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\issueinfo{5}{2}{December}{2002} \copyrightinfo{2002}{Information
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\begin{document}
\title[Almost-K\"ahler metrics]{A Note on the Scalar Curvature of Almost-K\"{a}hler Metrics }
\author[J. Kim]{Jongsu Kim and Chanyoung Sung}
\thanks{Supported by grant No. R01-1999-00004 from the
KOSEF}
\thanks{Received October 1, 2002}
\subjclass{53C55, 53C15, 32G05, 53C21} \keywords{almost-K\"ahler
metric, constant scalar curvature}
\address{{J. Kim :} Sogang University, Seoul, Korea}
\email{jskim@ccs.sogang.ac.kr} \address{{Chanyoung Sung : } Korea
Institute for Advanced Studies, Seoul, Korea}
\email{cysung@kias.re.kr}
%Supported by grant No. R01-1999-00004 from the Korea Science and Engineering Foundation.
%MS Classification(2000): 53C55,53C15,53C21
\begin{abstract}
In this note we explain on a construction, on certain compact manifolds, of infinite dimensional families of (non-K\"ahler) almost-K\"ahler metrics with constant scalar curvature, which share some properties of \k metrics. These families of metrics are obtained by deforming constant-scalar-curvature \k metrics on suitable compact complex manifolds.
%We prove several other similar results concerning the scalar curvature and/or the {\it $*$-scalar} curvature.
%We also discuss the scalar curvature functions of almost-\k metrics.
%We show that near an almost \k metric with proper conditions the space of almost \k metrics with constant scalar curvature is an ILH submanifold of of the manifold almost \k metrics.
\end{abstract}
%\vfill
%\pagebreak
\maketitle
\section{Introduction}
\indent An almost-K\"{a}hler metric is a Riemannian metric
$g$ compatible with a symplectic structure $\omega$ on a smooth
manifold, i.e. $ \omega(X, Y)= g(JX, Y) $ for an almost complex
structure $J$, where $X, Y$ are tangent vectors at a point of the
manifold. We often denote it by the triple $(g, \omega, J)$ or
the couple $(g, \omega)$ to specify what $\omega$ or $J$ is. We
may call this triple or couple an {\it almost-K\"{a}hler
structure}. Notice that any one of the pairs $(g, \omega)$,
$(\omega, J)$, or $(g, J)$ determines the other two. For
simplicity $g$ is called $\omega$-almost-\k if $(g, \omega)$ is an
almost-\k structure. It is well known that an almost-\k structure
$(g, \omega, J)$ is \k if and only if $J$ is integrable. When $J$
is not integrable then we call $(g, \omega, J)$ {\it strictly}
almost-\k.
Since the first example of a symplectic, non-K\"{a}hler compact manifold was given by Thurston \cite{Th}, an abundance of similar examples was later discovered by McDuff and others, e.g. \cite{Go}. Now that we have so many symplectic manifolds,
one may well be interested in studying scalar curvatures of almost-\k metrics,
in particular those with constant scalar curvature.
Recall that in \k geometry the existence of \k metrics with constant scalar curvature on compact manifolds
has been one of the central objects of research.
It is natural to first look for such metrics near \k metrics with constant scalar curvature. In \cite{LSim} LeBrun-Simanca showed the existence of deformations of \k metrics with constant scalar curvature by applying the Banach-Space Implicit Function Theorem for the scalar curvature map; see also Fujiki-Schmacher \cite{FS}.
The relevant surjectivity condition is formulated in terms of holomorphic vector fields on the manifold.
In this paper we first look for similar deformation arguments, based on the same sufficient condition.
We refine the scalar curvature map to apply to
our almost-\k deformations.
We set up almost-\k deformations
in a specific and efficient form, by varying \k potential functions and harmonic forms
globally and almost complex structures on a small open subset of $M$.
One benefit from this way of specific setting is that we can extract extra information
such as `prescribed' scalar curvature, see \cite{KS} to get zero scalar
curvature metrics.
Another is that we keep the resulting metrics to share some Riemannian geometric properties with \k metrics as shown in the last statement of the theorem.
\medskip
The paper is organized as follows.
In section 2 we discuss the deformation of almost-\k metrics.
In section 3 we briefly the scalar curvature deformation theory of Kazdan-Warner
and adapt to almost-\k metrics.
\medskip
\noindent {\bf Notations and Conventions} \mbox{ }
Let $E$ be a smooth real vector bundle endowed with a metric on a
compact real $m$-dimensional Riemannian manifold $M$. Then
$L^p_{k}(E)$ denotes the real Banach space of sections of $E$ whose
first $k$ derivatives have bounded $L^p$-norms. The Sobolev
imbedding theorem states that if $k> \frac{m}{p} + l$, then
$L^p_{k}(E)\subset C^l(E)$, the space of continuous sections whose
derivatives of order up to $l$, are continuous. When $E$ is a
trivial line bundle, it is usually denoted as $L^p_{k}(M)$ or more
briefly $L^p_{k}$, which is a Banach algebra for $k> \frac{m}{p}$.
In section 2 and 3, $k> \frac{m}{p}$ will be assumed. Finally by
$L^p_{k}(M)/ \Bbb R$, we mean the quotient Banach space of
$L^p_{k}(M)$ by the closed subspace of constant functions. By abuse of
notation, any element of
$L^p_{k}(M)/ \Bbb R$ will be denoted with or without $[\cdot]$, e.g.
$[\varphi]$ or $\varphi$.
\section{The Construction}
\medskip
In thie section we consider the deformation problem of constant scalar
curvature almost-K\"ahler metrics on compact \k manifolds.
First we need to recal on the {\it Futaki} invariant;
\indent
Let $(M,J)$ be a compact complex manifold and ${\mathfrak h}(M)$ be the Lie algebra
of holomorphic vector fields on it. It's been well known that the existence of
K\"ahler metric of constant scalar curvature gives a restriction on the structure
of ${\mathfrak h}(M)$. For a given K\"ahler metric within a K\"ahler
class $[\omega]$, let $G$ and $s$ denote the Green's operator of the Laplace-Beltrami
operator and the scalar curvature respectively.
The Futaki invariant is defined to be the map $${\mathfrak F} : {\mathfrak h}(M) \times
{H^{1,1}(M,\Bbb R)}^+ \longrightarrow \Bbb C$$
$$(\Xi,[\omega]) \longmapsto -2\int_M \Xi(Gs) d\mu,$$ where the K\"ahler cone
${H^{1,1}(M,\Bbb R)}^+ $ is the set of all cohomology classes of
K\"ahler forms on $(M,J)$. It is a non-obvious fact that
${\mathfrak F}$ depends only on the K\"ahler class rather than the
particular K\"ahler form. The vanishing of the Futaki invariant at
$[\omega]$ is a necessary condition for $[\omega]$ to be
represented by a K\"ahler metric of constant scalar curvature. Let
the Futaki invariant be zero at $[\omega]$. Then we say that the
linearization of the Futaki invariant at $[\omega]$ considered as
a map $d{\mathfrak F}|_{[\omega]} : {\mathfrak h}(M) \rightarrow
{(H^{1,1}(M))}^*$ is non-degenerate at $[\omega]$ if it is
injective on the subspace ${\mathfrak h}_0(M)$ of holomorphic
vector fields with zeros. For example, it is non-degenerate if
every global holomorphic vector field is parallel. For more about
the Futaki invariant, the readers are referred to \cite{LSim}.
This non-degeneracy of Futaki invariant served as a sufficient condition for deformability of constant-scalar-curvature \k metrics and will be used in our deformation too.
Now we state;
\begin{thm} \label{th01}
Let $(M,g_0,\omega_0,J_0)$ be a compact K\"ahler manifold with constant scalar curvature. Suppose
that the linearized Futaki invariant is non-degenerate at the
cohomology class $[\omega_0]$.
\indent
Then there exists an infinite dimensional
family, modulo diffeomorphisms, of strictly almost-K\"ahler metrics of constant scalar curvature near $(g_0,\omega_0,J_0)$.
\indent
More strongly, for any open subset $U$ of $M$ over which the tangent bundle is trivial there exists an infinite dimensional
family, modulo diffeomorphisms, of strictly almost-K\"ahler metrics of constant scalar curvature
all of which are \k away from $U$.
\end{thm}
%Let $(M,g_0,\omega_0,J_0)$ be a compact K\"ahler manifold of complex dimension $\dim_{\Bbb C}(M)=n \geq 2$ with constant scalar curvature.
We will vary the metric $g_0$ in a family of almost-\k metrics $g_{\varphi,\alpha,t}$ in $3$ parameters, $(\varphi,\alpha,t)$ to be explained below, such that $g_0 = g_{0,0,0}$.
We shall explain how to vary $g_0$ in three steps.
\medskip \indent
{\bf Step I}:
Before we actually vary the metrics, here we pre-fix some data, which consist of
$p, U, f^1, f^2, \cdots, f^{2n}, a_1, a_2, \cdots, a_n$.
We shall explain these now.
Choose any point $p\in
M$. Take any trivializing neighborhood $U \ni p$ of the tangent
bundle $TM$ and a local orthonormal frame
$f^1,f^2,$ $\cdots,$ $f^{2n}$ for $(TM,g_0)$ on $U$ such that
$J_0f^{2i-1}=f^{2i}$ for $i=1,\cdots,n$.
Then take any
smooth functions $a_1,a_2,\cdots,a_{n}$ on $M$ compactly
supported in $U$ satisfying a generic condition which involves only
$g_0, p, U, f^i $. Indeed the condition is exactly
that the quantity of (\ref{star}), appearing below
in the proof of Theorem \ref{th01}, is nonzero.
(For instance one may choose $a_i$'s to satisfy that
$a_1 (p) = a_2 (p)= 0$ and $f^3(a_1) (p) \neq 0.$)
\medskip \indent
{\bf Step II} : We fix $J_0$ and deform the metric and the \k
form. We denote the space of real $\omega_0$-harmonic
$(1,1)$-forms by $ {\mathcal H}^{1,1}$. For $[\varphi] \in
L^2_{k+4}/\Bbb R$ and $\alpha \in {\mathcal H}^{1,1}$ with
sufficiently small norms,
$\omega_{\varphi,\alpha}=\omega_0+\alpha+i\partial\overline{\partial}
\varphi$ is also a K\"ahler form with respect to $J_0$. Let
$g_{\varphi,\alpha}$ be the corresponding metric.
\medskip \indent
{\bf Step III}: We fix the form $\omega_{\varphi,\alpha}$ and deform
the almost complex structure and the metric to get $g_{\varphi,\alpha,t}$.
Now to define $g_{\varphi,\alpha,t}$, we need an
orthonormal frame $\{f^1_{\varphi,\alpha},\cdots,
f^{2n}_{\varphi,\alpha}\}$ for $g_{\varphi,\alpha}$ on $U$
compatible with $J_0$ i.e.
$J_0f^{2i-1}_{\varphi,\alpha}=f^{2i}_{\varphi,\alpha}$. Take
$f^1_{\varphi,\alpha}=\frac{f^1}{\|f^1\|_{g_{\varphi,\alpha}}}$
and $f^2_{\varphi,\alpha}=J_0f^1_{\varphi,\alpha}$. Let
${\tilde{f}}^3$ be the orthogonal projection of $f^3$ to the
$(n-2)$-plane orthogonal to $f^i_{\varphi,\alpha}$'s for $i<3$,
i.e. ${\tilde{f}}^3=f^3- \langle
f^3,f^1_{\varphi,\alpha}\rangle_{g_{\varphi,\alpha}}
f^1_{\varphi,\alpha} - \langle
f^3,f^2_{\varphi,\alpha}\rangle_{g_{\varphi,\alpha}}
f^2_{\varphi,\alpha}$. Now define,
$f^3_{\varphi,\alpha}=\frac{{\tilde{f}}^3}{\|{\tilde{f}}^3\|_
{g_{\varphi,\alpha}}}$ and
$f^4_{\varphi,\alpha}=J_0f^3_{\varphi,\alpha}$. Other
$f^i_{\varphi,\alpha}$'s are defined inductively.
For a real number $t$ define
$$g_{\varphi,\alpha,t}=
\left \{ \begin{array}{ll} \sum^{n}_{i=1}e^{ta_i}{(f^{2i-1}_
{\varphi,\alpha})}^*\otimes
{(f^{2i-1}_{\varphi,\alpha})}^*+e^{-ta_i}{(f^{2i}_{\varphi,\alpha})}^*
\otimes & {(f^{2i}_{\varphi,\alpha})}^* \; \;{\rm on} \ U \\
g_{\varphi,\alpha} & {\rm elsewhere},
\end{array}
\right.
$$
where ${(f^{i}_{\varphi,\alpha})}^*$ denotes the dual of
$f^{i}_{\varphi,\alpha}$. Since $a_i$'s are compactly supported in $U$,
$g_{\varphi,\alpha,t}$ is a $L^2_{k+2}$-metric on $M$.
An orthonormal frame field
for $g_{\varphi,\alpha,t}$ on $U$ is
$f^{2i-1}_{\varphi,\alpha,t}=e^{-\frac{ta_i}{2}}f^{2i-1}_
{\varphi,\alpha}$ and $f^{2i}_{\varphi,\alpha,t}=e^{+
\frac{ta_i}{2}}f^{2i}_ {\varphi,\alpha}$, $i = 1, 2, \cdots, n$. Thus
$g_{\varphi,\alpha,t}$ is almost-K\"ahler with almost-K\"ahler
form ${\omega}_{\varphi,\alpha}$ and the almost complex structure
$J_t$ defined by
$$ \begin{array}{ll} J_t(f^{2i-1}_{\varphi,\alpha,t})= f^{2i}_{\varphi,\alpha,t}, \ \
J_t(f^{2i}_{\varphi,\alpha,t})=-f^{2i-1}_{\varphi,\alpha,t} & {\rm on} \ U \\
J_t=J_0 & {\rm elsewhere},
\end{array}
$$
for $i=1,\cdots,n$.
This finishes the step III.
\bigskip
Now we shall prove Theorem \ref{th01} which claims that a
deformation of this type produces strictly almost-K\"ahler metric
of constant scalar curvature.
\bigskip
{\bf Proof of Theorem \ref{th01}. }
Let $(\vartheta^{ij}_{\varphi,\alpha,t})$ be the Riemannian
connection form of the metric $g_{\varphi,\alpha,t}$ with respect
to the frame
$f^{1}_{\varphi,\alpha,t},\cdots,f^{2n}_{\varphi,\alpha,t}$
defined above. Consider the map between Banach spaces given by
$$ {\mathcal F} : L^2_{k+4}/\Bbb R \times
{\mathcal H}^{1,1} \times \Bbb R \supset {\mathcal V}
\longrightarrow L^2_{k}/\Bbb R \times \Bbb R$$
$$([\varphi],\alpha,t) \longmapsto (S(\varphi,\alpha,t),w(\varphi,\alpha,t)),$$
where ${\mathcal V}$ is a sufficiently small neighborhood of
$([0],0,0)$, $S(\varphi,\alpha,t)$ is the scalar curvature mod
constants of $g_{\varphi,\alpha,t}$, and $w(\varphi,\alpha,t)$ is
defined as
$[(\vartheta^{13}_{\varphi,\alpha,t}-\vartheta^{24}_{\varphi,\alpha,t})(f^1)]_{x=p}$,
where $[ \cdot ]_{x=p}$ means the evaluation at $x=p$. This
$w(\varphi,\alpha,t)$ map is designed to detect the
non-integrabilty of the corresponding almost complex structure
$J_t$ of $g_{\varphi,\alpha,t}$. In fact, for any K\"ahler metric
$(g,\omega,J)$ with the Riemannian connection
$\nabla,(\vartheta^{ij})$ and a $J$-compatible frame
$\{e^1,\cdots,e^{2n}\}$,
\begin{eqnarray*}
\vartheta^{13}-\vartheta^{24}&=&
g(\nabla e^{3},e^{1})-g(\nabla e^{4},e^{2})\\
&=&g(\nabla e^{3},e^{1})-g(\nabla Je^{3},Je^{1})\\
&=&g(\nabla e^{3},e^{1})-g(J\nabla e^{3},J e^{1})\\
&=&0.
\end{eqnarray*}
We now show that $\mathcal F$ is a smooth map between Banach
spaces. For $(\varphi,\alpha,t)$ in ${\mathcal V}$,
$S(\varphi,\alpha,t)$ can be written as $H(x,\partial^\beta
\varphi, \alpha,t), x \in M, |\beta| \leq 4,$ where $H$ is a
smooth function defined on the range of $(x,\partial^\beta
\varphi,\alpha,t)$. Note that here $\alpha$ can be considered as a
real parameter in ${\Bbb R}^{h^{1,1}} \simeq {\mathcal H}^{1,1}$.
Clearly the linear map $f \mapsto \partial f$ is a smooth map from
$L^2_{k+1}/\Bbb R$ to $L^2_{k}$. By \cite{P} it follows that $S$
is a smooth map. The other map $w(\varphi,\alpha,t)$ is written as
the composition of the evaluation map ${\bf eval}_p:L^2_k
\rightarrow \Bbb R$ and $\tilde{H}(x,\partial^\beta \varphi,
\alpha,t), x\in M,|\beta| \leq 3$, where $\tilde{H}$ is also a
smooth function defined on the range of
$(x,\partial^\beta,\varphi,\alpha,t)$ with $\alpha$ considered
again as a real parameter in ${\Bbb R}^{h^{1,1}} \simeq {\mathcal
H}^{1,1}$. The evaluation map $f \mapsto f(p)$ is bounded linear
and hence a smooth map from $L^2_{k}$ to $\Bbb R$. Thus $w$ is
also smooth.
Let's compute the Fr\'{e}chet derivative $D{\mathcal F}$ of
$\mathcal F$ at $([0],0,0)$. First, $DS|_{([0],0,0)}$ was already
computed by LeBrun and Simanca in \cite{LSim2}. Let ${\mathcal
L}:L^2_{k+4}/\Bbb R \rightarrow L^2_{k}/\Bbb R$ be the
differential operator $-\frac{1}{2}({\Delta^2_{g_0}}+ ric_{g_0}
\cdot {\nabla}_{g_0} {\nabla}_{g_0})$ where ${\Delta}_{g_0}$,
$ric_{g_0}$, and ${\nabla}_{g_0}$ are the Laplace-Beltrami
operator, the Ricci tensor, and the Riemannian connection of $g_0$
respectively. Then $DS|_{([0],0,0)}$ may be expressed as a
$1$-by-$3$ matrix:
$$
DS|_{([0],0,0)} = \left( \begin{array}{ccc} {\mathcal L} &
-2\langle\rho,\cdot \rangle_{g_0} & *
\end{array} \right),
$$
where $\rho$ is the Ricci form of $g_0$.
They showed \cite{LSim} that $DS|_{([0],0,0)}$ is surjective when the linearized Futaki
invariant is non-degenerate at $[\omega_0]$.
For $Dw|_{([0],0,0)}$, we claim that
$$
Dw|_{([0],0,0)} = \left( \begin{array}{ccc} 0 & 0 & b
\end{array} \right),
$$
where $b$ is a real number which is nonzero for our generic choice
of $a_i$'s (in step I). The first two entries are obviously zero because
$g_{\varphi,\alpha}$ is still K\"ahler with the complex structure
$J_0$. Let $\nabla^t$ be the Riemannian connection of $g_{0,0,t}$,
and define $$h:=\frac{dg_{0,0,t}}{dt}|_0=
\sum^{n}_{i=1}{a_i}{(f^{2i-1})}^*\otimes {(f^{2i-1})}^*
{-a_i}{(f^{2i})}^* \otimes{(f^{2i})}^*.$$ Note that
\begin{eqnarray*}
\omega^{13}_{0,0,t}(f^1)=
g_{0,0,t}(\nabla^t_{f^1}f^3_{0,0,t},f^1_{0,0,t})
&=&e^{ta_1}g_0(\nabla^t_{f^1}(e^{-\frac{ta_2}{2}}f^3),e^{-\frac{ta_1}{2}}f^1)\\
&=&e^{\frac{t(a_1-a_2)}{2}}g_0(\nabla^t_{f^1}f^3,f^1),
\end{eqnarray*}
\begin{eqnarray*}
\omega^{24}_{0,0,t}(f^1)=
g_{0,0,t}(\nabla^t_{f^1}f^4_{0,0,t},f^2_{0,0,t})
&=&e^{-ta_1}g_0(\nabla^t_{f^1}(e^{\frac{ta_2}{2}}f^4),e^{\frac{ta_1}{2}}f^2)\\
&=&e^{-\frac{t(a_1-a_2)}{2}}g_0(\nabla^t_{f^1}f^4,f^2).
\end{eqnarray*}
Recall the formula in \cite{Be}
$$g_0({(\nabla^t_X Y)}^{\prime}_{t=0},Z)=\frac{1}{2}
\{ (\nabla^0_X h)(Y,Z) + (\nabla^0_Y h)(X,Z) - (\nabla^0_Z h)(X,Y)
\}.$$ For clarity's sake, let $(\vartheta^{ij})$ be the Riemannian
connection form of $g_0$. Then
\begin{eqnarray*}
\frac{\partial w}{ \partial t}|_{([0],0,0)} &=&
[\frac{d}{dt}|_0(\vartheta^{13}_{0,0,t}-\vartheta^{24}_{0,0,t})(f^1)]_{x=p}
\\ &=& [\frac{a_1-a_2}{2}g_0(\nabla^0_{f^1}f^3,f^1) \\ &{
+ } & \frac{1}{2}
\{(\nabla^0_{f^1}h)(f^3,f^1)+(\nabla^0_{f^3}h)(f^1,f^1)-(\nabla^0_{f^1}h)(f^1,f^3)
\}\\ &{ - } & (- \frac{a_1-a_2}{2})g_0(\nabla^0_{f^1}f^4,f^2) \\
&{ - } &
\frac{1}{2}\{(\nabla^0_{f^1}h)(f^4,f^2)+(\nabla^0_{f^4}h)(f^1,f^2)-
(\nabla^0_{f^2}h)(f^1,f^4)\}]_{x=p}
\end{eqnarray*}
which is computed to
$$
[\frac{a_1-a_2}{2} \vartheta^{13}(f^1)+
\frac{1}{2}\{f^3(h(f^1,f^1))-h(\nabla^0_{f^3}{f^1},f^1)-h(f^1,\nabla^0_{f^3}{f^1})\}$$
$$
+ \frac{a_1-a_2}{2} \vartheta^{24}(f^1)-
\frac{1}{2}\{-h(\nabla^0_{f^1}{f^4},f^2)-h(f^4,\nabla^0_{f^1}{f^2})$$
$$-h(\nabla^0_{f^4}{f^1},f^2)-h(f^1,\nabla^0_{f^4}{f^2})
+h(\nabla^0_{f^2}{f^1},f^4)+h(f^1,\nabla^0_{f^2}{f^4})\}]_{x=p}$$
\begin{equation}\label{star}
=[\frac{a_1-a_2}{2} \vartheta^{13}(f^1)+ \frac{1}{2}f^3(a_1)
+\frac{1}{2}\{2a_1\vartheta^{12}(f^4)
-(a_1+a_2)\vartheta^{14}(f^2)\}]_{x=p},
\end{equation}
where $f^3(\cdot)$ denotes the directional derivative in the
direction of $f^3$. Thus $\frac{\partial w}{
\partial t}|_{([0],0,0)}$ is nonzero for generic $a_1$ and $a_2$, (already
chosen so in step I).
Summarizing the above, we now have
$$
D{\mathcal F}|_{([0],0,0)} = \left( \begin{array}{ccc} {\mathcal
L} &
-2\langle\rho,\cdot \rangle_{g_0} & * \\
0 & 0 & b
\end{array} \right),
$$ which is surjective. From the choice of our $a_i$ functions it is easy to see that it has $h^{1,1}$-dimensional kernel. Applying the
Implicit Function Theorem, for a sufficiently small neighborhood
${\mathcal V}$ of $0$, $({\mathcal F}^{-1}(\{[0]\} \times \Bbb R)
\bigcap {\mathcal V})- \{([0],0,0)\}$ gives a
$(h^{1,1}+1)$-dimensional submanifold consisting of strictly
almost-K\"ahler $L^2_{k+4}$-metrics of constant scalar curvature.
To show the smoothness of these metrics, we resort to the elliptic
regularity. Consider the smooth $4$-th order non-linear
differential operator $\tilde{S}$ smoothly parameterized by
$r=(r_1,\cdots,r_{h^{1,1}}) \in {\Bbb R}^{h^{1,.1}}$ and $t\in
\Bbb R$
$$\tilde{S} : L^2_{k+4} \supset {\tilde {\mathcal V}} \longrightarrow L^2_{k}$$
$$\varphi \longmapsto {\rm the \ scalar \ curvature \ of \ }g_{\varphi,\sum_{i} r_i\alpha_i,t}$$
where ${\tilde {\mathcal V}}$ is a sufficiently small neighborhood
of $0$ in $L^2_{k+4}$, and $\alpha_1,$ $\cdots,$
$\alpha_{h^{1,1}}$ is a basis for ${\mathcal H}^{1,1}$. The
derivative at $0$ when $r=t=0$ is $-\frac{1}{2}({{\Delta}^2_{g_0}}
+ ric_{g_0} \cdot {\nabla}_{g_0} {\nabla}_{g_0})$ which is
elliptic. Observe that $\tilde{S}=\tilde{S}(x,\partial^\beta
\varphi,r,t), |\beta| \leq 4$ is smooth in all its arguments.
Since $\tilde{S}$ is elliptic at $0\in L^2_{k+4} \subset C^4$,
$\tilde{S}$ is also elliptic at $\varphi$ when
$\|\varphi\|_{L^2_{k+4}} + \|\ r \| + |t|$ is sufficiently small
by Lemma \ref{lem1} below. Then the elliptic regularity theorem
6.8.1 in \cite{Mo} gives the smoothness of our solution satisfying
$\tilde{S}(\varphi)= {\rm constant}$. For simplicity we skip the
proof of the infinite dimensionality modulo diffeomorphisms and
just refer to \cite{KS} for details. \hfill $\Box$
\begin{lemma}\label{lem1}
Let $E$ and $F$ be smooth vector bundles of the same dimension on a
compact manifold $M$. Let $G(x,\partial^\beta\varphi,t), |\beta|
\leq k, t\in {\Bbb R}^m$ be a smooth map of all of its arguments so that $G:C^\infty(E)
\rightarrow C^\infty(F)$ defines a smooth $m$-parameter family of smooth differential
operator of order $k$. Suppose $G(x,{\partial}^\beta \varphi,\tilde{t})$ is elliptic at
$\tilde{\varphi} \in C^k$. Then for $\varphi \in C^k$ and $t\in {\Bbb R}^m$ with $\|
\varphi - \tilde{\varphi} \|_{C^k} + \|t-\tilde{t}\|$ sufficiently small,
$G(x,\partial^\beta\varphi,t)$ is also elliptic at $\varphi$.
\end{lemma}
\begin{proof}
The linear operator $P(\psi)= \frac{d}{ds}|_0 G(x,\partial^\beta(\varphi + s\psi),t)$ is the linearization at $\varphi$ . To compute the symbol
$\sigma_\xi(P;x) : E_x \rightarrow F_x$ at $x\in M$ and for $\xi \in T^*_x M$,
choose a smooth function $g$ satisfying that g is zero at $x$ and $dg(x)=\xi$.
Then for $\psi \in C^\infty(E)$ with $\psi(x)=v\neq 0$,
$$\sigma_\xi(P;x)v=\frac{i^k}{k!}P(g^k\psi)|_x=\frac{i^k}{k!}\frac{d}{ds}|_0
G(x,\partial^\beta(\varphi + sg^k\psi),t)|_x,$$
which is also nonzero when $\|
\varphi - \tilde{\varphi} \|_{C^k} + \|t-\tilde{t}\|$ is sufficiently small.
Since $M$ is compact, the symbol is invertible for all $x\in M$.
\end{proof}
\bigskip
\noindent {\bf Example 1}
As the primary application of Theorem \ref{th01}, we get infinitely many strictly almost-\k metrics
with constant scalar curvature near most of the \k Einstein metrics whose existence were generally shown by Yau \cite{Ya} and near most of constant-scalar-curvature \k metrics in \cite{KLP, LSim, LSin}.
In particular such metrics with negative constant scalar curvature exist
on complex tori or $K3$ surfaces.
It is interesting to find the metrics of Theorem \ref{th01} explicitly. See Example 2 for some explicit metrics on complex tori.
\bigskip
\noindent {\bf Example 2 }
%\begin{xpl}
In \cite{Je}, Jelonek constructed explicitly an infinite
dimensional family of strictly almost-K\"ahler metrics with
non-positive constant scalar curvature on the real
$2n$-dimensional torus $T^{2n}$ for $n \geq 3$. Let $u,v\in
C^\infty(\Bbb R)$ be any smooth real-valued functions satisfying
$u(1+x)=u(x), v(1+x)=v(x)$, and ${(u^\prime)}^2+{(v^\prime)}^2=1$.
Think of $T^{2n}$ as $T^2 \times T^{2n-2}$. On $T^2$, define two
functions $f(x,y)=e^{tu(lx+my)},h(x,y)=e^{tv(lx+my)}$ where $t\in
\Bbb R$ is a nonzero constant, and $l,m \in \Bbb Z$ are integer
constants satisfying $l^2+m^2>0$. Let's denote the flat metric on
$T^2$ by $g_{T^2}$ and the standard metric on $S^1=\Bbb R / \Bbb
Z$ by $g_{S^1}$. Then for example on $T^6$,
$$g_{f,h}=g_{T^2} + f g_{S^1} + f^{-1}g_{S^1} + h g_{S^1} + h^{-1}g_{S^1}$$
gives strictly almost-K\"ahler metric of constant scalar curvature
$-2t^2(l^2+m^2)$ with the same symplectic form as that of $g_{1,1}$.
This
may be viewed as a special case of deformations of Theorem \ref{th01}: with the notation in section 2, $n=3$, $U = M$, $a_1 = 1$, $\phi=0$.
\bigskip
\noindent {\bf Example 3 }
%\begin{xpl}
In \cite{Su}, the second-named author constructed explicitly an infinite
dimensional family of strictly almost-K\"ahler metrics on $\Sigma
\times S^1 \times S^1$, where $\Sigma$ is any compact Riemann
surface of genus bigger than $1$. On $\Sigma$, put the metric
$g_c$ with negative constant scalar curvature c. Let $f$ be any
nonzero smooth function on $\Sigma$. Then
$$g_{c,f}=g_c + e^f g_{S^1} + e^{-f} g_{S^1}$$
gives strictly almost-K\"ahler metrics with $s+s^*=2c$ and has the same
symplectic form as that of $g_c + g_{S^1} + g_{S^1}$.
\remark{Remark}
\k metrics with zero scalar curvature on compact complex surfaces $M$
are absolute minima of the squared $L^2$-norm functional of the Riemannian curvature tensor over the space of smooth Riemannian metrics on $M$, \cite{LSin}.
A refinement of \ref{th01} can produce infinitely many strictly almost-\k metrics
with zero scalar curvature near most of \k metrics with zero scalar
curvature on some blow-ups of minimal ruled surfaces over any compact smooth Riemann surface.
\endremark
\begin{thebibliography}{99}
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\bibitem{Be} A.L. Besse, {Einstein manifolds}, Ergebnisse der Mathematik, 3. Folge, Band 10, Springer-Verlag, 1987.
\bibitem{FS} A. Fujiki, G. Schumacher,
{\em The moduli space of extremal compact \k manifolds and generalized Weil-Petersson metric},
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\bibitem{Go} R. Gompf,
{\em A new construction of symplectic manifolds},
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{\em Some simple examples of almost-K\"ahler non-K\"ahler
structures},
Math. Ann. 305 (1996) 639-649.
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{ A direct approach to the determination of gaussian and scalar curvature functions,}
Invent. Math. 28 (1975) 227-230.
\bibitem{KW1} J.L. Kazdan, F.W. Warner,
{ Existence and conformal deformation of metrics with prescribed gaussian and scalar curvatures,}
Ann. of Math. (2) 101 (1975) 317-331.
\bibitem{KLP} J. Kim, C. LeBrun, and M. Pontecorvo,
{\em Scalar flat K\"ahler surfaces of all genera},
Jour. Reine Angew. Math. 486 (1997) 69-95.
\bibitem{KS} J. Kim, C. Sung,
{ Deformations of Almost-K\"{a}hler Metrics with Constant Scalar Curvature on Compact \k Manifolds}, to appear in Ann. Global Annal. Geom.
\bibitem{LSim} C. LeBrun, and S.R. Simanca,
{\em Extremal K\"ahler metrics and complex deformation theory},
GAFA 4 (1994) 179-200.
\bibitem{LSim2}\leavevmode\vrule height 2pt depth -1.6pt width 23pt,
{\em On K\"ahler surfaces of constant positive scalar curvature},
J.\ Geom.\ Anal.\ 5 (1995) 115-127.
\bibitem{LSin} C. LeBrun and M. Singer,
{ Existence and deformation theory for scalar-flat \k metrics
on compact complex surfaces,}
Invent. Math. 112 (1993) 273-313.
\bibitem{dusa} D. McDuff, and D. Salamon, Introduction to symplectic
topology, Oxford, 1998.
\bibitem{Mo} C.B. Morrey,
{ Multiple integrals in the calculus of variations,}
Springer-Verlag, Berlin, Heidelberg, New York 1966.
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Benjamin, 1968.
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{\em Extremal Almost-K\"ahler metrics and Seiberg-Witten theory},
Stony Brook Preprint (2001).
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{\em Some simple examples of symplectic manifolds},
Proc. Am. Math. Soc. 55 (1976), no. 2, 467--468.
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{\em On the Ricci curvature of a compact K\"ahler manifold and the
complex Monge-Amp\`{e}re equation I},
Comm. Pure Appl. Math. 31 (1978), no. 3, 339--411.
\end{thebibliography}
%\begin{flushleft}
%Jongsu Kim, Department of Mathematics, Sogang University,
%Sinsu-dong 1, Mapo-gu, Seoul 121-742, KOREA \\
%jskim@ccs.sogang.ac.kr\\
%\bigskip
%Chanyoung Sung, Department of Mathematics, SUNY at Stony Brook,
%Stony Brook, NY 11794, U.S.A.\\
%cysung@math.sunysb.edu
%\end{flushleft}
\end{document}
\indent Here we review some basic facts on DeRham and Dolbeault
cohomology of compact \k manifolds, \cite{BPV}. For a compact
complex manifold $M$ of complex dimension $n$, the decomposition
${\Bbb C} \otimes TM = T^{1,0}M \oplus T^{0,1}M$ into the $\pm i$
eigenspace of $J$ induces a decomposition $\Lambda^r = \otimes_{p
+q =r} \Lambda^{p,q}$ of the bundle of $r$-forms into direct sum
of bundles of $(p,q)$-forms $\Lambda^{p,q} = (\Lambda^pT^{1,0})
\otimes (\Lambda^qT^{1,0})^*$. We denote by ${\mathcal E}^r$ and
${\mathcal E}^{p,q}$ the space of sections of the bundles
$\Lambda^r$ and $\Lambda^{p,q}$ on $M$, respectively. Let
$H^{p,q}(M,\Bbb C)$ be the Dolbeault cohomology group $ \{ \alpha
\in {\mathcal E}^{p,q} | \bar{\partial} \alpha = 0 \} /
\bar{\partial} {\mathcal E}^{p,q-1} $. Note that $H^{p,0}(M,\Bbb
C)$ is the space of holomorphic $p$-forms on $M$. We set $h^{p,
q}:= \dim_{\Bbb C} H^{p,q}(M,\Bbb C)$.
There is so-called Hodge decomposition of the DeRham cohomology group: $H^k(M, {\Bbb C}) =
\oplus_{p + q =k} H^{p,q}(M,\Bbb C)$ for a compact \k manifold
$M$. So we have $b_k = \sum_{p+ q =k} h^{p, q}$, where $b_k$ is
the $k$-th {\it Betti} number $\dim_{\Bbb R} H^k (M, {\Bbb R})$.
Furthermore $H^{p,q}(M,\Bbb C)$ can be identified with the space
of harmonic $(p,q)$-forms ${\mathcal H}^{p,q}_{\Bbb C}$ and hence
$h^{p,q} = h^{q,p} $.
\medskip
Recall from \cite{Be} that for an oriented real $4$-dimensional manifold,
the Hodge-star operator $*$ acting on the space of real 2-forms satisfies $*^2 := * \circ * = Id.$
So $\Lambda^2$ decomposes into $\pm 1$ eigenspace of $*$, so called self-dual and anti-self-dual $2$-forms:
$\Lambda = \Lambda^+ \oplus \Lambda^-$.
For a \k manifold of complex dimension $2$ with its \k form $\omega$,
we have
$\Lambda_{\Bbb C}^+ = {\Bbb C} \omega \oplus \Lambda^{0,2} \oplus \Lambda^{2,0}$.
This, together with the Hodge decomposition, implies that the real part of a holomorphic $2$-form is self-dual harmonic.
\subsection{Almost-\k metric}
The almost complex structure $J$ gives rise to a type
decomposition of symmetric (2,0)-tensors. For any symmetric (2,0)-tensor field $h$, we
have the splitting $h = h^+ + h^-$, where $h^+ (X, Y)= \frac{1}{2}
\{ h (X, Y) + h (JX, JY) \}$ and $h^- (X, Y)=\frac{1}{2} \{ h (X, Y) - h (JX, JY) \}$.
\noindent
A symmetric (2,0)-tensor field $h$ is called $J$-invariant (or $J$-anti-invariant) if $h^- =0$ (or $ h^+=0$), respectively.
\par
For a (2,0)-tensor $h$ , we define two new tensors $Jh$ and $hJ$ by
$$ Jh (X,Y) = -h(JX,Y) , \quad hJ (X,Y) = h(X,JY) $$
According to Blair \cite{Bl}, for a smooth curve $g_t$ in $\Omega$ with the corresponding curve $J_t := J_{{g_t}}$ of almost complex structures, $h = \frac{d g_t}{dt} |_{t=0}$ is $J_0$-anti-invariant i.e. $h(x,y)= - h(J_0 x, J_0 y)$,
and conversely for $g$ in $\Omega_\omega$ with the corresponding $J$,
any $J$-anti-invariant symmetric $2$-tensor field $h$ on $M$ is tangent to a smooth curve in $\Omega_{\omega}$.
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