\documentstyle[12pt]{amsart} \headheight=8.5pt \textheight=574pt \textwidth=432pt \oddsidemargin=18.88pt \evensidemargin=18.88pt \topmargin=14.21pt % restore \matrix from regular latex, not amsart \makeatletter \def\matrix#1{\null\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \makeatother \newtheorem{theorem}{THEOREM}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{problem}[theorem]{Problem} \newtheorem{principle}[theorem]{Decomposition Principle} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}{EXAMPLE} \newtheorem{remark}{REMARK} \newtheorem{proposition}[theorem]{Proposition} \parindent=20pt \def\wbar{\overline{w}} \def\C{\cal C} \def\dist{\hbox{dist}} \def\V{\lambda} \def\th{\vartheta} \def\R{\cal R} \def\D{\cal D} \def\lp{\lambda_{\varphi}} \def\lbar{\overline{\ell}} \def\B{\overline B } \def\fbar{\overline{f}} \def\kbar{\overline{k}} \def\xibar{\overline{\xi}} \def\L{{\cal L}} \def\hbar{\overline{h}} \def \bomeg{\partial \Omega} \def\Omegabar{\overline{\Omega}} \def \S{\cal S} \def\qbar{\overline{q}} \font\teneufm=eufm10 \font\seveneufm=eufm7 \font\fiveeufm=eufm5 \newfam\eufmfam \textfont\eufmfam=\teneufm \scriptfont\eufmfam=\seveneufm \scriptscriptfont\eufmfam=\fiveeufm \def\frak#1{{\fam\eufmfam\relax#1}} \newfam\msbfam \font\tenmsb=msbm10 scaled \magstep1 \textfont\msbfam=\tenmsb \font\sevenmsb=msbm7 scaled \magstep1 \scriptfont\msbfam=\sevenmsb \font\fivemsb=msbm5 scaled \magstep1 \scriptscriptfont\msbfam=\fivemsb \def\Bbb{\fam\msbfam \tenmsb} \def\RR{{\Bbb R}} \def\CC{{\Bbb C}} \def\QQ{{\Bbb Q}} \def\NN{{\Bbb N}} \def\ZZ{{\Bbb Z}} \def\II{{\Bbb I}} \def\PP{{\Bbb P}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\hexdigit#1{\ifnum#1<10 \number#1\else \ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else \ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi} \chardef\\="5C %% Typesets \ in \tt mode \chardef\{="7B \chardef\}="7D %% also left and right braces %% \def\HollowBoxx #1#2#3{{\dimen0=#1 \advance\dimen0 by -#2 \dimen1=#1 \advance\dimen1 by #3 \vrule height 0pt depth #3 width #2 \hskip -#3 \vrule height #1 depth #3 width #3}} \def\LeftContraction{\mathord{\kern1.45pt \HollowBoxx{6pt}{3.5pt}{.4pt}}\, } \def\HollowBox #1#2#3{{\dimen0=#1 \advance\dimen0 by -#3 \dimen1=#1 \advance\dimen1 by #3 \vrule height #1 depth #3 width #3 \vrule height 0pt depth #3 width #2 \hskip -#3}} \def\RightContraction{\mathord{\, \HollowBox{6pt}{3.1pt}{.4pt}} \kern1.6pt} \def\lbar{\overline{\ell}} \def\B{\overline B } \def\jbar{\overline{j}} \def\kbar{\overline{k}} \def\xibar{\overline{\xi}} \def\L{{\cal L}} \def\hbar{\overline{h}} \def \bomeg{\partial \Omega} \def\Omegabar{\overline{\Omega}} \def \S{\cal S} \def\ra{\rightarrow} \def\ss{\subseteq} \def\sss{\subset \subset} \def\endpf{\hfill \rule{.1in}{.1in}} \def\sq{\Box} \def\tr{\hbox{tr}} \def\sm{\setminus} \def\Re{\mbox{\rm Re}\,} \def\Im{\mbox{\rm Im}\,} \def\etabar{\overline{\eta}} \def\epf{\hskip.2in\vrule width.4pt height6.65pt depth.15pt\vrule width2.5pt height6.65pt depth-6.25pt\hskip-2.5pt\vrule width2.5pt height.25pt depth.15pt\vrule width.4pt height6.65pt depth.15pt\ } \def\ra{\rightarrow} \def\ss{\subseteq} \def\sss{\subset \subset} \def\endpf{\hfill \rule{.1in}{.1in}} \def\sq{\Box} \def\sm{\setminus} \def\Re{\mbox{\rm Re}\,} \def\Im{\mbox{\rm Im}\,} \def\etabar{\overline{\eta}} \def\epf{\hskip.2in\vrule width.4pt height6.65pt depth.15pt\vrule width2.5pt height6.65pt depth-6.25pt\hskip-2.5pt\vrule width2.5pt height.25pt depth.15pt\vrule width.4pt height6.65pt depth.15pt\ } \def\proof{\noindent {\bf Proof. }} \def\dotf{\stackrel{\cdot}{F}} \def\qed{\hskip.2in\vrule width.4pt height6.65pt depth.15pt\vrule width2.5pt height6.65pt depth-6.25pt\hskip-2.5pt\vrule width2.5pt height.25pt depth.15pt\vrule width.4pt height6.65pt depth.15pt\ } \def\nbar{\overline{n}} \def \Omegabar{\overline \Omega} \def \bomega{\partial \Omega} \def\o{\omega} \def \d{\partial} \def\fbar{\overline {f}} \def\jbar{\overline{j}} \def\xibar{\overline{\xi}} \def\wbar{\overline{w}} \def \S{\widetilde{S}} \def \X{\cal{ X} } \def \M{\cal{ M} } \def \F{\cal{ F} } \def \P{\cal P } \def\zbar{\overline{z}} \def\dbar{\overline{\partial}} \def \hbar{\overline{h}} \def\Fbar{\overline{F}} \def\ibar{\overline{i}} \def\lbar{\overline{\ell}} \def\jbar{\overline{j}} \def\A{\cal A} \def\H {\cal H} \def\bH{\bf H} \def\\GBB{\cal B} \def\supp{\hbox{\rm supp}} \tolerance = 3000 \begin{document} \title[Application of the Complex Monge-Amp\`{e}re Equation] {Application of the Complex Monge-Amp\`{e}re Equation to the Study of Proper Holomorphic Mappings of Strictly Pseudoconvex Domains} \author{Steven G. Krantz} \address{\hskip-\parindent Steven G. Krantz\\ Department of Mathematics, Box 1146, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130} \email{sk@@math.wustl.edu} \author{Song-Ying Li} \address{\hskip-\parindent Song-Ying Li\\ Department of Mathematics, University of California, Irvine, CA 92697} \email{sli@@math.uci.edu} \thanks{Krantz was partially supported by NSF Grant DMS--9531967, and at MSRI by NSF Grant DMS--9022140. Li was partially supported by NSF Grant DMS--9500758.} \date{December 2, 1996} \begin{abstract} We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity of solutions of complex Monge-Amp\'ere equation and the reflection principle, enables us to give a new proof of the Fefferman mapping theorem. \end{abstract} \maketitle \section{Introduction} In classical analysis, an important theorem of Painlev\'{e} [Pai] and Kellogg [Kel] states that any conformal mapping between two smoothly bounded domains in the complex plane $\CC$ can be extended to be a diffeomorphism on the closures of the domains. This theorem was generalized by C.\ Fefferman [Fef] in 1974 to strictly pseudoconvex domains in $\CC^n$. Fefferman's original proof of this theorem is very technical, relying as it does on deep work on the boundary asymptotics of the Bergman kernel and on the regularity of $\dbar$-Neumann operator that is due to J.\ J.\ Kohn [Ko]. Bell/Ligocka [BeL], and later Bell [Be], gave a simpler proof which deals with more general domains, including pseudoconvex domains of finite type, by using regularity of the Bergman projection and the $\dbar$-Neumann operator as studied by Kohn [Ko], Catlin [Ca], Boas/Straube [BS], and others. From the results in [NWY], [Web], [Lem], [PH] and [For], we know that the proof of the Fefferman mapping theorem can be reduced to proving that $\det(\varphi'(z))$ extends continuously across the boundary if $\varphi:\Omega_1\to \Omega_2$ is a biholomorphic mapping and the $\Omega_j, j = 1,2, $ are smoothly bounded strictly pseudoconvex domains in $\CC^n$; such an argument uses the reflection principle as developed by Nirenberg/Webster/Yang [NWY], Webster [Web], Pinchuk/Hasanov [PH], Coupet [Cou], and more recently F.\ Forstneric [For]. We know from [Ker] that Painlev\'{e} and Kellogg's theorem can be proved by using the regularity of the Dirichlet problem for the Laplacian in a smoothly bounded planar domain, where the property of the Laplacian being conformally invariant plays an important role in the proof. The natural generalization of the Laplacian in one complex variable to several complex variables, with these considerations in mind, is the complex Monge-Amp\`{e}re equation. In [Ker], Kerzman observed that the proof of the Fefferman mapping theorem would follow from the $C^{\infty}$ global regularity of the Dirichlet problem of a degenerate complex Monge-Amp\`ere equation. However, counterexamples in Bedford/Forn\ae ss [BF] as well as in Gamelin/Sibony [GS] show that, in general, the degenerate Dirichlet problem for the complex Monge-Amp\`ere equation does not have $C^2$ boundary regularity. Thus Kerzman's idea does not work in the sense of its original formulation. The main purpose of the present paper is to give a new method for using the complex Monge-Amp\`{e}re equation to study the boundary regularity of biholomorphic mappings of strongly pseudoconvex domains; thus, in effect, we validate the program initiated by Kerzman/Kohn/Nirenberg [Ker]. We achieve this goal by proving our Theorem 2.2 and then combining our result with a result of Caffarelli/Kohn/Nirenberg/Spruck [CKNS] on the Dirichlet problem for the complex Monge-Amp\`{e}re equation. Specifically, we shall prove the following theorem. \begin{theorem} \sl Let $\Omega_j, j = 1,2$ be two bounded strictly pseudoconvex domains in $\CC^n$ with $C^{\infty}$ boundary. Let $\varphi :\Omega_1 \to \Omega_2$ be a proper holomorphic mapping. Then for any $\epsilon > 0$ we have \smallskip (i) $\varphi$ can be extended as a $\hbox{Lip}_{1}(\Omegabar_1)$ mapping; (ii) $\det(\varphi')\in \hbox{Lip}_{1/2}(\Omegabar_1)$; (iii) There is a $C^2$ defining function $\rho$ for $\Omega_2$ so that, for any $k = 1, \dots, n$, we have $\sum_{j=1}^n \left [ {\d \rho \over \d w_j} \circ \varphi \right ] \cdot {\d \varphi_j \over \d z_k} \in \hbox{Lip}_{1}(\Omegabar_1)$. \end{theorem} As a corollary of (ii) and of a theorem in [PH] and [Web], we obtain a new proof of the Fefferman mapping theorem. \medskip This paper is organized as follows. In Section 2, our main theorem (Theorem 2.2) is stated and proved. Theorem 1.1 is proved in the second part of Section 2 by combining Theorem 2.2 and Theorem 2.1 (a result in [CKNS]). We shall then include an explanation (in Section 3) of how to combine theorems in [PH] and [Web] together with Theorem 1.1 to obtain a new proof of the Fefferman mapping theorem. \medskip The authors would like to thank Xiaojun Huang and Peter Li for helpful conversations. \section{An Application of the Complex Monge-Amp\`ere Equation} Let us recall a theorem of Caffarelli, Kohn, Nirenberg and Spruck [CKNS]. \begin{theorem}\sl Let $\Omega$ be a bounded, strictly pseudoconvex domain in $\CC^n$ with $C^3$ boundary $\d \Omega$. Let $f$ be a non-negative function on $\Omegabar$ such that $f(z)^{1/n}\in C^{1,1}(\Omegabar)$. Let $H(u)$ denote the complex Hessian matrix of the function $u$. Then there is a unique plurisubharmonic function $v\in C^{1,1}(\Omegabar)$ satisfying \begin{eqnarray*} \det H(v)(z) & = & f(z) \quad \hbox{for}\ \ z \in \Omega; \\ v & = & 0 \ \quad \ \ \mbox{for} \ \ z \in \d \Omega. \qquad \qquad \qquad \qquad (2.1) \end{eqnarray*} \end{theorem} \noindent {\bf Note:} The theorem stated in [CKNS] is for $\d \Omega$ being $C^{3,1}$ but, we may approximate $\d \Omega$ by a sequence of bounded strictly pseudoconvex domains with $C^4$ boundaries, and do a careful count of which derivatives are actually used in their proof, to obtain the result stated here. \smallskip As we mentioned in the introduction, the following theorem plays an essential role in the proof of Theorem 1.1. \begin{theorem}\sl Let $\Omega$ be a bounded strictly pseudoconvex domain in $\CC^n$ with $C^{\infty}$ boundary $\d \Omega$. For any $0<\epsilon << 1$ and any positive integer $q$ there is a plurisubharmonic defining function $\rho_q\in C^{\infty}(\Omegabar)$ for $\Omega$ so that $$ \det H(\rho_q)(z) \le C\dist(z,\d \Omega)^q,\quad z\in \Omega. \leqno(2.3) $$ \end{theorem} \proof Let $\delta(z)$ denote the (signed) distance function from $z$ to $\d \Omega$. Since $\Omega$ is a bounded domain in $\CC^n$ with $C^{\infty}$ boundary $\d \Omega$, then $\delta(z)\in C^{\infty}(\Omegabar)$ (after modification of the distance function on a compact set in the interior). By a rotation we see that, for any fixed point $z^0\in \Omega$ near the boundary, we may assume that the $z_n$ direction is the direction $({\d \delta(z)\over \d \zbar_1},\cdots,{\d \delta\over \d \zbar_n})$ at the point $z^0$. Let $$ H(\delta)_{n-1}(z)=\left[{\d^2 \delta \over \d z_\ell \d \zbar_q}(z) \right]_{1\le \ell,q\le n-1} $$ Since $\Omega$ is strictly pseudoconvex, there is an $\epsilon > 0$ so that $$ -H(\delta)_{n-1}(z^0)\ge \epsilon I_{n-1} $$ for all $z\in \Omegabar$ with $\delta(z)\le \epsilon$. We may assume that $H(\delta)_{n-1}$ is diagonal at $z^0$. Note that \begin{eqnarray*} H(- \delta)(z) &=& \left[\matrix{- H(\delta)_{n-1}(z) & 0\cr 0 & -{\d^2 \delta \over \d z_ n \d \zbar_n } \cr}\right] \end{eqnarray*} If it happens that $\sum_{i j}{\d \delta\over \d z_i }{\d \delta \over \d \zbar_j} {\d^2 \delta \over \d z_i \d \zbar_j} = 0$, then $\rho_q=-\delta(z)+ \delta(z)^{2+q}$ is the desired defining function. [In fact it is these terms that distinguish the study of the real Hessian from the more subtle study of the complex Hessian. In particular, we know that ${\d^2 \delta\over \d x_n^2}$ equals $0$, but the term ${\d^2 \delta \d z_n \d \zbar_n}$ may not be zero. Therefore estimate (2.3) is easy to check for the determinant of the real Hessian of $\delta$; matters are much trickier for the complex Hessian.] Now we let $$ \rho(z)=-\delta(z) $$ and $$ r^{[m]}(z)=\rho(z)+\sum_{k=2}^m a_k(z) \delta(z)^k \quad z\in \Omega. $$ We will prove inductively that $$ \hbox{det} (H(r^{[m]})(z)) = b_{m-1}(z) \cdot \rho^{m-1} , $$ where $b_{m-1}$ is some smooth function. In particular, $\hbox{det} (H(r^{[m]}))$ vanishes to order $m-1$ at the boundary. Now \begin{eqnarray*} \d_{i\jbar} r^{[m]}(z) &=& \d_{i\jbar} \rho(z) + \sum_{k=2}^m [\d_{i\jbar} a_k \, \rho(z)^2+k (k-1) a_k \d_i \rho \d_{\jbar} \rho)] \rho^{k-2}\\ && +\sum_{k=2}^m k[\d_i a_k \d_{\jbar} \rho +\d_i \rho \d_{\jbar} a_k + a_k\rho_{i\jbar}] \rho^{k-1}\\ \end{eqnarray*} Let $H(r^{[m]})_{n-1}(z)=[{\d^2 r^{[m]} \over \d z_i \d \zbar_j}]_{(n-1)\times (n-1)}$, and let $B(z)^*=[r_{n \bar{1}},\cdots, r_{n\overline{n-1}}]$ be a row-vector. Then \begin{eqnarray*} H(r^{[m]})(z) &=& \left[\matrix{H(r^{[m]})_{n-1}(z) & B(z)\cr B(z)^* & r^{[m]}_{n\nbar}(z ) \cr}\right] \end{eqnarray*} Then $$ \det( H(r^{[m]})(z))=\det (H(r^{[m]})_{n-1}(z)) [r^{[m]}_{n\nbar} - \{ B^* H(r^{[m]})_{n-1}(z)\}^{-1} B(z)] $$ We know that $$ H(r^{[m]})_{n-1}(z)\ge \epsilon I_{n-1} $$ for all $z\in \Omega_{\epsilon}$, where $\epsilon$ is a positive number depending only on $\Omega$ and $\|a_j\|_{C^2(\Omegabar)}$. First, we let $$ (d\rho)(z)= \sum_{i,j}\rho_i \rho_{\jbar} \rho_{i\jbar} \, ; $$ then we define $$ a_2(z)=-{(d \rho)(z) \over 2 ((d \rho)(z) +|\d \rho|^4)}. $$ Thus we have \begin{eqnarray*} r_{n\nbar}(z) &=&\rho_{n\nbar} +\sum_{k=2}^m [ \d_{n \nbar} a_k \rho^k + k(k-1) a_k |\d \rho|^2 \rho^{k-2}]\\ &&+\sum_{k=2}^m k[\d_n a_k \d_{\nbar} \rho + \d_n \rho \d_{\nbar} a_k +a_k \rho_{n\nbar}] \rho^{k-1}\\ &=&\sum_{k=2}^m [ \d_{n \nbar} a_k \rho^k + k(k+1) a_{k+1} |\d \rho|^2 \rho^{k-1}]\\ &&+\sum_{k=2}^m k[\d_n a_k \d_{\nbar} \rho + \d_n \rho \d_{\nbar} a_k +a_k \rho_{n\nbar}] \rho^{k-1}\\ &=& O(\rho) \end{eqnarray*} Therefore $$ \det(H(r^{[2]})(z))=b_1(z) \rho(z) $$ Assume that we have constructed $r^{[m]}=\rho(z)+\sum_{k=2}^m a_k \rho(z)^k$ such that $$ \det (H(r^{[m]})(z))=b_{m-1}(z)\rho^{m-1},\quad z\in \Omega. $$ We consider $$ r^{[m+1]}=r^{[m]} +a_{m+1} \rho(z)^{m+1}. $$ Since \begin{eqnarray*} \lefteqn{H(r^{[m+1]})(z)}\\ &=&H(r^{[m]})+\rho(z)^{m+1} H(a_{m+1})(z) +(m+1) a_{m+1} \rho(z)^m H(\rho)\\ && +(m+1) m a_{m+1}\rho^{m-1} \d \rho(z) \otimes \overline{\d \rho} \\ && +(m +1)\rho(z)^m [\d \rho \otimes \overline{ \d a_{m+1}}+\d a_{m+1} \otimes \overline{\d \rho}] , \end{eqnarray*} it is easy to see that $$ \det(H(r^{[m+1]})(z)) =b_m(z) \rho(z)^m)+\det \bigl [ H(r^{[m]})+ (m+1) m a_{m+1}\rho^{m-1}) \bigr ] . $$ By a rotation, we may let $z_n$ be the complex normal direction of $\d \Omega_{\delta(z)}$ at $z$. Thus \begin{eqnarray*} \det(H(r^{[m]})(z)+ (m+1) m \rho(z)^{m-1}\d \rho(z)\otimes \overline{\d \rho}) && \\ &=&\det\, M , \end{eqnarray*} where $$ M = \left [\matrix{ H(r^{[m]})_{n-1}(z) & B(m)(z)\cr B(m)(z)^* & r^{[m]}_{n \nbar}(z)+ (m+1) m \rho^{m-1} \rho_n \rho_{\nbar}\cr} \right ] . $$ But this \begin{eqnarray*} &=& \det(H(r^{[m]})_{n-1}(z) [r^{[m]}_{n \nbar}+ B(m)^* H(r^{[m]})_{n-1}(z)^{-1} B(m)(z) \\ && \qquad +(m+1) m \rho^{m-1} \rho_n \rho_{\nbar}] \\ &=& \det (H(r^{[m]})(z))+\det(H(r^{[m]})_{n-1}(z)) (m+1) m \rho_n \rho_{\nbar} \rho(z)^{m-1}\\ &=& b_{m-1}(z) \rho(z)^{m-1}+ \det(H(r^{[m]})_{n-1}(z)) (m+1) m \rho_n \rho_{\nbar} \rho(z)^{m-1} . \end{eqnarray*} We therefore choose $a_{m+1}$ such that $$ b_{m-1}(z)+ (m+1) m \det(H(r^{[m]})_{n-1}(z)) \rho_n \rho_{\nbar}(z)=0 $$ From this it will follow that $$ \det(H(r^{[m+1]})(z))=b_m (z) \rho(z)^m. $$ By our construction, we know that $H(r^{[m]})_{n-1}(z)$ is positive definite with least positive eigenvalue $\epsilon_m$ for all $z\in \Omega\setminus \Omega_{\epsilon_m}$. Thus if $c_m$ is large enough so that $|\nabla \delta|^2 c_m \ge 2|b_m(z)|$ then, by choosing $\epsilon_m$ small enough, we will easily see that the function $$ \rho_m(z) \equiv r^{[m+1]}(z)+ c_m \delta(z)^{m+2} \leqno(2.4) $$ is positive definite in $\Omega\setminus \Omega_{\epsilon_m}$ and (2.3) holds on $\Omega\setminus \Omega_{\epsilon_m}$. Here $\Omega_t=\{z\in \Omega: \delta(z) 0$ so that $$ H(\exp(\rho_{4n})(w) \ge \epsilon I_n, \quad \hbox{\rm for\ all }\ \ w\in \Omega_2 . $$ Thus \begin{eqnarray*} {\d^2 \exp( r(z))\over \d z_\ell\d \zbar_\ell} &=& \sum_{p,\ell=1}^n {\d^2 \exp(\rho_{4n})\over \d w_p\d \wbar_{\ell}}(\varphi(z)) {\d \varphi_p\over \d z_\ell} \overline{ {\d \varphi_{\ell}\over \d z_\ell}}(z)\\ &\ge& \epsilon \sum_{p=1}^n \Big| {\d \varphi_p\over \d z_\ell}(z)\Big|^2. \end{eqnarray*} This shows that $$ \|\varphi\|^2_{{\rm Lip}(\Omegabar_1)} \le C\lambda_0^{-1}\|\exp(v)\|_{C^{1,1}(\Omegabar_1)}. $$ Thus we have $\det (\varphi'(z))\in L^{\infty}(\Omega_1)$. If we apply ${\d^2\over \d z_\ell \d z_m}$ to $ r(z) $ and use the above result, then we have $$ \left |\sum_{m=1}^n {\d \rho_{4n}\over \d w_p}(\varphi(z)) {\d^2 \varphi_p\over \d z_\ell \d z_m} \right | \le C. $$ Let $z^0\in \Omega_1$ be sufficiently near to $\d \Omega_1$. Without loss of generality, by applying a rotation, we may assume that $z_1^0,\cdots, z_{n-1}^0$ are complex tangential at $z^0$ and also that at the point $\varphi(z_0)$ the directions $ w_1,\cdots, w_{n-1}$ are complex tangential. Thus \begin{eqnarray*} \lefteqn{{\d \over \d z_p} \log( \det(\varphi'(z_0)))(z)}\\ &=& \sum_{\ell, m=1}^n \varphi^{\ell m} {\d ^2\varphi_m \over \d z_\ell \d z_p}(z)\\ &=& \sum_{\ell 1$). Let $D$ be a domain in $\CC^n$, with boundary passing through the origin, that is smoothly bounded and pseudoconvex along $M \subseteq \partial \Omega$, and let $D'$ be a similar such domain bounded in part by $M'$. We assume that we have a biholomorphic mapping $f: D\to D'$ that extends to a diffeomorphism on $D\cup M$ near the origin and such that this extended map sends $M$ to $M'$. Since the problem is a local one, we may shrink our sets toward the origin, and we shall freely do so in the sequel. Let $a\in \CC^n\setminus \{0\}$ and let $$ [a]=\{z\in \CC^n :\sum_{m=1}^n z_m a_m=0\}\in \CC\PP^{n-1} . $$ If $f$ is a holomorphic mapping, then we let $$ F(z, [a])=(f(z), [a f'(z)^{-1}]) $$ where $f'(z)$ is the Jacobian matrix of $f$ at $z$ and $a f'(z)^{-1}$ is the matrix product of a row vector $a$ with the matrix $\bigl [ f'(z)^{-1} \bigr ] $. Recall that $H_z M \subset T_z M$ is the maximal complex subspace of the~$(2n - 1)$-dimensional real tangent space $T_z M$. We associate to $M$ the smooth submanifold $\tilde{M}$ of $X=\CC^n \times \CC P^{n-1}$ defined by $$ \tilde{M}=\{ (z, H_z M)\in X: z\in M\}, $$ and $$ \tilde{M'}=\{ (z, H_z M')\in X: z\in M'\}. $$ The following theorem is due to Webster [Web]. \begin{theorem} \sl If $M\subset \CC^n$ is a strictly pseudoconvex hypersurface, then the associated manifold $\tilde{M}\subset \CC^n\times \CC P^{n-1}$ is totally real (whence maximally real). \end{theorem} If $f$ and $\det (f'(z))$ are continuous up to $D\cup M$, then for each $z\in M$ the derivative $f'(z)$ maps $H_z M$ isomorphically onto $H_{f(z)} M'$. Hence the associated mapping $F$ extends continuously from the domain $\tilde{D}$ to $\tilde{D} \cup \tilde{M}$ and maps $\tilde{M}$ to $\tilde{M'}$. Its restriction to $\tilde{M}$ is given by $$ \tilde{f}(z, H_z M)=(f(z), H_{f(z)} M') $$ When $M=\d \Omega_1$ and $\d \Omega_2$ are strictly pseudoconvex and smooth then, by Theorem 1.1, we have that $f(z)$ and $\det (f'(z))$ are continuous on $\Omegabar_1$. Combining this result with Theorem 3.2 and then with and 3.1, we obtain a new proof of the following theorem of C. Fefferman: \begin{theorem}[C.\ Fefferman] \sl Let $\Omega_1$, $\Omega_2$ be strongly pseudoconvex domains in $\CC^n$ with $C^\infty$ boundaries. Assume that $\Phi: \Omega_1 \rightarrow \Omega_2$ is a biholomorphic (proper) mapping. 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