%--------------------- CUT HERE ------------------------------ % % AUTHORS: GREGERY T. BUZZARD AND FRANC FORSTNERIC % % TITLE: A CARLEMAN TYPE THEOREM FOR PROPER HOLOMORPHIC EMBEDDINGS % % % Last edited: April 3, 1996 % % Plain TeX % % % % % This is the Macro file. % % \scrollmode \magnification=\magstep1 \parskip=\smallskipamount \advance\vsize-.25in \advance\voffset.25in \def\Id{\mathop{\rm Id}\nolimits} \headline={\ifnum\pageno>1\smaller\ifodd\pageno\hfill A CARLEMAN TYPE THEOREM FOR PROPER HOLOMORPHIC EMBEDDINGS \hfill\the\pageno \else \the\pageno\hfill\uppercase{Gregery T. Buzzard and Franc Forstneric}\hfill\fi\else\hss\fi} \footline={\hss} \def\demo#1:{\par\medskip\noindent\it{#1}. \rm} \def\ni{\noindent} % noindent \def\np{\hfill\break\indent} % new indented par in horizontal mode \def\npni{\hfill\break\noindent} % new unindented par in horizontal mode \def\ll{\leftline} \def\cl{\centerline} \def\rl{\rightline} % % \beginsection % \outer\def\beginsection#1\par{\bigskip \message{#1}\leftline{\bf\} \nobreak\smallskip\vskip-\parskip\noindent} % % \proclaim % \outer\def\proclaim#1:#2\par{\medbreak\vskip-\parskip \noindent{\bf#1.\enspace}{\sl#2} \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\ba{\medskip\rm} \def\endpr{\hfill $\spadesuit$ \medskip} \def\inner#1#2{\langle{#1},{#2}\rangle} % % % Roman capital characters % % \def\rD{{\rm D}} % roman D in math mode \def\rH{{\rm H}} \def\rT{{\rm T}} % roman T \def\rN{{\rm N}} % roman N in math \def\rH{{\rm H}} % % % Boldface capital characters % % \def\B{{\bf B}} \def\C{{\bf C}} \def\D{{\bf D}} \def\H{{\bf H}} \def\N{{\bf N}} \def\P{{\bf P}} \def\Q{{\bf Q}} \def\R{{\bf R}} \def\T{{\bf T}} \def\Z{{\bf Z}} \def\CP{{\bf CP}} \def\GL{{\bf GL}} \def\U{{\bf U}} \def\SU{{\bf SU}} % \def\O{{\bf O}} \def\SO{{\bf SO}} \def\SL{{\bf SL}} \def\CP{{\bf CP}} \def\Sp{{\bf Sp}} % % % Caligraphic capital characters % % \def\cA{{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}} \def\cD{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}} \def\cG{{\cal G}} \def\cH{{\cal H}} \def\cI{{\cal I}} \def\cJ{{\cal J}} \def\cK{{\cal K}} \def\cL{{\cal L}} \def\cM{{\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}} \def\cP{{\cal P}} \def\cQ{{\cal Q}} \def\cR{{\cal R}} \def\cS{{\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}} \def\cV{{\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}} \def\cY{{\cal Y}} \def\cZ{{\cal Z}} % % % Small Greek letters in Math mode % \def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\epsilon} \def\z{\zeta} \def\i{\iota} \def\k{\kappa} \def\l{\lambda} \def\r{\rho} \def\s{\sigma} \def\u{\upsilon} \def\c{\chi} %\def\o{\omega} \def\t{\theta} \def\x{\xi} \def\S{{\Sigma}} \def\L{\Lambda} %\def\O{\Omega} % % % Miscellaneous % % \def\bar{\overline} % conjugate \def\bs{\backslash} % backslash \def\disc{\triangle} % disc \def\cdisc{\bar\triangle} % closed disc \def\di{\partial} % partial derivative \def\dibar{\bar\partial} % di-bar derivative % % Abbreviations % \def\dim{{\rm dim}\,} % dimension \def\holo{holomorphic} % holomorphic \def\auto{automorphism} % automorphism \def\homo{homomorphism} % homomorphism \def\as{analytic subset\ } % analytic subset \def\homeo{homeomorphism} % homeomorphism \def\nbd{neighborhood} % neighborhood \def\psc{pseudoconvex} % pseudoconvex \def\spsc{strongly\ pseudoconvex} % strongly psc \def\ra{real-analytic} % real-analytic \def\psh{plurisubharmonic} % plurisubharmonic \def\spsh{strongly\ plurisubharmonic} \def\tr{totally real} % totally real \def\pc{polynomially convex} % polynomially convex \def\hc{holomorphically convex} % holo convex \def\hf{holomorphic\ function} % holomorphic function \def\ss{\subset\!\subset} % relatively compact subset \def\supp{{\rm supp}\,} % support \def\cne{$\C^n$-equivalent} % C^n equivalent \def\wlog{without\ loss\ of\ generality\ } \def\tcp{This\ concludes\ the\ proof\ of\ } \def\iff{if and only if} \def\hvf{holomorphic vector field} \def\vf{vector field} \def\div{{\rm div}} \def\phe{proper holomorphic embedding} \def\ha{holomorphic automorphism} \def\hra{\hookrightarrow} \def\dist{{\rm dist}} \def\Aut{{\rm Aut}} % roman Aut in math mode \def\Autsp{{\rm Aut}_{\rm sp}} % sypmlectic automorphisms \def\begin{\ll{}\vskip 10mm \nopagenumbers} % beginning of the paper \def\pn{\footline={\hss\tenrm\folio\hss}} % pagenumbers at bottom \def\ii#1{\itemitem{#1}} \begin \cl{\bf A CARLEMAN TYPE THEOREM}\medskip \cl{\bf FOR PROPER HOLOMORPHIC EMBEDDINGS} \bigskip\medskip \cl{\bf Gregery T. Buzzard\footnote{} {\rm \hskip-\parindent Buzzard's research at MSRI is supported in part by NSF grant DMS-9022140. Forstneric is supported in part by an NSF grant and by a grant from the Ministry of Science of the Republic of Slovenia.} and Franc Forstneric} \font\small=cmr9 \font\smaller=cmr8 \font\ninecsc=cmcsc9 \midinsert \narrower\narrower \cl{\bf Abstract} \baselineskip=10pt {\small In 1927, Carleman showed that a continuous, complex-valued function on the real line can be approximated in the Whitney topology by an entire function restricted to the real line. In this paper, we prove a similar result for proper holomorphic embeddings. Namely, we show that a proper $\cC^r$ embedding of the real line into $\C^n$ can be approximated in the strong $\cC^r$ topology by a proper holomorphic embedding of $\C$ into $\C^n$. } \endinsert \baselineskip=\normalbaselineskip % % % \beginsection 1. Introduction We denote the complex numbers by $\C$ and the real numbers by $\R$. Let $\C^n$ be the complex Euclidean space of dimension $n$ with complex coordinates $z=(z_1,\ldots,z_n)$. To motivate our main result we recall the Carleman approximation theorem [3], [10]: {\it For each continuous function $\l\colon \R\to\C$ and positive continuous function $\eta\colon \R\to (0,\infty)$ there exists an entire function $f$ on $\C$ such that $|f(t)-\l(t)|<\eta(t)$ for all $t\in \R$.} This extends the Weierstrass approximation theorem in which the approximation takes place on compact intervals in $\R$. If $\l$ is smooth, we can also approximate its derivatives by those of $f$. A more general result was proved by Arakelian [1] (see [13] for a simple proof). Our main result is an extension of Carleman's theorem to \it \phe s \rm of $\C$ into $\C^n$ for $n>1$: \proclaim 1.1 Theorem: Let $n>1$ and $r\ge 0$ be integers. Given a proper embedding $\l\colon \R \hra \C^n$ of class $\cC^r$ and a continuous positive function $\eta\colon \R \to (0, \infty)$, there exists a \phe\ $f\colon \C \hra \C^n$ such that $$ |f^{(s)}(t) - \l^{(s)}(t)| < \eta(t), \quad t \in \R, \quad 0 \le s \le r. $$ If in addition $T = \{t_j\} \subset \R$ is discrete, there exists $f$ as above such that $$ f^{(s)}(t) = \l^{(s)}(t), \quad t \in T, \quad 0 \le s \le r. $$ Here $f^{(s)}(t)$ denotes the derivative of $f$ order $s$, and similarly for $\l$. We emphasize that, in general, one can not expect to extend $\l$ to a holomorphic embedding of $\C$ into $\C^n$, even if $\l$ is \ra. In this case $\l$ will extend holomorphically to some open set in $\C$, but in general not to all of $\C$; and even if $\l$ extends to $\C$, the extension need not be a proper map into $\C^n$. So our result is the best possible in this context. Another motivation for Theorem 1.1 are the recent developments on embedding Stein manifolds in $\C^n$; see the papers [2], [6], [7], [8]. (For Stein manifolds and other topics in several complex variables mentioned here we refer the reader to H\"ormander [11].) In these papers it was shown that a Stein manifold $M$ which admits a \phe\ in $\C^n$ for some $n>1$ also admits an embedding $f\colon M\hra \C^n$ whose image $f(M)$ contains a given discrete subset $E\subset \C^n$ (with prescribed finite order jet of $f(M)$ at any point of $E$), and such that $f(M)$ intersects the image of any entire map $g\colon \C^d\to\C^n$ of rank $d=n-\dim M$ at infinitely many points [6, Theorem 5.1]. This implies in particular that $f(M)$ cannot be mapped into any hyperplane in $\C^n$ by an automorphism of $\C^n$. In this context we recall that any Stein manifold $M$ embeds in $\C^n$ for $n>(3\,\dim M+1)/2$ according to Eliashberg and Gromov [4]. In light of these results it is a natural question whether one can in some sense prescribe the embedding $M\hra\C^n$ on a positive dimensional submanifold $N\subset M$. Our result provides an affirmative answer in the simplest case when $M=\C$ and $N=\R\times \{i0\} \subset \C$. The details of our construction are considerable even in this simplest case, and the full scope of the method remains to be seen. If $\l$ is of class $\cC^\infty$, our method can be modified so that we approximate to increasingly high order on complements of compact subsets of $\R$. We shall not go into details of this. Another possible extension is to approximate a proper smooth embedding by a \phe\ on a finite set of real lines (or certain other real curves) in $\C$. A seemingly more difficult question is whether a result of this type holds when $N$ is a {\it complex\/} submanifold of positive dimension in a complex manifold $M$. For instance, \it does every \phe\ $\l\colon \C\hra \C^n$ for $n\ge 3$ extend to a \phe\ $f\colon \C^2 \hra\C^n$ ? \rm One of the main tools in our construction is the following result from [9]. The case $r=0$ was obtained earlier in [8]. This result can also be obtained by methods in [5]. % % % MOVING HAIR % % \proclaim 1.2 Proposition: Let $K \subset \subset \C^n$ ($n \ge 2$) be a compact, polynomially convex set, and let $C \subset \C^n$ be a smooth embedded arc of class $\cC^\infty$ which is attached to $K$ in a single point of $K$. Given a $\cC^\infty$ diffeomorphism $F\colon K \cup C \to K \cup C' \subset \C^n$ such that $F$ is the identity on $(K \cup C) \cap U$ for some open neighborhood $U$ of $K$, and given numbers $r\ge 0$, $\e>0$, there exist a neighborhood $W$ of $K$ and an automorphism $\Phi \in \Aut\C^n$ satisfying $$ \|\Phi - \Id\|_{\cC^r(W)} < \e, \qquad \|\Phi-F\|_{\cC^r(C)} < \e. $$ (Here $\Id$ denotes the identity map.) Moreover, for each finite subset $Z \subset K \cup C$ we can choose $\Phi$ such that it agrees with the identity to order $r$ at each point $z \in Z \cap K$ and $\Phi|_{C}$ agrees with $F$ to order $r$ at each point of $Z \cap C$. The same result holds with any finite number of disjoint hairs attached to $K$. We explain our notation. $\Delta_\rho$ is the closed disk in $\C$ of radius $\rho$ and center $0$. $\B$ is the open unit ball in $\C^n$ with center $0$, and $R\B$ is the ball of radius $R$. For a set $A \subset \C^n$ and $\rho>0$, let $A + \rho\overline{\B} = \{ a+z \colon a \in A, |z| \le \rho\}$. We identify $\C$ and $\R$ with their images in $\C^n$ under the embedding $\z \to (\z, 0, \ldots, 0)$. For $1\le j\le n$ we denote by $\pi_j$ the coordinate projection $\pi_j(z_1,\ldots,z_n)=z_j$. In the proof we shall use special automorphisms of $\C^n$ of the form $$ \Psi(z)=z+f(\pi z)v,\quad z\in\C^n,$$ where $v\in\C^n$, $\pi\colon \C^n\to \C^k$ is a linear map for some $k \max\{|P(w)|\colon w\in A\}$. We refer the reader to [11] for properties of such sets. \proclaim 2.2 Lemma: Let $A \subset \C^n$ be compact and \pc\ and $\rho>0$. Let $I \subset \R$ be an interval whose endpoints lie in $\C^n \bs (A \cup \Delta_\rho)$, and let $r, \e>0$. Then there exists an automorphism $\Psi(z) = z + g(z_1) e_2$ of $\C^n$ such that \item{(i)} $|\Psi(z) - z| < \e$ for $z \in \Delta_\rho$, \item{(ii)} $\|\Psi|_\R (t) - t\|_{\cC^r(I)}< \e$, and \item{(iii)} $\Psi(t) \notin A$ for $t \in \overline{\R\bs I}$. \npni If $Z \subset I$ is finite, we can choose $\Psi$ as above so that $g^{(s)}(t) = 0$ for $t \in Z$, $0 \le s \le r$. \demo Proof: Let $\mu_1< \mu_2$ denote the endpoints of $I$ in $\R$, and let $\Gamma_j = \{(\mu_j, \zeta, 0, \ldots, 0)\colon \zeta \in \C\}$ for $j=1, 2$. Let $R> \max\{|\mu_1|, |\mu_2|\}+1$ such that $A\subset R\B$. Consider the set $E_j =A \cap \Gamma_j$. Since $A$ is \pc, $E_j$ is \pc\ in $\Gamma_j$ and hence $\Gamma_j\bs E_j$ is connected. Since the endpoints of $I$ lie in $\Gamma_j\bs E_j$, there exists a smooth curve $\gamma_j \colon [0,1] \to \Gamma_j\bs E_j$ with $\gamma_j(0) = (\mu_j, 0, \ldots, 0)$ and $|\pi_2\gamma_j(1)|>R+1$ for $j=1,2$. Since $A$ is compact, there exists $\d>0$ such that $\gamma_j([0,1]) + 3 \d \bar{\B} \subset \C^n \bs A$. Let $\pi_2(z) = z_2$. Let $K=\{x+iy\in \C \colon \mu_1-\d/2 \le x\le \mu_2+\d/2,\ |y|\le \rho+1\}$. Define a function $h\colon K\cup [-R,R] \to \C$ by $$ h(t) = \cases{ \pi_2 \gamma_{1}(1)&if $t \in [-R, \mu_{1}-2\d]$; \cr \pi_2 \gamma_{1}((\mu_1-\d-t)/\d)&if $t \in [\mu_1-2\d, \mu_1-\d]$; \cr \pi_2 \gamma_{2}((t-\mu_2-\d)/\d)& if $t \in [\mu_2+\d,\mu_2+2\d]$; \cr \pi_2 \gamma_{2}(1)& if $t \in [\mu_2+2\d, R]$; \cr 0& otherwise.} $$ Choose $\eta$, $0< \eta < \min\{\e,\d\}$. By Mergelyan's theorem [14, p.386] there is an entire function $g$ on $\C$ such that $|h(z)-g(z)|< \eta$ for $z\in K\cup [-R,R]$. The shear $\Psi(z) = z + g(z_1) e_2$ then satisfies (i) and (iii). Since $I \subset {\rm Int}K$, Cauchy's estimates imply that it also satisfies (ii) provided that $\eta>0$ is chosen sufficiently small. The last condition on $g$ is a trivial addition to Mergelyan's theorem. \endpr \medskip \proclaim 2.3 Lemma: Let $\l\colon\R \hra \C^n$ be a proper, $\cC^\infty$ embedding, $K \subset \C^n$ compact, $\e>0$, and $r\in\Z_+$. Let $Z \subset \R$ be finite, and suppose $\l(t) \in \C = \C \times \{0\}^{n-1}$ for each $t \in Z$. Then there exists a shear $\Psi(z) = z + h(z_1) v$ for some $v \in \C^n$ with $\pi_1 v = 0$ such that \item{(i)} $\Psi(\C) \cap \l(\R) = \l(Z)$, \item{(ii)} $|\Psi(z) - z| < \e$ for $z \in K$, and \item{(iii)} $\Psi(z) = z + O(|z-\l(t)|^{r+1})$ as $z \to \l(t)$, for all $t \in Z$. \demo Proof: Let $Z=\{t_j\}_{j=1}^s$. Define a polynomial on $\C$ by $h(\z) = \Pi_{1\le j\le s}(\z - \pi_1 \l(t_j))^{r+1}$. Consider the map $\Phi\colon \C \times \C^{n-1} \to \C^n$ given by $$ \Phi(z_1, \alpha_2, \ldots, \alpha_n) = (z_1, 0, \ldots,0) + h(z_1)(0, \alpha_2, \ldots, \alpha_n). $$ Clearly $\Phi$ is an automorphism of $(\C\bs \l(Z))\times\C^{n-1}$. Let $\Delta_{R,j}$ denote the closed disk of radius $R$ in $\C$ with center $\pi_1 \l(t_j)$ for $j=1,2,\ldots,s$. Choose $R >0$ such that the discs $\Delta_{R,j}$ for $1\le j\le s$ are pairwise disjoint. Choose a $\rho$, $0<\rho0$, and $T \subset \R$ discrete. Suppose that $\l(t), f(t) \notin K$ for $t \in \overline{\R\bs I}$. Then there exists $\Phi \in \Aut\C^n$ such that if $g = \Phi \circ f$, then \item{(i)} $|g^{(s)}(t) - \l^{(s)}(t)| < \e$ for $t \in [-a,a]$, $0 \le s \le r$, \item{(ii)} $g^{(s)}(t) = \l^{(s)}(t)$ for $t \in T \cap [-a,a]$, $0\le s \le r$, and \item{(iii)} $|\Phi(z) - z| < \e$ for $z \in K$. \demo Proof: We may assume that $I \subset (-a,a)$. Let $I_1$, $I_2$ be the two connected components of $\{\z \in I\colon f(\z) \in \C^n \bs K\}$ containing the respective endpoints of $I$, and let $I_0 = I \bs (I_1 \cup I_2)$. Let $A$ be the polynomial hull of $K \cup f(I_0)$. Then $A$ is the union of $K \cup f(I_0)$ and the bounded connected components of $f(\C)\bs (K \cup f(I_0))$. Note that $f(I_1)$ and $f(I_2)$ lie in $f(\C) \bs A$ since $f(t)\notin K$ for all $t\in \bar{\R\bs I}$. Let $L = A \cup f([-a,a])$. Then $C = \overline{L \bs A}$ is the union of two embedded arcs, each containing an endpoint of $f([-a,a])$. Define $F$ on $L$ by $F(z) = z$ if $z \in A$, and $F(z) = \l f^{-1}(z)$ if $z \in f([-a,a])$. Then $F$ is a $\cC^\infty$ diffeomorphism of $L$ which extends as the identity map on $(A \cup C) \cap U$ for some neighborhood $U$ of $A$. Apply proposition 1.2 to get $\Phi \in \Aut \C^n$ such that $|\Phi(z)-z| < \e$ for $z\in K$ and such that $g = \Phi \circ f$ satisfies (i) and (ii). \endpr % % % % % \beginsection 3. Proof of theorem 1.1. Choose a smooth cutoff function $\chi$ on $\R$ such that $\chi(t)=1$ for $|t|$ small and $\supp\chi \subset (-1,1)$. Define the constant $C=C_r > 1$ such that $\|\chi h\|_{\cC^r} \leq C \|h\|_{\cC^r}$ for each $h\in \cC^r(\R)$. We fix such $C$ for the entire proof. By approximation we may assume that $\l\colon \R \hra \C^n$ in theorem 1.1 is a proper $\cC^\infty$ embedding. Decreasing $\eta$ if necessary we may also assume that $\eta$ satisfies lemma 2.1 for $\l$ and $\eta(t) < 1/2$ for all $t\in \R$. We use an inductive procedure to obtain a sequence of \phe s $f_k\colon \C\hra\C^n$ such that $f=\lim_{k\to\infty} f_k$ exists on $\C$ and satisfies theorem 1.1. Each $f_k$ will be a restriction to $\C=\C\times\{0\}^{n-1}$ of a holomorphic automorphism of $\C^n$. The next map $f_{k+1}$ will be of the form $f_{k+1}=\Phi_{k+1}\circ f_k \circ \Psi_{k+1}$ for suitably chosen $\Phi_{k+1},\Psi_{k+1} \in\Aut\C^n$. We will describe the case $k=1$ after the inductive step is given. Recall that $\Delta_k$ is the closed disc in $\C=\C\times\{0\}^{n-1}$ with center $0$ and radius $k$, and $\B$ is the unit ball in $\C^n$. For the induction at step $k$, suppose we have the following: \item{(a)} closed balls $B_j = R_j \bar \B \subset \C^n$ with $R_j \ge \max\{j+1, R_{j-1}+1\}$, $j= 1, \ldots, k$, \item{(b)} automorphisms $\Phi_1, \ldots, \Phi_k$ of $\C^n$ with $|\Phi_j(z) - z|< 2^{-j}$ for $z \in B_{j-1}$, $j = 2, \ldots, k$, \item{(c)} numbers $\e_j>0$ such that $\e_1< 2^{-1}$ and $\e_j < \e_{j-1}/2 < 2^{-j}$ for $j = 1, \ldots, k$, \item{(d)} automorphisms $\Psi_1, \ldots, \Psi_k$ of $\C^n$ of the form $\Psi_j(z)= z + g_j(z_1) e_2 + h_j(z_1)v_j$, where $\pi_1(v_j) = 0$ and $|\Psi_j(z) - z|< \e_j$ for $|z| \le j$, \item{(e)} closed intervals $I_j = [-a_j, a_j]$, $j = 1, \ldots, k$, with $a_j > \max\{a_{j-1}+2, j+2\}$, and \item{(f)} numbers $0<\d_j < C^{-1} \inf\{\eta(t)\colon t \in I_j\}$, $j = 1, \ldots, k$, \noindent such that the automorphism $$ f_k = \Phi_k \circ \cdots \Phi_1 \circ \Psi_1 \circ \cdots \circ \Psi_k \in\Aut\C^n $$ (whose restriction to $\C=\C\times \{0\}^{n-1}$ provides an embedding $\C\hra\C^n$) satisfies: \item{($1_k$)} $f_k(\Delta_j + \e_k\bar\B) \subset {\rm Int}B_j$ \ for $j = 1, \ldots, k$, \item{($2_k$)} $|f_k^{(s)}(t) - \l^{(s)}(t)| < \eta(t)$ for $t \in I_k$ and $0 \le s \le r$, \item{($3_k$)} $|f_k^{(s)}(t) - \l^{(s)}(t)| < \d_k$ for $t \in I_k \bs (-a_k+1, a_k-1)$ and $0 \le s \le r$, \item{($4_k$)} $f_k^{(s)}(t)=\l^{(s)}(t)$ for $t \in T \cap I_k$, $0 \le s \le r$, \item{($5_k$)} $f_k(\C) \cap \l(\R) = \l(T \cap I_k)$, \item{($6_k$)} $|\l(t)| > R_k+1$ for $|t|\ge a_k-1$, \item{($7_k$)} $|f_k(t)| > R_k$ for $|t|\ge a_k-1$. We will now show how to obtain these hypotheses at step $k+1$. Let $I_k^1$ and $I_k^2$ be the two connected components of the set $\{\z \in I_k\bs \Delta_{k+1} \colon |f_k(\z)| > R_k\}$ containing the respective endpoints of the interval $I_k$. Let $I_k^0=I_k\bs(I_k^1 \cup I_k^2)$ be the middle interval. By $(7_k)$ we have $I_k^0 \subset (-a_k+1,a_k-1)$. Let $K_k$ be the polynomial hull of the set $B_k \cup f_k(\Delta_{k+1} \cup I_k^0)$. Since $f_k(\C)$ is a complex submanifold of $\C^n$ and $B_k$ is \pc, it is seen easily that $K_k$ is contained in $B_k\cup f_k(\C)$, and it is the union of $B_k \cup f_k(\Delta_{k+1} \cup I_k^0)$ and the bounded connected components of the complement $f_k(\C)\bs (B_k \cup f_k(\Delta_{k+1} \cup I_k^0))$ (see Lemma 5.4 in [6]). Note that $(7_k)$ and (e) imply that $f_k(\R\bs (-a_k+1, a_k-1)) \subset \C^n \bs K_k$. Choose $R_{k+1} > R_k + 1$ such that $K_k \subset (R_{k+1} -1)\B$, and let $B_{k+1} = R_{k+1} \bar \B$. Choose $a_{k+1}>a_k +2$ to get $(6_{k+1})$. We now want to approximate $\l$ on the larger interval $I_{k+1} = [-a_{k+1}, a_{k+1}]$ by the image of the next embedding $\C\hra\C^n$ (to be constructed). In order to apply lemma 2.4 we first approximate $\l$ as follows: \proclaim 3.1 Lemma: There exists a proper $\cC^\infty$ embedding $\l_k\colon\R \hra \C^n$ satisfying \item{(i)} $\l_k = f_k$ on $[-a_k+1, a_k-1]$, \item{(ii)} $\l_k = \l$ on $\R\bs I_k$, \item{(iii)} $|\l_k^{(s)}(t) - \l^{(s)}(t)| < \eta(t)$ for $t \in I_k \bs (-a_k+1, a_k-1)$, $0 \le s \le r$, \item{(iv)} $\l_k^{(s)}(t) = \l^{(s)}(t)$ for $t \in T$, $0\le s\le r$, and \item{(v)} $\l_k(t)\notin K_k$ when $|t|\ge a_k-1$. \demo Proof: We define the cutoff function $\chi_k$ on $\R$ using $\chi$, so that $\chi_k =1$ on $[-a_k+1, a_k-1]$, $\chi_k =0$ on $\R \bs I_k$, and $\|\chi_k h\|_{\cC^r} < C \|h\|_{\cC^r}$ as before. Let $$ \hat{\l}_k(t) = f_k(t) \chi_k(t) + \l(t) (1-\chi_k(t)), \quad t\in \R. $$ By lemma 2.1, $(3_k)$, $(4_k)$, and choice of $\eta$ and $\d_k$, we see that (i)--(iv) are satisfied for $\hat{\l}_k$ in place of $\l_k$. To obtain (v) we use a transversality argument to perturb $\hat{\l}_k$ on $I_k\bs (-a_k+1,a_k-1)$. First note that if $|t|> a_k$, then $|\hat{\l}_k(t)| = |\l(t)| > R_k+1$ by $(6_k)$, so $\hat{\l}_k(t) \notin B_k$. Also, by $(5_k)$, we see that $\hat{\l}_k(t) \notin f_k(\C)$, so $\hat{\l}_k(t) \notin K_k$. Next, if $t \in T \cap (I_k \bs (-a_k+1, a_k-1))$, then by $(4_k)$, $(7_k)$, and (e) we see that $\hat{\l}_k(t) = f_k(t) \notin K_k$. Hence there exists a neighborhood $V$ of $T \cap (I_k \bs (-a_k+1, a_k-1))$ such that $\hat{\l}_k(\overline{V}) \cap K_k = \emptyset$. Thus we need only perturb $\hat{\l}_k$ on $I_k \bs (V \cup (-a_k+1, a_k-1))$ to get (v). Note that if $t \in I_k \bs (-a_k+1, a_k-1)$, then from $(6_k)$ and $(2_k)$ we see that $|\hat{\l}_k(t)| > R_k+1/2$, so $\hat{\l}_k(t) \notin B_k$. Finally, a simple transversality argument implies that we can make an arbitrarily small $\cC^\infty$ perturbation of $\hat{\l}_k$ to avoid $f_k(\C)$, and hence we get $\l_k$ with $\l_k = \hat{\l}_k$ outside $I_k \bs (V \cup (-a_k+1, a_k-1))$ and $\l_k$ satisfying (i)--(v). \endpr Now we can use lemma 2.4 to approximate $\l_k$, hence to approximate $\l$. Set $$ \eqalign{ \d_{k+1} &= \min\{\eta(t)\colon t \in I_{k+1}\}/(2C), \cr \sigma_{k+1} &= \min\{\eta(t) - |\l_k^{(s)}(t) - \l^{(s)}(t)| \colon t \in I_{k+1}, 0 \le s \le r\} >0. \cr} $$ Choose $\e>0$ so small that $$ \e < \min\{ 2^{-(k+1)}, \d_{k+1}, \sigma_{k+1} \}, \qquad f_k(\Delta_j + \e_k \bar{\B}) + \e \bar{\B} \subset {\rm Int}B_j, \ \ 1\le j\le k. $$ Apply lemma 2.4 with $\l = \l_k$, $f = f_k$, $I = [-a_k+1, a_k-1]$, $K = K_k$, $a = a_{k+1}$, $r$ and $T$ unchanged, and $\e$ as above. This provides $\Phi_{k+1} \in \Aut\C^n$ and $G = \Phi_{k+1} \circ f_k \in \Aut\C^n$ satisfying $$ \cases{ |\Phi_{k+1}(z) - z|< \e & for $z \in K_k$, hence on $B_k$; \cr |G^{(s)}(t) - \l_k^{(s)}(t)| < \e & for $t \in I_{k+1}$, $0 \le s \le r$; \cr G^{(s)}(t)=\l_k^{(s)}(t) & for $t \in T \cap I_{k+1}$, $0 \le s \le r$. \cr} $$ In particular, $(2_{k+1})$--$(4_{k+1})$ hold with $G$ in place of $f_{k+1}$. Since $f_k(\Delta_{k+1}) \subset K_k \subset (R_{k+1}-1)\B$, we can choose $\e_{k+1}'< \e_k$ small enough that $(1_{k+1})$ holds with $G$ in place of $f_{k+1}$ and $\e_{k+1}'$ in place of $\e_{k+1}$, and such that if $\psi \in \Aut\C^n$ with $\|\psi(t) - t\|_{\cC^r(I_{k+1})} < \e_{k+1}'$, then $(2_{k+1})$ and $(3_{k+1})$ hold with $G \circ \psi$ in place of $f_{k+1}$. Let $\e_{k+1} = \e_{k+1}'/2$. Then with $G$ in place of $f_{k+1}$, we have $(1_{k+1})$--$(4_{k+1})$, $(6_{k+1})$, and $G(-a_{k+1}), G(a_{k+1}) \notin B_{k+1}$ by $(6_{k+1})$ and $(2_{k+1})$. \medskip Next we want to obtain $(7_{k+1})$. We do this using lemma 2.2 to change the embedding so that the image of $\R \bs I_{k+1}$ misses $B_{k+1}$ while leaving the embedding essentially unchanged on $\Delta_{k+1} \cup I_{k+1}$. Apply lemma 2.2 with $A = G^{-1}(B_{k+1})$, $\rho = k+1$, $I=I_{k+1}$, $r$ unchanged, $Z = T \cap I_{k+1}$ and $\e=\e_{k+1}/2$. This gives a shear $$ \psi_{k+1}(z) = z + g_{k+1}(z_1) e_2 $$ with $$ \cases{ |\psi_{k+1}(z) - z| < \e_{k+1}/2,\ \ z \in \Delta_{k+1}; \cr \|\psi_{k+1}|_\R(t) - t \|_{\cC^r(I_{k+1})} < \e_{k+1}/2; \cr g_{k+1}^{(s)}(t) = 0, \ \ t \in T \cap I_{k+1}, 0 \le s \le r; \cr \psi_{k+1}(t) \notin G^{-1}(B_{k+1}), \ \ t \in \bar{\R \bs I_{k+1}}. \cr} $$ Let $H = G \psi_{k+1}$. Then with $H$ in place of $f_{k+1}$, we have $(1_{k+1})$--$(4_{k+1})$, $(6_{k+1})$, and $(7_{k+1})$. For the final correction, we use lemma 2.3 to obtain $(5_{k+1})$ while maintaining the other properties. Let $R > a_{k+1}$ be such that $A=G^{-1}(B_{k+1}) \subset R\B$. Let $\d>0$ be such that $$ \psi_{k+1} \left( [-R,R]\bs (-a_{k+1},a_{k+1}) + \d \bar\B \right) \cap A = \emptyset, $$ and such that if $\theta \in \Aut\C^n$, with $|\theta(z) - z| < \delta$ on $R \overline{\B}$, then $$ \|\psi_{k+1} \theta|_\R(t) - t\|_{\cC^r(I_{k+1})} < \e_{k+1}. \eqno(1) $$ Apply lemma 2.3 with $\l$ replaced by $H^{-1} \l$, $K=R\bar \B$, $r$ unchanged, $Z = T \cap I_{k+1}$, and $\e = \min\{\d, \e_{k+1}/2\}$. This gives a shear $\theta_{k+1}(z) = z + h_{k+1}(z_1) v_{k+1}$ with $\pi_1 v_{k+1} = 0$ such that $$ \cases{ |\theta_{k+1}(z)-z| < \min\{\d, \e_{k+1}/2\},\ \ z \in\Delta_{k+1}; \cr \theta_{k+1}(\C) \cap H^{-1} \l(\R) = H^{-1} \l(T \cap I_{k+1}); \cr h_{k+1}^{(s)}(t) = 0,\ \ t \in T \cap I_{k+1},\ 0 \le s \le r,} $$ and such that (1) holds with $\theta = \theta_{k+1}$. Also, by the choice of $R$ and $\d$, $$ \psi_{k+1} \theta_{k+1}(\bar{\R \bs I_{k+1}}) \cap A = \emptyset. $$ Taking $\Psi_{k+1} = \psi_{k+1} \theta_{k+1}$ and $$ f_{k+1} = H\circ \theta_{k+1}= \Phi_{k+1}\circ f_k\circ \Psi_{k+1} $$ we obtain $(5_{k+1})$ and preserve the remaining hypotheses. Hence we obtain $(1_{k+1})$--$(7_{k+1})$. Note that $(k+1)\B \subset B_{k+1}$ so we also obtain (a)--(f), thus finishing the inductive step. \medskip The case $k=1$ is similar to the general step. First apply proposition 1.2 with $K = \emptyset$, $C = [-3,3] \subset \C$, $F = \l$, $\e = C^{-1} \inf\{\eta(t) \colon t \in [-3,3]\}$, and $Z = T \cap [-3,3]$ to get $\phi_1^1 \in \Aut\C^n$ satisfying the conclusions of that proposition. Choose $R_1 \ge 2$ such that $\phi_1^1(\Delta_1) \subset (R_1-1)\B$, choose $a_1 > 4$ to get $(6_1)$, and let $I_1 = [-a_1, a_1]$. Choose $\d_1$ to satisfy (f) for $j=1$. Define a proper $\cC^\infty$ embedding $\l_0$ as in lemma 3.1 so that (i)--(v) are satisfied with $\l_0$ in place of $\l_k$, $\phi_1^1$ in place of $f_k$, $3$ in place of $a_k$, $[-3,3]$ in place of $I_k$, and $\phi_1^1(\Delta_1)$ in place of $K_k$. Apply lemma 2.4 with $\l = \l_0$, $f = \phi_1^1$, $I = [-2,2]$, $K = \phi_1^1(\Delta_1)$, $a = a_1$, $\e = \delta_1$, and $T$ and $r$ unchanged. This gives $\phi_1^2 \in \Aut\C^n$ such that $$ |\phi_1^2(z) - z| < \d_1 \le 1/2,\ \ z \in \phi_1^1(\Delta_1), $$ and such that $\Phi_1 = \phi_1^2 \phi_1^1$ satisfies $$ \cases{ \|\Phi_1 - \l_0\|_{\cC^r(I_1)} < \e; \cr \Phi_1^{(s)}(t) = \l_0^{(s)}(t),\ \ t \in T \cap I_1,\ 0 \le s \le r.} $$ As before, we can apply lemmas 2.2 and 2.3 to obtain $\e_1>0$ and a shear $\Psi_1$ such that the hypotheses $(1_1)$--$(7_1)$ hold for $f_1 = \Phi_1 \Psi_1$, and (a)--(f) hold for $k=1$. This completes the base case. \medskip To finish the proof of theorem 1.1, note that $$ \Psi_1 \cdots \Psi_k(z) = z + \sum_{j=1}^k (g_j(z_1) e_2 + h_j(z_1) v_j) $$ and that (c) implies $$ |g_j(z_1) e_2 + h_j(z_1) v_j| < 2^{-j},\quad |z_1|1$. By $(1_k)$ we have $$ \Psi_1 \cdots \Psi_k(\Delta_{k-1} + \e_k \bar \B) \subset (\Phi_k \cdots \Phi_1)^{-1}(B_{k-1}). $$ Since $|\Psi_j(z) - z| < \e_j$ on $\Delta_{k-1}$ for $j \ge k$, and $\sum_{j=k+1}^\infty \e_j < \e_k$ by (c), we see that $$ \lim_{m \to \infty} \Psi_{k+1} \cdots \Psi_m(z) \in \Delta_{k-1} + \e_k \bar \B $$ for each $z \in \Delta_{k-1}$. Hence $$ \Psi(\Delta_{k-1}) \subset (\Phi_k \cdots \Phi_1)^{-1}(B_{k-1}) \subset \Omega, \qquad k>1, $$ so the claim holds. In particular, $\Phi \Psi \colon \C \to \C^n$ is a \phe. Finally, using the conditions $(1_k)$--$(7_k)$, we see that $\Phi \Psi \colon \C \hra \C^n$ is a proper holomorphic embedding with the desired properties. \endpr %------------------------------------------------------ % % % REFERENCES % % \bigskip \ni\bf References. \rm \medskip \item{1.} Arakelian, N.\ U.: Uniform approximation on closed sets by entire functions (Russian). {\it Izv.\ Akad.\ Nauk SSSR} {\bf 28}, 1187--1206 (1964) \item{2.} Buzzard, G., Forn\ae ss, J.E.: An embedding of $\C$ into $\C^2$ with hyperbolic complement. {\it Math.\ Ann.}, to appear \item{3.} Carleman, T.: Sur un theorem de Weierstrass. {\it Arkiv Mat., Astr.\ och Fysik} {\bf 20B}, no.\ 4 (1927) \item{4.} Eliashberg, Y., Gromov, M.: Embeddings of Stein manifolds of dimension $n$ into the affine space of dimension $3n/2+1$. {\it Ann.\ of Math.} {\bf 136}, 123--135 (1992) \item{5.} Forstneric, F.: Approximation by automorphisms on smooth submanifolds of $\C^n$. {\it Math.\ Ann.} {\bf 300}, 719--738 (1994) \item{6.} Forstneric, F.: Interpolation by holomorphic automorphisms and embeddings in $C^n$. 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North-Hol\-land: Amsterdam 1985 \item{13.} Rosay, J.-P., Rudin, W.: Arakelian's approximation theorem. {\it Amer.\ Math.\ Monthly} {\bf 96}, 432--434 (1989) \item{14.} Rudin, W.: \it Real and Complex Analysis. \rm McGraw Hill: New York 1970 \bigskip\rm \noindent{\ninecsc Gregery T. Buzzard, Department of Mathematics, Indiana University, Bloomington, IN 47405, USA} \smallskip \noindent{\ninecsc Franc Forstneric, Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA} \bigskip \bye