\documentclass[research]{raex}
\FirstPageNumber{421}
\Volume{36} % publication information for your paper
\IssueNumber{2} % publication information for your paper
\Year{2010/2011} % publication information for your paper
\Editor{Alexander Olevskii} % for your paper
\Received{November 10, 2010} % publication information for your paper
%\CoverAuthor{Steven G. Krantz\\}
%\CoverTitle{Convergence of Automorphisms and Semicontinuity of Automorphism Groups}
% use the same format for each author, listing each author in a separate \begin{author}--\end{author} environment
\begin{Author}
% First and last name of author
\FirstName{Steven G.}\LastName{Krantz}
% full postal address including postal code and country
\PostalAddress{Department of Mathematics, Washington University in St. Louis, St. Louis, MO, U.S.A}
% email address for author
\Email{sk@math.wustl.edu}
% grant support is indicated by giving \Thanks
\Thanks{Author supported in part
by the National Science Foundation and by the Dean of the Graduate
School at Washington University.}
\end{Author}
% Put primary an secondary math reviews numbers here, using one line per number. Multiple numbers are OK.
\begin{MathReviews}
\primary{32H02}
\primary{32M05}
\secondary{32H99}
\end{MathReviews}
% Put key words and phrases here, one keyword at a time.
\begin{KeyWords}
\keyword{automorphism group}
\keyword{semicontinuity}
\keyword{holomorphic mapping}
\keyword{convergence of mappings}
\end{KeyWords}
\title{Convergence of Automorphisms and Semicontinuity of Automorphism Groups}
% The raex class uses the myheadings option by default, so we specify the even and
% odd running heads. You may have to abbreviate one or both of them to fit.
\markboth{Steven G. Krantz}{Semicontinuity of Automorphism Groups}
%%% Author's macros and definitions go below this line %%%
% The standard LaTeX environments should be set up for theorems, corollaries, definitions, etc.
% The raex class predefines macros for the standard spaces:
% \naturalnumbers, \integers, \rationalnumbers, \realnumbers, \complexnumbers.
% You should use these to define your own macros, as \R is defined below.
\newcommand{\R}{\realnumbers}
\newcommand{\rae}{\textsl{Real Analysis Exchange}}
\def\sm{\setminus}
\def\ra{\rightarrow}
\def\sm{\setminus}
\def\ss{\subseteq}
\def\card{\hbox{card}}
\def\e{\epsilon}
\def\d{\delta}
\def\Re{\hbox{\rm Re}\,}
\def\Im{\hbox{\rm Im}\,}
\def\series{\sum_{j = 1}^\infty}
\def\parseries{\sum_{j = 1}^N}
\def\powser{\sum_{j=0}^\infty a_j (x - c)^j}
\def\wta{\widetilde{a}}
\def\wtb{\widetilde{b}}
\def\K{{\cal K}}
\def\C{{\cal C}}
\def\M{{\cal M}}
\def\O{{\cal O}}
\def\avgint{\frac{1}{2\pi} \int_0^{2\pi}}
\def\dbar{\overline{\partial}}
\def\Aut{{\rm Aut}}
\def\boac{{\rm boundary orbit accumulation point}}
\def\Boac{{\rm Boundary orbit accumulation point}}
\def\sboac{{\rm special boundary orbit accumulation point}}
\font\titlebf = cmbxsl10 at 24 pt
\font\smit=cmti10
\font\smitt=cmti8
%%%%% This is Hollowbox
\def\HollowBox #1#2{{\dimen0=#1 \advance\dimen0 by -#2
\dimen1=#1 \advance\dimen1 by #2
\vrule height #1 depth #2 width #2
\vrule height 0pt depth #2 width #1
\llap{\vrule height #1 depth -\dimen0 width \dimen1}%
\hskip -#2
\vrule height #1 depth #2 width #2}}
\def\BoxOpTwo{\mathord{\HollowBox{6pt}{.4pt}}\;}
%%%%% This is Hollowbox
%%%%% This is BoxOpTwo
\def\endpf{\hfill $\BoxOpTwo$}
\def\bomega{\partial \Omega}
\def\sss{\subset \, \, \subset}
\def\zbar{\overline{z}}
\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 \textfont\msbfam=\tenmsb
\font\sevenmsb=msbm7 \scriptfont\msbfam=\sevenmsb
\font\fivemsb=msbm5 \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\TT{{\Bbb T}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\makeindex
\usepackage{graphicx}
\usepackage{amsmath}
\begin{document}
\maketitle
\begin{abstract}
We study the compactness of the automorphism group of a domain
in $\CC^n$, and in particular the convergence properties of mappings.
We supply an application to the semicontinuity of automorphism groups under
perturbation of the underlying domain. Relevant examples are provided.
\end{abstract}
\section{Introduction}
A {\it domain} $\Omega$ in $\CC^n$ is a connected, open set. An {\it automorphism}
of $\Omega$ is a biholomorphic self-map. The collection of automorphisms forms
a group under the binary operation of composition of mappings. The standard topology
on this group is uniform convergence on compact sets, or the compact-open topology.
We denote the automorphism group by $\Aut(\Omega)$. When $\Omega$ is a bounded domain,
the group $\Aut(\Omega)$ is a real (never a complex) Lie group.
Although domains with {\it transitive automorphism group} are of some interest,
they are relatively rare (see [9, Section III.3]). A geometrically more natural condition
to consider, and one that gives rise to a more robust and broader class of domains,
is that of having {\it non-compact automorphism group}. Clearly a domain has
non-compact automorphism group if there are automorphisms $\{\varphi_j\}$ which
have no subsequence that converges to an automorphism. The following proposition
of Henri Cartan is of particular utility in the study of these domains:
\begin{proposition} \sl
Let $\Omega \ss \CC^n$ be a bounded domain. Then $\Omega$ has non-compact automorphism
group if and only if there are a point $X \in \Omega$, a point $P \in \partial \Omega$, and
automorphisms $\varphi_j$ of $\Omega$ such that $\varphi_j(X) \ra P$ as $j \ra \infty$.
\end{proposition}
\noindent We refer the reader to [14, p.\ 65] for discussion and proof of Cartan's result.
We say that a domain $\Omega \ss \CC^n$ has $C^k$ boundary, $k \geq 1$ an integer,
if it is possible to write
$$
\Omega = \{z \in \CC^n: \rho(z) <0\}
$$
for a function $\rho$ that is $C^k$ and which satisfies
$\nabla \rho \ne 0$ on $\partial \Omega$. This definition is equivalent
to a number of other natural definitions of $C^k$ boundary for a domain
(see the Appendices in [11]). Below we shall define a topology
on the collection of domains with $C^k$ boundary.
Domains with compact automorphism group exhibit certain rigidities which
are of interest for our studies. We begin this paper by showing
that, for certain smoothly bounded domains with compact automorphism group, the convergence
of automorphisms will take place in a much stronger topology than
the standard one specified in the first paragraph. This fact has
intrinsic interest, but is also of considerable use for further
studies in complex function theory. It is even new in the context
of one complex variable.
As an application of the ideas in the last paragraph, we offer a new
result about the semicontinuity of the automorphism group under perturbation
of the underlying domain. This generalizes results of [7].
We also offer a direct generalization of the result of [7, Theorem 0.1] to
finite type domains. Some of the proof techniques presented here are new.
It is a pleasure to thank the referee for many useful suggestions.
\section{Convergence of holomorphic mappings}
Throughout this section, and in subsequent parts of the paper,
we shall use the concept of finite type as developed by
Kohn/Catlin/D'Angelo. See [11, Section 11.5] for an explication of these
ideas. For completeness we supply the relevant definitions here.
\begin{definition} \rm
Let $\Omega = \{z \in \CC^2: \rho(z) < 0\}$ be a smoothly bounded domain and
$P \in \partial \Omega.$ Let $m$ be a non-negative integer.
We say that $\partial \Omega$ is of finite
type $m$ at $P$ if the following condition holds:
there is a non-singular complex analytic disc $\varphi$
tangent to $\bomega$ at $P$ (with $\varphi(0) = P$ and $\varphi'(0) \ne 0$)
such that, for small $\zeta,$
$$
|\rho \circ \phi(\zeta)| \leq C |\zeta|^m.
$$
But there is no non-singular disc $\psi$
tangent to $\bomega$ at $P$ such that $\psi(0) = P$, $\psi'(0) \ne 0$, and, for small $\zeta,$
$$
|\rho \circ \phi(\zeta)| \leq C |\zeta|^{(m+1)} .
$$
\end{definition}
We note that the definition just given, which is sometimes called {\it geometric finite type},
is equivalent to another definition involving commutators of vector fields (and which
is called {\it analytic finite type}). Details may be found in [11, Section 11.5].
The definition of finite type in higher dimensions (due to J. P. D'Angelo) is more complex.
We give it in three steps.
\begin{definition} \rm
Let $f$ be a scalar-valued holomorphic function of a complex variable and $P$ a point
of its domain. The {\em multiplicity} of $f$ at $P$ is defined to be the least
positive integer $k$ such that the $k^{\rm th}$ derivative of $f$ does not
vanish at $P.$ If $m$ is that multiplicity then we write
$v_P(f) = v(f) = m.$
If $\phi$ is instead a vector-valued holomorphic function of a complex
variable then its multiplicity
at $P$ is defined to be the minimum of the multiplicities of its
entries. If that minimum is $m$ then we write $v_P(\phi) = v(\phi) = m.$
\end{definition}
\begin{definition} \rm
Let $\phi:D \ra \CC^n$ be a holomorphic
curve and $\rho$ the defining function for a smoothly bounded
domain $\Omega$. Then the {\em pullback} of $\rho$ under $\phi$ is
the function $\phi^* \rho(\zeta) = \rho\circ \phi(\zeta).$
\end{definition}
\begin{definition} \rm
Let $\Omega$ be a smoothly bounded domain in $\CC^n$ and
$\partial \Omega$ its boundary. Let $P \in \partial \Omega.$ Let $\rho$ be a defining
function for $\Omega$ in a neighborhood of $P.$ We say that $P$ is a point
of finite type (or finite 1-type) if there is a constant $C > 0$ such
that
$$
\frac{v(\phi^*\rho)}{v(\phi)} \leq C
$$
whenever $\phi$ is a non-constant (possibly singular) one-dimensional holomorphic curve
through $P$ such that $\phi(0) = P.$
The infimum of all such constants $C$ is called the {\em type} (or {\em1- type})
of $P.$ It is denoted by $\Delta(M,P) = \Delta_1(M,P).$
\end{definition}
Again, the reference [11, Section 11.5] provides a thorough treatment, with examples,
of the concept of point of finite type.
It is a basic fact---see, for instance, [2, Main Theorem, p.\ 103] and the
discussion in [11, Section 11.5]---that any automorphism of a smoothly bounded, finite type
domain $\Omega$ extends to be a $C^\infty$ diffeomorphism of
the closure of the domain $\Omega$ to itself.\footnote{In fact the standard
condition to guarantee such an extension to a diffeomorphism of the closures
is Bell's Condition $R$---see [11, Section 11.5]. Condition $R$ is guaranteed by a
subelliptic estimate for the $\overline{\partial}$-Neumann problem, and that
condition is known to hold on domains of finite type.} Thus it is
natural in the present context to equip the automorphism group
with a different topology which we shall call the $C^k$ topology.
Fix $k$ a positive integer. Let
$\epsilon > 0$. If $\varphi_0 \in \Aut(\Omega)$ then a
subbasic neighborhood of $\varphi_0$ is one of the form
%\smallskip \\
$$
{\cal U}_{k, \epsilon}(\varphi_0) \equiv \left \{ \varphi \in \Aut(\Omega):
\left | \frac{\partial^\alpha}{\partial z^\alpha} \left ( \varphi - \varphi_0 \right ) (z) \right | < \epsilon \
\hbox{for all} \ z \in \Omega \right.
$$
$$
\qquad \qquad \qquad \qquad \hbox{and all multi-indices} \ \alpha \ \hbox{with} \ |\alpha| \leq k \biggr \} \, .
$$
%\vspace*{.15in}
\noindent It is easy to see that, with this topology, $\Aut(\Omega)$ is still a real Lie group
(see [10, Section V.2]) when $\Omega$ is a bounded domain.
Our first result of this section is as follows:
\begin{proposition} \sl
Let $\Omega \ss \CC^n$ be a bounded finite type domain with compact
automorphism group in the $C^k$ topology, $k >0$ an integer. Let $\alpha$ be a multi-index
such that $|\alpha| \leq k$. Then
there is a positive, finite constant $K_\alpha$ such that
$$
\sup_{z \in \Omega} \left | \frac{\partial^\alpha}{\partial z^\alpha} \varphi (z) \right | \leq K_\alpha \eqno (1.5.1)
$$
for all $\varphi \in \Aut(\Omega)$.
\end{proposition}
The point here is that we have a uniform bound on the
$\alpha$th derivative of {\it all} automorphisms of $\Omega$, that bound
being valid {\it up to the boundary}. A result of this kind
was proved in [7, Proposition 5.1] for the automorphism group of a {\it strongly
pseudoconvex} domain considered in the
compact-open topology. That proof was rather complicated, using
Fefferman's asymptotic expansion for the Bergman kernel of a
strongly pseudoconvex domain [4, Theorem 2] as well as the concept of
Bergman representative coordinates [6, Section 4.2]. The proof presented
here---for the $C^k$ topology---is much simpler, and works in
considerably greater generality.
%\smallskip \\
\begin{proof} Suppose to the contrary that, for some
fixed multi-index $\alpha$, there is no bound $K_\alpha$. Then
there are a sequence $\varphi_j$ of automorphisms of $\Omega$ and
points $P_j \in \Omega$ such that
$$
\left | \frac{\partial^\alpha}{\partial z^\alpha} \varphi_j (P_j) \right | \ra + \infty \, .
$$
But $\Aut(\Omega)$ is compact, so there is a subsequence $\varphi_{j_k}$ that
converges in the $C^k$ topology to a limit automorphism $\varphi_0$.
Let
$$
L_0 \equiv \sup_{z \in \Omega} \left | \frac{\partial^\alpha}{\partial z^\alpha} \varphi_0 (z) \right | \, ,
$$
which is finite because $\Omega$ is finite type.
Let $\epsilon > 0$. Choose $K$ so large that
$$
\left |\frac{\partial^\alpha}{\partial z^\alpha} \varphi_{j_k}(P_{j_k}) \right | > L_0 + 2\epsilon
$$
for $k > K$. Choose $M$ so large that
$$
\left |\frac{\partial^\alpha}{\partial z^\alpha} \varphi_{j_m}(z) - \frac{\partial^\alpha}{\partial z^\alpha} \varphi_0(z) \right | < \epsilon
$$
for all $m > M$, $z \in \Omega$. It then follows that, for $\ell > \max(K, M)$,
$$
\left |\frac{\partial^\alpha}{\partial z^\alpha} \varphi_0(P_{j_\ell}) \right | > L_0 + \epsilon \, .
$$
This is impossible.
\end{proof}
%\smallskip \\
The next result relates our different topologies on the automorphism group in an important
new way.
\begin{proposition} \sl
Let $k$ be a positive integer. Let $\Omega$ be a smoothly bounded domain on which
$$
\left | \frac{\partial^\alpha}{\partial z^\alpha} \varphi (z) \right | \leq K_\alpha \eqno (1.6.1)
$$
for all $\varphi \in \Aut(\Omega)$, all $z \in \Omega$, and all multi-indices $\alpha$ such that $|\alpha| \leq k$.
Then any sequence $\varphi_j$ of automorphisms that converges uniformly on
compact sets to a limit automorphism $\varphi_0$ in fact converges in the $C^{k-1}$ topology
to $\varphi_0$.
\end{proposition}
\begin{remark} \rm
As the previous result shows, the converse of this proposition is true as well for finite type domains.
\end{remark}
\begin{proof}[Proof of the Proposition] From (1.6.1), there is a constant $K_1$ so that
$$
\left | \nabla \varphi_j(z) \right | \leq K_1
$$
for all $\varphi \in \Aut(\Omega)$, all $j$, and all $z \in \Omega$. Let $\epsilon > 0$. Choose a compact
set $K \ss \Omega$ so large that if $w \in \Omega \setminus K$ then there
is a line segment $\ell_w$ connecting $w$ to an element $k_w \in K$ (and
parametrized by $\gamma_w(t) = (1-t)w + t k_w$) which has
length less than $\epsilon/K_1$.
Now choose $j$ so large that
$$
\left | \varphi_j(z) - \varphi_0(z) \right | < \epsilon \eqno (1.6.2)
$$
for all $z \in K$. Choose a point $w \in \Omega \setminus K$. Then
\begin{eqnarray*}
\left | \varphi_j(w) - \varphi_0(w) \right | & \leq &
\left | \varphi_j(w) - \varphi_j(k_w) \right | +
\left | \varphi_j(k_w) - \varphi_0(k_w) \right | +
\left | \varphi_0(k_w) - \varphi_0(w) \right | \\
& \equiv & I + II + III \, .
\end{eqnarray*}
Now we know that $II < \epsilon$ by (1.6.2). For $I$, notice that
\begin{eqnarray*}
\left | \varphi_j(w) - \varphi_j(k_w) \right | & = & \left | \int_0^1 \frac{d}{dt}
\left [ \varphi_j \circ \ell_w(t) \right ] \, dt \right | \\
& \leq & K_1 \cdot \frac{\epsilon}{K_1} \\
& = & \epsilon \, .
\end{eqnarray*}
A similar estimate obtains for $III$.
In summary,
$$
\left | \varphi_j(w) - \varphi_0(w) \right | < 3\epsilon \, .
$$
This gives the uniform convergence estimate that we want for all points
of $\Omega$. That proves the result for $k = 1$.
Of course similar estimates may be applied to $|(\partial^\alpha/\partial z^\alpha)\varphi_j(w)
- (\partial^\alpha/\partial z^\alpha)\varphi_0(w)|$
for any $|\alpha| < k$. Thus we get convergence in the $C^{k-1}$ topology.
\end{proof}
%\smallskip \\
\begin{corollary} \sl
Let $\Omega \ss \CC^n$ be a smoothly bounded domain on which automorphisms
satisfy uniform bounds on derivatives as in (1.6.1). Let
$\varphi_j \in \Aut(\Omega)$ be a sequence of automorphisms that
converges uniformly on compact sets to a limit automorphism $\varphi_0$.
Then in fact $\varphi_j \ra \varphi_0$ uniformly on $\overline{\Omega}$.
\end{corollary}
\begin{proof} This is a special case of the preceding result.
\end{proof}
%\smallskip \\
\begin{remark} \rm
Let $\Omega$ be a strongly pseudoconvex domain with
real analytic boundary which is not biholomorphic to the ball. Then the results on uniform
bounds of derivatives of automorphisms are particularly
easy to prove. For $\Aut(\Omega)$ must be compact (see [15, Main Theorem, p.\ 253]). It is further known---see [8]---that there
is an open neighborhood $U$ of $\overline{\Omega}$ such that
every automorphism (and its inverse, of course) analytically
continues to $U$. It then follows directly from Cauchy estimates
that, if $\alpha$ is a multi-index, then
$$
\left | \frac{\partial^\alpha}{\partial z^\alpha} \varphi (z) \right | \leq K_\alpha
$$
for all $\varphi \in \Aut(\Omega)$ and all $z \in \Omega$.
\end{remark}
It is possible to use Bergman representative coordinates (see [6, Section 4.2]) in a new
fashion to obtain the uniform-bounds-on-derivatives result for finite
type domains in $\CC^2$ {\it in the compact-open topology}. More precisely,
\begin{theorem} \sl
Let $\Omega \ss \CC^2$ be a smoothly bounded, finite type domain in $\CC^2$ with
compact automorphism group in the compact-open topology.
Let $\alpha$ be a multi-index. Then there is a constant $K_\alpha > 0$ so that
$$
\left | \frac{\partial^\alpha}{\partial z^\alpha} \varphi (z) \right | \leq K_\alpha
$$
for all $\varphi \in \Aut(\Omega)$ and all $z \in \Omega$.
\end{theorem}
\begin{proof} For a fixed $w \in \Omega$, let $\delta_w$ denote the
Dirac delta mass at $w$. Then of course
$$
K(z,w) = P(\delta_w)(z) \eqno (1.10.1)
$$
for all $z \in \Omega$, where $K$ is the Bergman kernel for $\Omega$
and $P$ the Bergman projection.
Now, by a well-known formula of Kohn (see [12, Section 7.9]),
$$
P = I - \overline{\partial}^* N \overline{\partial} \, .
$$
Here $N$ is the $\overline{\partial}$-Neumann operator. It follow that
$P$ is hypoelliptic up to the boundary (again see [12, Sections 7.8, 7.9]).
Let $U$ be a tubular neighborhood of $\partial \Omega$. Let $L \subset \!\subset \Omega$
be a compact set so that $\partial L \ss U$. Now pick $w \in \partial L$. So there will be an $r >0$, with
$r$ greater than the radius of $U$, so that
$K( \, \cdot \, , w)$ is smooth on $\overline{\Omega} \cap B(w, r)$.
Now assume that $w \in U \cap \Omega$. Let $\widetilde{w}$ be the point of $\partial \Omega$ that is nearest to $w$.
Then, because we are in complex dimension 2, (see [1, Theorem 3.1]) there is a holomorphic peak function\footnote{The
construction of peaking functions in [1, Theorem 3.1] is quite difficult and technical.
It amounts to a delicate scaling procedure. An alternative approach to the matter, using
entire functions that grow at a certain rate at infinity, appears in [5]. The paper [1]
proves the peak point result for domains with real analytic boundary. The paper [5] proves
the result for finite type domains.}
$f_{\widetilde{w}}$ for $\widetilde{w}$.
We may replace $f_{\widetilde{w}}(z)$ with $[9 + f_{\widetilde{w}}(z)]/10$ so that
our peak function does not vanish on $\overline{\Omega}$. Continue to denote the peak function by $f_{\widetilde{w}}$.
Then we may write
\begin{eqnarray*}
K(z,w) & = & P(\delta_w)(z) \\
& = & \int_\Omega K(z,\zeta) \delta_w(\zeta) \\
& = & \int_\Omega K(z, \zeta) \sum_j (\alpha_j) \cdot \left ( \frac{1}{\eta_j^4 \cdot \Omega_4} \right ) \cdot \chi_{B(w,\eta_j)}(\zeta) \, dV(\zeta) \, .
\end{eqnarray*}
Here $\chi_S$ denotes the characteristic function of the set $S$.
In the right-hand part of this last sequence
of equalities, the $\alpha_j$ are positive numbers that sum to 1 and $\Omega_4$ is the volume of
the unit ball in $\RR^4 \approx \CC^2$ (see [11, Section 1.4]). [We are simply invoking
here the mean value property of a holomorphic function on balls.] Also the $\eta_j$ are an increasing sequence of finitely
many positive radii with the largest of them equalling the distance $\tau$ of $w$ to $\partial \Omega$.
Now this last equals
$$
\int_\Omega K(z,\zeta) c \cdot f^j_{\widetilde{w}}(\zeta) \, dV(\zeta) + {\cal E}(z,w)
= f^j_{\widetilde{w}}(z) + {\cal E}(z,w) \, ,
$$
where $c > 0$ is a constant, $j$ (interpreted as a {\it power}) is a suitably chosen positive integer, and ${\cal E}(z,w)$ is an error term.
Now we know that the first term in this last displayed expression {\it does not vanish} on $\Omega$ intersect
a ball about $w$ that has radius larger than $\tau$ and the error term is negligible in
this regard---because the Bergman projection of $\sum_j \alpha_j \left ( \frac{1}{\eta_j^4 \cdot \Omega_4} \right ) \chi_{B(w,\eta_j)}(\zeta)$ is,
by inspection, approximated closely in the uniform topology by the dilated peaking function.
Thus Bergman representative coordinates (see [6, Section 4.2] for this concept), which are given by
$$
b_{j, w}(z) = \frac{\partial}{\partial \overline{\zeta}_j} \log \frac{K(z,\zeta)}{K(\zeta, \zeta)} \biggr |_{\zeta = w} \, ,
$$
are well defined on $\beta_w \cap \Omega$ with $\beta_w = B(w, \tau')$ for some $\tau' > \tau$. And the size of $b_{j,w}$ may be taken to be
uniformly bounded, independent of $w$, just by the noted regularity properties of the Bergman kernel. Of course $L \cap \beta_w \ne \emptyset$.
Now fix a multi-index $\alpha$. Then certainly $|(\partial^\alpha/\partial z^\alpha) \varphi(z)|$ is bounded
by some $M_\alpha$ for all $\varphi \in \Aut(\Omega)$ and all $z \in L$. But then the Bergman representative
coordinates enable us to realize each automorphism as a linear map (namely, the Jacobian---again see [6, Section 4.2])
on $\beta_w \cap \Omega$. And the size
of the coefficients of these linear maps depends only on the Jacobian of the automorphism at the center
of the ball. Of course the center of the ball lies in a compact subset of $\Omega$, so these Jacobians
have uniformly bounded coefficients. The conclusion then is that
$|(\partial^\alpha/\partial z^\alpha) \varphi(z)|$ is uniformly bounded on $L \cup \beta_w$.
And the bound is independent of $w$. Remembering that $w$ is an arbitrary element of
$\partial L$, we see that $|(\partial^\alpha/\partial z^\alpha)\varphi(z)|$ is uniformly
bounded on all of $\Omega$, uniformly for all $\varphi \in \Aut(\Omega)$.
\end{proof}%\smallskip \\
\section{Topologies on domains}
Let $\Omega = \{z \in \CC^n: \rho(z) <0\}$ be a domain
with $C^k$ boundary. Let $\epsilon > 0$. We define
an $\epsilon$-neighborhood of $\Omega$ in the $C^k$ topology
to be a set of the form
$$
{\cal E}_{\Omega, \epsilon} = \biggl \{ \Omega^* \subseteq \CC^n: \Omega^* = \{z \in \CC^n: \rho^*(z) <0\} \
\hbox{and} \ \|\rho - \rho^*\|_{C^k} < \epsilon \biggr \} \, .
$$
Note particularly that ${\cal E}_{\Omega, \epsilon}$ is a {\it set of domains}. Our semicontinuity
results below will be formulated in terms of this topology on the collection
of domains with $C^k$ boundary. In particular, when we speak of a ``small $C^k$ perturbation
of $\Omega$,'' we mean a domain selected from ${\cal E}_{\Omega, \epsilon}$ with $\epsilon > 0$
small. For convenience, when $\Omega'$ is an element of ${\cal E}_{\Omega, \epsilon}$, then
we say that $\Omega'$ has $C^k$ distance less than $\epsilon$ from $\Omega$.
\section{The semicontinuity theorem}
Now one of the main results of this paper is the following:
\begin{theorem} \sl
Let $\Omega$ be a smoothly bounded, finite type domain in $\CC^2$
which has compact automorphism group in the compact-open topology.
Let $k$ an integer be sufficiently large.
Then there is an $\epsilon > 0$ so that if $\Omega'$ is a smoothly
bounded, finite type domain with $C^k$ distance less than $\epsilon$
from $\Omega$, then $\Aut(\Omega')$ can be realized as a subgroup of $\Aut(\Omega)$.
By this we mean that there is a smooth diffeomorphism $\Phi: \Omega' \ra \Omega$ so that
$$
\varphi \longmapsto \Phi \circ \varphi \circ \Phi^{-1}
$$
is a univalent homomorphism of $\Aut(\Omega')$ into $\Aut(\Omega)$.
\end{theorem}
\begin{proof} The proof of this result is standard (see [7, Theorem 0.1], so we only sketch the
steps.
\smallskip% \\
\noindent {\bf Step 1:} There is a Riemannian metric, smooth on $\overline{\Omega}$,
which is invariant under any automorphism of $\Omega$. We construct this
metric simply by averaging the Euclidean metric with respect to Haar
measure on the automorphism group of $\Omega$. In order for the resulting
metric to be smooth to the boundary, we must invoke the uniform bounds
on automorphism derivatives that we proved in Section 2.
\smallskip% \\
\noindent {\bf Step 2:} The metric in Step 1 can be modified so
that it is a product metric near the boundary, and still invariant.
This is a standard construction from Riemannian geometry, and we
omit the details.
\smallskip% \\
\noindent {\bf Step 3:} We may form the metric double $\widehat{\Omega}$ of $\overline{\Omega}$,
and the resulting metric is smooth on $\widehat{\Omega}$.
\smallskip% \\
\noindent {\bf Step 4:} Any automorphism of $\Omega$ can now be realized as an
isometry of $\widehat{\Omega}$.
\smallskip% \\
\noindent {\bf Step 5:} By a classical result of David Ebin [3, Section 1], there is
a semicontinuity result for isometries of compact Riemannian manifolds.
We may apply this result to the isometry group of $\widehat{\Omega}$. In
particular, any smooth deformation $\Omega'$ of $\Omega$ gives rise
to a smooth deformation $\widehat{\Omega'}$ of $\widehat{\Omega}$ and
hence to a deformation of the invariant metric on $\widehat{\Omega}$.
Thus we may compare the isometry group of the perturbed metric to the
isometry group of the original metric.
\smallskip% \\
\noindent {\bf Step 6:} We may unravel the construction to see that
Step 5 may be interpreted to say that the automorphism group of $\Omega'$
is a subgroup of the automorphism group of $\Omega$, and we may extract
the conjugation map $\Phi$ from the conjugation map provided by Ebin's
theorem.
\smallskip% \\
That completes the argument.
\end{proof}
%\endpf
%\smallskip \\
Since we introduced the $C^k$ metric for the space of automorphisms,
it is worthwhile to formulate a result for that topology. We have:
\begin{theorem} \sl
Let $\Omega$ be a smoothly bounded, finite type domain in $\CC^n$.
Equip $\Aut(\Omega)$ with the $C^k$ topology, some integer $k \geq 0$.
Assume that $\Omega$ has compact automorphism group in the $C^k$ topology.
Then there is an $\epsilon > 0$ so that if $\Omega'$ is a smoothly
bounded, finite type domain with $C^m$ distance less than $\epsilon$
from $\Omega$ (with $m \leq k$), then $\Aut(\Omega')$ can be realized as a subgroup of $\Aut(\Omega)$.
By this we mean that there is a smooth mapping $\Phi: \Omega' \ra \Omega$ so that
$$
\varphi \longmapsto \Phi \circ \varphi \circ \Phi^{-1}
$$
is a univalent homomorphism of $\Aut(\Omega')$ into $\Aut(\Omega)$.
\end{theorem}
\begin{proof} The proof is just the same as that for the
last theorem. The main point is to have a uniform bound
for derivatives of automorphisms (Proposition 2.5), so that the smooth-to-the-boundary
invariant metric can be constructed.
\end{proof}
%\endpf
%\smallskip \\
\section{Some examples}
In this section we provide some examples which bear on the context of Theorems 3.1 and 3.2.
\begin{example} \rm
Let
$$
\Omega= B(0,2) \setminus \overline{B}(0,1) \, .
$$
Then $\Omega$ is a bounded domain, but it is not pseudoconvex.
Of course any automorphism of $\Omega$ continues analytically to $B(0,2)$. But
it also must preserve $S_1 \equiv \{z: |z| = 1\}$ and $S_2 \equiv \{z: |z| = 2\}$. It follows
that $\Aut(\Omega) = U(n)$. Now an obvious Lie subgroup of $U(n)$ is $SU(n)$. But
$SU(n)$ has precisely the same orbits as $U(n)$---in fact the orbit of any point
in $S_2$ is $S_2$ itself and the orbit of any point in $S_1$ is $S_1$ itself.
It follows that there is no domain that is ``near'' to $\Omega$ in any $C^k$ topology
and with automorphism group that is precisely $SU(n)$. Therefore an obvious sort
of converse to Theorems 4.1, 4.2 fails in this case. That is to say, not every closed
subgroup of the automorphism group of $\Omega$ arises as the automorphism group
of a nearby domain.
We note, however, that with suitable hypotheses (including strong pseudoconvexity), there
is a sort of converse to Theorem 4.1---see [13, Section 1].
\end{example}
\begin{example} \rm
If we do not mandate that the domain $\Omega$ have smooth boundary, then Theorems 3.1 and 3.2
need not be true. As a simple example, consider
$$
\Omega = \{z \in \CC^n: 0 < |z| < 1\} \, .
$$
Of course this $\Omega$ is not pseudoconvex and does not have a smooth
defining function (so does not have smooth boundary by our reckoning). The automorphism group
of $\Omega$ is $U(n)$. A ``small'' perturbation of $\Omega$
is $\Omega' = B = \{z \in \CC^n: |z| < 1\}$. But the automorphism
group of $\Omega'$ is much larger than $U(n)$ (it includes $U(n)$, but
it also includes the M\"{o}bius transformations). So semicontinuity
of automorphism groups fails.
\end{example}
\section{Closing remarks} \rm
The idea of semicontinuity for automorphism groups is an important paradigm that
has far-reaching applicability in geometry. In any situation where symmetries
are considered, one may formulate the idea of semicontinuity. The basic
idea is that symmetry is hard to create but easy to destroy: small perturbations
can and will reduce symmetry, but it takes a large perturbation to create symmetry.
In the present paper we have taken a fundamental theorem of [GK1, Theorem 0.1] in the strongly
pseudoconvex setting and extended it in various ways to the finite type setting.
It would be interesting to know whether the result is true in complete generality.
Even more interesting would be an example---say in the infinite type context---in
which semicontinuity fails.
We hope to explore these matters further in future papers.
\bigskip \bigskip \\
\begin{thebibliography} \rm
\bibitem{BEF} E. Bedford and J. E. Forn\ae ss, A
construction of peak functions on weakly pseudoconvex domains,
{\it Ann. Math.} 107(1978), 555--568.
\bibitem{BEL} S. R. Bell, Biholomorphic mappings and the
$\dbar$ problem, {\it Ann. Math.}, 114(1981), 103--113.
\bibitem{EBI} D. Ebin, The manifold of Riemannian metrics,
1970 Global Analysis ({\it Proc.\ Sympos.\ Pure Math.}, Vol. XV,
Berkeley, Calif., 1968), pp.\ 11--40, Amer.\ Math.\ Soc.,
Providence, R.I.
\bibitem{FEF} C. Fefferman, The Bergman kernel and biholomorphic mappings
of pseudoconvex domains, {\it Invent. Math.} 26(1974), 1--65.
\bibitem{FOM} J. E. Forn\ae ss and J. McNeal, A
construction of peak functions on some finite type domains.
{\it Amer.\ J.\ Math.} 116(1994), no. 3, 737--755.
\bibitem{GKK} R. E. Greene, K.--T. Kim, and S. G. Krantz,
{\it The Geometry of Complex Domains}, Birkh\"{a}user
Publishing, Boston, MA, 2011, to appear.
\bibitem{GRK1} R. E. Greene and S. G. Krantz, The
automorphism groups of strongly pseudoconvex domains, {\it
Math. Annalen} 261(1982), 425--446.
\bibitem{GRK2} R. E. Greene and S. G. Krantz,
Biholomorphic self--maps of domains, {\it Complex Analysis}, II
(College Park, Md., 1985--86), 136--207, Lecture Notes in
Math., 1276, Springer, Berlin, 1987.
\bibitem{HEL} S. Helgason, {\it Differential Geometry and Symmetric Spaces},
Academic Press, New York, 1962.
\bibitem{KOB} S. Kobayashi, {\it Hyperbolic Manifolds and Holomorphic
Mappings}, Dekker, New York, 1970.
\bibitem{KRA1} S. G. Krantz, {\it Function Theory of Several Complex Variables},
$2^{\rm nd}$ ed., American Mathematical Society, Providence, RI, 2001.
\bibitem{KRA2} S. G. Krantz, {\it Partial Differential
Equations and Complex Analysis}, CRC Press, Boca Raton, FL,
1992.
\bibitem{MIN} B.--L. Min, Domains with prescribed
automorphism group, {\it J.\ Geom.\ Anal.} 19 (2009),
911--928.
\bibitem{NAR} R. Narasimhan, {\it Several Complex
Variables}, University of Chicago Press, Chicago, 1971.
\bibitem{WON} B. Wong, Characterizations of the ball in
$\CC^n$ by its automorphism group, {\it Invent. Math.}
41(1977), 253--257.
\end{thebibliography}
\vspace*{.95in}
%\small
%\noindent \begin{quote}
%Department of Mathematics \\
%Washington University in St. Louis \\
%St.\ Louis, Missouri 63130 \\
%{\tt sk@math.wustl.edu}
%\end{quote}
\par\newpage$\quad$\end{document}
\end{document}