and by the reality of $\rho_2$, we have $\rho_2(\H{f}(\o),\overline {q})=0$ and therefore, $ \wt{\rho_2}(\H{f}(\o),\overline {q})=0$ for $\o\in Q_{z}\cap O(z^*)$. Now, since Lemma 2.1 indicates that $\wt{\rho_2}({\cal F}(\o),\overline {q})$ is well defined and holomorphic for $\o\in {\cal U}_3$, we conclude that $\wt{\rho_2}({\cal F}(\o),\overline {q})=0$ for $\o\in {\cal U}_2\cap Q_z$. This implies that ${\cal F}(Q_z)\subset Q'_q$. In a similar manner, we can also show that ${\cal F}(Q_{z^*})\subset Q'_{q^*}$ for any $q^*\in {\cal 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\in {\cal E}\cup {\cal R}({\cal E})\cup 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\ra z$ so that ${\cal F}(z_j)\subset A'_{\wt{f}(z_j)}$ for any $\wt{f}(z_j)\in {\cal F}(z_j)$. In the other words, $Q'_{\wt{f}(z_j)}=Q'_{\H{f}(z_j)}$ for any $\wt{f}(z_j), \H{f}(z_j)\in {\cal F}(z_j)$. By Lemma (3.0) (ii) and (I.d), it then follows that $Q'_{\wt{f}(z)}=Q'_{\H{f}(z)}$ for any $\wt{f}(z), \H{f}(z)\in {\cal F}(z)$, i.e, ${\cal F}(z)\in Q'_{\wt f(z)}$ for any ${\wt f}(z)\in {\cal F}(z)$. Similarly, we also have ${\cal F}(Q_z)\subset Q'_{\wt f(z)}$ for any ${\wt f(z)}\in {\cal F}(z)$. Hence, after shrinking ${\cal U}_1$ one more time if necessary, we see the proof of Lemma 3.1. $\endpf$ \bigbreak {\bf Remark 3.2} (a) By Lemma 3.1, we can now well-define $Q'_{{\cal F}(z)}$ to be $Q'_{q}$ and $A'_{{\cal F}(z)}=A'_q$ for some $q\in {\cal F}(z)$. Then Lemma 3.1 can be written as ${\cal F}(z)\subset A'_{{\cal F}(z)}$ and ${\cal F}(Q_z)\subset Q'_{{\cal F}(z)}$. (b) As an application of Lemma 3.1, we conclude that when $A'_{w}$ is just a single point for $w\approx 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 ${\cal A}'$ by sending each point $w$ to $A'_w$. One can see that if $f$ does not allow holomorphic extension, then ${\cal A}'$ branches at $0$ (we will make this more precise in $\S 5$). It is this fact that links the branching points of ${\cal F}$ with the singular points of ${\cal 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 ${\cal G}=(\sqrt{z_1},z_2)$ be the multiple-valued map from $D_1$ to $D_2$. Then the branching locus of ${\cal G}$ is given by $Z=\{(0,z_2)\}$, and ${\cal G}(Z)=\{(0,z_2)\}$. We observe that ${\cal G}(Z)$ is exactly the branching locus of the ${\cal A}'$-map of $\PP D_2$. \bigbreak {\bf Lemma 3.3}: (i) Let $M_2$ be as given in (2.2). Let $\O^{\pm}$ be defined by $\pm{\rho_2}<0$ respectively. Write $\H{n}^{\pm}=\{b= (0',b_{n+1}): \pm b_{n+1}<0\}$ with $|b_{n+1}|$ small. Then $A'_b\subset \O^{\pm}$ when $b\in \H{n}^{\pm}$, respectively. \noindent (ii) If for some $q\in {\cal F}(z)$, it holds that $q\in M_2$, then ${\cal F}(z)\subset M_2$. Hence, ${\cal F}(M_1\cap {\cal U}_1)\subset M_2$. \noindent (iii) $f_{n+1}(z)=z^k_{n+1}g(z)$ for some holomorphic function $g(z)$ defined near $0$. \noindent (iv) $\{{\cal F}^{-1}(0)\}\cap {\cal U}_2=\{0\}$ after shrinking ${\cal U}_2$, i.e , $\wt{f}(z)\not =0$ for any $z(\in {\cal U}_2)\not =0$ and $\wt{f}(z)\in {\cal F}(z)$. Moreover ${\cal F}^{-1}(w)\cap {\cal U}_2$ is a finite set for any $w\approx 0$; and for any analytic variety ${ E}$ passing through $0\in {\cal U}_1$, ${\cal F}({ E})$ also gives the germ of an analytic variety at $0\in {\cal U}'_1$ with $\h{dim}_0 {\ E}=\h{dim}_0({\cal F}({ E}))$. \noindent (v) After shrinking ${\cal U}_1$ if necessary, then for any $z\in {\cal U}_1$, either ${\cal F}(z)\subset \O^+$ or ${\cal F}(z)\subset \O^-$, or ${\cal F}(z)\subset M_2$. \bigbreak {\it Proof of Lemma 3.3}: (i): Let $\eta=(\eta',\eta_{n+1})\in A'_{b}$, i.e, $Q'_{ \eta}\cap {\cal U}'_2 =Q'_{b}\cap {\cal U}'_2$. We see that $$\{(w',w_{n+1})\in {\cal U}'_2: w_{n+1}+\overline {\eta_{n+1}}+\rho^*_{2}(w',\overline \eta', {w_{n+1}-\overline { \eta_{n+1}}\OO\ 2i})=0\}$$ $$ =\{(w',w_{n+1})\in {\cal U}'_2: w_{n+1}+\overline {b_{n+1}}+\rho^*_{2}(w',\overline b', {w_{n+1}-\overline { b_{n+1}}\OO\ 2i})=0\}. ,$$ where $\rho^*=\sum_{j\ge 0}\psi_j(w',\overline {w'})(\h{Im}w_{n+1})^j$. Letting $w'=0$, we see that $\eta_{n+1}=b_{n+1}$. Meanwhile, since $\{(w',-\overline {\eta_{n+1}})\}\cap {\cal U}'_2= Q'_{\eta}$, it follows that $\rho^*_{2}(w',\overline {\eta'},i\overline {\eta_{n+1}})=0$ for all $w'$ with $(w',b_{n+1})\in {\cal U}'_2$. %After making ${\cal U}'_1$ sufficiently small, We see, in particular, that $\rho^*_{2}(\eta',\overline {\eta'},i\overline {\eta_{n+1}})=0$. Therefore, ${\psi_0}(\eta',\overline {\eta'})=-\sum_{j>0}^{\infty}{\psi_j}(\eta',\overline {\eta'}) (i\overline {\eta_{n+1}})^j=o(|\eta_{n+1}|)$. Now, $\rho_2(\eta,\overline {\eta})=\eta_{n+1}+\overline {\eta_{n+1}}+ \rho^*_{2}(\eta',\overline {\eta'}, {\eta_{n+1}-\overline { \eta_{n+1}}\OO\ 2i})$ = $\eta_{n+1}+\overline {\eta_{n+1}} +{\psi_0}(\eta',\overline {\eta'})+o(|\eta_{n+1}|) =2\eta_{n+1}+o(\eta_{n+1})$, which is positive when $b_{n+1}=\eta_{n+1}>0$ and small; and negative when $b_{n+1}<0$. This gives the proof of part (i). (ii). Let $q\in M_2\cap {\cal F}(z)$. Then $q\in Q'_{q}=Q'_{\wt f(z)}$ for each $\wt{f}(z)\in {\cal F}(z)$. Thus $\wt{f}(z)\in Q'_{q}=Q'_{\wt f(z)}$, by Lemma 3.1. So, $\wt{f(z)}\in M_2$ (see [DF1]). (iii). By Lemma 3.1, it follows that ${\cal F}(Q_0)\subset Q'_{{\cal F}(0)}= Q'_{0}$. Thus, we see that $f_{n+1}(z',0)\equiv 0$. Since $f_{n+1}$ is holomorphic near $0$, we conclude that $f_{n+1}(z',z_{n+1})=z_{n+1}^kg(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 ${\wt V}=\{(z, w)\approx (0,0): w_{n+1}=f_{n+1}(z), {\cal P}_j(z,w_j)=0, j=1,\cdots, n\}$ and let $V$ be its irreducible component at $(0,0)$ which contains ${\Gamma_f}=\{(z,w): w=f(z), z (\approx 0)\in D\}$. Let $\pi$ and $\pi'$ be the natural projections from $V$ to the first and second copies of $\bf C^{n+1}$, respectively. Since ${\cal P}_j(0;X)=0$ has only zero solutions, i.e, $\pi ^{-1}(0)=0$, $\pi$ is locally proper. Thus, we can choose a small neighborhood $U$ of $0$ such that (a) $\pi^{-1}(U)$ is irreducible; (b) $\pi$ is proper from $\pi^{-1}(U)$; and (c) $\pi$ is a sheeted covering map from $\pi^{-1}(U)\sm \pi^{-1}({\cal E})$ to $U\sm {\cal E}$ (see, for example , Theorem 1 of [Chi], pp 122). In the following discussion, we make ${\cal U}_2\supset U\supset {\cal U}_1$ and restrict $\pi$ to $\pi^{-1}(U)$. Now, by using some standard arguments as appeared in the proof of Lemma 3.0 (iii) and using the irreducibility of $\pi^{-1}(U)$, one can see that ${\cal F}(z)=\pi'\circ\pi^{-1}(z)$, for $z\in {\cal U}_2$. (Otherwise, we can form a proper analytic subvariety of $V$ (of the same dimension), which is defined by $\{(z,w):\ w\in {\cal F}(z)\}$). On the other hand, it obviously holds that ${\cal F}^{-1}(0)\cap U= \pi(\pi'{}^{-1}(0)\cap \pi^{-1}(U))$. Thus, if $0$ is an accumulation point of $Y={\cal F}^{-1}(0)$, then $Y\cap 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\cap U$ contains some holomorphic curve $Y^*\subset {\cal U}_1$ parametrized by $z'=\phi(t), \zn=\psi(t)$, ($t\in \D$, the unit disk in $\bf C^1$) with $\phi(0)=\psi(0)=0$, $\|d\phi\|+\|d\psi\|\not =0$ for $t\not =0$. For each $z\in Y\cap {\cal U}_1$, ${\cal F}(Q_z)\subset Q'_0$, by Lemma 3.1. We claim that $\cup Q_{z}$ with $z\in Y^*$ fills in an open subset in $\bf C^{n+1}$. This then gives us a contradiction; for we assumed that $J_f\not \equiv 0$ and $Q_0'$ is a complex hypersurface. To see the size of $U^*=\cup Q_{z}$ with $z\in Y^*$, we note that $M_1$ is also defined by $\overline {\wt{\rho_1}(z,\overline z)}=0$. Therefore, it can be seen that $Q_z$ can be parametrized by $$\zn=-\overline {\psi(t)}-\sum_{j=0}^{\infty}\overline {\wt{\phi_j}}(\overline {\phi(t)},z') {\overline {\psi(t)}^j} \equiv g(z',\overline {t}).$$ Here and in what follows, we use the notation $\overline {h}(z)$ for the function $\overline {h(\overline z)}$. So, $U^*$ can be parametrized by the map $${\cal T}(z',\overline {t}):\ {\bf C^n}\times \D\ra {\bf C}^{n+1}$$ ${\cal T}(z',\overline {t})=(z',-\overline {\psi(t)}-\sum_{j=0}^{\infty}\overline {\wt{\phi}_j}( \overline {\phi(t)},z')\overline {\psi(t)}^j)$. To see that ${\cal T}(z,t)$ is a biholomorphism at certain point $(z',\overline t)(\approx (0,0))$, it suffices to show that ${\PP {g}\OO\PP \overline {t}}\not =0$ at $(z',\overline t)$. Indeed, this can be argued as follows: When $\psi(t)\not \equiv 0$, we can simply choose $(0',t)$ for some $t$ with $\psi'(t)\not =0$; when $\psi\equiv 0$, if ${\PP g\OO {\PP \overline t}}\equiv 0$, then for any given $z'$ with $|z'|$ small, $\overline {\wt{\phi_0}}(\overline {\phi(t)},z')$ is independent of $t$. Hence it has to be $0$; for $\phi(0)=0$ and thus $\overline {\wt{\phi_0}}(\phi(0),z')=0$. Letting $z'=\phi(t)$, it follows that $\overline {\wt{\phi_0}}({\overline {\phi(t)}},{\phi(t)})\equiv 0$. This contradicts the finite type assumption on $M_1$; for it implies that $Y^*=(\phi(t),0)$ stays inside $ M_1$. So, by making ${\cal U}_2$ small, it holds that ${\cal F}^{-1}(0)\cap {\cal U}_2=\{0\}$. Therefore, we conclude that $0$ is an isolated point of $\pi'{}^{-1}(0)$. Hence, $\pi'$ is locally finite near $0$. Moreover, ${\cal F}^{-1}$, when restricted as a map from $O(0)$ to ${\cal U}_2$, maps a small neighborhood $0\in {\cal U}'_1$ into a small neighborhood of $0\in {\cal U}_1$. Now, shrinking ${\cal U}_2$ if necessary, we assume that $q\approx 0$, ${\cal F}^{-1}(q)\cap {\cal U}_2= \pi(\pi'{}^{-1}(q))$ is finite. By the elimination theorem ([Chi], Theorem 1 of pp122), there exists an open neighborhood $U'$ of $0$ such that $\pi'$ is proper from $\pi'{}^{-1}(U')$ to $U'$; and away from a proper analytic set, $\pi'$ gives a sheeted covering map. In particular, $\pi'$ 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 ${\cal F}(U)\subset U'$, and we restrict $\pi'$ to $\pi'{}^{-1}(U')$. By the above discussion, we also observe the fact: $\pi^{-1}(U)\subset {\pi'} ^{-1}(U')$. Using again the elimination theorem and noting that $\pi$, $\pi'$ are local analytic covering maps, it follows that for any analytic variety $ E\subset U$, % of codimension 1, ${\cal F}( E)\cap \pi'(\pi^{-1}(U)) = \pi'({\pi}^{-1}(E)\cap \pi'(\pi^{-1}(U)))$ is an analytic variety in $\pi'(\pi^{-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 $\pi$ 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 $${\cal F}(B)\cap {U}=\pi\left ({\pi'} ^{-1}(B)\cap\pi^{-1}(U)\right).$$ Since ${\pi'}^{-1}(B)$ is closed in ${\pi'} ^{-1}(B)\cap {\pi'} ^{-1}(U')$, it is closed in ${\pi'} ^{-1}(B)\cap \pi ^{-1}(U)$ by the above arrangement. By using the closeness of $\pi$, it follows that ${\cal F}^{-1}(B)$ is closed in $U$. In particular, we see that $\wt{M}_1\cap U={\cal F}^{-1}(M_2\cap U')$ is closed in $U$. Clearly, ${\cal F}(\wt{M_1})\subset M_2$ by part (ii) of this lemma. Now, let $U^0=U\sm {\wt M}_1$. Then it is open. We notice that $\PP U^0$ is contained in $\PP U \cup {\wt M}_1$. Since ${\cal F}=\pi'\circ\pi^{-1}$, the openness of $\pi'$ implies the openness of ${\cal F}$ from $U$ as a multiple-valued map. Therefore it sends interior points to interior points. On the other hand, using the weak continuity of ${\cal F}$ (Lemma 3.0 (ii)), one sees that ${\cal F}$ maps the closure of $U^0$ to the closure of ${\cal F}(U^0)$. Hence, a simple topological argument shows that the boundary of ${\cal F}(U^0)$ is contained in $ {\cal F}(\PP U)\cup M_2$. Since $ {\cal F}^{-1}(0)\cap {\cal U}_2 =\{0\}$, it follows easily that $0$ is not contained in the closure of ${\cal F}(\PP U)$. Hence, we see that there is a small ball, denoted by $B_0$, centered at the origin such that $B_0\cap {\cal F}(\PP U)=\emptyset$. Let $U^*$ be a small neighborhood of $0$ such that ${\cal F}(U^*)\subset\subset B_0$. If $U^0\cap U^* =\emptyset$, then Part (ii) of this lemma shows that ${\cal F}(U^*)\subset {\cal F}(\wt {M_1})\subset M_2$ and thus we are done. So, without loss of generality, we assume that $U^*\cap U^0\not =\emptyset$. Let $D^*$ be a connected component of $ U^0$ with $U^*\cap U^0\not =\emptyset$. Note that $\PP {\cal F}(D^*)\cap B_0=\emptyset$. Then a simple topological argument indicates that %with ${\cal F}(D^*)\cap B_0\not =\emptyset$, then % we see that that $\wt{M_1}=\pi(\pi'{}^{-1}(M_2))$ is a closed subset %$U$. % Let $D^*$ be a connected component of ${\cal U}_1\sm\wt{M_1}$. %Then $\pi'(\pi^{-1}(D^*))$ is an open subset of ${\cal U}_1'$. We also claim %that its boundary near $0$ is contained % in $M_2$. Thus, by an argument similar to [BHR2], %one sees that there is an open neighborhood %${\cal U}'_0\subset\subset {\cal U}'_1$ such that either ${\cal F}(D^*)\supset B_0\cap\O^+$ or ${\cal F}(D^*)\supset \O^-\cap B_0$ (see, for example, [BHR2]). %(If there does not exist such an $D^*$, then as discussed above, %${\cal F}$ maps a small neighborhood of $0$ to $M_2$ and we are done.) Without loss of generality, we assume the first case. Then, there exists a point $p_0\in D^*$ and certain $\wt{f}(p_0)\in {\cal F}(p_0)$ such that $\wt{f}(p_0)\in {\H n}^+$ and $\wt{f}(p_0)$ is close to $0$.\ By the first part of this lemma and Lemma 3.1, we see that ${\cal F}(p_0)\subset \O^+$. Now, for any $z\in D^*\sm {\cal E}$, let $\g$ be a curve in $D^*$ with $\g(0)=p_0$, $\g(1)=z$, and $\g((0,1))\cap {\cal E}=\emptyset$. Then for any $\wt{f}(z)\in {\cal F}(z)$, there is a branch $\H f$ of ${\cal F}$, which is continuous on $\g$ and $\H f(z)$ coincides with $\wt{f}(z)$. Now, since $\H{f}\circ\g$ does not meet $M_2$ by our choice of $U^0$, and $\H{f}\circ\g(t)\in \O^+$ for $t$ close enough to $0$ (this can be seen by the fact that all limit points of $\H{f}(\g (t))$ ($t\ra 0)$ are in ${\cal F}(p_0)$), we conclude that $\H f(z)$ and thus $\wt{f}(z)$ have to be in $\O^+$. For $z\in {\cal E}\cap D^*$, by passing to a limit and noting ${\cal F}(z)\cap M_2=\emptyset$, we can also see that ${\cal F}(z)\subset \O^+$. % Now we notice that when $z$ is close enough to $0$, %${\cal F}(z)\subset B_0$. So, we just proved that either ${\acl F}(z)\in M_2$ %or it stays in some connected component of $D^*$ with the property mentioned a%above. Thus the proof of the last part of Lemma 3.3 is complete if we make ${\cal U}_1$ small. $\endpf$ \bigbreak {\bf 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 ${\cal F}$: There exists an open neighborhood ${\cal U}'_0$ of $0$ such that ${\cal U}_0={\cal F}^{-1}({\cal U}'_0)\cap {\cal U}_2$ is an open subset of ${\cal U}_1$. Moreover, ${\cal F}|_{{\cal U}_1\ra {\cal F}({\cal U}_1)}$ and ${\cal F}^{-1}|_{{\cal U}_0'\ra {\cal U}_0}$ are multiple-valued open mappings. (b) Let $(V,\pi,\pi(V))$ be the m-sheeted analytic covering space, as in Lemma 3.3 (iv). Let $z_0\in {\cal E}\cap \pi(V)$ be close to the origin. Then, it is easy to see that in case $\# \pi^{-1}(z_0)0}\wt{\psi}_j(w_1,\l)w_2^j$ and $q_1=(q_0)'_{\l}$. We see that the equations $G(w_1,w_2,\l)=0$, $G'_{\l}(w_1,w_2,\l)=0$ can be solved as $w_2=q_0h^*(q_0,w_1,\l))$ and $q_1=q_0\psi^*(w_1,\l,q_0)$ for certain holomorphic functions $h^*$, $\psi^*$. In the following discussions, we always let $\tau=\mu (z_1,\xi)$. Let $w_1(z_1,\xi)=\H{f_1}(\overline {\xi},\overline {\tau})$ with $\H{f_1}(\overline {\xi},\overline {\tau})\in {\cal F}(\overline {\xi},\overline {\tau})$, and let $\l(z_1) =\overline {\wt f_1}(\overline {z_1},0)$ in the above formulas, we obtain the following equality % with $\tau=\mu(z_1,\xi)$: $$q_1( \H{f_1}(\overline {\xi},\overline {\tau}), \overline {\wt f_1}(\overline {z_1},0))= q_0(\H{f_1}(\overline {\xi},\overline {\tau}),\overline {\wt f_1}(\overline {z_1},0)) \times\psi^*(\H{f_1}(\overline {\xi},\overline {\tau}),\overline {\wt f_1}(\overline {z_1},0),q_0).\leqno (4.4)$$ Here for clarity, we summarize the situations in which $(4.4)$ makes sense and holds: (a) $z_1$ is an arbitrarily given non zero complex number near the origin. (b) ${\wt f}_1$ is some holomorphic branch of ${\cal F}$ over some small neighborhood $O((z_1,0))$ of $(z_1,0)$, whose size depends on $z_1$. (c) $\xi$ is taken in a small neighborhood $O(z_1,\wt{f}_1)$ of $0$, whose size depends on the choice of $O((z_1,0))$ and the choice of $\wt{f}_1$. (d) $\H{f}(\overline {\xi},\overline {\tau})$ is any point in ${\cal F}_1(\overline {\xi},\overline {\tau})$. Since $f_2\not\equiv 0$ and ${\PP \mu\OO\PP z_1} \not \equiv 0$, $Z^*=\{(z_1,\xi):\ f_2(\overline \xi,\overline {\mu(z_1,\xi)}) =0\}$ is an analytic variety of dimension 1. So, we can assume that $Z^*$ has only one component of the form $\{(a,\xi)\}$ near the origin, i.e, the one with $a=0$. We therefore see that $f_2(\overline \xi,\overline {\mu(z_1,\xi)})$, when regarded as a function in $\overline \xi$, is not constant for each fixed $z_1\not =0$; for if it were constant, then it would be identically $0$. Meanwhile, noticing that $(\H{f_1}({\overline \xi},{\overline \tau}),{f_2}(\overline {\xi},\overline {\tau}))\in Q'_{(\wt{f_1}( z_1,0),0)}$, we see that $$f_2(\overline \xi,\overline {\mu(z_1,\xi)})=-\wt{\psi_0}(\H{f_1}(\overline \xi,\overline {\mu(z_1,\xi)}), \overline {\wt {f_1 }}(\overline {z_1},0)).\leqno (4.5)$$ Let $Z^{**}\subset {\bf C^2}$ be the zero set of $q_1(w_1,\l)-q_0(w_1,\l)\psi^*(w_1,\l,q_0(w_1,\l))$. If it is not of dimension 2, then it has only finitely many irreducible one dimensional components near the origin. Thus, we can assume that for any $\l(\not =0)$ with $|\l|$ small, $Z^{**}\cap \{(w_1,\l)\}$ is of dimension $0$. Returning to (4.4), one sees that for each $z_1\not =0$ close to the origin, $\H{f_1}(\overline \xi, \overline {\mu(z_1,\xi)})$ takes only finitely many values for $\xi\in O(z_1,\wt{f})$. This obviously contradicts (4.5); for $f_2(\overline \xi,\overline \mu(z_1,\xi))$ can not take only finitely many values when $\xi\in O(z_1,\wt{f}_1)$ by the above argument. Therefore, we showed that $(q_0)'_{\l}=q_1(w_1,\l)\equiv q_0(w_1,\l)\psi^*(w_1,\l,q_0(w_1,\l))$. Now, applying the same argument as in [BR2], we can see, by the basic theory of ODE and by the initial condition $q_0(w_1,0)=G(w_1,0,0)=0$, that $q_0\equiv 0$. This contradicts the finite type assumption of $M_1$. $\endpf$ \bigbreak % We also observe that in case ${\cal F}$ is %known to be holomorphic near $0$, much more properties involving invariant %objects can be obtained (see [We], [DW], % and (Theorem 1 of [BR4]), for 0) {\bf Remark 4.1$'$}: The above proof can be adapted to the general dimensions as in [BR2], almost without any change. However, we would like to include here some details just for convenience of the reader: We first replace $z_1 $ by $z'\in {\bf C}^n$ and define, in the same manner, the functions $\tau$, $\mu$, $R$. We then have, by Lemma 3.1, the following $$\overline {f_{n+1}(z',R(z',\xi,\tau))}= G({\cal F}(\overline {\xi},\overline {\tau}),\overline {\wt {f^*}(z',R(z',\xi,\tau)});$$ for $\overline {(\xi,\tau)}\in Q_{(z',R(z',\xi,\tau))}$ with $\xi\in {\bf C^n}$. Let ${\cal E}'={\cal E}_0\cap Q_0$. By Lemma 5.1 (a), it is a proper analytic variety of $Q_0$. In the same manner, we let $z'\in Q_0\sm {\cal E}'$ be close to $0$ and choose a small neighborhood $O((z',0))$ of $(z',0)$ such that ${\cal F}$ can be stratified into several (non-constant) holomorphic branches over $O((z',0))$. Arguing in the same way, we have for each $l\le n$, $${{\PP \overline {f_{n+1}}}\over \PP{z_l}}+{{\PP \overline {f_{n+1}}}\over \PP {z_{n+1}}}\overline {R'_{z_l}(z',\xi,\tau)} =\sum_{j=1,\cdots,n}{\PP G\over \PP{\l_j}}({\PP \overline { \wt {f_j}}\over \PP {z_l}}+ {{\PP \overline {\wt f_j}}\over \PP {z_{n+1}}} \overline {R'_{z_l}(z', \xi,\tau)}).$$ Let $\tau=\mu(z',\xi)$. Assume that $f_{n+1}=z_{n+1}^kg$ with $k>1$. One then has, for $\xi$ with $\|\xi\|$ sufficiently small, that $$\sum_{j=1,\cdots,n}{\PP G\over \PP{\l_j}}({\PP \overline { \wt {f_j}}\over \PP {z_l}}(\overline {z'},0)+ {{\PP \overline {\wt f_j}}\over \PP {z_{n+1}}}(z',0) \overline {R'_{z_l}(z', \xi,\tau)})=0.$$ Since ${\cal F}^{-1}$ is a finite-to-finite map, $f^*(z',0)$ is a finite-to-one map from $O(z',0)\cap Q_0$ too. Moving to a nearby point if necessary, we can assume that $$\h{det}( {\PP \overline {\wt f_j}\over \PP {z_l}}(\overline {z'},0))_{1\le j,l\le n}\not =0.$$ Reasoning as in the proof of the two dimensional case and making $|\xi|$ sufficiently small, we have for each $l0$ when ${\cal F}(z)\subset \O ^-$; and $=0$ when ${\cal F}(z)\subset M_2$ (see Lemma 3.3 (a) for related notations). In fact, for each fixed $\wt{f}(z)\in {\cal F}(z)$, we have $$\h{Re}\left (\overline {f_{n+1}(z)}-G({\wt f},\overline {\wt f(z)})\right) =\h{Re}\wt{\rho_2}\left({\wt f}, \overline {\wt f(z)}\right)$$ $$=\rho_2\left({\wt f}, \overline {\wt f(z)}\right) \h{Re}h_2({\wt f(z)}, \overline {\wt f(z)}))= \rho_2({\wt f}, \overline {\wt f(z)}))(1+o(\|\wt{f(z)}\|).$$ Thus, the claim follows from Lemma 3.3 (v). Next, applying Lemma 4.1, one sees that $$\rho^*=2\R (f_{n+1}(z))+o(|z|)= \R (g(0)z_{n+1})+o(|z|)$$ with $g(0)\not = 0$. Thus, $\h{d}_0\rho^*\not =0$. Therefore, $\rho^*$ serves as a real analytic defining function of $M_1$ near $0$. Without loss of generality, let us assume that $\rho^*(z,\overline z)<0$ for $z\in D$. Then $\rho(z,\overline z)>0$ for $z\in D^c$. Now, by making ${\cal U}_1$ small, the above argument shows that ${\cal F}(D\cap {\cal U}_1)\subset \O^+$ and ${\cal F}(D^c\cap {\cal U}_1) \subset \O^-$. So, we can let the $\O$ in Lemma 4.2 be $\O^+$. Of course, $\O^c$ is then $\O^{-}$. $\endpf$ \bigbreak {\bf Remark 4.3} (a). In what follows, we will assume, without loss of generality, that $\O=\O^+$. Then we observe that the above results give the fact $g(0)>0$. To see this, we write $f_{n+1}(z)=\a z_{n+1}+o(\|z\|)$ with $\a=\a_1+\sqrt{-1}\a_2\not =0$. Letting $t=x+\sqrt{-1}kx$, then $\rho^*(t)=(\a_1-\a_2k)x+o(|x|)$. Notice that $\rho^*(x)<0$ for any given real number $k$ and $x(\approx 0)<0$. We conclude that $(\a_1-\a_2k)>0$. Thus, we see that $\a_2=0$ and $\a_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^*\cap D$ to ${U^*}'\cap \O$. 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\in B\cap D:\ f(z)\not\in \PP {B\cap D}\}$ such that $f(U^*)$ contains $\O\cap O(0)$ with $O(0)$ sufficiently small. \section{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 ${\cal A}$ the map, which sends: $w\in {\cal U}_1$ to the finite set $A_{w}$. $w_0$ is called a {\it separable} point of ${\cal 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,\cdots, N^*$) such that ${\cal A}(w)\cap {O}_{w_j}=\{w\}$ for any $w\in { O}_{w_j}$. % (See also [DW] for certain related notions). We write ${\cal W}=\{ w\in {\cal U}_1:\ w \ \h{is not a separable point of}\ {\cal A}\}$. %We remark that ${\cal W}$ is an analytic variety near $0$ ([DF1]). We say that $M_1$ satisfies {\it 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: \noindent (a) $\cal W$ is contained in an analytic variety $E$ spread over $(z_2,\cdots,z_{n+1})$-space, i.e, $E$ is defined by an equation of the form: $z_1^{N^*}+\sum_{j=0}^{N^*-1}a_j(z_2,\cdots,z_{n+1})z_1^{j}=0$ with $a_j(z_2,\cdots,z_{n+1})$ holomorphic near the origin. \noindent (b) For each small $t>0$, there is a sufficiently small positive number $\epsilon(t)$ such that for each $b^*=(b_2,\cdots,b_n)$ with $|b^*|<\epsilon(t)$, the set ${E}\cap \{(z_1,b^*,-t): z_1\in {\bf C}^1\}$ is contained in some connected component of $D\cap \{(z_1,b^*,-t): z_1\in {\bf C}^1\}$. % and the set % ${\cal W}\cap \{(z_1,b^*,-t): z_1\in {\bf C}^1\}$ is contained % in some connected component of % $D^c\cap \{(z_1,b^*,-t): z_1\in {\bf C}^1\}$ %in $D^c$. Similarly, we can define ${\cal A}'$ and ${\cal W}'$ and speak of {\it Condition S} for $M_2$. In this case, we use $\O$ to replace the role of $D$ (see Remark 4.3(a)). We will see in the following lemma that ${\cal W}$ and ${\cal W}'$ can be used to control the branching locus of ${\cal F}$. \bigbreak {\bf Lemma 5.1}: \noindent (a) Suppose that $M_1$ and $M_2$ satisfy Condition S at the origin. Then ${\cal F}({\cal E}_0\cap O(0))\subset {\cal W}'$, ${\cal E}_0\cap O(0)\subset {E}$, and ${\cal E}_0\cap O(0)\cap \{(z_1,\cdots,z_n, 0)\}$ is a proper variety near the origin of the $\{(z_1,\cdots,z_n,0)\}$-space. Here $E$ is chosen as in the definition of Condition S, and ${\cal E}_0$ is as defined in Remark 3.4 (c). (When $n=1$, we always have ${\cal F}({\cal E})\subset {\cal W}'$ and thus $0$ is an isolated point of ${\cal E}\cap Q_0$). \noindent (b) $M_1$ and $M_2$ satisfy Condition S, when $n=1$ or when $M_1$ and $M_2$ are rigid at the origin. \bigbreak {\it Proof of Lemma 5.1}: We first prove Part (a) of the lemma: Let $z\in {\cal E}_0$ be sufficiently close to $0$. Then, by the definition of ${\cal E}_0$ and Remarks 3.4 (b) (c), it follows that there is a sequence $z_j\ra z$ such that we can find two sequences $\{\eta_j\}$ and $\{\xi_j\}$ with $\xi_j,\eta_j\in {\cal F}(z_j)$ $\eta_j\not=\xi_j$, but $\xi_j,\eta_j\ra w_0$ for some $w_0$. By Lemma 3.1, it then follows that $\xi_j\in A'_{\eta_j}$. Hence, $w_0$ is not a separable point of ${\cal A}'$. Again by Lemma 3.1, we conclude that ${\cal F}(z)\subset {\cal W}'$. {\it Notice that for this part, no Condition S is necessary}. We next prove that ${\cal E}_0\cap O(0)\subset {E}$. To this aim, we assume, without loss of generality, that ${\cal E}_0\not=\emptyset$. Let $\cup_j{\cal E}_j$ be the union of the irreducible components of ${\cal E}_0$ near $0$. We recall by Remark 3.4 (d), that each ${\cal E}_j$ then must be of codimension 1 at $0$. Now, since ${\cal F}({\cal E}_j) \subset {\cal 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 ${\cal F}({\cal E}_j)\cap \O\not =\emptyset$ near $0$. Therefore, it follows from Lemma 4.2, that ${\cal E}_j\cap D$ is not empty in any neighborhood of $0$. Now, pick any point $z\in {\cal E}_j\cap D$, which is sufficiently close to $0$. Then, as above, we can find a sequence $z_j\ra z$ and $\{\eta_j\}$, $\{\xi_j\}$ with $\xi_j,\eta_j\in {\cal F}(z_j)$ $\eta_j\not=\xi_j$, but $\xi_j,\eta_j\ra w_0\in \O$. 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}\cap f^{-1}(w_0)=\{z_0\}$. Since $f$ is single valued, there are two distinct points $a_j$, $b_j\in U$ such that $f(a_j)=\xi_j$ and $f(b_j)=\eta_j$. Now, the following Claim 5.2 indicates that $a_j\in A_{b_j}$. Since $a_j,b_j$ clearly converge to $z_0$ as $j\ra \infty$, we conclude that $z_0\in {\cal W}$. Again by Lemma 3.1 and Claim 5.2, we see that $z_0,z\in {\cal W}$. By the arbitrariness of the choice of $z$ and the assumption of Condition S, this fact implies that ${\cal E}_j$ has an open subset contained in ${\cal W}$ and thus in $E$. From the uniqueness of analytic varieties, it follows that ${\cal E}_j\subset { E}$ for any $j$. \bigbreak {\bf Claim 5.2}: Let $M_1$, $M_2$, and $f$ be as before. We have that $b\in A_a$ if $Q'_{{\cal F}(a)}=Q'_{{\cal F}(b)}$, where $a,b\in {\bf C}^{n+1}$ are sufficiently close to $0$. (For this claim, no Condition $S$ is required). \bigbreak {\it Proof of Claim 5.2}: Let $a$ and $b$ be as in the claim. %\in {\cal F}^{-1}(f(a))$. Since $f_2(z)=z_2$, We then need to show that $Q_a\cap {\cal U}_2=Q_b\cap {\cal U}_2$. To this aim, we consider the function $\wt{\rho_2}({\cal F}(z), \overline {\wt{f}(\o)})$, for any given $\wt{f}(\o)\in {\cal F}(\o)$. By Lemma 2.1, $\wt{\rho_2}({\cal F}(z),\overline {\wt{f}(\o)})$ is well defined and holomorphic in $(z,\l)$ with $\l=\overline {\wt{f}(\o)}$ for $(z,\l)\in O_{z}(0)\times O_{\l}(0)$. %for $(z,\o)\in {\cal O}_z\times D$ and is Holder continuous %up to $(z,\o)\in {\cal O}_z\times D\cup M_1$. Meanwhile $\wt{\rho_2}({\cal F}(z),\overline {\wt f(\o)}) =cz_{n+1}+c\overline {\o}_{n+1}+O(|z||\l|+o(\o))$ for some $c>0$, by Remark 4.3 (a). Thus, for each $\o$ sufficiently close to $0$, since $|\wt{f}(\o)|\approx 0$, the implicit function theorem indicates that $\wt{\rho_2}({\cal F}(z),\overline {\wt f(\o)})=0$ defines a connected complex hypersurface $W_{\o}$ in some small neighborhood $U^{\#}$ of $0$, whose size is independent of the parameter $\o$ with $|\o|<<1$. Notice, by Lemma 3.1, that ${\cal F}(Q_{\o})\subset Q'_{\wt{f}(\o)}$ near $0$. We see that $Q_{\o}\cap U^{\#}\subset W_{\o}$. Thus it follows that $W_{\o}=Q_{\o}$ in $ U^{\#}$ when $|\o|<<1$; for $Q_{\o}$ is also a connected complex submanifold of codimension 1 in ${\cal U}_2$. Now, if $Q'_{{\cal F}(b)}= Q'_{{\cal F}(a)}$, % with $\wt{f}(z)\in {\cal F}(z)$ and $\H{f}(a)\in {\cal F}(a)$, we then conclude that $$\wt{\rho_2}({\cal F}(z),\overline {{\cal F}(b)})=0$$ and ${\wt \rho_2}({\cal F}(z), \overline {{\cal F}(a)})=0$ define the same variety. Hence, when $a,b\approx 0$, we conclude, by Lemma 3.0 (iv), that $Q_{a}=Q_{b}$; for both of them have a small open subset of $W_{a}\cap U^{\#}=W_{b}\cap U^{\#}$ in common. This gives the proof of Claim 5.2. (We mention that for the proof Claim 5.2, no assumption of Condition S is required). \bigbreak The last statement in (a) follows clearly from the above facts and the assumption of Condition $S$ for the hypersurfaces. More precisely, we notice that ${\cal F}({\cal E}_0\cap O(0))\subset {\cal W}'$ and $ {\cal F}(Q_0)\subset Q'_0$. From Lemma 3.3 (iv) and the fact that ${\cal W}'\cap \{(z_1,\cdots,z_n,0)\}$ is contained in a proper subvariety (of codimension at least 1) in $Q'_0$, it follows that ${\cal E}_0\cap Q_0$ is also of codimension at least $1$ in $Q_0$. {\it Actually, using the same proof as in Part (b) for the rigid case, and using a simple projection lemma, we notice that this part does not require Condition S neither}. Also, we notice that the argument for this part does not use Claim 5.2, i.e, the use of Lemma 4.1, neither. \bigbreak Now, we proceed to the proof of (b): We first consider the case of $n=1$. %(a) We first let $n=1$. As before, we let $M_2$ be defined by $$z_{2}+\overline {z_{2}}+\sum_{j=0}^{\infty}\wt{\psi_j}(z_1,\overline {z_1})z_{2}^j=0; \h{or}\ z_{2}+\overline {z_{2}}+\sum_{j=0}^{\infty}\overline {\wt{\psi_j}}(\overline {z_1},{z_1}) \overline {z_{2}^j}=0.$$ For each $b=(b_1,b_{2})\approx 0$, $A'_b=\{(w_1,w_{2}):\ Q'_{w}\cap {\cal U}'_2=Q'_{b}\cap {\cal U}'_2\}$. As did in Lemma 3.3, by letting $z_1=0$ in the defining equations of $Q'_w$ and $Q'_b$, one sees that $w_{2}=b_{2}$. Now, $Q_{w}$ can also be defined by $$z_2=-\overline { b_2}-\sum_{j=0}^{\infty}\overline {{\wt \psi_j}}(\overline w_1,z_1)\overline {b_2}^j,$$ where the notation $\overline {\wt \psi}$ is the same as explained before. Write $$\overline {\wt \psi_j}(\overline {w_1},z_1)= \sum_{\a>0}\overline {\Xi_{j\a}}(\overline {w_1})z_1^{\a}.$$ Then $z_2=-\overline {b_2}-\sum_{\a}\sum_{j}\overline {\Xi_{j\a}}(\overline w_1)\overline {b_2}^j z_1^{\a}$. So, the equation: $Q_{w}'=Q'_{b}$ can be written as $$\sum_{j}\overline {\Xi_{j\a}}(\overline w_1)\overline {b_2}^j=\sum_{j}\overline {\Xi_{j\a}}(\overline b_1) \overline {b_2}^j$$ or $$\sum_{j}{\Xi_{j\a}}( w_1){b_2}^j=\sum_{j}{\Xi_{j\a}}( b_1){b_2}^j,$$ for any $\a$. Choose $k_0$ to be the smallest integer so that for certain $\a_0$, $\Xi_{0\a_0}(w_1)=a^*w_1^{k_0}+o(|w_1|^{k_0})$ with $a^*\not =0$. % It is not difficult to see % that this $k_0$ coincides with the integer $k_0$ appeared in Remark % (2.2) b. Write $P(w_1,b_2)=\Xi_{0,\a_0}(w_1)+\sum_{j>0}\Xi_{j,\a_0}(w_1)b_2^j$. For $(b_1,b_2)$ close to the origin, we define $$E=\{(b_1,b_2):\ \h {for some}\ w_1^0,\ \ P(w_1^0,b_2)=P(b_1,b_2),\ {\PP P\over \PP w_1}(w_1^0,b_2)=0\}.$$ Then we first claim that near the origin, ${\cal W}\subset E$. For this, let $w=(\xi,b_2)\not \in E$, and let $C(w)=\{(z_1,b_2):\ P(z_1,b_2)=P(\xi,b_2)\}$. Then by the definition of $P$, one has $A'_w\subset C(w)$. Write $A'_w=\{(\tau_j,b_2)\}_j$ and suppose that $w\in W_0$. Then for some $j^*$, by the definition of $\cal W$, there are two sequences $\{(\eta_j, c_j)\}$ and $\{(\xi_j,c_j)\}$ such that (a) $\eta_j\not =\xi_j$; (b) $\eta_j, \xi_j \ra \tau_{j^*}$ for some index $j^*$, $c_j\ra b_2$; and (c) $A'_{(\eta_j, c_j)}=A'_{(\xi_j,c_j)}$. From (c) and the definition of $P$, it follows that $P(\eta_j,c_j)=P(\xi_j,c_j)$. Moreover, by the Taylor expansion: $P(\eta_j,c_j)=P(\xi_j,c_j)+{\PP P\OO \PP w_1}(\xi_j,c_j)(\eta_j -\xi_j) +o(|\eta_j-\xi_j|)$; we obtain ${\PP P\OO \PP w_1}(\xi_j,c_j)=o(1)$. Letting $j\ra 0$, we therefore conclude that ${\PP P\OO \PP w_1}(\tau_{j^*},b_2)=0$, which together with the fact $P(\tau_{j^*},b_2)=P(\xi,b_2)$ implies that $w\in E$. This is a contradiction and thus proves our claim. We next claim that $E$ is an analytic variety spread over $b_2$-axis. To see that, we let $$E^{*}=\{(b_1,b_2,w_1)\in O(0): P(w_1,b_2)=P(b_1,b_2),\ {\PP P\over \PP w_1}(w_1,b_2)=0\}. $$ Then $E$ is the projection of $E^{*}$ to the first two copies of $\bf 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 ${\PP P(w_1,b_2)\over \PP 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|\ale |b_2|^{{1\over 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 ${\PP P(w_1,b_2)\over \PP w_1}=0$. Furthermore, one can easily see that $|b_1|\ale |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\cap ({\bf C^1}\times \{b_2\})\subset \D_{|b_2|^{\a^*}}\times \{b_2\}$ for $|b_2|$ sufficiently small, where $\D_{r}$ is used to stand for the disk in $\bf C^1$ with center at the origin and with radius $r$, $\a^*$ is a constant between $1\over k_{0}+1$ and $1\over k_0$. By our choice of $k_0$, it follows easily for sufficiently small $|b_2|$, that $ \D_{|b_2|^{\a^*}}\times \{b_2\}\subset D$ when $b_2<0$ , and $\D_{|b_2|^{\a^*}}\times \{b_2\}\subset D^c$ when $b_2>0$. This completes the proof of Part (a) in case $n=1$. % Clearly, from the way $E$ was constructed, we see that %$E\cap \{(w_1,0)\}\cap O(0)=\{0\}$. Thus, the Weiestrass preparation theorem %indicates $E$ can be defined by an equation with the form as in the statement %of lemma. % Now, since ${\cal W}\subset E$, we see that $\{(w_1,0)\}\cap W_0=\{0\}$%. % 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}}+\sum_{|\a|,|\b|>0}a_{\a\b}z'{}^{\a}\overline {z'{}^{\b}}=0$. Let $P_{\a}(z')=\sum_{\b}\overline {a_{\a\b}}z'{}^{\b}$. 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_{\a}(a')=P_{\a}(b')$. By the finite type assumption, it follows that the common zero of $P_{\a}'$s is $0$ near the origin. Using the Nullstellantz theorem, we see that for some finitely many indices $\{\a_j\}_{j=1}^{m}$, the locus of $\{P_{\a_j}\}_{j=1}^{m}$ is also zero. Now, for a small neighborhood $U$ of 0, let $U'=U\cap \{(z_1,\cdots,z_n,0)\}$. Define $\Lambda$ from a small neighborhood $U'$ of $0'$ to $\bf C^{m}$ by $\Lambda(z')=(P_{\a_j}(z'))_{j=1}^{m}$. Then $\Lambda$ is finite to $0$ and proper from $U'$, after suitably shrinking of $U$. So, by the Remmert theorem, we conclude that the set $E'=\Lambda^{-1}\left(\L(\{z\in U: \ \h{d} {\Lambda}\ \h{does not have maximal rank at} \ z\})\right)$ is a proper variety of $U'$ (see [DF1] for a similar argument). It is easy to see that ${\cal W}\subset E$, with $E=(E'\times {\bf C^1})\cap {\cal U}_1$. Now, after a linear change of coordinates in the $(z_1,\cdots,z_{n})$-space, we see that $E$ is contained in an analytic variety defined by an equation of the form: $z_1^{N^*}+\sum_{j0$, i.e, there exist holomorphic functions $a_j\in \h{Hol}(\D_{\d_0}\times \D_{\d}^{n})$ such that $P(z,g(z))=g^N(z)+\sum_{j=0}^{N-1}a_jg^j(z)=0$ for $z\in D$. Here we assume that $P(z,X)$ is irreducible in $X$, and $\D_{\d_0}\times\D_{\d}^{n}\cap D\subset\subset 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 $\wt{P}(z_2,\cdots,z_{n+1};z_1)=z_1^{N_1}+\sum_{j=0}^{N_1-1} d_j(z_2, \cdots,z_{n+1})z_1^j=0$ for certain $d_j(z_2,\cdots,z_{n+1})\in \h{Hol}(\D ^{n}_{\d})$ with $d_j(0)=0$. Also, suppose that for each small positive number $t$, there is a positive number $\epsilon(t)$ such that when $|b_j|<\epsilon(t)$ with $j=2,\cdots, n$, $Z\cap D^*(b^*,t)$ is contained in some connected open subset of $D\cap D^*(b^*,t)$, where $D^*(b^*,t)=\{z:\ z=(z_1,b_2,\cdots,b_n,-t): z_1\in \bf C^1\}$. % and $\D_{a}$ % denotes a disk in $\bf C^1$ centered at the origin and with radius $a$. Then $g$ admits a holomorphic extension near $0$. \bigbreak {\it Proof of Lemma 5.3}: Without loss of generality, we assume that $$\wt{P}(z_2,\cdots,z_{n+1};X)=0$$ has no double roots, i.e, $\wt{P}(z_2,\cdots,z_{n+1};X)$ and ${\PP \wt{P}\OO \PP X}$ are relatively prime. Denote by $\wt{Z}$ the branching locus of $\wt{P}$, which is a subvariety of $ \D_{\d}^{n}$. Now, we choose a small $t>0$ and $b^*=(b_2,\cdots,b_{n})$ with $|b_j|<\epsilon(t)$ such that $(b^*,-t)\not\in \wt{Z}$ and $Z^*\cap {\bf C^1}\times \{(b^*,-t)\}$ is discrete. Write $B_{\eta}(b^*,-t)$ for the {\it closed} ball in $\bf C^{n}$ centered at $(b^*,-t)$ with radius $\eta$. Then, when $\eta$ is sufficiently small, $B_{\eta}(b^*,-t)\cap \wt{Z}=\emptyset$ and $0\not\in B_{\eta}(b^*,-t)$. Construct a pseudoconvex domain $\wt{B}(t)\subset\subset \D_{\d}^{n}$ such that the following holds: (i) $0\in \wt{B}(t)$, $\wt{B}(t)\supset B_{\eta}(b^*,-t)$; and (ii) there is a contraction map $T(\cdot,\tau): \wt{B}(t)\ra \wt{B}(t)$ ($\tau\in [0,1]$) such that $T(\cdot,0)=\h{id}$, $T(\wt{B}(t),1)= B_{\eta}(b^*,-t)$, and for any $\tau$, $T(\cdot,\tau)$ is the identical map when restricted to $B_{\eta}(b^*,-t)$. Now, using the hypothesis on $Z$ and shrinking $\d$ if necessary, we can find $ R>0$ with $R<\d _0$ such that $ (\D_{R}\sm \D_{{1\over 2}R})\times \D_{\delta}^n\cap Z=\emptyset$. Write $\O_0=\left(\D_{R}\times\wt{B}(t)\right)\sm \overline { \left(\D_{{1\over 2}R}\times (\wt{B}(t)\sm B_{\eta}(b^*,-t))\right)}$, which can also be written as $$\left(\D_{R}\times \h{int}(B_{\eta}(b^*,-t))\right) \cup \left((\D_{R}\sm \overline {\D_{{1\over 2}R}})\times \wt{B}(t)\right).$$ Here, for a subset $K$, we write $\h{int}(K)$ for the collection of the interior points of $K$. Then, it is easy to see that any (closed) loop in $\O_0\sm Z$ based at certain $z_0\in \left (\D_{R}\times \h{int}(B_{\eta}(b^*,-t))\right )\sm Z^*$ can be deformed, relative to the base point and without cutting $Z$, to a loop in $\left(\D_{R}\times B_{\eta}(b^*,-t)\right)\sm Z$. Indeed, let $\g=(\g_1,\g^*)$ be a loop in $\O_0\sm Z$ based at $z_0$. Then the above described deformation can be performed by $\left(\g_1(t),T(\g^*(t),\tau)\right)$. We next set up the following notation: For a given connected subset $D'$ (near the origin) with $D'\sm Z$ pathwise-connected, a point $z_0\in D'\sm Z^*$, and a complex number $w_0$ with $P(z_0,w_0)=0$, we define $\h{Val}(z_0,w_0;D')$ to be the collection of $w'$s, which is obtained as follows: There are a loop $\g\subset D'\sm Z,$ based at $z_0$, and a continuous section $\cal G$ from $[0,1]$ to $({\cal O},\pi,{\bf C})$ with $P(z;{\cal G})\equiv 0$ such that each representation of ${\cal G}$(0) takes value $w_0$ and each representation of ${\cal G}(1)$ takes value $w$ at $z_0$. We write ${\cal P}(z)=\{w:\ P(z,w)=0\}$. When $\V(z_0,w_0;D')$ is independent of the choice of $w_0$, we simply use the notation $\V(z_0;D')$. In particular, this is the case when $\#\V(z_0,w_0;D')=N$. \bigbreak {\bf Claim 5.4}: $\#\V(z_0; \D_{R}\times B_{\eta}(b^*,-t))=N$, i.e, $$\V(z_0; \D_{R}\times(B_{\eta}(b^*,-t))=\{\h{all solutions of}\ P(z_0,X)=0\}.$$ \bigbreak {\it Proof of Claim 5.4}: Let $\O_0$ be as defined above. We then notice that the holomorphic hull of $\O_0$ is $\D_{R} \times \wt{B(t)}$. In fact, for any $\phi\in\h{Hol}(\O_0)$, the following Cauchy integral gives the holomorphic extension of $\phi$ to $\D_{R}\times \wt{B}(t)$: $$ {1\over 2\pi i}\int_{|\xi|=\wt R}{\phi(\xi,z_2,\cdots,z_{n+1}) \over \xi-z_1}\h{d}\xi,\ \h{with}\ \wt R\ra R.$$ Now, we fix a point $\ z_0\in\O_0\sm Z^*$ and a number $w_0$ with $P(z_0,w_0)=0$. For $z\in \O_0\sm Z^*$, let $\g\subset \O_0\sm Z$ be a curve connecting $z_0$ to $z$. Denote by $w$ be the value of the branch of ${\cal P}$ at $z$ which is continuous along $\g$ and takes the initial value $w_0$. Write $\V(z,w;\O_0)=\{w_1,\cdots,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, $\V(z,w;\O_0)$ is independent of the choice of $\gamma$, though $w$ does. Now, $P_0(z,X)=\prod_{j=1}^{k'}(X-w_j(z))$ will give a well-defined polynomial in $X$ with coefficients holomorphic and bounded in $\O_0\sm Z^*$. Applying the Riemann extension theorem, we conclude that $P_0(z,X)$ is a polynomial in $X$ with coefficients holomorphic in $\O_0$. The above observation then indicates that the coefficients of $P_0$ are holomorphic in $\D_{R}\times \wt{B}(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. $\endpf$ \bigbreak {\bf Claim 5.5}: Let $b^*$ and $t$ be as chosen above and write $p_0=(b^*,-t)$. Let $\epsilon(t)$ be sufficiently small. Let $z_0=(z_1,p_0)\in \D_{R}\times \{p_0\}\sm Z^*$ for a certain $z_1$. Then $\#\V(z_0;\D_{R}\times\{p_0\})=N$. \bigbreak We first mention that the proof of Lemma 5.3 follows easily from Claim 5.5. Indeed, by the hypothesis, the finite set $Z\cap \{\D_R\times \{p_0\}\}$ stays in some {\it connected} open subset of $D\cap \{{\bf C^1}\times \{p_0\}\}$, say $D^*$. Since $\# (Z\cap \{{\bf C^1} \times \{p_0\}\})$ is finite, we can choose a simply connected smooth domain $D^{**}\subset\subset D^*$ with $Z\cap ({\bf C^1}\times \{p_0\})\subset D^{**}\cap ({\bf C^1}\times \{p_0\})$ (see Remark 5.6 (a) below). Therefore, by the Schoenflies theorem, there is a retraction from $\D_R\times \{p_0\}$ to $\overline {D^{**}}$ that deforms $\D_{R}\times \{p_0\}\sm D^{**}$ to the boundary of $D^{**}$. Thus any loop in $\D_{R}$ that is based at $z^*\in D^{**}\sm Z^*$ can be deformed, relative to the base point, to $\overline {D^{**}}$ without cutting $Z\cap {\D_R}\times \{p_0\}$. Hence, by the monodromy theorem, it follows that $\#\V({z^*};D^*)=N$ for any given point $ z^*\in 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_{\eta}(b^*,-t)$, we know that $$\wt{P}(z_2,\cdots,z_{n+1};z_1)=z_1^{N_1}+\sum _{j0.$$ By using a M\"obius transform, we can assume that $p_1(p_0)=p_1^0=0$. %After a rotation, depending continuously on $z_2$, we %can make $p_2(z_2)$ and $p_2^0$ on the same ray emanating from %the origin. Applying certain modified dilation, %which depends continuously on $z_2$, we can map $p_2(z_2)$ to $p_2^0$. Suppose that for an index $j0$. We then wish to find a homeomorphic self map $\wt{\sigma_{j+1}}(\cdot,z^*)$ of $\D_R$, depending continuously on $z^*\in B_{\eta}(b^*,-t)$, such that $\wt{\sigma_{j+1}}(p_l^0,z^*)=p_l^0$ for $l\le j$ and $\wt{\sigma_{j+1}}(p_{j+1}(z^*),z^*)=p^0_{j+1}.$ If this can be done, letting $ {\sigma_{j+1}}(\cdot,z^*)=\wt{\sigma_{j+1}}(\cdot,z^*)\circ \sigma_j(\cdot,z^*)$, and applying the induction argument, we see the existence of the aforementioned map $\pi_{z^*}$. Indeed, the existence of $\wt{\sigma_{j+1}}(\cdot,z^*)$ clearly follows from the following Remark 5.6 (b) and by making $\epsilon(t)$ sufficiently small: \bigbreak {\bf 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 $\bf C^1$. Then, for any given finitely many points $\{p_j\}_{j=1}^{k}\subset D$, there is a smoothly bounded simply connected domain $D^*\subset\subset D$ such that $\{p_j\}_{j=1}^{k} \subset D^*$. Indeed, we can find a curve $\g$ 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 $\g$ if necessary, we can assume that $C=\g ([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 $\wt C$ 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 $\wt D(\subset D^*$) of $\wt C$, that can be retracted to ${\wt C}$ and thus is simply connected, too. (See for example, J. F. Hudson: {\it Piecewise Linear Topology}, W. A. Benjamin, Inc, 1969; in particular, Theorem 2.11, pp57 ). Now, we have a Riemann mapping $\sigma$ from $\wt D$ to $\D$. Then obviously, $\sigma^{-1}(\D_{r})$ does our job when $r<1$ is sufficiently close to $1$. $\endpf$. \bigbreak (b) Let $\{a,a_1,\cdots, a_k\}$ be $k+1$ distinct points in $\D$. Suppose that $p(z)$ is a continuous map from the ball the closed unit ball $ B_n\subset {\bf C^n}$ to $\D$ such that $\epsilon=\min_{j=1,\cdots,k;z\in B_n}|p(z)-a_j|$ is a sufficiently small positive number, and $|p(z)-p(z_0)|<{1\over 4}\epsilon$. Then there exists a homeomorphic self map $\sigma(\cdot,z)$ of $\D_{R}$, which depends continuously on $z$, such that $\sigma(p(z),z)=a$ and $\sigma(a_j,z)=a_j$ for $j=1,\cdots, k$. \bigbreak The proof of this fact can be seen as follows: Pick a continuous retraction $\pi$ from $B_n$ to $z_0$. We will then construct a homeomorphic self map $\sigma(\cdot,z)$ of $\D$, that depends continuously on $z\in B_n$, such that the following holds: (a) $\sigma(a_j,z)=a_j$ $(j=1,\cdots, k)$; (b) $\sigma(\cdot,z_0)=\h{id}$; (c) $\sigma(p(z),z)=p(z_0)$ for $z\in B_n$. To this aim, we assume that $p(z_{0})=0$ to simplify the notation. Notice that $|p(z)-p({z_0})|<\epsilon/4$ and thus $p(z)$ is contained in the disk $\D_{\epsilon/4}$. We also observe, by the hypothesis, that $a_j\not\in \D_{\epsilon}$. Now, we let $\tau(\cdot,z)$ be the M\"obius transform from $\D_{\epsilon/4}$ to itself that maps $p(z)$ for $z$ to $p(z_0)$. Meanwhile, we impose the condition that $\tau'(p(z),z)>0$ to make it unique. From the explicit formula of $\tau$, it is clear that $\tau(\cdot,z)$ depends continuously on $z\in B_n$. For $w=\epsilon/4 e^{\sqrt{-1}\theta}$ with $0\le \theta<2\pi$, write $\tau(w,z)={\epsilon\OO 4}e^{\sqrt{-1}r(\theta,z)}$, where $r(0,z)\in [0,2\pi)$, $r(2\pi,z)-r(0,z)=2\pi$, and $r(\theta,z)$ is a strictly increasing function of $\theta$ for fixed $z$. Moreover, %by making $N>>1$ if necessary, we can also make $r(\theta,z)$ depend continuously on $(\theta,z)$. The homeomorphism $\sigma(\cdot,z)$ from $\D$ to itself can be defined as follows: (i) $\sigma(\cdot,z_0)=\h{id}$. For $z\in B_n$, we let $\sigma(w,z)=w$, for $w\not\in \D_{\epsilon}$; $\sigma(w,z)=\tau(w,z)$, for $w\in \D_{\epsilon/4}$; and otherwise $$\sigma(w,z)= |w|\h{exp}\left({\sqrt{-1}{4\OO 3\epsilon}\left( (|w|-\epsilon/4)\theta+(\epsilon-|w|)h(\theta,z)\right)}\right),$$ where $w=|w|\h{e}^{\sqrt{-1}\theta}$ with $|w|\in [\epsilon/4,\epsilon]$. It can be easily verified that $\sigma$ possesses all the properties we imposed. \bigbreak {\it 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). \bigbreak {\bf Theorem 1.1$'$}: Let $M_1\subset \bf C^{n+1}$ and $M_2\subset \bf 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\in 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 $\O$ as introduced before, respectively. Then $f$ admits a holomorphic extension near $p$. \bigbreak {\it 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 ${\cal E}_0^{(j)}$, is contained in $ {\cal E}_0$. By Lemma 5.1, it then follows that for sufficiently small $t>0$, one can always find $\epsilon(t)>0$ such that when $|b_j|<\epsilon(t)$ ($j=2,\cdots, n)$, ${\cal E}^{(j)}_0\cap \{(z_1,b_2,\cdots, b_n,-t)\}$ is contained in some connected component of $D\cap \{(z_1,b_2,\cdots, b_n,-t)\}$. Now, applying Lemma 5.3, we conclude that $f_j$ extends holomorphically across $0$ for each $j$. $\endpf$ \bigbreak {\bf 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]). \section {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^{\pm}$, $S_2$, $S_1$, and $S_0$. Here $M^{\pm}_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\sm M_2^{\pm}$; $S_1$ is a locally finite union of 1-dimensional real analytic submanifolds in $M_2\sm (M^{\pm}\cap 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\in 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^{\pm}$, where $M_1^{\pm}$ denotes the set of points in $M_1$ where the Levi form do not vanish. When $f(U_0)\cap M_2^{\pm}\not =\emptyset$, 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\not\equiv 0$ near $p$. Now, the algebraicity follows from a theorem of Webster [We]. When $f(U_0)\subset 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\subset U_0$ and a wedge $W^+\subset 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\equiv 0$ near $E$. This contradicts the non constant assumption for $f$, by [BR3]. Similarly, we can exclude the case: $f(U_0) \subset S_1\cup 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 $\bf 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: \bigbreak {\bf Lemma 6.1}: Let $M_1$ and $M_2$ be two connected algebraic hypersurfaces of finite D-type in $\bf 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 $\bf C^n$, that is continuous up to $D\cup M_1$, and maps $M_1$ into $M_2$. Then for each point $p\in M_1$, $f$ extends as a holomorphic correspondence to a neighborhood of $p$. \bigbreak {\it Proof of Lemma 6.1}: Let $M_1$ and $M_2$ be defined by two real polynomials $\rho_1(z,\overline z)$ and $\rho_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^{\rm th}$ component $f_j$ of $f$ satisfies the irreducible polynomial equation $P_j(z,X)=\sum_{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 $\CC^n\times\CC ^n$, defined by $P_j(z,w)=0$ for $j=1,\cdots,n$, which contains the graph $\Gamma _f$ of $f$ over $D$. Write $\pi$, $\pi'$ for the nature projections from $V$ to the first and the second copies of $\bf 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 $\ge 1$. \bigbreak {\bf Claim 6.2}: Fix a point $z_0\in (M_1\sm E)\cap {\cal U}_1$ and choose a sufficiently small ball $B$ centered at $z_0$. Consider the multiple-valued extension ${\cal F}_j$ of $f_j$ from $B\cap D$ to $B\cup (D^c\cap {\cal U}_1)\sm E$ with ${\cal U}_1$ as chosen before (see $\S 3$ for a precise definition of ${\cal F}_j$) . If there is a constant $C$ such that $|\wt{f}(z)|\le C$ for any $z\in B\cup (D^c\cap {\cal U}_1)$ and $\wt{f}(z)\in {\cal F}_j(z)$, %If for any simple curve $\g\in (D^c \cup M_1)\cap {\cal U}_1 \sm E$ %connecting $z_0$ to $z$, and for the holomorphic branch $\H{f_j}$ of the %roots of $P_j(z,X)=0$ with $\H{f}(z_0)=f_j(z_0)$, %it holds that %$|\H{f}(z)|$ is bounded by a constant depending only on ${\cal U}_1$. then $f_j$, near $0$, satisfies a polynomial equation with leading coefficient 1. \bigbreak {\it Proof of Claim 6.2}: % Similar to the argument in Claim 5.3, For each $z\in B\cup (D^c\cap {\cal U}_1)\sm E$, let $${\cal F}_j(z)=\{f_j^{(1)}(z),\cdots, f_j^{(k(z))}(z)\}$$. % be all possible $\H{f}(z)$ by %varying the curve as described in the claim. %We first show that % $k$ is independent of the choice of $z$ with $z\in {\cal U}_1\cap D^c\sm %E$. For this, by using the monodromy theorem, %we need only to show that any curve $\g_0\subset D^c\cap {\cal U}_1\cap\sm E$ % with $\g_0(0)=z_0$, $\g_0(1)=z$, and $g_0((0,1])\subset D^c$, %can be deformed in $D^c\cap {\cal U}_1\cap\sm E$ to a curve as described %in the claim. Indeed, this follows easily % from the Whitney approximation theorem %(see Lemma 1.29 and Theorem 1.29 of [Mil]). % Indeed, let $B_(z_0)$ and $B(z)$ be two small balls centered %at $z_0$ and $z$, respectively. Then %$D^c\cap {\cal U}_1\cap\sm E$ can be retracted to %$D^c\cap {\cal U}_1\cap\sm (E\cup B(z_0)\cup B(z))$ First, by the monodromy theorem, $k(z)$ is constant. %is % independent of the choices of $z_0$ and $z$, by the monodromy %theorem. Now, suppose that all elements in ${\cal F}_j(z)$ is bounded by a constant independent of $z$. As we did before, using the Riemann extension theorem, we conclude that $\wt{P_j}(z,X)=\prod_{l}(X-f^{(l)}_j(z))$ is a polynomial in $X$ with coefficients holomorphic and bounded in $D^c\cap {\cal U}_1$. Now, by the aforementioned result of Baouendi-Treves and Trepreau, one sees that the coefficients of $\wt{P_j}$ can be extended holomorphically to a neighborhood of $0$. % if all $f^{(l)}$ $'$s are bounded. Obviously, Obviously, $\wt{P_j}$ has leading coefficient 1 and annihilates $f_j(z)$ when $z\in {\cal U}_1$ . $\endpf$ \bigbreak Now, seeking a contradiction, suppose that Lemma 6.1 is false. Then Claim 6.2 indicates that no matter how we shrink ${\cal 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\in B\cup (D^c\cap {\cal U}_1)\sm E$ with $|\H{f_{j_0}}(z^1)|>1$ for certain ${\H f}_{j_0}(z)\in {\cal F}_{j_0}(z)$. Observe that for a given curve $\g\subset B\cup (D^c\cap {\cal U}_1)\sm E$ with $\g(0)=z_0$ and $\g(1)=z^1$, there is a certain holomorphic branch $\wt{f_{j_0}}$ of ${\cal F}_{j_0}$ along $\g$, which takes value $f_{j_0}(z_0)$ at $z_0$ and takes value $\H{f_{j_0}}(z^1)$ at $z^1$. On the other hand, by the Whitney approximation theorem (see J. Milnor: {\it Differential Topology}, Lecture Notes by Munkress; in particular, Theorem 1.28 and Lemma 1.29) and by noting that $B\cup (D^c\cap {\cal U}_1)\sm E$ can be retracted to $D^c\cap {\cal U}_1\sm E$, we notice the existence of %for each point $z\in %B\cup (D^c\cap {\cal U}_1)\sm E$, a smooth simple curve $\g^*$ connecting $z_0$ to $z$ with $\g((0,1])\subset (D^c\cap {\cal U}_1)\sm E$ and $\g$ transversal to $M_1$ at $z_0$. Moreover $\g^*$ is homorpotic to $\g$ (relative to the base point) in $B\cup (D^c\cap {\cal U}_1)\sm E$. So, using the Monodromy theorem %by noting $f(0)=0$, we can simply assume that $\g=\g^*$. As what we did in Lemma 3.1, we can thicken $\wt{g}=\g\cup {\cal R}(\g)$ and then apply the uniqueness theorem of holomorphic functions and the invariant property of Segre varieties to conclude that $f(Q_z\cap O(z^*))\subset Q'_{\wt{f}(z)}$, for any $z\in\g$, where the branch $\wt{f}$ of $\cal F$ is determined by the initial value condition ${\wt f}(z_0)=f(z_0)$. %extending ${\cal F}$ from $\g\cup {\cal R}(\g)$, %which coincides with $f$ near ${\cal R}(\g)$. %So, if Lemma 6.1 is false, %by Claim 6.2 Since $f(0)=0$, for any small $\epsilon>0$, by changing the size of ${\cal U}_1$, we see the existence of a sequence $\wt{p}_j\ra 0$ such that $|\wt{f}(\wt p_j)|=\epsilon$. In particular, we see that the cluster set $Cl_{\wt f}(0)$ of $\wt{f}$ at $0$ contains points with norm $\epsilon$. Here, the branch ${\wt f}$ 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 $\H{V}$ be the algebraic closure of $V$ in $\bf P^{n}\times \bf P^{n}$ (see [Wh]). Without loss of generality, we can further assume that $\H{V}$ is irreducible and contains $\Gamma_f$. Let $\H{\pi}$ and $\H{\pi'}$ be the nature projection from $\H{V}$ %$\bf P^{n}\times \bf P^{n}$ to the first and the second copies of $\bf P^n$, respectively. Since $\H{V}$ is of dimension $n$, away from a proper subvariety, we see that $\H\pi$ and $\H\pi'$ are local biholomorphisms. For each $z$, let $Q_z =\{w\in {\bf C^n}\cap {\cal U}_1:\ \rho_1(w,\overline z)=0\}$ as before, and let $\H{Q}_z$ be the compactification of $Q_z$ in $\bf P^n$. Now, for each $w\in Cl_{ \wt{f}}(0)$, the cluster set of ${ {\wt f}}$ at $0$, let $z_j\ra 0$ with $\wt{f}(z_j)= w_j\ra w_0$. Then $z_j\in D^c$ and $f(Q_{z_j}\cap O(z_j^*))\subset Q'_{w_j}$. %Here, as before, ${\cal U}_1$ denotes a small 0 of $0$. This gives that $\H{\pi'}\circ\H{\pi}^{-1}(\H{Q_{z_j}})\supset Q'_{w_j}\cap {\cal U}'_2$, where ${\cal U}'_2$ is as defined in $\S 2$. Now, letting $j\ra \infty$, we see that $\H{\pi'}\circ\H{\pi}^{-1}(\H{Q_{0}}) \supset Q_{w'}\cap {\cal U}'_1$. Notice that ${\pi'}\circ\H{\pi}^{-1}(\H{Q_0})$ is an analytic variety of ${\bf P}^n$ of codimension $1$, by the Remmert mapping theorem. Hence, it follows that $\H{Q}_{w'}\cap {\cal U}'_2$ has to be contained in one of the finite irreducible components of $\pi'\circ\pi^{-1}(\H{Q_0})$. Since $\#A_{w}'<\infty$ ($w\in {\cal U}'_2$), for each connected component of $\pi'\circ\pi^{-1}(\H{Q_0})$, there are only finitely many $w$'s with $|w|$ small such that the above property holds. % there are only finitely many choices of such an $w'$. This implies that $\#( Cl_{\wt f}(0)\cap O(0))<\infty$ and contradicts our assumption. The proof of Lemma 6.2 is complete. $\endpf$ \section {Extending proper holomorphic mappings in $\bf 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. \bigbreak {\bf Theorem $1.3'$}: Let $D$ and $D'$ be two smoothly bounded domains with real analytic boundaries in $\bf C^2$. Let $f$ be a proper holomorphic map from $D$ to $D'$. Suppose that $f$ extends continuously up to $\PP D$. Then $f$ extends as a holomorphic correspondence across $\PP D$. By Theorem 1.1, it thus follows that $f$ admits a holomorphic extension to $\overline {D}$. \bigbreak To start with, we let $M$ be a real analytic hypersurface of finite type in $\bf C^2$, and let $\O$ 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^+\cup M^-\cup T_2^+\cup T_2^-\cup T_2^{\pm}\cup T_1\cup T_0.$$ Here \noindent (a) $M^+$ ($M^-$) is the set of strongly pseudoconvex (strongly pseudoconcave, respectively) boundary points of $\O\cup M$; \noindent (b) $T_2^+$ ($T^-_2$) is a locally finite union of 2-dimensional totally real analytic submanifolds, where the vanishing order of the Levi form of $M$, as the boundary of $\O\cup M$, is even and positive (negative, respectively). Therefore, $\O$ is pseudoconvex near $T^+_2$ and pseudoconcave near $T_2^-$; \noindent (c) $T_2^{\pm}$ is a locally finite union of 2-dimensional totally real submanifolds, where the vanishing order of the Levi form is odd. It is known (see [DF3] or [BCT]) that any CR functions defined near $T^{\pm}_2$ can be holomorphically extended to both sides of $M$ near $T_2^{\pm}$; \noindent (d) $T_1$ is a locally finite union of one dimensional real analytic curves; and $T_0$ is a locally finite subset of $M$. Now, let $D$ and $D'$ be as in Theorem 1.3$'$. After making the above type of the stratifications, we have the following disjoint unions: $$D=\PP D^+\cup\PP D^-\cup T_2^+\cup T_2^-\cup T_2^{\pm}\cup T_1\cup T_0,$$ $$D'=\PP {D'}^+\cup\PP {D'}^-\cup {T'}_2^+\cup {T'}_2^-\cup {T'}_2^{\pm}\cup T'_1\cup T'_0.$$ In what follows, we write $\Sigma'= \PP {D'}^+\cup\PP {D'}^-\cup {T'}_2^+\cup {T'}_2^-\cup {T'}_2^{\pm}$, and write $\Sigma$ for the boundary points in $\PP D$ where $f$ extends holomorphically. \bigbreak {\bf Lemma 7.1}: (a) For any boundary point $p\in \PP D$, if $f(p)\in \Sigma'$, then $f$ extends holomorphically across $p$. (b) $f$ extends almost everywhere in $\PP D$ and $\Sigma\cap T_2^+$ is dense in $T_2^+$. \bigbreak {\it Proof of Lemma 7.1}: We first note that in case $q=f(p)\in \PP D'{}^{-}\cup {T'}_2^{\pm}\cup {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)\in \PP {D'}^+\cup {T'}_2^+$. Pick a small pseudoconvex piece $M'\subset \PP D$ of $q$, and let $M$ be the connected component of $f^{-1}(M')$ which contains $p$. Then since $f(\PP D)\subset \PP D'$, it is clear that $f^{-1}(q)\subset\subset 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\subset\PP D$ near $p$, $f(T_2^+\cap U_p)\subset T_1'\cup T'_0$, then it is easy to see that there is a wedge $W^+$ with edge $U_p\cap T^+_2$ such that $W^+\subset D$. Indeed, after a local change of coordinates, one can assume that $p=0$, $T^+_2=i{\bf R}^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)\approx (0,0):\ x_1<0, |x_j|>|y_j|\}$. Notice that $T'_1\cup 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\in U_p\cap T_2^+$. Now, notice that $J_f\not\equiv 0$ for $w\in U_p\cap T_2^+$ ($\approx z$). This gives a contradiction; for $f$ maps the two dimensional manifold $U_p\cap T^+_2$ into a one dimensional semi-analytic subset. Now, from the above argument, it follows that $\Sigma\cap T^+_2$ is dense in $T^+_2$. Similarly, one can see that $f$ extends holomorphically across almost every point in $\PP D$, as did in the beginning of $\S 6$. $\endpf$ \bigbreak Now, we fix $p_0\in\PP 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 $\PP D$ near $p_0$ can be extended holomorphically to the side in $D$. For each $p\approx p_0$ and $q\approx q_0$ with $q_0=f(p_0)$, we introduce Segre neighborhood systems near $p$ and $q$, as did in $\S 2$: ${\cal U}_3(p)\supset\supset {\cal U}_2(p)\supset\supset {\cal U}_1(p) \ni p$; ${\cal U}'_3(q)\supset\supset {\cal U}'_2(q)\supset\supset {\cal U}'_1(q)\ni q$. % To simplify the notation, we will drop the reference %point when there is no confusing arising. As in [DFY] and [DP] (in particular, see [DP]), we define the ``potential" correspondence of $f$ near $p_0$ as follows: $${\cal V}(p_0,q_0)=\{(w,w')\in {\cal U}_1(p_0)\times {\cal U}_1'(q_0):\ f(Q_w(p_0)\cap {\cal U}_2(p_0)\cap D)\supset\ _{w^{'*}}Q'_{w'}(q_0)\}.$$ Here, as in $\S 2$, we write $w'{}^*$ for the reflection point ${\cal R}'(w')$ of $w'$; and we use $ _{{w'*}}Q'_{w'}$ to denote the germ of the variety $Q'_{w'}$ at $w^{'*}$. Also, the Segre varieties are chosen in the indicated Segre systems. In what follows, we will drop the reference points for the Segre systems and Segre varieties to simplify the notation, when there is no confusion arising. By the invariant property of Segre surfaces, it can be seen that if $f$ extends holomorphically in a small neighborhood $G$ of some point $p\in {\cal U}_1(p_0)\cap \PP D$, then ${\cal V}(p_0,q_0)\supset \{(w, w')\approx (p,f(p)):\ w'=f(w),\ w\in G\sm{\overline D}\}$. %(In fact, %${\cal V}\cap G\times {\cal U}'_1=\{ %(w,w'):\ w\in G\ \h{and}\ w'=A_{f(w)}\cap {\cal U}'_1\}$, which is an analytic %variety in $G\times{\cal U}_1'$) (see the definition of ${\cal R}'$ and %the following (7.f)). The key point in proving Theorem 1.3$'$ is to show that ${\cal V}(p_0,q_0)$ extends as an analytic variety in ${\cal U}_1(p_0)\times{\cal U}_1'(q_0)$ with the first projection map $\pi: {\cal V}(p_0,q_0)\rightarrow {\cal U}_1(p_0)$ locally proper. For this purpose, we will use the strategy of Diederich-Pinchuk [DP] of applying the extension lemma of Bishop ( [Chi]). Before arguing further, we need recall some useful facts: \noindent (7.a) $$\PP D\sm({\cal U}_2(p_0)\cap \PP D\cap Q_T)$$ is dense in $\PP D\cap {\cal U}_1(p_0)$, where $Q_T=\cup_{z\in (T_1\cup T_0)\cap {\cal U}_1(p_0)}Q_z(p_0).$ \noindent (7.b) ([DP]): Making ${\cal U}_1'$ small, then $(T'_1\cup T'_0)\cap {\cal U}'_1(q_0)$ is contained in a pluripolar set $\wt{T}$ in ${\cal U}'_1$. That is, there is a plurisubharmonic function $h\ (\not\equiv -\infty)$ such that $T'_0\cup T'_0\subset {\wt T}=\{z\in {\cal U}'_1:\ h(z)=-\infty\}$. (This follows from the fact that the semi-analytic subset $(T'_1\cup T'_0)\cap {\cal U}'_1$ is contained in a one dimensional real analytic subset [BM], which can be complexified.) % ( Here, we remark that even if ${\cal U}'_1$ is not small %the assertion still holds.) \noindent (7.c) Fix ${\cal U}_1'$ . Using the properness and the boundary continuity of $f$, one can always choose ${\cal U}_2$ small enough so that $\cal V$ has no limit points in $({\cal U}_1\sm {\overline D})\times \PP {\cal U}_1'$. \noindent (7.d) ([DP]) For each $p\in T_2^+$, after making ${\cal U}_2(p)$ small, it holds that $Q_p\cap {\cal U}_2(p)\cap\PP D\subset \PP D^+\cup \{p\}$. \noindent (7.e) For each $p\in \PP D^+$, after making ${\cal U}_2(p)$ small and then making ${\cal U}_1(p)$ sufficiently small, it then holds that $Q_{\wt p}\cap {\cal U}_2(p)\cap\PP D=\{\wt p\}$ for any ${\wt p}\in {\cal U}_1(p)$. \noindent (7.f) ([BR2]) Let $M_1$ and $M_2$ be two connected oriented real analytic hypersurfaces in $\bf C^n$ of finite D-type. Let $f$ be a non constant holomorphic map from $M_1$ into $M_2$ with $f(M_1)\subset 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. \noindent (7.h) (Bishop's Lemma [Chi]): Let $E$ be a (complete) pluripolar set in the domain $U\times U'\subset {\bf C}^4$, where $U$ and $U'$ are domains in $\bf C^2$. Let $A\subset (U\times U')\sm E$ be an analytic set of pure dimension $2$ with no limit points in $U\times \PP U'$. Suppose that there exists an open subset $V\subset U$ such that $\overline {A}\cap((V\times U'))$ is an analytic variety in $V\times U'$ of pure dimension 2. Then $\overline {A}$ is an analytic set in $U\times U'$. \bigbreak {\bf Remark} (7.1)$'$(a): We observe that (7.d) can be strengthened as follows: For any point $\wt{p}(\in T_2^+)\approx p$, it also holds that $Q_{\wt p}\cap {\cal U}_2(p)\cap\PP D\subset \PP D^+\cup {\wt p}$. 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 $i{\bf R}^2=\{(iy_1,iy_2)\}$. We also choose $z_1$-axis for the complex tangent space of $\PP D$ at $0$. By the way we chose $T^+_2$, we can assume that $D$ is defined near $0$ by $\rho=2x_2+(2x_1)^{k}a(z_1,\overline {z_1},y_2)$, where $m>1$ and $a|_{i{\bf R}^2}\not\equiv 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)\in i{\bf R}^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}\cap i{\bf R}^2=\{w\}$. This completes the proof of the above fact. (b) To see $(7.a)$, we notice the facts that $Q_z\cap \PP D$ is at most a real analytic subset of dimension 1 and $T_1\cup T_0$ is contained in a real analytic set of at most dimension 1 [BM]. Therefore, ${\cal U}_2(p_0)\cap \PP D\cap 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, ${\cal U}_2(p_0)\cap \PP D\cap Q_{T}$ has at most dimension 2. (c) (7.e) can be seen as follows: For each $\wt p$ close to $p$, after a normal holomorphic change of coordinates (see related discussions in $\S 2$) and using the strong pseudoconvexity of $\PP D$ at $\wt p$, one sees clearly that $Q_{\wt p}$ cuts $\PP D$ only at $\wt p$. Since this process can be done uniformly, we can achieve (7.e) after shrinking the size of ${\cal U}_2$. \bigbreak {\bf Lemma 7.2}: Shrinking ${\cal U}_1(p)$ if necessary, % and ${\cal U}_1'$, one can make ${\cal V}(p,q)$ having no limit points in $({\cal U}_1(p)\sm {\overline {D}})\times \Sigma'$. \bigbreak {\it Proof of Lemma 7.2}: Suppose that $(w_v,w'_v)\in {\cal V}$ with $(w_v,w'_v)\ra (w_0,w_0')\in({\cal U}_1\sm \overline {D})\times \Sigma'$. Then $f(Q_{w_v}\cap {\cal U}_2\cap D)\supset\ _{w'{}^*_v}Q'_{w'_v}$. In particular, from the properness of $f$, it can be seen that there exists a point $\xi^*_{v}\in Q_{w_v}\cap \overline {\cal U}_2\cap D$ such that $f(\xi^*_v)= w^{'*}_v$ and $f(Q_{w_v}\cap O(\xi^*_v))\supset _{w'{}^*_v}Q'_{w'_v}$ Assume, without loss of generality, that $\xi^*_{v}\ra \xi_0\in \overline {\cal U}_2\sm D$ and $w^*_v\ra w'_0$. Then $f(\xi_0)=w'_0$ and $\xi_0\in\PP D$. By Lemma 7.1, it follows that $f$ extends holomorphically to a neighborhood of $\xi_0$. Therefore, there is a small open subset $\O$ near $\xi_0$ such that $f$ is a proper holomorphic map from $\O$ to $f(\O)$ and ${f|_{\overline {\O}}}^{-1}(f(\xi_0))=\xi_0$ (see (7.f)). Meanwhile, $f(\O\cap D)\subset D'$, $f(\O\cap D^c)\subset D'{}^c\cap {\cal U}_2'$. Now, let $\eta_v\in \O$ be such that $f(\eta_v)=w_v'$. Then, for $v\gg 1$, it holds that $f(Q_{\eta_v}\cap O({\eta_v^*}))\subset Q'_{w_v'}$. Obviously, $\eta_v\ra \xi_0$ and $f(Q_{\eta_v}\cap {\O})\supset f(Q_{w_v}\cap {O}({\xi_0}))$. Write ${f|_{\O}}^{-1}=\{\sigma_1,\cdots,\sigma_k\}$. Hence, we see that $Q_{w_v}\cap O({\xi_0})\subset \cup_{j=1}^{k}\sigma_j(f(Q_{\eta_v}\cap\O))$. Similar to Claim 5.2, we have the following \bigbreak {\bf Claim 7.2$'$}: For $v$ sufficiently large, $Q_{\eta_v}\cap {\cal U}_2=Q_{w_v}\cap {\cal U}_2$. Thus, by passing to the limit, it holds that $w_0\in A_{\xi_0}$. \bigbreak {\it Proof of Claim 7.2$'$}: Let $\rho$ and $\rho'$ be the defining functions of $D$ and $D'$, respectively. Then we notice that $\rho'(f(z),\overline f(w)) =\rho(z,\overline w)h(z,\overline w)$ for $z,w\in \O$ (see (7.f)) and $h\not =0$ (we might need to shrink $\O$ here). For $z \in\cup_{j=1}^{k}\sigma_j(f(Q_{\eta_v}\cap\O))$, there is a point $\wt{z} \in Q_{\eta_v}\cap\O$ such that $f(z)=f(\wt{z})$. Now, from $$\rho'(f(z),\overline {f(\eta_v)}))=\rho(z,\overline {\eta_v})h(z,\overline {\eta_v})$$ and $\rho'(f(\wt{z}),\overline {f(\eta_v)})=\rho(\wt{z},\overline {\eta_v})h(\wt{z}, \overline {\eta_v})=0$, it follows that $\rho(z,\overline {\eta_v})=0$. Thus, we conclude that $z\in Q_{\eta_v}$. Since both $Q_{w_v}$ and $Q_{\eta_v}$ are connected complex curves in ${\cal U}_2$, we see the $Q_{\eta_v}\cap {O}({\xi_0})=Q_{w_v}\cap {O}({\xi_0})$ Passing to a limit, we now conclude that $Q_{\xi_0}\cap {\cal U}_2=Q_{w_0}\cap {\cal U}_2$. \bigbreak Next, $w_0\in A_{\xi_0}$. Since $\xi_0\in\PP D$, $w_0\in\PP D$. This is a contradiction and thus completes the proof of Lemma 7.2. $\endpf$ \bigbreak In what follows, we always make ${\cal U}_1$ small such that Lemma 7.2 holds. Following the strategy of Diederich-Pinchuk, we now prove the following lemma. \bigbreak {\bf Lemma 7.3}: Suppose that for certain ${\cal U}_2(p)$ small enough, it holds that $(Q_{p}\cap \PP D\cap {\cal U}_2(p))\sm \{p\}\subset\Sigma$. Then after shrinking ${\cal U}_1$ and ${\cal U}'_1$ if necessary, ${\cal V}(p,q)$ is an analytic sub-variety of $({\cal U}_1(p)\sm \overline {D})\times ({\cal U}'_1(p) \sm \overline {D'})$ which has no limit points in $\left(({\cal U}_1\sm \overline {D})\times \PP {\cal U}_1'\right)\cup \left( ({\cal U}_1\sm \overline {D}) \times \Sigma'\right)$. % and thus $f$ extends as %a holomorphic correspondence at $z_0$. We mention that in this lemma, $(Q_{p}\cap \PP D\cap {\cal U}_2(p))\sm \{p\}$ can be empty, and all Segre surfaces are defined in the Segre system as chosen here. % is a neighborhood system near $z_0$. \bigbreak {\it Proof of Lemma 7.3}: Part of the proof is greatly motivated by an argument in [DP] for the study of biholomorphic mappings. Let ${\cal U}_2$ be sufficiently small so that $(Q_{p}\cap \overline { D}\cap \overline {\cal U}_2)\sm \{p\}\subset\Sigma$. We also assume the property in (7.c). When $(Q_{p}\cap \overline { D}\cap \PP{\cal U}_2)\sm \{p\}\not =\emptyset$, then by the hypothesis, there is a small $\epsilon >0$ such that $f$ extends holomorphically to $\overline {B_{3\epsilon}(E_p)}$, where $E_p=Q_p\cap \overline {D}\cap \PP {\cal U}_2$ and $B_{\delta}(E_p)=\{z:\ \h{dist}(z,E_p)<\delta\}$ for $\delta>0$. Moreover, by (7.f), we can assume that $f$ is finite to one from $B_{3\epsilon}(E_p)$ and $p\not\in B_{4\epsilon}(E_p)$. Now, we % for any $w\in Q_{p}\cap \PP {\cal U}_2\cap \overline {D}$, %$f$ extends holomorphically to a ball centered at $w$ and with a %radius independent of w. %Let $B(p)$ be a sufficiently %small closed ball with center at $p$. let $R'(p)=\{w'\in {\cal U}_1'\sm { D'}:\ f(Q_{p}\cap B_{2\epsilon}(E_p))\supset \ _{w'{}^*}Q'_{w'}\}$. Then we claim that $R'(p)$ is a finite set. Indeed, this can be seen as follows: Let $\{w'_v(\in R'(p))\}$ with $w'{}^*_v\ra w'{}^*$. Choose $w_v\in Q_{p}\cap B_{2\epsilon}(E_p)$ such that $f(w_v)= {w'}_v^*$. Assume also that $w_v\ra w\in Q_{p}\cap \overline {B_{2\epsilon}(E_p)}$. Then as above, $f$ is a proper holomorphic map from a small neighborhood of $w$. This implies that $f(Q_{p}\cap O(w))$ is an analytic curve near $w'{}^*$. Thus, it follows that any $Q'_{w'_v}$ coincides with $Q'_{w'}$ when $v\gg 1$. So, by the finiteness of $A'_{w'}$, it follows that $\{w'_v\}$ is a finite sequence. Let $R(p)=\{w\in Q_{p}\cap B_{2\epsilon}(E_p): \ f(w)=w'{}^*\ \h{for some}\ w'\in R'(p)\}$. Then $R(p)$ is also finite. Slightly shrinking ${\cal U}_2$ and $\epsilon$ if necessary, we can assume that $R(p)\cap \PP {\cal U}_2\cap \overline {D}=\emptyset$. %Shrinking ${\cal U}_1$, we can %assume that for each $w\in {\cal U}_1\cap \overline {D}$, %the set $R(w)$ does not intersect a small neighborhood of the %set $Q_{w}\cap \overline {D}\cap \PP {\cal U}_2$ (This can be seen by noting %that $\lim_{z\ra p}R(z)\cap \PP {\cal U}_2\subset R(p)\cap \PP {\cal U}_2$). %In particular, from the above argument, one concludes that %$\pi$ is a finite to one map from $\cal V$ to ${\cal U}_1\sm\overline {D}$. To complete the proof of Lemma 7.3, we need to show that after shrinking ${\cal U}_1$, ${\cal V}$ is an analytic sub-variety in $({\cal U}_1\sm \overline {D})\times ({\cal U}'_1\sm \overline {D'})$, which has no limit points in $\left(({\cal U}_1\sm \overline {D})\times \PP {\cal U}_1'\right)\cup \left( ({\cal U}_1\sm \overline {D}) \times \Sigma'\right)$. We first claim that $\cal V$ is a local variety. To this aim, we will prove that after shrinking the size of ${\cal U}_1$, for any $(w,w')\in {\cal V}$, then $E^*(w,w')\cap \PP{\cal U}_2\cap Q_w \cap \overline {D}=\emptyset$, where $E^*(w,w')=\{\xi\in ({\cal U}_2\cap D)\cup B_{\epsilon}(E_w)\cap Q_{w}: f( _{\xi}Q_{w})\supset _{w^{'*}}Q'_{w'}\}$. Here $E_{w}=Q_w\cap \overline {D}\cap\PP {\cal U}_2$ and $B_{\epsilon}(E_{w})$ is defined in a similar way. Also, by $f( _{\xi}Q_{w})\supset _{w^{'*}}Q'_{w'}$, we mean that for each small neighborhood $O(\xi)$ of $\xi$, it holds that $f(Q_{w}\cap O(\xi))\supset _{w^{'*}}Q'_{w'}$. We also make $\overline {B_{\epsilon}(E_w)}\subset B_{2\epsilon}(E_p)$ by shrinking ${\cal U}_1$. %\{a_1,a_2,\cdots, a_m\}$. Indeed, when $E_p=(Q_{p}\cap \overline { D}\cap \PP{\cal U}_2) =\emptyset$, a continuity argument then shows that $E_w=(Q_{w}\cap \overline { D}\cap \PP{\cal U}_2) =\emptyset$ for $w\approx p$. Now, if the above assertion does not hold in case $E_p\not =\emptyset$, after shrinking ${\cal U}_1$ again, we can then find several sequences $\{z_j\}\subset \PP {\cal U}_2\cap \overline {D}\cap E^*(w_j,w'_j)$ with $z_j\ra z_0$, $\{w_j\}\subset {\cal U}_1\sm \overline {D}$ with $w_j\ra p$, and $\{w_j'\}\subset {\cal U}_1'\sm\overline {D}'$ with $w_j'\ra w_0'\in {\cal U}_1'$ such that the following holds: \indent (a) $z_j\in Q_{w_j}$, (b) $f(z_j)=w_j'$, and (c) $f(_{z_j}Q_{w_j}\cap B_{\epsilon}(E_{w_j}))\supset _{w_j^{'*}}Q_{w_j'}$ when $j>>1$. We observe that $f$ extends to a proper holomorphic map from $O(z_0)\cap B_{\epsilon}(E_p)$, Hence, by passing to the limit, we can see that $f( _{z_0}Q_p\cap B_{\epsilon}(E_p))\supset _{w_0^{'*}}Q'_{w'_0}$, $f(z_0)=w'_0$, and $z_0\in Q_p\cap\PP {\cal U}_2\cap\overline {D}$. Hence, we have a contradiction; for it implies that $z_0\in R(p)\cap \PP {\cal U}_2$. Now, by making the size of ${\cal U}_1$ sufficiently small, we can assume that for each $(w,w')\in {\cal V}$, $E^*(w,w')\cap D\cap \overline {\cal U}_2\subset {\cal U}_2\cap D$. Write $E^*(w,w')\cap D\cap \overline {\cal U}_2=\{a_1,\cdots,a_m\}\subset D\cap {\cal U}_2$. Let $(\wt w, \wt w') (\in {\cal V})\approx (w,w')$. Then, since $E(\wt{w},\wt{w}')\cap D\cap \PP {\cal U}_2=\emptyset$, it can be seen that there exists a point $\wt \xi$ so that $\wt {\xi}\in Q_{\wt w}\cap {\cal U}_2\cap D$, $f(_{\wt \xi}Q_{\wt w}) \supset _{\wt{w'}^*}Q'_{w'}$, and $f(\wt \xi)=\wt{w'}^*$. Indeed, let $\{\wt \xi_j\}=f^{-1}(\wt {w^{'*}})\cap \overline {\cal U}_2\cap D$. Then since $f$ is locally proper near each $\wt{\xi_j}$, the choice of our $w'$ indicates that $f(O(\wt{\xi_l})\cap Q_{\wt w})\supset _{\wt{w'}^*}Q'_{w'}$ for a certain $l$. So, we can take $\wt \xi$ to be this $\wt \xi_l$, which has to stay in ${\cal U}_2\cap D$. By the properness of $f$ and by the continuous dependence of Segre surfaces on the base points, it follows that $\wt{\xi}$ has to be close to one of the $a_j's$ when $\wt w$ is close to $w$; for otherwise we would have $E^*(w,w')\cap \overline {D}\cap\PP {\cal U}_2\not =\emptyset$. We let $\rho_2$ be a real analytic defining function of $\PP D'$ near $q=f(p)$, and we choose a coordinates system near $p$ so that the tangent space $T^{(1,0)}_{p} \PP D$ is the $z_1$-axis. Then $Q_{\wt w}$ is given by $z_2=h(z_1,\overline {\wt w})$ for some holomorphic function $h$. We claim that the condition that $f(Q_{\wt w}\cap D\cap {\cal U}_2)\supset \ _{\wt{w'}^*}Q'_{\wt w'}$ for $(\wt{w},\wt{w'})\approx (w,w')$, can be expressed by the equation $\rho_2(f(z),\overline {\wt w}')=0$ with $z\in Q_{\wt w}\cap O(a_j)$ for some $j$. Indeed, $\rho_2 (f(z),\overline {\wt w}')=0$ implies that $f(Q_{\wt w}\cap O(a_j))\subset Q'_{\wt 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\cap 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'\cap Q'_{\wt w'}$ is connected for $\wt{w'}\approx w'$, thus $f(Q_{\wt w}\cap O(a_j))=Q_{\wt w'}\cap O_j'$. Now, if we make $\wt{w'}$ close enough to $w^{'}$, then by the continuity of the ${\cal R}$-operator, $\wt{w}^{'*}\in f(Q_{\wt w}\cap 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$: $$\rho_2(f(z),\overline {\wt w}')=\sum_jb^l_j(\overline {\wt w},\overline {\wt w'})(z_1-a_l^{(1)})^j$$ Let $V_l$ be the common zeros of $b_j^{l}(\overline {\wt w},\overline {\wt w'})$ for $j=0,1,\cdots$. Then clearly near $(w,w')$, ${\cal V}$ is the finite union of $V_l$'s. This shows that ${\cal V}$ is a local variety. We next let $(w_v,w'_v)(\in {\cal V})\ra (w_0,w'_0)\in ({\cal U}_1\sm \overline D)\times ({\cal U}_1'\sm \overline {D}')$. We then show that $(w_0,w_0')\in {\cal V}$, too. Notice that $f(Q_{w_v}\cap {\cal U}_2)\supset\ _{{w'}_v^*}Q'_{w_v'}$. So, as argued before, there is a point $\xi_v\in Q_{w_v}\cap {\cal U}_2\cap D$ such that $f( _{\xi_{v}}Q_{w_v}) \supset_{w^{'*}_v}Q'_{w'_v}$. % Notice also that % $\xi_v\in R(w_v)\cap Q_{w_v}$. Assume, without loss of generality, that $\xi_0=\lim \xi_v$. % Then $\xi_0\in E^*(w_0,w'_0)\cap \overline {D}$. %(R(w_0)\cap Q_{w_0}\cap\overline {{\cal U}_2})\cup B(p)$. Letting $v\ra \infty$, since $\xi_0\in Q_{w_0}\cap \overline {\cal U}_2\cap {D}$ and since $f$ is locally proper near $\xi_0$, it follows that $f(O(\xi_0)\cap Q_{w_0}\cap (B_{\epsilon}(E_{w_0})\cup ({\cal U}_2\cap D)) \supset _{w_0^{'*}}Q'_{w_0'}$. Thus, by the above choice of ${\cal U}_2$, it follows that $\xi_0\in {\cal U}_2$. We therefore conclude that $f(Q_{w_0}\cap {\cal U}_2\cap D)\supset\ _{{w'}_0^*}Q'_{w_0'}$. This shows that $(w_0,w_0')\in ({\cal U}_1\sm \overline {D})\times ({\cal U}'_1\sm \overline {D'})$. Next, applying Lemma 7.2 and (7.b), we conclude that ${\cal V}$ has no limit points in $ ({\cal U}_1\sm \overline {D})\times \PP {\cal U}'_1\cup ({\cal U}_1\sm \overline {D})\times \Sigma'$. $\endpf$ \bigbreak We still keep the above notation and assume the hypothesis in Lemma 7.3. % Notice that $\pi$ is finite to one and Notice that $\cal V$ contains points of dimension 2 (see the observation made after Lemma 7.1). We see that $ {\cal V}$ is an analytic variety of dimension 2 in $\left(({\cal U}_1\sm {\overline D})\times {\cal U}'_1\right)\sm {A})$. Here $A=({\cal U}_1\sm \overline {D}) \times (T'_1\cup T'_0)$. Decompose $\cal 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 $\cal V$, for the brevity of notations. (The new $\cal V$ can be formed by considering all smooth points of dimension 2 in the old $\cal V$ and then adding new points by taking the closure). Now, we can say that ${\cal V}$ is an analytic variety in $(({\cal U}_1\sm \overline {D})\times {\cal U}'_1))\sm {A}$, which has pure dimension 2. Next, we let $\wt{A}=({\cal U}_1\sm \overline {D})\times \wt{T}(\supset A)$, which is a pluripolar set in $({\cal U}_1\sm \overline {D})\times {\cal U}'_1$ by (7.b). We also write %$A=({\cal U}_1\sm \overline {D})\times (T'_1\cup T'_0)$, $\wt{\cal V}={\cal V}\sm \left(({\cal U}_1\sm \overline {D})\times \wt{T})\right)$. We now think of ${\cal V}$ and $\wt{\cal V}$ as subsets in $({\cal U}_1\sm {\overline D})\times {\cal U}'_1$. Then, we see that $\wt {\cal V}$ is an analytic variety of pure dimension 2 in $\left(({\cal U}_1\sm {\overline D})\times {\cal U}'_1\right)\sm \wt {A}$. \bigbreak {\bf Lemma 7.4}: As before, let $\{{\cal 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\}\subset \Sigma$ with $p_j\ra p$, such that $Q_{p_j}\cap \PP D\cap {\cal U}_2\subset \Sigma$, then ${\cal V}$ extends as an analytic variety of pure dimension $2$ in ${\cal U}_1\times {\cal U}'_1$, after shrinking ${\cal U}_1$ and ${\cal U}'_1$. Thus, by Theorem 1.1, $f$ admits a holomorphic extension at $p$. \bigbreak {\it 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\in \PP D$ can be holomorphically extended to $D$ (near $p$) (see [BT] and [Tr]). We let $ z$ be a certain $p_j$ with $j\gg 1$. Slightly shrinking ${\cal U}_2$, we observe that $Q_{ z}\cap \PP D\cap \overline {\cal U}_2\subset \Sigma$. It thus follows that for a small neighborhood $G$ of $z$, one also has $Q_{\wt z}\cap \PP D\cap \overline {\cal U}_2\subset \Sigma$ for any $\wt z\in\overline { G}$. Now, we consider ${\cal V}^* =\left({\cal V}\cap ((G\sm \overline {D})\times {\cal U}'_1)\right)$. Then as did in Lemma 7.2, one sees that ${\cal V}^*$ has no limit points in $(G\sm \overline {D})\times \Sigma'$. We claim that ${\cal V}^*$ has no limit points in $(G\sm \overline {D})\times (T'_1\cup T'_0)$, neither. To see this, we let $(w_v,w_v') (\in {\cal V}^*)\ra (w_0,w_0')\in (G\sm \overline {D})\times (T'_1\cup T'_0)$. Then, as in Claim 7.2 $'$, one sees that for certain $\xi_0\in Q_{w_0}\cap \PP D\cap \overline {\cal U}_2$, it holds that $f(\xi_0)=w_0'$. However, by our choice of $G$, we know that $\xi_0\in\Sigma$. Thus, a similar argument as in Claim 7.2$'$ indicates that $w_0\in A_{\xi_0}\in\PP D$. This is a contradiction. Hence, from the above argument, it follows that $\overline {{\cal V}^*}\cap ((G\sm \overline {D})\times {\cal U}'_1)={\cal V}^*$ is an analytic variety of pure dimension 2 in $(G\sm \overline {D})\times {\cal U}'_1$ without limit points in ${\cal U}_1\times \PP {\cal U}'_1$. Now, we let $\wt{\cal V}^*={\cal V}^*\sm \wt{A}$. Then $\overline {\wt{\cal V}^*}\cap((G\sm \overline {D})\times{\cal U}'_1) = \overline {\wt{\cal V}}\cap((G\sm \overline {D})\times{\cal U}'_1)$. \bigbreak {\bf Claim 7.5}: $\overline {\wt{\cal V}^*}\cap((G\sm \overline {D})\times{\cal U}'_1) ={\cal V}^*$. Thus, $\overline {\wt{\cal V}}\cap((G\sm \overline {D})\times{\cal U}'_1) $ is an analytic variety of pure dimension 2 in $(G\sm\overline {D})\times {\cal U}'_1$. \bigbreak {\it Proof of Claim 7.5}: Let $\pi_{G}: {\cal V}^*\ra G$ and $\pi'_G:{\cal V}^*\ra {\cal U}'_1$. We first show that $\pi'_G$ is finite to one, after shrinking $G$ if necessary. To this aim, assume that $w'\in {\cal U}'_1$ and let $\{z_j\}$ be a sequence in $\pi_G({\pi'_G}^{-1}(w'))$ with $z_j\ra z_0(\in \overline {G})$. As in Lemma 7.3, we can then assume that for a certain sequence $\{\xi_j(\in Q_{z_j})\}$, it holds that $\xi_j(\in Q_{z_j}\cap\overline {\cal U}_2)\ra \xi_0\in Q_{z_0}\cap \overline {\cal U}_2\cap\overline {D}$, $f(\xi_j)\ra w'{}^*$, and $f(_{\xi_j}Q_{z_j}) \supset \ _{{w'}^*}Q'_{w'}$. As before, by our choice of $G$, if follows that there is a small neighborhood of $\xi_0$ which is mapped properly by $f$ to a neighborhood of $w'{}^*$. Thus, $f^{-1}(\ _{w'{}^*}Q_{w'})$ has only finitely many components near $\xi_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\in G$, we see that $\pi_{G}^{-1}(w')$ is a finite set and thus $\pi_{G}$ is a local analytic covering map. In particular, $\pi'_{G}$ is an open mapping. Next, we consider ${{\cal V}^*}\cap((G\sm \overline {D})\times \wt{T}) $. Then the above fact indicates that it does not contain any non-empty open subset of $ {\cal V}^*$. In fact, if this is not the case, $\pi'_{G}( {{\cal V}^*}\cap((G\sm \overline {D})\times\wt{T})\subset \wt{T} $ would contain open subsets of ${\cal U}'$, which contradicts the property of $\wt T$ (see (7.b)). Hence, we see that $\overline {\wt{\cal V}^*}\cap((G\sm \overline {D})\times{\cal U}'_1) ={\cal V}^*$. $\endpf$ \bigbreak Now, we can apply the Bishop extension lemma (7.h) to conclude that $\wt{\cal V}$ extends as an analytic variety of dimension 2 in $({\cal U}_1\sm \overline {D})\times {\cal U}'_1$, which has no limit points in ${\cal U}_1\times \PP {\cal U}'_1$. Let us still write the extended variety as ${\cal V}$. Since any compact analytic variety in $\bf C^2$ has to be a finite set, we can see, by using the above mentioned property of $\cal V$, that $\pi^{-1}(z)$ is finite for each $z\in {\cal U}_1\sm\overline {D}$, and %V$ has no limit points in %${\cal U}_1\times \PP {\cal U}'_1$, we %see that $\pi$ is proper. Hence, ${\cal V}$ can have only finitely many irreducible components, $\{{\cal V}\}_{j=1}^m$, (of dimension 2); and each of them has to be a sheeted analytic covering space over ${\cal U}_1\sm \overline {D}$ ([Wh]). So, each ${\cal V}_i$ can be defined by equations of the form ([Wh]): $$P_{ik}(w,w'_k)=w_k'{}^{N_{ik}}+\sum_{0\le j