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\ \vskip -1.5cm
{\noindent Complex Geometry,
Lect. Notes in Pure and Appl.
Math. {\bf 143}, 67-76, Dekker, 1993}
\vskip .7cm
\topmatter
\title
Scalar pseudo-hermitian invariants and the Szeg\"o kernel\\
on three-dimensional CR manifolds
\endtitle
\author
Kengo Hirachi
\endauthor
\affil
Department of Mathematics,
Osaka University\\
Toyonaka, Osaka, 560, Japan\endaffil
\footnote"{}"{This work was supported by Grant-in-Aid
for Scientific Research, The Ministry of Education,
Science and Culture.}
\endtopmatter
\document
\noindent{\bf Introduction}\vskip 5pt\noindent
Let $M$ be a three-dimensional strictly pseudoconvex CR
manifold which bounds a relatively compact domain in
$\Bbb C^2$. We fix a Levi metric on $M$, which is called a
pseudo-hermitian structure, and define the Szeg\"o
kernel with respect to the volume element associated with
the metric. In this note, we give an invariant-theoric
characterization of $\psi_0$ the first invariant in the
logarithmic term of Fefferman's asymptotic expansion of
the Szeg\"o kernel, and write down the invariant in
terms of geometric local pseudo-hermitian invariants.
In computing the invariant, we also find that
the transformation law of $\psi_0$, under the
change of pseudo-hermitian structure, can be expressed by
using a fourth-order linear differential operator $\Cal
C_\th$ on $M$,
which was introduced in [GL] as the
compatibility operator for a degenerate Dirichlet
problem (for example, in case $M$ is the sphere $\Cal
C_\th=\square_b\c\square_b$, where $\square_b$ is the
Kohn Laplacian). See Corollary 1 below. This formula
enables us to reduce the analysis of $\psi_0$ to that of
the differential equation $\Cal C_\th f=g$. In
particular, by studying the global solutions to $\Cal
C_\th f=0$, we can show that $\psi_0$ vanishes globally
on $M$ if and only if the Szeg\"o kernel is defined with
respect to an invariant volume element introduced in
[F2], see also [H], under the assumption that $M$ has a
transversal symmetry. See Corollary 2 below.
For the Szeg\"o kernel defined with respect to the
invariant volume element, we have obtained, in
[HKN], some detailed results on its
asymptotic expansion. Thus, if we combine the results on the
invariant Szeg\"o kernel and the above characterization
of the invariant volume element, we can prove that the
Szeg\"o kernel has no logarithmic singularity if and
only if $M$ is spherical,
without assuming any condition on the choice of volume
element. See Corollary 3. In this characterization of
spherical surfaces, the assumption that the logarithmic
term vanishes {\it globally\/} is essential, while in
the case of the Bergman kernel, this type of result
holds under a local vanishing condition of the
logarithmic term on a piece of the boundary, see [G,
Theorem 3.2] and [B]. We can not expect such a localized
result for the Szeg\"o kernel, without specifying the
choice of volume element as in [HKN, Remark 2] or
[Ha], see Remark 1.1 below. A counter example can
be found in [Fu], see Remark 1.2 below.
The plan of this note is as follows: after a review of
some geometric identities on pseudo-hermitian manifolds
in \S 2, we first use in \S 3 Weyl's classical invariant
theory for the unitary group $U(1)$ to see that the first
invariant in the logarithmic term $\psi_0$ is written as
a linear combination of the ``Weyl invariants"
constructed from the curvature and the torsion of the
Tanaka-Webster connection. To determine the coefficients
in the linear combination, we derive in \S 4 the
transformation law of $\psi_0$ under a change of
pseudo-hermitian structure, which is forced by the
transformation law of the Szeg\"o kernel, and show in \S
5 that the transformation law uniquely characterizes
$\psi_0$ up to a constant multiple. The evaluation of the
constant will be done in \S 6. Finally, in \S 7, we discuss
the equation $\Cal C_\th f=0$ and give the proofs of
corollaries.
\beginsection{1. Results}
\noindent
Let $\Omega\subset \Bbb C^2$ be a
strictly pseudoconvex domain with smooth boundary $M$. We
fix a pseudo-hermitian structure on $M$ by giving a contact
form $\th$. Then it was shown in [F1] and [B-S] that the
Szeg\"o kernel defined with respect to the volume element
$\th\wedge d\th$ has the asymptotic expansion
$$
S(z,\c z)=\varphi (z)\rho
(z)^{-2}+\psi (z)\log \rho (z)
\quad\text{with}\quad\varphi,\psi\in C^\infty(\c \Omega),
$$
where $\rho$ is a defining function of $\Omega$ with
$\rho>0$ in $\Omega$.
\proclaim{Main Theorem}
The boundary value of the log term coefficient
$\psi_0=\psi|_M$ is
given by
$$
\psi_0=\frac1{24\pi^2}
(\Delta_bR-2\, \text{\rm Im}A_{11,}^{\ \ \ 11}),
$$
where $\Delta_b$ is the sublaplacian, $R$ is the Webster
scalar curvature, and $A_{11,}^{\ \ \ 11}$ is the
contraction of the second covariant derivative of the
Webster torsion.
\endproclaim
In proving this formula, we also get the transformation law
of $\psi_0$:
\proclaim{Corollary 1}
If $\widetilde\th=e^{2f}\th$ is another pseudo-hermitian
structure, then $\psi_0$ transforms according to
$$
\widetilde \psi_0=
e^{-4f}(\psi_0+\frac1{2\pi^2} \Cal C_\th f),
\quad\text{where}\quad
\Cal C_\th f=f_{\c 1\ 1}^{\ \c1 \
1}+i(A_{11}f^1)^{,1}.
$$
Here the indices $1$ and $\c1$ indicate the covariant
derivatives, see \S2 for definition.
\endproclaim
By evaluating the explicit formula of $\psi_0$, we can prove
$\psi_0=0$ if $\th\wedge d\th$ is the invariant volume
element up to multiplication by a CR-pluriharmonic function,
see \S 7 below; this fact also follows from Graham's result
[G],
see [HKN]. Moreover, in case $\Omega$ has a transversal
symmetry, we can also prove its converse.
\proclaim{Corollary 2}
Assume that $\Omega$ has a transversal symmetry
{\rm (}see {\rm [GL])}.
Then $\psi_0=0$ if and only if $\th\wedge d\th$ is
the invariant volume element
up to multiplication by a CR-pluriharmonic function.
\endproclaim
Combining this corollary and [HKN, Remark 2], we get
\proclaim{Corollary 3}
Assume that $\Omega$ has a transversal symmetry and that
$\psi=O(\rho^3)$. Then the boundary $M$ is spherical,
i.e. locally CR-isomorphic to the sphere.
\endproclaim
\newpage
\demo{Remark 1.1}
In Hanges [Ha], he computed $\psi_0$ for the domains in
$\Bbb C^2$ with transversal symmetries. The computation was
done with respect to a specific choice of volume element. By
using the formula, he proved a similar result to our
corollary 3 in case the Szeg\"o kernel is defined with
respect to that specific volume element.
\enddemo
\demo{Remark 1.2}
Fuks [Fu] derived the explicit formula for the Szeg\"o
kernel of the tube domain
$\{(z,w)\in\Bbb C^2:\text{Im}z\,\text{Im}w>1,\
\text{Im}z>0\}$
with a suitable choice of volume element on the boundary,
see the formula (4.126) in [Fu].
This Szeg\"o kernel contains no logarithmic singularity.
Since we can check that this Szeg\"o kernel satisfies the
holonomic system given by Kashiwara's theorem, see
[HKN, \S2], the singularity of the Szeg\"o kernel can
be localized, and thus we can make an example
of a bounded domain for which the logarithmic term of the
Szeg\"o kernel vanishes on a non-spherical piece of its
boundary.
\enddemo
\beginsection{2. Pseudo-hermitian geometry}
\noindent
Let $M$ be a $C^\infty$ real three-dimensional
manifold. A {\it CR structure\/} on $M$ is a complex
one-dimensional subbundle $T^{1,0}\subset\Bbb C TM$
satisfying $T^{1,0}\cap \overline{T^{1,0}}=\{ 0\}$.
We will always assume that the structure is
{\it strictly pseudoconvex\/}: for some choice of a real
one-form $\th$ annihilating $T^{1,0}$, the {\it Levi
form\/} $L_\th (V,\c W)=-id\th(V\wedge \c W)$
gives a hermitian metric on $T^{1,0}$.
A choice of a such one-form
$\th$ defines a {\it pseudo-hermitian structure\/} on $M$,
and it induces a natural linear
connection called the {\it Tanaka-Webster connection\/}
[T], [W].
We shall quickly review the definition.
Let $\{T,Z_1,Z_{\c1}\}$ be a frame of $\Bbb C T$, where
$Z_1$ is any local frame of $T^{1,0}$,
$Z_{\c 1}=\c{Z_1}$
and $T$
is the {\it characteristic vector field\/}, that is, the
unique real vector field such that
$\th (T)=1,\ T\rfloor d\th =0$.
Then
$\{\th,\th^1,\th^{\c1}\}$,
the coframe dual to
$\{T,Z_1,Z_{\c1}\}$,
satisfies
$$
d\th =i h_{1\c1}\th^1\wedge\th^{\c1}
$$
for some positive function
$h_{1\c1}$.
We call the one-form
$\th^1$
an {\it admissible coframe}.
In terms of this frame, the Tanaka-Webster connection
$\nabla$ is defined by the relations
$$
\gathered
\nabla Z_1={\omega_1}^1\otimes Z_1,\ \ \
\nabla Z_\c1={\omega_\c1}^\c1\otimes Z_\c1,\ \ \
\nabla T=0,\\
d\th^1=\th^1\wedge{\omega_1}^1+
{A_\c1}^1\th\wedge\th^\c1,\ \ \
\omega_{1\c1}+\omega_{\c1 1}=dh_{1\c1}
\endgathered
$$
for a one-form ${\omega_1}^1$,
with ${\omega_\c1}^\c1=\overline{{\omega_1}^1}$,
and a function ${A_\c1}^1$, called the
{\it Webster torsion}\/.
\par
We denote the components of covariant derivatives of a
tensor by indices proceeded by a comma; as in
$
\nabla (A_1^{\,\ \c1}Z_{\c 1}\otimes\th^1 )=
(A_{1\,\ ,1}^{\,\ \c1}\th^1+
A_{1\,\ ,\c1}^{\,\ \c1}\th^{\c 1}+
A_{1\,\ ,0}^{\,\ \c1}\th)\otimes
Z_{\c 1}\otimes\th^1 .
$
For a scalar
functions, we usually omit the comma.
The structure equation
for the Tanaka-Webster connection [L2] is then given by
$$
d{\omega_1}^1=R\ h_{1\c1}\th^1\wedge\th^\c1+
A_{1\,\ ,\c1}^{\,\ \c1}\th^1\wedge\th-
A_{\c1\,\ ,1}^{\,\ 1}\th^\c1\wedge\th,
\tag2.1
$$
where $R$ is a function called the {\it Webster
curvature\/}, and the {\it Bianchi identity\/} is
$$
R_{,0}={A_{11,}}^{11}+{A_{\c1\,\c1,}}^{\c1\,\c1}.
\tag2.2
$$
Here we use $h_{1\c1}$ and its
inverse
$h^{1\c1}$
in the usual way to raise and lower indices.
\par
Now we shall collect some fundamental formulas used below.
First recall [L2, Lemma 2.3] that the covariant derivatives
of a function $u$ satisfy
$$
u_1^{\ \, 1}-u^1_{\ 1}=iu_0,\quad
u_{01}-u_{10}=A_{11}u^1,\quad
{u_{11}}^1-u_{1\ 1}^{\ 1}=iu_{10}+R\ u_1.
\tag2.3
$$
The transformation law of the connection
under a change of pseudo-hermitian structure was
computed in
[L1, \S5]. Let
$\widetilde\th=e^{2f}\th$
be a new pseudo-hermitian structure. Then we can define
an admissible coframe by
${\widetilde\th}^1=e^f(\th^1+2if^1\th)$.
With this coframe,
the connection form and the torsion tensor are given by
$$
\gather
{\widetilde \omega}_1^{\ 1}
={\omega_1}^1+3(f_1\th^1-f_\c1\th^\c1)+
i({f_1}^1+{f_\c1}^\c1+8f_1f^1)\th,
\tag2.4
\\
{\widetilde A}_{11}=e^{-2f}(A_{11}+2if_{11}-4if_1f_1),
\tag2.5
\endgather
$$
and thus the Webster curvature transforms as
$$
\widetilde R=e^{-2f}(R-4f_1^{\ 1}-4f_\c1^{\ \,\c1}-8f_1f^1).
\tag2.6
$$
Here covariant derivatives in the right sides are
taken with respect to the pseudo-hermitian structure
$\th$ and an admissible coframe
$\th^1$.
Note also that the dual frame of
$\{\widetilde\th,{\widetilde\th}^{\;1},
{\widetilde\th}^{\;\c1}\}$
is given by
$\{\widetilde T,{\widetilde Z}_1,{\widetilde Z}_\c1\}$,
where
$$
\widetilde T=e^{-2f}(T+2if^\c1Z_\c1-2if^1Z_1),
\quad {\widetilde Z}_1=e^{-f}Z_1.
$$
Finally we shall recall [S, Lemma 1.8] the transformation
law of the {\it sublaplacian} $\Delta_b$ on functions defined
by $\Delta_b u=-(u_1^{\ \,1}+u_\c1^{\ \,\c1})$.
If we denote by ${\widetilde\Delta}_b$ the sublaplacian
associated with $\widetilde\th$, then we get
$$
{\widetilde\Delta}_b u=e^{-2f}(\Delta_b u-2f^1u_1-
2f^{\c1}u_{\c1}).
\tag2.7
$$
\beginsection{3. Scalar pseudo-hermitian invariants}
\noindent
Our proof of Main Theorem begins with a study of scalar
pseudo-hermitian invariants; as we see in the lemma below,
our object $\psi_0$ is a scalar pseudo-hermitian invariant.
By a {\it scalar pseudo-hermitian invariant\/} we mean a
polynomial $Q$ in the components of the curvature and the
torsion of the Tanaka-Webster connection and their
covariant derivatives which is invariant under a change
of frame of $T^{1,0}$. A scalar pseudo-hermitian invariant
defines a $C^\infty$ function $Q(\th)$ on each
pseudo-hermitian manifold $(M,\th)$ by evaluating the
polynomial at each point. We also call this assignment of
functions a scalar pseudo-hermitian invariant and will
identify two invariant polynomials if they define the same
assignment.
The simplest scalar pseudo-hermitian invariant is of
course the Webster curvature $R$. Other examples of scalar
pseudo-hermitian invariants can be constructed out of $R,
A_{11},A_{\c1 \;\c1}$ and their covariant derivatives in
$Z_1$ and $Z_\c1$ by taking tensor products and contracting:
$$
\text{\rm contraction}(R_{,\alpha_1\cdots\alpha_n}
\otimes\cdots\otimes
R_{,\alpha'_1\cdots\alpha'_m}\otimes
A_{\beta_1\beta_2,\beta_3\cdots\beta_p}
\otimes\cdots\otimes
A_{\beta'_1\beta'_2,\beta'_3\cdots\beta'_q})
$$
is a scalar pseudo-hermitian invariant, called a {\it Weyl
invariant}\/, for any choice of indices such that the numbers
of $1$ and $\c1$ are the same. The contraction is taken with
respect to the Levi metric $h_{1\c1}$ for some pairing of
holomorphic and antiholomorphic indices.
Since a scalar pseudo-hermitian invariant is an
invariant polynomial under
the action of the unitary group $U(1)$, the classical
invariant theory identifies all such invariant polynomials.
This leads to the conclusion that every scalar
pseudo-hermitian invariant is a linear combination of the
Weyl invariants. Note that the Weyl invariants contain no
terms containing covariant derivatives in $T$; such terms
can be expressed as the polynomials in the components of the
tensors which contain no $T$ covariant derivatives, by
using, e.g. (2.2) and (2.3).
\proclaim{Lemma 3.1}
The first invariant in the log term of the Szeg\"o
kernel $\psi_0$ is a scalar pseudo-hermitian
invariant, and thus $\psi_0$ is written as a linear
combination of the Weyl invariants.
\endproclaim
\demo{\it Proof}
In [BGS, Theorem 7.30] the same type of result was
proved in the case of the asymptotic expansion of
the heat kernel for $\square_b$.
The argument used there can be also applied to our case, if
we fix a defining function (or a phase function) and employ
the algorithm of computing the Szeg\"o kernel given in
[B-S, \S 4] or [HKN].
This implies that $\psi_0$ is written as a polynomial in
the components of the curvature, the torsion tensor, and
their covariant derivatives.
Since we know that $\psi_0$ is independent of a choice of
defining function, $\psi_0$ must be an invariant polynomial.
\qed
\enddemo
\beginsection{4. Transformation law of the Szeg\"o kernel}
\noindent
This section derives the following transformation law.
\proclaim{Proposition 4.1} Let $S$ and $\widetilde S$ be the
Szeg\"o kernels defined with respect to pseudo-hermitian
structures $\th$ and $\widetilde \th$ on $M$ respectively.
If we have $\widetilde \th =e^{2f}\th$ for a CR-pluriharmonic
function in a neighborhood of $p$ in $M$,
then there exists a neighborhood $U$ of $p$ in $\Bbb C^2$
such that
$$
\widetilde S (z,\c z)\equiv e^{-4f(z)}S(z,\c z)
\quad\text{\rm mod}\quad C^\infty (U\cap\c\Omega),
\tag4.2
$$
where $f(z)$ is identified with its
pluriharmonic extension to $U\cap\Omega$.
In particular, the log term coefficients of $S$ and
$\widetilde S$ satisfy
$$
\widetilde\psi (z)=e^{-4f(z)}\psi (z) \quad
\text{\rm on}\quad U\cap M.
\tag4.3
$$
\endproclaim
\demo{\it Proof}
It was shown in [B-S] that the Szeg\"o projector on
$(M,\th)$ is micro-locally characterized as
the unique Fourier integral operator $\Bbb S$, which we
call the {\it local Szeg\"o projector}\/, satisfying
$$
\Bbb S\sim\Bbb S^*\sim \Bbb S^2,\quad
\c \pa_b \Bbb S\sim 0,\quad
\text{\rm Id}\sim\Bbb S+\Bbb L\c \pa_b
\quad
\text{for some regular operator }\Bbb L.
\tag4.4
$$
Here ``$\sim$" means
that the operators of each side differs
by an operator of degree $-\infty$ (i.e. an operator with
$C^\infty$ kernel function).
We shall construct operators $\Bbb S'$ and $\Bbb L'$
which satisfy (4.4), near $p$, with respect to the
volume element $\widetilde\th\wedge d\widetilde\th$
from $\Bbb S$ and $f$.
\par
Let $F$ be a function on $M$ which is
CR-holomorphic in a neighborhood $p$ and satisfies
$\text{\rm Re}F=2f$, and we regard $e^{-F}$ as the operator,
on functions and one-forms,
defined by the multiplication $u\mapsto e^{-F}u$. Then
$
e^{-F}:L^2(M,\th\wedge d\th)\to
L^2(M,\widetilde\th\wedge d\widetilde\th)
$
is unitary, and we have
$
\c\pa_b(e^{F}u)=e^{F}\,\c\pa_b u
$
for any function $u$ with support in a small neighborhood of
$p$.
Thus we see that
$\Bbb S'=e^{-F}\ \Bbb S\ e^{F}$ and
$\Bbb L'=e^{-F}\ \Bbb L\ e^{F}$ satisfy (4.4), near $p$,
with respect to the volume element
$\widetilde\th\wedge d\widetilde\th$.
Therefore, we get
$\widetilde{\Bbb S}\sim e^{-F}\ \Bbb S\ e^{F}$, near $p$,
by the uniqueness of the local Szeg\"o projector.
If we rewrite this formula in terms of kernel functions, we
get (4.2). \qed
\enddemo
\beginsection{5. Invariant-theoric characterization of
$\psi_0$}
\noindent
In \S\S3 and 4 we have shown that $\psi_0$ is a scalar
pseudo-hermitian invariant and satisfies the transformation
law (4.3). In this section we show that the transformation
law uniquely determines the scalar
pseudo-hermitian invariant $\psi_0$ up to a constant
multiple.
\proclaim{Theorem 5.1}
Let $Q$ be a scalar pseudo-hermitian invariant on
three-dimensional CR manifolds which satisfies the
transformation law
$$
Q(e^{2f}\th )=e^{-4f}Q(\th )\quad
\text{for any CR-pluriharmonic function }f.
\tag5.2
$$
Then there exists a constant $c$ such that
$$
Q=c \ (\Delta_b R-2\,\text{\rm Im}{A_{11,}}^{11}).
\tag5.3
$$
\endproclaim
We begin by showing that the right side of (5.3) satisfies
the transformation law (5.2).
\proclaim{Lemma 5.4}
The divergence of the one-form
$
W_1\th^1=(R_{,1}-i{A_{11,}}^{1})\th^1
$
is written as
$$
W_{1,}^{\ \; 1}=-\frac12\Delta_b R+
\text{\rm Im}{A_{11,}}^{11}
\tag5.5
$$
and, if $\widetilde \th=e^{2f}\th$, then $W_1$ and
${W_{1,}}^1$ transform as follows:
$$
\gather
\widetilde W_1=e^{-3f}(W_1-6P_1f), \quad \text{where} \quad
P_1f=f_{\c1 \ \, 1}^{\ \c1}+iA_{11}f^1,
\tag5.6
\\
\widetilde W_{1,}^{\ \;1}=
e^{-4f}(W_{1,}^{\ \;1}-6 C_\th f),\quad \text{where}\quad
C_\th f={(P_1f),}^1.
\tag5.7
\endgather
$$
\endproclaim
In [L2, Proposition 3.4] it was shown that a $C^\infty$ real
function on an open set $U$ satisfies $P_1f=0$ if and only
if $f$ is CR-pluriharmonic on $U$.
Thus (5.7) implies that $W_{1,}^{\ \;1}$ satisfies the
transformation law (5.2).
\demo{\it Proof of Lemma 5.4}
In view of (2.2) and
$R_{,1}^{\ \ 1}-R^{\ 1}_{, \ \; 1}=iR_{,0}$
which follows from (2.3),
we have
$$
\align
W_{1,}^{\ \;1}
&=R_{,1}^{\ \ 1}-i{A_{11,}}^{11}
=\frac12(R_{,1}^{\ \ 1}+R^{\ 1}_{, \ \; 1}
-i{A_{11,}}^{11}+i{A_{\c1\,\c1,}}^{\c1\,\c1})\\
&=-\frac12\Delta_b R+\text{\rm Im}{A_{11,}}^{11}.
\endalign
$$
This proves (5.5).
To simplify the computation of the transformation laws, we
shall work with an admissible coframe $\th^1$ for which
$\omega_1^{\ 1}=0$ at a point $p\in M$, so that the first
covariant derivatives at $p$ are equal to ordinary
derivatives, see [L2, Lemma 2.1]. At the point $p$,
we compute
$$
\align
{\widetilde R}_{,1}
&=\widetilde Z_1 \widetilde R=
e^{-f}Z_1e^{-2f}(R-4f_1^{\ 1}-4f_\c1^{\ \c1}-8f_1f^1)\\
&=e^{-3f}(R_{,1}-2Rf_1-4f_{1\ 1}^{\ 1}-4f_{\c1 \ 1}^{\ \c1}
-8f_{11}f^1+8f_1^{\ 1}f_1+16f_1f_1f^1),\\
i\widetilde A_{11,\c1}
&=i\left(
\widetilde Z_\c1-2\widetilde\omega_1^{\ 1}
(\widetilde Z_\c1)\right)
\widetilde A_{11}\\
&=ie^{-f}(Z_\c1+6f_\c1)e^{-2f}(A_{11}+2if_{11}-4if_1f_1)\\
&=e^{-3f}(iA_{11,\c1}+4iA_{11}f_\c1-2f_{11\c1}+8f_{1\c1}f_1-8f_{11}f_\c1
+16f_1f_1f_\c1).
\endalign
$$
Contracting the second equation with respect to the
Levi metric $\widetilde h_{1\c1}=h_{1\c1}$, we get
$$
\widetilde R_{,1}-i\widetilde A_{11,}^{\ \ 1}=
e^{-3f}(R_{,1}-i{A_{11,}}^1 -4f_{1\ \,1}^{\ 1} -4f_{\c1\
\,1}^{\ \c1}+2f_{11}^{\ \ 1} -2Rf_1-4iA_{11}f^1).
$$
So using
$
-4f_{1\ 1}^{\ 1}
+2f_{11}^{\ \ 1}
-2Rf_1=
-2f_{\c1\ \,1}^{\ \c1}-2iA_{11}f^1,
$
which follows from (2.3), we obtain (5.6). To prove (5.7),
by using (2.4), we compute
$$
\align
\widetilde W_{1,\c1}
&=\left(\widetilde Z_\c1-\widetilde\omega_1^{\ 1}
(\widetilde Z_\c1)\right)\widetilde W_1
=e^{-f}(Z_\c1+3f_\c1)e^{-3f}(W_1-6P_1f)\\
&=e^{-4f}Z_\c1(W_1-6P_1f)
=e^{-4f}\left(W_{1,\c1}-6(P_1f)_{,\c1}\right).
\endalign
$$
If we contract this equation, we get (5.7).
\qed
\enddemo
\demo{\it Proof of Theorem 5.1}
Suppose we have a scalar pseudo-hermitian invariant $Q$
satisfying (5.2).
First we consider the effect of a {\it change of scale}
in the Levi metric, that is, to consider the case $f$ is a
constant function.
Then, as in [BGS, \S 8] and [S, \S5], we see that the
possible Weyl invariants in $Q$ are $$
R_{,1}^{\ \ \,1},\ \
R_{,\c1}^{\ \ \ \,\c1},\ \
{A_{11,}}^{11},\ \
{A_{\c1\,\c1,}}^{\c1\,\c1},\ \
|R|^2,\ \
|A_{11}|^2.
$$
Since we know
$
\text{\rm Re}{A_{11,}}^{11}=
\text{\rm Im}R_{,1}^{\ \ 1}=R_{,0} $
from (2.2) and (2.3), $Q$ can be
written in the form $$ Q=c_1\Delta_bR+c_2\text{\rm
Im}{A_{11,}}^{11} +c_3|R|^2+c_4|A_{11}|^2+c_5 R_{,0}.
$$
\par
Now we shall determine the coefficients.
To simplify computation, we work on the Heisenberg group
$\Bbb H^3=\Bbb C\times \Bbb R$ with the CR structure given
by $Z_1=\pa/{\pa z}+i\c z\pa/{\pa t}$.
On $\Bbb H^3$ with the standard pseudo-hermitian structure
$
\th_0=\dfrac12(dt+izd\c z-i\c zdz),
$
the curvature and the torsion vanish, and thus we
can compute $R$ and $A_{11}$ for the pseudo-hermitian
structure $\th=e^{2f}\th_0$ from (2.5) and (2.6):
$$
\gather
R=-4e^{-2f}\left(Z_1Z_\c1f+Z_\c1Z_1f+2(Z_1f)(Z_\c1f)\right),\\
A_{11}=2i\,e^{-2f}\left(Z_1Z_1f-2(Z_1f)(Z_1f)\right).
\endgather
$$
Note that $Z_1$ is normalized by the Levi metric
associated with $\th_0$. For the
determination of the coefficients, it is enough to compute
$Q$ for some simple examples of CR-pluriharmonic functions
$f$.
First, we consider the case
$f=a\,\text{\rm Re}(z+z^2)$, where $a$ is a real constant.
Then we get
$$
|R|^2=4a^4+O(1), \quad
|A_{11}|^2=(a^2-2a)^2+O(1),
$$
where $O(1)$ indicates some smooth function which vanishes
at the origin $(0,0)\in\Bbb H^3$.
To verify $\Delta_bR$ we use (2.7):
$$
\align
\Delta_bR
&=e^{-2f}\left(-Z_1Z_\c1-Z_\c1Z_1-2(Z_1f)Z_\c1
-2(Z_\c1f)Z_1\right)R
\\
&=2(Z_1Z_\c1+Z_\c1Z_1+aZ_\c1+aZ_1)e^{-2f}
|a+2az|^2+O(1)
\\
&=-8a^2(a-2)+O(1).
\endalign
$$
Since we have shown in Lemma 5.4 that
$\Delta_b R-2\,\text{\rm Im}{A_{11,}}^{11}$ satisfies (5.2),
we also get
$$
\text{\rm Im}{A_{11,}}^{11}=\frac12\Delta_b
R=-4a^2(a-2)+O(1).
$$
For the evaluation of $R_{,0}$ we use
$T=e^{-2f}\left(\pa/\pa
t+2i(Z_1f)Z_\c1-2i(Z_\c1f)Z_1\right)$ and get
$$
R_{,0}=TR=-2\left(\frac\pa{\pa
t}+iaZ_{\c1}-iaZ_1\right)e^{-2f} |a+2az|^2+O(1)=O(1).
$$
To sum up, we have shown that the value of
$Q(\exp(a\text{\rm Re}(z+z^2))\th_0)$
at the origin is
$$
-4a^2(a-2)(2c_1+c_2)+4a^4c_3+(a^2-2a)^2c_4,
$$
which must
vanishes for every $a\in\Bbb R$. Thus we get
$2c_1+c_2=0$ and $c_3=c_4=0$.
So $Q$ is written as
$c_1(\Delta_bR-
2\,\text{\rm Im}A_{11,}^{\ \ \ 11})+c_5 R_{,0}.
$
To determine $c_5$ we compute $R_{,0}$ for the
CR-pluriharmonic
function
$f=t+|z|^2$
and get
$R_{,0}=32+O(1)$.
On the other hand, we know
$\Delta_bR-
2\,\text{\rm Im}A_{11,}^{\ \ \ 11}=0
$
for the pseudo-hermitian structure $\exp (2(t+|z|^2))\th_0$.
Thus $c_5$ must vanish.
\qed
\enddemo
\beginsection{6. Determination of the universal constant}
\noindent
By Theorem 5.1, we have
$\psi_0=c\; (\Delta_b R-2\text{\rm
Im}{A_{11,}}^{11})$
for some constant $c$ which depends on neither a choice of
domain nor a choice of pseudo-hermitian structure. In
order to determine the constant, we shall compute the
Szeg\"o kernel $S$ on the Heisenberg group with the
pseudo-hermitian structure $\th=\exp(2|z|^4)(dt+izd\c z-i\c
z dz)$. We embed $\Bbb H^3$ by
$(z,t)\mapsto (z,(|z|^2-it)/2)\in \Bbb C^2$,
so that $\Bbb H^3$ is defined by $\rho=w+\c w-z\c z=0$.
\proclaim{Lemma 6.1}
Set
$\gamma_\varepsilon=(0,\varepsilon /2)$. Then, as
$\varepsilon\to +0$, we have
$$
S(\gamma_\varepsilon,\overline{\gamma_\varepsilon})=
\frac1{4\pi^2} \varepsilon^{-2}
+
\left(\frac2{\pi^2}+O(\varepsilon)\right)\log \varepsilon.
\tag6.2
$$
\endproclaim
\pagebreak
\demo{\it Proof}
We employ the method used in [HKN, \S2].
Since the volume element $\th\wedge d\th$
corresponds to the $\delta$-function
$4\exp(4|z|^4)\delta(\rho)$, we get
$$
4\exp(4|z|^4)\delta(\rho)
=4\sum_{n=0}^\infty\frac{4^n}{n!}z^{2n}(D_zD_w^{-1})^{2n}
\delta(\rho).
$$
Thus we have
$$
\align
S=
&
\Bigl(
4\sum_{n=0}^\infty\frac{4^n}{n!}z^{2n}(D_zD_w^{-1})^{2n}
\Bigr)^{*-1}
\frac1{\pi^2}\rho^{-2}\\
=&\frac1{4\pi^2}(1-4D_z^2z^2D_w^{-2})\rho^{-2}
+(\text{terms of weight$>0$}).
\endalign
$$
Since
$
(D_z^2z^2D_w^{-2}\rho^{-2})|_{\gamma_\varepsilon}
=-2\log \varepsilon,
$
we get (6.2).
\qed
\enddemo
On the other hand, we see from (5.7) that the value of
$
\Delta_b R-2\,\text{\rm Im}{A_{11,}}^{11}
$
at the origin is
$
12{(P_1f),}^1=12f_{\c1\ 1}^{\ \c1\ 1}=12Z_{\c1}Z_1Z_1Z_{\c1}
|z|^4=48.
$
Therefore we find $c=1/{24\pi^2}$.
This concludes the proof of Main Theorem.
\beginsection{7. Proofs of corollaries}
\noindent
Corollary 1 has been proved in Lemma 5.4, so we begin with
the proof of Corollary 2.
We first recall that the invariant
volume element $\th\wedge d\th$ is defined by the
normalization $$ \th\wedge d\th
=i\th\wedge(T\rfloor\zeta)\wedge (T\rfloor\c\zeta)
\tag7.1
$$
with the closed $(2,0)$-form $\zeta=dz_1\wedge dz_2$.
This definition is equivalent to the one given in [F2] and
[HKN], see [H].
We denote the contact form satisfying this condition by
$\th_0$. To evaluate $\psi_0$ for this volume element, we
use the following lemma, which is an analogy of
[L2, Theorem 4.2].
\proclaim{Lemma 7.2}
Let $\th$ be a pseudo-hermitian structure on a
three-dimensional CR manifold. Then
$W_1=R_{,1}+i{A_{11,}}^1=0$ in a neighborhood
of a point $p\in M$
if and only if there exists a closed
$(2,0)$-form $\zeta$ in a neighborhood of $p$ satisfying
$(7.1)$.
\endproclaim
\demo{\it Proof}
If we take the exterior differential of the one-form
$\omega_1^{\ 1}+iR\ \th$, and use (2.1), we obtain
$$
d(\omega_1^{\ 1}+iR\
\th)=i\,(W_1\th^1+W_\c1\th^\c1)\wedge\th.
$$
Thus we see that $W_1=0$ if and only if
$\omega_1^{\ 1}+iR\ \th$ is closed.
This fact corresponds to [L2, Lemma 4.1], and thus the
arguments in the proof of [L2, Theorem 4.2] also hold
just as well in this case.
\qed
\enddemo
In particular, $\psi_0=-\dfrac1{12\pi^2} {W_{1,}}^1=0$ for
the invariant volume element. Thus, for the volume element
$e^f \th_0\wedge d\th_0$ with CR-pluriharmonic function $f$,
we also get $\psi_0=0$ by the transformation law (4.3).
In order to prove the only if part of Corollary 2, we use the
following:
\proclaim{Proposition 7.3}
Let $M$ be a compact three-dimensional CR manifold which has
a transversal symmetry, and $\th$ be any
pseudo-hermitian structure on $M$. Then a $C^\infty$ real
function $f$ satisfies $C_\th f=0$ on $M$ if and only if $f$
is CR-pluriharmonic.
\endproclaim
\pagebreak
\demo{\it Proof}
In [GL, Proposition 3.2], they proved this statement under
the assumption that $\th$ is normalized by the symmetry.
In case $\th$ is not normalized, we write
$\th=e^{2g}\widetilde \th$ with a normalized
pseudo-hermitian structure $\widetilde\th$.
Then the lemma below shows
$C_{\widetilde\th}f=e^{4g}C_\th f=0$,
which implies that $f$ is CR-pluriharmonic.
\par
Conversely, if $f$ is CR-pluriharmonic, then we have
$C_\th f=(P_1f),^1=0$.
\qed
\enddemo
\proclaim{Lemma 7.4}
Let $\th$ and $\widetilde\th$ be pseudo-hermitian structures on
a three-dimensional CR manifold.
If $\widetilde\th=e^{2g}\th$, then we have
$C_{\widetilde\th}=e^{-4g}C_\th$.
\endproclaim
\demo{\it Proof}
Take a real function $f$ and define another contact
form by $\widehat\th=e^{2f}\widetilde\th=e^{2(f+g)}\th$.
Then we have the transformation formulas of $W_{1,}^{\ \;1}$
$$
\gather
\widehat W_{1,}^{\ \;1}
=e^{-4f}(\widetilde W_{1,}^{\ \;1}-6 C_{\widetilde\th}f),\\
\widehat W_{1,}^{\ \;1} =e^{-4(f+g)}(W_{1,}^{\ \;1}
-6 C_{\th}f-6 C_{\th}g),
\\
\widetilde W_{1,}^{\ \;1}=e^{-4g}(W_{1,}^{\ \;1}-6
C_\th g).
\endgather
$$
These equations imply $C_{\widetilde\th}f=e^{-4g}C_\th f$.
$\square$
\enddemo
If $\psi_0$ vanishes for a volume element $\th\wedge d\th$,
then, writing $\th\wedge d\th =e^{4f}\th_0\wedge d \th_0$,
we get $\psi_0=\dfrac1{2\pi^2}e^{-4f}C_{\th_0}f=0$. Thus
Proposition 7.3 implies that $f$ is CR-pluriharmonic, which
proves Corollary 2.
Finally we shall prove Corollary 3.
By Corollary 2, we see that the volume element
is a CR-pluriharmonic function multiple of the invariant
volume element.
Thus the transformation law (4.2) implies that the
Szeg\"o kernel defined with respect to the invariant
volume element also has the logarithmic term which
vanishes to the third order at the boundary.
Thus [HKN, Remark 2] implies that the boundary is
spherical.
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\enddocument