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\begin{document}
\title{Heat Flows for Extremal K\"ahler Metrics}
\author{Santiago R. Simanca}
\address{Institute for Mathematical Sciences, Stony Brook, NY 11794}
\email{santiago@math.sunysb.edu}

\begin{abstract}
Let $(M,J,\Omega)$ be a polarized complex manifold of K\"ahler type. 
Let $G$ be the maximal compact subgroup of the automorphism group of
$(M,J)$. On the space of K\"ahler metrics that are invariant under 
$G$ and represent the cohomology class $\Omega$, we define a flow
equation whose critical points are extremal metrics, those that  
minimize the square of the $L^2$-norm of the scalar curvature. We prove 
that the dynamical system in this space of metrics defined by the said 
flow does not have periodic orbits, and that its only fixed points, or 
extremal solitons, are extremal metrics. We prove local time existence of
the flow, and conclude that if the lifespan of the solution is finite, then
the supremum of the norm of its curvature tensor must blow-up as time
approaches it. We end up with some conjectures concerning the plausible
existence and convergence of global solutions under suitable geometric 
conditions. 
\end{abstract}


\maketitle
\thispagestyle{empty} \input{imsmark}
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\SBIMSMark{2003/02}{October 2003}{}

\section{Introduction}
We define and study a geometrically motivated dynamical system in the 
space of K\"ahler metrics that represent a fixed cohomology class of a given
closed complex manifold of K\"ahler type. The critical points of this flow
are extremal metrics, that is to say, minimizers of the functional defined
by the $L^2$-norm of the scalar curvature. We derive the equation, describe 
some of its 
general properties, and prove that given an initial data, the equation 
has a unique classical solution on some time interval. It would be of great
interest to know if the solution exist for all time, or whether it
develops some singularities in finite time. We have no general answer to 
this yet. However, we show some evidence indicating that  
in some specific cases, the solution should exist for all 
time and converge to an extremal metric as time goes to infinity.

In order to put our equation in the proper perspective, we begin by
recalling a different but related one, the Ricci flow. Let $M$ be a compact 
manifold $M$ of dimension $n$. Given a metric $g$, we denote its Ricci 
tensor by $Ricci_g$ and its
average scalar curvature by $r_g$. The Ricci flow 
$$\frac{d}{dt}g=2\left( \frac{r_g}{n}g-Ricci_g\right) \, ,$$
was introduced by R. Hamilton \cite{ha} as a mechanism to improve the 
properties of its initial data. It is a non-linear heat equation in the
metric, which hopefully becomes better as time passes by in the same way as 
the heat equation improves an initial distribution of heat in a given 
region, and makes it uniform all throughout as time goes to infinity. 
Hamilton used it to show that on a three dimensional manifold, an initial
metric of positive Ricci curvature flows according to this equation 
towards a limit that has constant positive sectional curvature.

In the case of a K\"ahler manifold, Hamilton's flow equation may be used
when seeking a K\"ahler-Einstein metric on the said manifold. Of course, 
this would a priori require that the first Chern class $c_1$ has a sign so
that it may be represented by K\"ahler-Einstein metrics, or their opposites.
Regardless of that consideration, the idea inspired Cao \cite{cao} to study 
the equation
$$\frac{d\o}{dt} =-\rho_t+\eta \, ,$$ 
for $\eta$ a fixed real closed $(1,1)$-form representing the class 
$c_1(M)$. Using Yau's work on the Calabi 
conjecture, he proved that solutions exists for all 
$t\geq 0$ and that the path of metrics so defined 
converges to a K\"ahler metric with prescribed Ricci form $ \eta$ as 
$t\rightarrow \infty$. He went on and, under the assumption that $c_1(M)<0$, 
replaced $\eta$ in the equation above by $-\o_t$ and proved that the 
corresponding solution to the initial value problem exists for all time
and converges to a K\"ahler-Einstein metric as $t\rightarrow \infty $. 

Given a polarized K\"ahler manifold $(M,J,\Omega)$, we now propose to 
study the equation
$$\d_{t}\omega=-\rho_t+\Pi_t \rho\, ,$$
with initial condition a given K\"ahler metric representing $\Omega$.
Here $\Pi_t$ is a metric dependent projection operator that intertwines 
the metric trace with the $L^2$-orthogonal projection $\pi_t$ onto the 
space of 
real holomorphy potentials, these being those real valued functions whose 
gradients are holomorphic vector fields. The projection $\Pi_t$ is such that
$\Pi_t \rho_t - \rho_t$ is cohomologous to zero, and so all metrics 
satisfying the equation represent $\Omega$ if the same is true of the
initial data. For a variety of technical reasons, we define 
this flow only on the space of metrics representing $\Omega$ that are 
invariant under a fixed maximal compact subgroup of the automorphism group of 
$(M,J)$. As such, its critical points will be precisely the metrics whose 
scalar curvatures have holomorphic 
gradients, or said differently, the extremal metrics of Calabi 
\cite{ca1}. This fact constitutes the guiding principle behind our 
consideration of this new flow equation. 

In general, our flow equation is different from the K\"ahler version of the
Ricci flow, even when the class $\Omega$ represents $\pm c_1$. 
This last assertion is illustrated, for instance, by the blow-up of 
${\mathbb C}{\mathbb P}^2$ at one or two points, and the reason is basically 
a simple one: extremal metrics, which is what we seek when we consider the 
new flow, is a more general concept than that of K\"ahler-Einstein metrics, 
and when $\Omega=c_1$, these two concepts agree only if we impose an
additional restriction on $c_1$ \cite{fu,ss}. It is worth mentioning, however,
that on compact connected Riemann surfaces, both the Ricci and extremal 
flow coincide with one another, and they also coincide with the 
two-dimensional version of the Yamabe flow. This is so because 
regardless of the metric $g$ you consider on the given Riemann surface, the
holomorphy potential $\pi_g s_g$, where $s_g$ the scalar curvature of $g$, 
turns out to be a topological constant. Even in this last case, it is still 
of some interest to point out that of these three equivalent flows,
only the extremal one can be interpreted as the gradient 
flow of a variational problem.

The main point of the present article will be to show that solutions 
to the extremal flow equation exists locally in time. However, even if these
were going to exist globally, we should not expect that they would 
converge as time approaches infinity in all possible cases. We already 
know of examples of K\"ahlerian manifolds that do not admit extremal metrics 
\cite{bb,le,ti}.

We do not have a satisfactory general picture that
explains why these examples exist. Those in \cite{le} fail 
to satisfy a necessary condition on the space of holomorphic vector fields,
while those in \cite{bb} and \cite{ti} are related to stability of the
manifold under deformations of the complex structure, and this property 
does not appear reflected by the Lie algebra of holomorphic vector fields. 

At the positive end, we had proven \cite{cs2} that the set of K\"ahler 
classes that can be represented by extremal metrics is open in the 
K\"ahler cone. The study of the extremal flow equation above, and its 
potential convergence to a limit extremal metric, can be seen as a general
method that could decide if the the extremal cone is ---or is not--- closed 
also.

In proving local time existence of the extremal flow, we also
show that if the lifespan is finite, then the pointwise norm of the 
curvature tensor must blow-up as times approaches it. We leave for later
the analysis of global solvability and convergence under suitable geometric 
conditions.

We organize the paper as follows: in \S\ref{ekm} we recall the notion and
basic facts about extremal metrics; in \S\ref{sheat} we explain in detail
the derivation of the extremal flow equation, and prove general results 
about it; in  \S\ref{sl} we linearized this flow equation, showing that
it results into a pseudo-differential perturbation of the standard time
dependent heat equation. This form of the linearization is an essential 
fact in our
proof of local time existence, done in \S\ref{loc} via a fixed point type
of argument. We end with some remarks justifying our hope that the
extremal flow will converge to an extremal metric under suitable general 
geometric conditions.

\section{Extremal K\"ahler Metrics}\label{ekm}
Let $(M,J,g)$ be a K\"ahler manifold of complex dimension $n$. This means that
$(M,J)$ is a complex manifold and that $\omega (X,Y) := g(JX,Y)$,
which is skew-symmetric because $g$ is a {\it Hermitian} Riemannian metric,
is a closed 2-form. The differential form $\omega$ is called 
the {\it K\"ahler form}, and its  cohomology class 
$[\omega ] \in H^{2}(M,{\mathbb R})$ is called the {\it  K\"{a}hler class}. 

By complex  multi-linearity, we may extend the metric
 $g$, the Levi-Civita connection $\nabla$ and the curvature 
tensor ${\mathcal R}$  to the complexified tangent bundle
${\mathbb C} \otimes TM$. Since 
 ${\mathbb C} \otimes TM$ decomposes
into the $\pm i$-eigenspaces of $J$,  ${\mathbb C}
 \otimes TM = T^{1,0}M \oplus T^{0,1}M$, 
we can express any tensor field or differential operator in terms of 
the corresponding decomposition.
For example, if $(z^{1}, \ldots ,z^{n})$ is a holomorphic coordinate system 
on $M$, we get induced bases  $\{ \d_{z^\j} \}$ and $
\{ \d_{z^{\b{\j}}} :=\d_{\b{z}^\j} \}$ for $T^{1,0}M$ 
and $T^{0,1}M$, respectively, and if we express the metric  $g$ in terms of
this basis by setting $g_{\mu\nu} :=g\left( \d_{z^{\mu}},
\partial _{z ^{\nu}}\right)$, 
where the indices 
$\mu$, $\nu$ range over  $\{ 1, \ldots ,n,\bar{1}, \ldots ,\bar{n}\}$,
it follows from  the Hermiticity condition 
 that $g_{\j k} = g_{\b{\j}\b{k}}=0$, and that
$\omega = \omega _{{j}\b{k}}dz^{j}\wedge d\b{z}^{k}= i
g_{\j \b{k}}dz^{\j}\wedge d\b{z}^{k}$. 

The complexification of the exterior algebra can be decomposed into
a direct sum eof of forms of type $(p,q)$. Indeed, we have
$\wedge ^r M =\bigoplus_{p+q=r} \wedge ^{p,q}M$. The integrability of
$J$ implies that the exterior derivative  
 $d$ splits as $d={\partial}+\bar{\partial}$,
where ${\partial}: \wedge ^{p,q}M\to \wedge ^{p+1,q}M$,
$\bar{\partial}: \wedge ^{p,q}M \to \wedge ^{p,q+1}M$,
${\partial}^2=\bar{\partial}^2=0$ and ${\partial}\bar{\partial}=
-\bar{\partial}{\partial}$. Complex conjugation also extends, and we define
a form to be real if it is invariant under this operation.
An important result in K\"ahler geometry is that,
given a $d$-exact real form $\beta $ of type $(p,p)$, there exists a real form
$\a $ of type $(p-1,p-1)$ such that $\beta = i\ddb \a $.

The {\it Ricci form} $\rho$ is defined in terms of the 
Ricci tensor $r$ of  $g$ by $\rho (X,Y) =  r (JX,Y)$. It is a closed form 
whose components are given by  
$$r_{j\overline{k}} = -i \rho_{j\overline{k}} = 
-\frac{\partial ^{2}}{\partial {z^{j}} \partial 
{\overline{z}^{k}}} \log{{\rm  det}(g_{p\overline{q}})}\, .$$ 
We have that $\rho/2\pi$ is 
the curvature of the canonical line bundle $\kappa= \Lambda^n (T^*M)^{1,0}$,
and it represents the first Chern class $c_1(M)$.

The {\it scalar curvature} $s$ is,  by definition,  the trace
$s = r^{\mu}_{\mu}= 2g^{\j\b{k}}r_{\j\b{k}}$ 
of the Ricci tensor, and can be conveniently calculated by using the
formula 
\begin{equation}
s \, \omega^{\w n} = 2n \, \rho \w \omega^{\w (n-1)}\, .\label{eq:sca}
\end{equation}
Since the volume form is given by $d\mu ={\displaystyle \frac{\omega^{\w n}}
{n!}}$, this formula implies that, in the compact case,
$$ \int_M s \, d\mu = \frac{4\pi}{(n-1)!}  c_1
 \cup [\omega]^{\cup (n-1)}\, ,$$
a quantity that only depends upon the complex structure $J$ and the 
cohomology class $[\o]$. Notice that ${\displaystyle \int_M d\mu =
 \frac{1}{n!} [\omega]^{\cup n}}$, and so {\it the average scalar curvature}
$s_0$ is also a quantity that depends only on the K\"ahler class $[\omega]$ 
and the homotopy class of the complex-structure tensor $J$.

Suppose that  $(M,J)$ is a closed complex manifold {\it polarized}
by a positive class $\Omega \in H^{1,1}(M, {\mathbb C})\cap  
H^{2}(M, {\mathbb R})$. Let ${\mathfrak M}_{\Omega}$ be the set of all 
K\"ahler forms representing $\Omega$. Since any two elements $\tilde{\o}$ and
$\o$ of ${\mathfrak M}_{\Omega}$ are such that $\tilde{\o}=\o+i\ddb \varphi$
for some real valued potential function $\v$, we can define a topology 
on ${\mathfrak M}_{\Omega}$ by defining a suitable topology on the space
of potentials. In what follows, we shall not distinguish between the 
K\"ahler metric and its K\"ahler form, passing from one to the other at
will.

Consider the functional  
\begin{equation}
 \begin{array}{rcl}
{\mathfrak M}_{\Omega} & \stackrel{E_{\Omega}}{\longrightarrow} & 
{\mathbb R} \vspace{.08in} \\
\o & \mapsto & {\displaystyle \int _M s_\o ^2 d{\mu}_\o } \end{array}\, ,
\label{ene}
\end{equation}
where the metric associated with the form $\o$
has scalar curvature $s_\o$  and  volume form $d{\mu}_\o$. A critical
point of this functional is by definition
an {\it extremal} K\"ahler metric \cite{ca1}, a notion introduced
with the idea of seeking canonical representatives of $\Omega$.

Given any K\"ahler metric $g$, a smooth complex-valued
function $f$ gives rise to the (1,0) vector field  $f\mapsto \partial^{\#} 
f=\partial^{\#}_g f$ defined by the expression
$$g(\d^{\#} f, \hspace{1mm}\cdot \hspace{1mm})=\db f \, .$$
 This vector field is {\it holomorphic} iff we require 
that $\b{\partial}\partial^{\#} f=0$, condition equivalent 
to $f$ being in the kernel of the operator
\begin{equation}
L_g f:= (\b{\d}{\d}^{\#})^{\ast}\b{\d}{\d}^{\#}f= \frac{1}{4}\lap^2 f + 
\frac{1}{2}r ^{\mu\nu}\n_{\mu}\n_{\nu}f+ \frac{1}{2}(\n^{\b{\ell}}\s )
\n_{\b{\ell}}f \, .\label{lic}
\end{equation}
We then have that
$$\frac{d}{dt}E_{\Omega}(\o+ti\ddb \v)\mid_{t=0}=-4\int s_{\o}L_{\o}\v 
d\mu_{\o} \, .$$
Hence, the Euler-Lagrange's equation for a critical point $g$ of (\ref{ene}) 
is just that the scalar curvature $s_g$ be in the kernel of $L_g$. In other
words, the vector field $\d^{\#}_g s_g$ must be holomorphic.

\section{Derivation of the evolution equation} \label{sheat}
Calabi \cite{ca2} showed that the identity component of the isometry group of
an extremal K\"ahler metric $g$ is a maximal compact subgroup of the
identity component of the biholomorphism group of  $(M,J)$. This 
implies that, up to conjugation, the identity components of the isometry
groups of extremal K\"ahler metrics coincide \cite{cs2}. Therefore, 
modulo biholomorphisms, the search for extremal K\"ahler metrics is 
completely equivalent to the search for extremal metrics among those
that are invariant under the action of a fixed maximal compact subgroup
of the connected biholomorphism group. This last problem, however, turns out
to be technically easier to analyze.

\subsection{Holomorphy potentials}
For any given K\"ahler metric $g$ on $(M,J)$, every complex-valued function 
$f$ in the kernel of (\ref{lic}) is associated with the holomorphic vector 
field $\Xi= {\partial}^{\#}f$, and since the operator is elliptic, the 
space of such
functions is finite dimensional. However, since $(\b{\d}{\d}
^{\#})^{\ast}\b{\d}{\d}^{\#}$ is not a real operator, in general,
the real and imaginary part of a solution will not be solutions. 
It has been proven elsewhere \cite{cs2} that if $f$ is a real valued 
solution of this
equation, then the imaginary part of $\partial^{\#} f$ is a Killing field of
$g$, and that a Killing field arises in this way if, and only if, its
zero set is not empty.

Let $G$ be a maximal compact subgroup of the biholomorphism group of
$(M,J)$, and $g$ be a K\"ahler metric on $M$ with K\"ahler class $\Omega$.
Without loss of generality, we assume that $g$ is $G$-invariant. 
We denote by $L^2_{k,G}$ the real Hilbert space of $G$-invariant 
real-valued functions of class $L^2_{k}$, and consider $G$-invariant 
deformations of this metric preserving the K\"ahler class:
 \begin{equation}
\tilde{\omega }=\o +i\ddb \varphi \, , \quad \varphi \in L_{k+4,G}^2\, , \;
k>n.
\label{met}\end{equation}
In this expression, the condition $k>n$ ensures that $L_{k,G}^2$ is a Banach
algebra, making the scalar curvature of 
$\tilde{\o}$ a well-defined function in the space. 

Let ${\mathfrak h}$ be the complex Lie algebra of holomorphic vector 
fields of
the complex manifold $(M,J)$; by compactness of $M$, this  is
precisely the Lie algebra of the group of biholomorphism of $(M,J)$.
We denote by $\mathfrak g$ the Lie algebra of $G$. If ${\mathfrak z}
\subset {\mathfrak g} \subset {\mathfrak h}$ is the center of ${\mathfrak g}$,
we let ${\mathfrak z}_0= {\mathfrak z}\cap {\mathfrak g}_0$,
where ${\mathfrak g}_0\subset {\mathfrak g}$ is the ideal of 
Killing  fields which have  zeroes. 
If $\tilde{g}$ is any $G$-invariant K\"ahler metric on $(M,J)$, then 
each element
of ${\mathfrak z}_0$ is of the form $J\,\nabla_{\tilde{g}}f$ for a real-valued
solution of (\ref{lic}). In fact, 
${\mathfrak z}_0$ corresponds  to the set of real solutions
$f$ which are {\em invariant under $G$}, since   
$${\partial}^{\#}: \ker
 [(\b{\d}{\d}_{\tilde{g}}^{\#})^{\ast}\b{\d}{\d}_{\tilde{g}}] \to 
{\mathfrak h}_0\, ,$$
${\mathfrak h}_0 \subset {\mathfrak h}$ the subset of holomorphic vector 
fields with zeroes, is a homomorphism of $G$-modules.

The restriction of $\ker (\b{\d}{\d}_{\tilde{g}}^{\#})^{\ast}\b{\d}
{\d}_{\tilde{g}}$
to $L^2_{k+4,G}$ depends smoothly on the $G$-invariant metric 
$\tilde{g}$. Indeed, 
choose  a basis $\{ X_1, \ldots, X_m\}$ for ${\mathfrak z}_0$,
and, for each  $(1,1)$-form $\chi$ on $(M,J)$, consider the set 
of functions 
$$\begin{array}{rcl}
p_0(\chi ) & = &  1 \\ p_j ( \chi ) & = &
2i G_g{\db}^{*}_{g}((JX_{\jmath}+iX_{\jmath})\hok \chi )
\, , \quad j=1, \ldots , m
\end{array}$$ where $G_g$ is the Green's operator of the
metric $g$. 
If $\tilde{\o }$  is the K\"ahler form of the $G$-invariant metric 
$\tilde{g}$, then $\d ^{\#}_{\tilde{g}}p_j(\tilde{\o})=JX_{\jmath}
 +iX_{\jmath}$,
and the set $\{ p_j(\tilde{\o })\} _{\jmath =0}^{m}$ consists of real-valued 
functions and forms a basis for 
$\ker (\b{\d}{\d}_{\tilde{g}}^{\#})^{\ast}\b{\d}
{\d}_{\tilde{g}}$. Furthermore, for metrics $\tilde{\o}$ as in (\ref{met}),
the map $\varphi \mapsto p_j(\o +i \ddb \varphi)$
is, for each $\jmath$, bounded as a linear map from $L^2_{k+4,G}$ to 
$L^2_{k+3, G}$.

With respect to the  fixed $L^2$ inner product, 
let $\{ f_{\tilde{\o }}^0, \ldots , f_{\tilde{\o }}^m\}$ be the 
orthonormal set extracted from $\{ p_j (\tilde{\o }) \}$ by 
the Gram-Schmidt procedure. We then let
\begin{eqnarray} \pi_{\tilde{\o }}: L^2_{k,G}&\to &L^2_{k,G}\nonumber \\
u&\mapsto & \sum_{j=0}^m \<  f_{\tilde{\o }}^j, u\>_{L^2} f_{\tilde{\o }}^j
\label{proj1}\end{eqnarray}
denote the associated projector. In fact, by the regularity of the 
functions $\{ p_1, \ldots , p_{m}\}$, this projection can be defined on
$L^2_{k+j,G}$ for $j=0,\, 1,\, 2,\, 3$, and for metrics as in 
(\ref{met}), the map $\varphi \mapsto \pi_{\tilde{\o }}$ is smooth 
from a suitable neighborhood
of the origin in $L^2_{k+4,G}$ to the real
Hilbert space $\mbox{End} (L^2_{k+j,G})\cong \bigotimes^2 L^2_{k+j,G}$. 

Given a K\"ahler metric $\o$, its normalized Ricci potential $\psi_\o$ is 
defined to be the only function orthogonal to the constants such that
$\rho=\rho_{H}+i\ddb \psi_{\o}$, where $\rho_{H}$ is the 
$\o$-harmonic component of $\rho$. In terms of the scalar curvature and 
its projection onto the constants, we have that 
$\psi_{\o}=-G_{\o}(s_{\o} -s_0)$. Given any cohomology class
$\Omega=[\o]$ in the K\"ahler cone of $(M,J)$, the {\it Futaki character} of
the class is defined to be the map 
\begin{equation}
\begin{array}{rcl}
{\mathfrak F}: {\mathfrak h} &  \longrightarrow &  
{\mathbb C} \\ \Xi & \mapsto & 
{\mathfrak F} (\Xi,[\o ]) =  {\displaystyle \int _{M}
\Xi(\psi _{\o})d\mu = -\int _{M} \Xi( G_\o (s_\o-s_0) ) d\mu_\o }\, .
\end{array}
\label{futc}
\end{equation}
It is independent of the particular metric $\o$ in 
${\mathfrak M}_{[\o]}$ chosen to calculate it \cite{fu,ca2}, and
when applied to a holomorphic vector field of the form $\Xi = \d ^{\#}f$, it
yields 
\begin{equation}
{\mathfrak F} (\Xi,[\o ]) = {\displaystyle -\int_Mf(s_\o-s_0) \, d\mu_\o }
\, .\label{fuss}
\end{equation}

A metric $g\in {\mathfrak M}_{\Omega}$ is extremal
iff $\d ^{\#}_gs_g$ is a holomorphic vector field. In other words, $g$
is extremal iff $s_g =\pi_g s_g$. These metrics
achieve the infimum of $E_{\Omega}$ over ${\mathfrak M}_{\Omega}$. Indeed, 
there is exists a number $E=E(\Omega)$ and a holomorphic vector field
$X_{\Omega}$ such that 
\begin{equation}
E_{\Omega}(\o)\geq E(\Omega):=
s _{0}^2 \frac {\Omega^n}{n!}-
{\mathfrak F} (X_{\Omega},\Omega) 
\label{lb}
\end{equation}
for all $\o \in {\mathfrak M}_\Omega$. The field $X_{\Omega}$ may depend on 
the
choice of a maximal compact subgroup $G$ of the automorphism group of 
$(M,J)$, but the value of ${\mathfrak F} (X_{\Omega},\Omega)$ does not. Given 
$G$ and a metric $g\in {\mathfrak M}_{\Omega}$ that is $G$-invariant, we 
may take 
$X_{\Omega}=\d_g^{\#}(\pi_g s_g-s_0)$, and we easily see that
$$E({\Omega})= \int (\pi_g s_g)^2 d\mu_g \, .$$
This way of computing the energy of the class through $G$-invariant metrics is
very convenient and has been used several times elsewhere \cite{si2,ss,ss2}.

\begin{remark}
Let $g$ be any K\"ahler metric on any Riemann surface $\Sigma$. By the 
Gauss-Bonnet theorem, the average scalar curvature of $g$ is given by
$4\pi \chi(M)/\mu_{g}(\Sigma)$. The invariance of (\ref{futc}) and
(\ref{fuss}) imply that if $g$ is a $G$-invariant metric on 
$\Sigma$, the projection $\pi_g s_g$ of $s_g$ onto the space 
of real holomorphy potentials is always equal to this constant. 
Consequently, the vector field $X_{\Omega}$ of a polarized 
Riemann surface is always trivial.
\end{remark}
\medskip

From now on, we shall denote by ${\mathfrak M}_{\Omega,G}$ the set of 
$G$-invariant K\"ahler metrics representing the class $\Omega$. When 
considering a path 
of metrics $\o_t \in {\mathfrak M}_{\Omega, G}$, 
the fact that the kernel of $(\b{\d}{\d}_{\tilde{g}}^{\#})^{\ast}\b{\d}
{\d}_{\tilde{g}}$ depends smoothly on $\tilde{g}$ allows us to rightfully
compute the differential of $\pi_t s_t$. Here, $\pi_t$ and 
$s_t$ are the projection and scalar curvature associated to $\o_t$, 
respectively. Since $\pi_t s_t$ is of order four in the potential of the 
metric, naively we would expect its differential to be an operator of order 
four on the tangent space to ${\mathfrak M}_{\Omega, G}$ at $\o_t$. However, 
we get something significantly better, and gain quite a bit of regularity.
This fact that will be very convenient later on.

\begin{lemma}
\label{le1}
Let $\o_t=\o+i\ddb \v_t $ be a path of metrics in 
${\mathfrak M}_{\Omega, G}$ with $\o_0=\o$. Consider the projection 
$\pi_t s_t$ of the scalar curvature $s_t$ onto the space of real 
holomorphy potentials, and let
$\dot{\v}_t = \frac{d}{dt}\v_t$. Then
$$\frac{d}{dt}(\pi_t s_t) = \d \dot{\v}_t \hok X_{\Omega}=(\d^{\#}
\dot{\v}_t, X_{\Omega})_t=(\d \dot{\v}_t, \d (\pi_t s_t) )_t\, ,$$
where $X_{\Omega}=\d_t^{\#} \pi_t s_t$ is the holomorphic vector field
of the class $\Omega$.
In particular, this derivative is a differential operator of 
order one in $\dot{\v}_t$ whose coefficients depend non-linearly on the 
coefficients of $\o_t$. 
\end{lemma}

{\it Proof}. By the invariance of the Futaki character, if 
$\pi_\o s_\o$ is constant then so will be $\pi_{\tilde{\o}}s_{\tilde{\o}}$
for any other metric $\tilde{\o}$ in ${\mathfrak M}_{\Omega, G}$ (see
\S4 of \cite{ss}). In that case, $X_{\Omega}$ is trivial
and both sides of the expression in the statement are zero. The result 
follows.

So let us assume that $\pi_t s_t$ is not constant. For convenience, we use 
the subscript $t$ to denote geometric
quantities associated with $\o_t$. Thus, the imaginary part
of $X_{\Omega}=\d_{\o_t}^{\#} \pi_t s_t $ is a non-trivial Killing vector 
field, and in the construction of the projection map above, we can choose
a basis $\{ X_j\}$ for ${\mathfrak z}_0$ such that 
$X_{\Omega}=\d_{\o_t}^{\#} \pi_t s_t = JX_1 +iX_1=X_{\Omega}$. Hence,
$$\pi_t s_t = 2i G_{t}\db_{t}^{*}(\o_t \hok X_{\Omega})+s_0\, ,$$
expression that we now know depends differentiably on $\o_t$. Here,
$s_0$ is the projection of $s$ onto the constants, constant that itself
only depends on $\O$, $J$ and the volume of metrics in ${\mathfrak M}_{\O,G}$.
By the K\"ahler identity $\db_{t}^{*}=-i[\Lambda_{t},\d]$, we conclude that
$$\pi_t s_t = 2G_{t}\Lambda_t \d (\o_t \hok X_{\Omega})+s_0\, ,$$
and therefore,
$$\frac{d}{dt}\pi_t s_t = 2G_{t}\Lambda_t \d (\dot{\o}_t \hok X_{\Omega})
+2G_t \dot{\Lambda}_t \d  (\o_t \hok X_{\Omega}) +
2\dot{G}_t \Lambda_t \d (\o_t \hok X_{\Omega})\, .$$

The last two terms in the expression above cancel each other out. 
Indeed, $\o_t \hok X_{\Omega}=-i\db(\pi_t s_t)$ and computing the derivative
of $\dot{\Lambda}_t$ in terms of $\dot{\v}_t$, we see that
$2G_t \dot{\Lambda}_t \d  (\o_t \hok X_{\Omega})=2G_t(i\ddb \dot{\v}_t,
i\ddb (\pi_t s_t))_t$. On the other hand, the differential of the Green's 
operator is given by $-G_t \dot{\Delta}_t G_t$, and we obtain that
$2\dot{G}_t \Lambda_t \d (\o_t \hok X_{\Omega})=
2G_t\dot{\Delta}_t G_t \Lambda_t i\ddb (\pi_t s_t) = -G_t\dot{\Delta}_t(\pi_t 
s_t) =-2G_t(i\ddb \dot{\v}_t,i\ddb (\pi_t s_t))_t$.

Since the real and imaginary parts of $X_{\Omega}$ are Killing vector fields
and the metric potential $\v_t$ is $G$-invariant, we have that 
$X_{\Omega}(\dot{\v}_t)=0$, and so $\d \dot{\v}_t \hok 
X_{\Omega}=(\d \dot{\v}_t,i\d (\pi_t s_t))_t$ is orthogonal to the 
constants. But notice also that
since $X_{\Omega}$ is holomorphic, we have that
$\dot{\o}_t=i\ddb \dot{\v}_t \hok X_{\Omega}= -i \db(\d \dot{\v}_t \hok 
X_{\Omega})$. Thus, we obtain 
 $2G_{t}\Lambda_t \d (\o_t \hok X_{\Omega})=-2G_t \Lambda_t i\ddb 
(\d \dot{\v}_t \hok X_{\Omega})$. The desired result follows now because
$G_t$ is the inverse of the Laplacian in the complement of the constants. \qed

Given any K\"ahler metric $g$ in ${\mathfrak M}_{\Omega,G}$, the extremal 
vector field $X_{\Omega}$ of the class can be written as 
$X_{\Omega}=\d_g^{\#} (\pi_g s_g)$. Thus, the critical points
of $\pi_g s_g$ corresponds to zeroes of $X_{\Omega}$, and are therefore,
independent of $g$. By the Lemma above, this can be strengthen a 
bit, and we have the following remarkable consequence. This result is
reminiscent of the convexity theorem on the image of moment mappings
\cite{at,gust}.

\begin{theorem} \label{tc}
Let $\o$ be any metric in ${\mathfrak M}_{\Omega, G}$ and consider the
function $\pi_{\o} s_{\o}$ obtained by projection of the scalar 
curvature onto the space of real holomorphy potentials. Then the range
of $\pi_\o s_\o$ is a closed interval on the real line that only
depends on the class $\Omega$ and not on the particular metric $\o
\in {\mathfrak M}_{\Omega, G}$ chosen to represent it.
\end{theorem}

{\it Proof}. Let $\o_t=\o+i\ddb \v_t$ be a path in 
${\mathfrak M}_{\Omega, G}$. If we use
the subscript $t$ to denote geometric quantities associated with $\o_t$, by
Lemma \ref{le1} we have that
$$\frac{d}{dt}\pi_t s_t = (\d \dot{\v}_t , \d (\pi_t s_t) )_t \, .$$
Since the maximum and minimum of $\pi_t s_t $ occur at critical points, this
expression shows that these extrema values do not change with $t$. The
result follows because ${\mathfrak M}_{\Omega, G}$ is path connected. \qed

We now proceed to lift the projection $\pi_g$ onto holomorphy potentials to
the level of $G$-invariant $(1,1)$ forms. This lift will be essential in
our definition of the extremal flow.

So let us denote by $\wedge ^{1,1}_{k,G}$ be the space of real forms 
of type $(1,1)$, invariant under $G$ and of class $L_{k}^2$. 

\begin{lemma}
\label{le2}
Given any $G$-invariant metric $\tilde{g}$, there exists a uniquely defined
continuous projection map 
\begin{equation}
\Pi _{\tilde{g}}:\wedge ^{1,1}_{k+2,G} \mapsto 
\wedge ^{1,1}_{k+2,G} \, , \label{proj} \end{equation}
which intertwines the trace and the projection map $\pi _{\tilde{\o}}$ in
{\rm (\ref{proj1})}, and 
such that $\eta -\Pi _{\tilde{g}}\eta $ is cohomologous to zero for all 
$\eta \in  \wedge ^{1,1}_{k+2,G}$. For metrics $\tilde{\o}$ as in
{\rm (\ref{met})}, the map $\varphi \mapsto \Pi _{\tilde{\o}}$ from
$L^2_{k+4,G}$ to ${\rm End} (\wedge ^{1,1}_{k+2,G})$ is smooth. 
\end{lemma}

{\it Proof}. Let $\eta \in \wedge ^{1,1}_{k+2,G}$. Since 
$\Pi _{\tilde
{\o}}\, \eta $ must be of the form $\eta + i\ddb f$ for some real valued 
function $f$, the intertwining property of the projection and trace gives that
$${\rm trace}_{\tilde{\o}}\, \eta -\frac{1}{2}\Delta _{\tilde{\o}}f=
\pi _{\tilde{\o}}\,  {\rm trace}_{\tilde{\o}}\, \eta \, ,$$ and so
$$\Delta _{\tilde{\o}}f=-2(\pi_{\tilde{\o}} -1){\rm trace}_{\tilde{\o}}\eta 
\, .$$ 
The right side of this expression is a $G$-invariant real valued function
in the complement of the constants. We can then solve the equation for $f$
and obtain a real valued function which is invariant under $G$. By the
continuity properties of the map $\pi _{\tilde{\omega}}$, for metrics as in
(\ref{met}) the map $\varphi \mapsto \Pi _{\tilde{\o}}$  
is a smooth map from a suitable neighborhood of the origin in 
$L^2_{k+4,G}$ to the real Hilbert space ${\rm End} (
\wedge ^{1,1}_{k+2,G})$. \qed

\subsection{Extremal flow equation}
Lemma \ref{le2} provides us with the tool needed to set-up the heat flow
equation adapted to the extremal metric problem. The idea of using 
{\it good} flows to better geometric quantities was 
originally used by Eells and Sampson \cite{es} in another context, and
reconsidered by Hamilton \cite{ha} in his definition of the Ricci flow.
In our case, we are given a metric in ${\mathfrak M}_{\Omega,G}$ and try to 
improve it by 
means of a non-linear {\it pseudo-differential} heat equation, requiring 
the velocity of the curve to equal the component of the Ricci
curvature that is perpendicular to the image of $\Pi$. 

More precisely, we fix a maximal compact subgroup $G$ of the automorphism 
group of $(M,J)$, and work on ${\mathfrak M}_{\Omega,G}$, the space of all 
$G$-invariant K\"ahler forms that represent $\Omega$. Given $\o \in 
{\mathfrak M}_{\Omega,G}$, we consider a path $\o_t$ of K\"ahler 
metrics that starts at $\o$ at $t=0$ and obeys the flow equation
$\d _t \o _t  =  -\rho_t+\Pi _t\rho_t$. 
Since $-\rho_t+\Pi _t\rho_t$ is cohomologous to zero and $G$-invariant, 
for as long as the solution exists, we will have that $\o_t \in 
{\mathfrak M}_{\Omega,G}$. Thus, our evolution equation is given by
the initial value problem 
\begin{equation}
\label{evol}
\begin{array}{rcl}
\d _t \o _t & = & -\rho_t+\Pi _t\rho_t \, . \\
\o _{0} & = & \o \, .\end{array}
\end{equation}
Critical points of this equation correspond to extremal metrics, that is 
to say, metrics such that $\rho = \Pi \rho$. 

It is easy to reformulate (\ref{evol}) as a scalar equation. For if
$\o_t = \o +i\ddb \v_t$, we have that
$\Pi _t\rho_t -\rho_t=i \ddb G_t( s_t - \pi_t s_t )$, where
$G_t$ is the Green's operator of the metric $\o_t$, and by compactness
of $M$, we see that the potential $\v_t$ evolves according to
\begin{equation}
\label{evol2}
\begin{array}{rcl}
\d _t \varphi _t & = & G_t( s_t -\pi _t s_t ) \, , \\
\varphi_{0} & = & 0 \, . \end{array}
\end{equation}
A critical point of this scalar version of the equation is given by a
metric for which $G_\o (s_\o-\pi_\o s_\o)=0$, and since 
$s_\o-\pi_\o s_\o$ is orthogonal to the constant, this condition is 
equivalent to saying that $s_\o=\pi_\o s_\o$. Thus, a critical point 
$\o$ is an extremal metric.

\subsection{General properties of the extremal flow}
We begin by making a rather expected observation.

\begin{proposition}
Let $\o_t$ be a solution of the initial value problem 
{\rm (\ref{evol})}. If $d\mu_t$ is the volume form, we have that
$$\frac{d}{dt} d\mu_t = \frac{1}{2}(\pi _t s_t -s_t)d\mu_t \, .$$
In particular, the volume of $\o_t$ is constant.
\end{proposition}
 
{\it Proof}. The volume form is given by
$$d\mu_t =\frac{\o_t^n}{n!}\, .$$
Differentiating with respect to $t$, we obtain:
$$\frac{d}{dt}d\mu_t = \frac{1}{(n-1)!}\o_t^{n-1}\wedge \dot{\o_t}=
\frac{1}{(n-1)!}\o_t^{n-1}\wedge (\Pi_t \rho_t-\rho_t)=
\frac{1}{2}(\pi _t s_t -s_t)d\mu_t \, ,$$
as desired. Notice that this form of maximal rank is exact. \qed

Our next results address the plausible existence of fixed points or 
periodic solutions of the flow equation. 

Observe that (\ref{evol}) is invariant under the group of diffeomorphims 
that preserve the complex structure $J$. An {\it extremal soliton} is a 
solution that changes only by such a diffeomorphism. Then, 
there must be a holomorphic vector field $V=(V^i)$ such that $V_{i,\bar{j}}+
V_{\bar{j},i}=\Pi \rho _{i\bar{j}}-\rho_{i\bar{j}}$. If the vector field
$V$ has a holomorphy potential $f$, we refer to the pair $(g,V)$ as 
a gradient extremal soliton.

\begin{proposition}
There are no extremal solitons other than extremal metrics.
\end{proposition}

{\it Proof}. Suppose we have an extremal gradient soliton $(g,V)$ 
defined by a 
holomorphic potential $f$. Then 
$$i\ddb f = \Pi \rho - \rho \, , $$
and therefore, 
$$f = G_g(s - \pi s) \, .$$
This implies that $\Delta f=s-\pi s$ and since $\Delta $ is a real operator, 
the holomorphy potential $f$ must be real. But $f$ is a holomorphy potential,
so it is $L^2$-orthogonal to $s-\pi s$. Hence,
$$\| \nabla f \| ^2 = \int f \Delta f d\mu_g = \int f(1-\pi)s\, d\mu=0 \, .$$
Thus, $f$ is constant, and therefore, necessarily zero.

Thus, a non-trivial soliton, if any, must be given by a holomorphic vector 
field $V$ that is not a gradient. The set of all such vector fields forms an 
Abelian 
subalgebra of the algebra of holomorphic vector fields. The group of 
diffeomorphism they generate must be in the maximal compact subgroup $G$ of
isometries of the metric. This vector field does not change the metric
and so $\o_t =({\rm exp}(tV))^{*}\o=\o$. Hence, $\dot{\o}_t=0=
\Pi \rho - \rho $, and the metric is extremal. \qed

We now show that the evolution equation (\ref{evol}) is {\it almost} 
the gradient flow of the $K$-energy that characterizes extremal K\"ahler 
metrics \cite{si1}. Given two elements $\omega_0$ and $\omega_1$ of
${\mathfrak M}_{\Omega,G}$,
there exists a $G$-invariant function $\varphi$, unique modulo constants, 
such that $$\omega_1 = \omega_0 +i{\partial  \overline{\partial}}\varphi \, .$$
Let $\v_t$ be a curve of $G$-invariant functions such that 
$\omega_t=\omega_0 +i{\partial  \overline{\partial}}\v_t \in 
{\mathfrak M}_{\Omega,G}$ and $\omega(0)=\omega_0$, $\omega(1)=\omega_1$. 
We set
$$M(\omega_{0},\omega_{1})= -\int _{0}^{1}\!
dt \int _{M} \! \dot{\v}_t(s_t-\pi_t s_t)d\mu_t\, ,$$
where $s_t$ and $d\mu_t$ are the scalar curvature and volume form of 
the metric $\omega_t$, $\pi_t$ is the projection (\ref{proj1}) onto the 
space of $G$-invariant holomorphic potentials associated with this metric, 
and $\dot{\varphi}_t={\displaystyle \frac{d\varphi_t}{dt}}$. This definition
is independent of the curve $t\rightarrow \v_t$ chosen. 

Fix $\o_0\in {\mathfrak M}_{\Omega,G}$. The $K$-energy 
is defined to be
\begin{equation}
\begin{array}{rcl}
{\mathfrak M}_{\Omega,G} & \stackrel{\kappa}{\longrightarrow} & 
{\mathbb R} \\
\omega & \rightarrow & M(\omega_0, \omega)\, .
\end{array}
\end{equation}
We have (see Proposition 2 in \cite{si1}) that
$$\frac{d}{dt}\kappa (\omega_t)=-\int_{M} \! \dot{\varphi}_t( s_t 
-\pi _t s_t)d\mu _t \, .$$
Thus, up to the action of the non-negative Green's operator, the gradient
of $\kappa$ is given by the right-side of (\ref{evol2}), and we have

\begin{proposition}
Let $\o_t$ be a solution of the initial value problem 
{\rm (\ref{evol})}. Then
$$\frac{d}{dt} \kappa(\o_t) = -\int_M (s_t-\pi _t s_t)
G_t(s_t-\pi _t s_t)
 d\mu_t \, .$$
\end{proposition}

The flow equation (\ref{evol}) is also invariant under the one-parameter 
group of homotheties, where time scales like the square of the distance. 
In principle, such an invariance could give rise to periodic orbits of the
flow. However,

\begin{proposition}
The only periodic orbits of the flow equation {\rm (\ref{evol})} are its
fixed points, that is to say, the extremal metrics {\rm (}if any{\rm )} in 
${\mathfrak M}_{\Omega,G}$.
\end{proposition} 

{\it Proof}. Consider the $K$-energy suitably normalized by a volume 
factor to make it scale invariant. If there is a loop solution $\o_t$ of 
(\ref{evol}) for $t\in [t_1,t_2]$, since the volume remains constant, we will
have that $\kappa(\o_{t_1})=\kappa(\o_{t_2})$. By the previous proposition,
since $G_t$ is a non-negative operator, we conclude that 
$G_t(s_t-\pi_t s_t)=0$ on this time interval. This says that $\o_t$ is 
extremal for each $t$ on the interval, and so the 
right side of the evolution equation is zero. Thus, the loop is trivial, a 
fixed point of the flow. \qed

We end this section by showing that the functional (\ref{ene}) decreases
along the flow (\ref{evol}). This should be clear from the way the equation
was set-up, or at the very least, expected. 

\begin{proposition}
Let $\o_t$ be a path in ${\mathfrak M}_{\Omega,G}$ that solves the flow
equation {\rm (\ref{evol})}. Then
$$\frac{d}{dt}E_{\Omega}(\o_t)= -4\int (s_t-\pi_t s_t ) 
L_{t}G_t(s-\pi_t s_t ) d\mu_t \leq 0 \, ,$$
and the equality is achieved if and only if $\o_t$ is extremal. In this
expression, $L_t=(\db \d^{\#})^{*}(\db \d^{\#})$ and $G_t$ is the Green's
operator. 
\end{proposition}

{\it Proof}. Given any variation of the metric with potential function
$\v$, we know that
$$\frac{d}{dt}E(\o_t)=-4 \int s L_t\dot{\v} d\mu_t \, .$$
But $\dot{\v}=G_t(s_t-\pi_t s_t )$, and since $\pi_t s_t$ is a holomorphy
potential and $L_t$ is self-adjoint, we see that
$$\frac{d}{dt}E(\o_t)=-4\int (s_t-\pi_t s_t ) L_{t}G_t(s-\pi_t s_t ) 
d\mu_t \, .$$

Both $L_t$ and $G_t$ are non-negative operators. Then so is $L_tG_t$, and
the expression above is non-negative. If it reaches the value zero at some
$t$, then we must have that $f_t=G_t(s_t-\pi_t s_t )$ is a holomorphy 
potential and $\Delta_t f=(1-\pi_t)s_t$ is an element of the image of 
$(1-\pi_t)$. Thus, $f_t$ is $L^2$-orthogonal to $(1-\pi_t)s_t$, and 
integration by parts yields that the gradient field $\nabla _t f_t$ is zero.
Thus, $f_t$ is a constant, which is necessarily zero. We then obtain
that $s_t=\pi_t s_t$ and the metric $\o_t$ is extremal. \qed

It is clear that we could have used also the energy $E$ in the r\^ole
that $\kappa$ played when proving that the flow does not have periodic 
orbits other than its fixed points. In fact, it is better to work with
$E_{\Omega}$ itself. For we do not know if $\kappa$ is in general bounded 
below on ${\mathfrak M}_{\Omega,G}$, but the energy 
functional $E_{\Omega}$ has this property indeed. If the solution
to the flow equation were to exist for all $t\in [0,\infty)$, the 
monotonicity result above would lead us to expect that, as 
$t\rightarrow \infty$, the sequence 
$\o_t$ should be getting closer and closer to an extremal metric. 
In any case, we discussed both $\kappa$ and $E_{\Omega}$ to show their
similar behaviour under the extremal flow.

\section{The linearized flow equation}\label{sl}
Consider a family of metrics in ${\mathfrak M}_{\Omega,G}$ of the form 
$\o_t(v)=\o_{\v}+i\ddb \a(t,v)$, with $\a(t,0)=0$.
We set $\be=\be_t={\displaystyle \frac{d\a(t,v)}{dv}\mid_{v=0}}$.
The linearization of (\ref{evol2}) at $\o_{\v}$ in the direction of
$\be$ is given by
$$\d _t \be_t =\frac{d}{dv}(G_{(t,v)}((1-\pi_{(t,v)})s_{(t,v)}))\mid_{v=0}\, .
$$
Of course, before taking the restriction to $t=0$, the argument of the 
$v$-differentiation in the right side involves quantities associated with 
the metric $\o_t(v)$.

In the remaining part of this section we use the subscript $\v$, or no 
subscript at all, to denote geometric quantities associated with the metric
$\o_{\v}$. We have that
$$\frac{d s_{(t,v)}}{dv}\mid_{v=0}=
-\frac{1}{2}\Delta^2_{\v} \be-2(\rho_{\v},i\ddb \be)_{\v}\, .$$
Since the variation of the Green's operator is $-G_\v(\frac{d}{dv}\Delta_{
(t,v)})G_\v$ (keep in mind that this operator needs to be applied only to 
$s-\pi s$, a function that is orthogonal to the constants), 
using the relation between $\rho_\v$ and $\Pi_\v \rho_\v$,
we obtain that 
$$\d _t \be =-\frac{1}{2}\Delta_\v \be -2G_\v(\Pi_\v \rho_\v, i\ddb \be)_\v-
G_\v\left( \frac{d}{dv}\pi_{(t,v)}s_{(t,v)}\mid_{v=0}\right) \, .
$$
By Lemma \ref{le1}, we may write this as
\begin{equation}
\label{lf}
\d _t \be =-\frac{1}{2}\Delta_\v \be -2G_\v(\Pi_\v \rho_\v, i\ddb \be)_\v-
G_\v (\d_{\v}^{\#}\be , X_{\Omega})_{\v}\, , 
\end{equation}
where $X_{\Omega}$ is the holomorphic vector field of the class
$\Omega$, vector field that can be expressed as 
$X_{\Omega}=\d_{\v}^{\#}(\pi_{\v}s_{\v})$.
Notice that 
$$P_{\v}(\be)=G_\v (\d_{\v}^{\#}\be , X_{\Omega})_{\v}$$ 
is a pseudo-differential operators of order $-1$ in $\be$ whose
coefficients depend non-linearly on the coefficients of the metric $\o_\v$.

We summarize our discussion into the following 

\begin{theorem}
Let $(M,J,\Omega)$ be a polarized K\"ahler manifold and let $G$ be a maximal 
compact subgroup of $Aut(M,J)$. The extremal flow equation 
{\rm (\ref{evol2})} 
{\rm (}or equivalently, {\rm (\ref{evol}))} in 
${\mathfrak M}_{\Omega,G}$ is a non-linear
pseudo-differential parabolic equation.
\end{theorem}

\begin{remark}
Generically, the manifold $(M,J)$ carries no non-trivial holomorphic
vector fields, and the space of holomorphic potentials reduces to the 
constants. (For example, this is the situation when the first Chern class
$c_1(M,J)$ is negative.) This is still so in the slightly
larger case of a manifold $(M,J)$ where all of its non-trivial 
holomorphic vector fields have no zeroes. Under this hypothesis, the 
pseudo-differential term of order $-1$ in the right side of the linearized 
flow equation (\ref{lf}) vanishes, and the equation reduces to 
$$\d _t \be_t =
-\frac{1}{2}\Delta_\v \be -2G_\v(\Pi_\v \rho_\v,i\ddb \be)_\v \, ,$$
still a pseudo-differential equation, in this case, a zeroth-order 
perturbation of pseudo-differential type of the standard time dependent heat
equation. Thus, even for generic complex manifolds of K\"ahler type, 
the pseudo-differential nature of our flow equation remains.
\end{remark}

\begin{remark}
Even if $\Omega$ is the canonical K\"ahler class $c_1$ (which a fortiori 
must then have a sign), the extremal flow equation (\ref{evol}) (or
equivalently, (\ref{evol2})) does
not necessarily coincide with the K\"ahler Ricci flow. This will only be the
case if we know a priori that $\pi s$ is a constant, which is a rather
non-trivial condition to impose and only happen if the Futaki character of
the canonical class vanishes. This fails to be so in general, as is the
case, for instance, of the blow-up of ${\mathbb C}{\mathbb P}^2$ at one or 
two points.
\end{remark}
\medskip

We now introduced an {\it approximate} linearized equation whose solution
is needed in our study of local solvability of (\ref{evol2}). In order to
do so, we make some preliminary observations. 

Let $T$ be a positive real number to be determined later and set
$I=[0,T]$. A scale ${\mathcal Y}=\{ {\mathcal Y}_{j}\}_{j\geq 0}$ of 
Banach spaces is a countable family of complete normed spaces such that  
${\mathcal Y}_{j}\supset {\mathcal Y}_{j+1}$ and each ${\mathcal Y}_{j}$
is dense in ${\mathcal Y}_0$. Given one such, we define
$$C_{(j,k)}(I;{\mathcal Y})=C^{0}(I;{\mathcal Y}_{j})\cap \cdots \cap
C^{j-k}(I;{\mathcal Y}_{k}) \, ,$$ 
and provide it with the norm
$$ \| v \| _{j,k}=\sup_{t\in I}\{ \sup_{0\leq r \leq j-k}
\{ \| \partial _{t}^{r}v(t)\|
_{j-r}\} \} \; .$$ 

In what follows, where we shall consider metrics of the form 
$\o_t=\o+i\ddb \v_t$ for path of functions $\v_t$ that begin at $0$ when
$t=0$, we shall always use the scale of Sobolev spaces  
$${\mathcal Y}_{j}=H^{2j}(M)$$
as defined by the background metric $\o$. When $t$ varies on 
the interval $[0,T]$, if we choose $T$ sufficiently small, all the metrics 
$\o_t$ will be equivalent, and the Sobolev spaces defined by them
will be equivalent to each other, with equivalent norms.
We let the Sobolev order jump by $2$ because the operator $F(\v)$ in the 
right side of 
(\ref{evol2}), $$F(\v):=G_{t}(s_t-\pi_t s_t)\, ,$$ 
is of second order. That is the reason
for the peculiar definition of the scale ${\mathcal Y}_j$. By the
Sobolev embedding theorem we know that $H^{k}(M)$ is a Banach algebra if
$k>n$. Thus, for as long as the metric $\o_t$ is equivalent to $\o$
and provided that $k>n$, 
we have a continuous mapping 
$$F: H^{s+4}(M) \mapsto H^{s+2}(M)\, .$$

\begin{proposition}\label{cd}
Assume that a solution $\v(t)$ of {\rm (\ref{evol2})} is in 
$C_{(s+1,0)}(I;{\mathcal Y})$ on the interval $I$ for some integer $s$ 
such that $2s>n+2$. Then all the values
of $\d_t^{r}\v(t)$ {\rm (}$1\leq r\leq s+1${\rm )} restricted to $t=0$ 
are completely determined and $\d_t^{r}\v(t)\mid_{t=0}:=\v_r \in {\mathcal
Y}_{s+1-r}=H^{2s+2-2r}(M)$.   
\end{proposition}

{\it Proof}. The initial condition $\v\mid_{t=0}$ is zero, and the equation 
itself sets the value of $\d_t \v \mid_{t=0}=F(0)=
G_{\o}(s_{\o}-\pi_{\o}s_{\o})$ that is evidently in $H^{2s}(M)$.

The relation (\ref{lf}) for $\be=\d_t \v_t$ says that
$$\frac{d}{dt}\be=-\frac{1}{2}\Delta_{\v}\be + P_0(\v)\be \, ,$$
where $P_0$ is a pseudo-differential operator of order zero whose 
coefficients
depend on the coefficients of the metric $\o_\v$ and its curvature
tensor. Since 
$\v_t\in C(I;H^{2s}(M))$ and $2s> n+2$, by the Sobolev embedding theorem,
these coefficients are continuous functions. By regularity of 
pseudo-differential operators on Sobolev spaces, we obtain that
$\d_t \be =\d_t^2 \v_t \in H^{2(s-1)}(M)$, which is still a continuous 
function because $2(s-1)>n$.

If we differentiate the expression above for $\be=\d_t \v_t$ with respect to 
$t$, we obtain 
$$\frac{d^2}{dt^2}\be=-\frac{1}{2}\Delta_{\v}\frac{d}{dt}\be 
 -\frac{1}{2}L_{\v}(\be)\be+P_0(\v)\frac{d}{dt}\be + P_{0,\v}(\be)\be \, ,$$ 
where $L_{\v}(\be)$ and $P_{0,\v}(\be)$ are the linearizations of 
$\Delta_{\v}$ and $P_{0}(\v)$ at $\v$ in the direction of
$\d_t \v$, respectively. The first is an operator of
order two whose coefficients are continuous. By the metric dependence of
$P_{0}(\v)$, the latter is a pseudo-differential operator of order zero
whose coefficients are also continuous functions. Hence, 
$\d_t^2\be =\d_t^3 \v_t\in H^{2(s-2)}(M)$. 

Iteration of the argument above yields that
$$\d_t^r \v=F_r(\v, \d_t \v, \ldots, \d_t^{r-1}\v) \, ,$$
where $F_r$ is some operator whose coefficients depend upon the 
coefficients of the metric $\o_\v=\o+i\ddb \v$. 
The desired result for the regularity of $\d_t^r \v$ follows again using 
the Sobolev embedding theorem and the known regularity of the lower
order time derivatives $\d_t^j \v$, $0\leq j\leq r-1$. \qed

Assume given Cauchy data $\v_{0}=0$ for $(\ref{evol2})$
and let $\v_{r}=\partial _{t}^{r}\v(t)\mid _{t=0}$ be the sequence of 
coefficients of the Taylor series  of $\v(t)$ given by the 
proposition above. The Cauchy data $\v_{0}$ determines the sequence 
$\v_{r}$, $1\leq r \leq s+1$. We consider the metric space:
\begin{equation}
W(I)=W^s(I)=\{ \psi(t)\in C_{(s+1,0)}(I;{\mathcal Y}): \; \partial _{t}^{r}\psi(t)
\mid _{t=0}=\v_{r}, \; 0\leq r\leq s+1\} \; . 
\label{spa}
\end{equation}
It is not empty, as can be seen by solving the
Cauchy problem for a suitable parabolic equation. 

By a continuity argument, for any $\psi(t)\in W(I)$ the form
$\o_{\psi}=\o + i\ddb \psi(t)$ is positive provided that $t$ is 
sufficiently small. Hence, $\o_{\psi}$ defines a K\"ahler metric.
This metric is not smooth in general. However, if $2s>n+2$, by the Sobolev 
embedding theorem, $\o_{\psi}$ is at least $C^2$, and the operator in the 
right side of (\ref{lf}) will make sense when $\psi$ plays the r\^ole of 
$\v$. Thus, we set
\begin{equation}
P_0(\psi)b=-2G_\psi(\Pi_\psi \rho_\psi, 
i\ddb b)_\psi-G_\psi (\d_{\psi}^{\#}b,X_{\Omega})_{\psi} \, .
\label{lo}
\end{equation}
Then $P_0(\psi)$ is a pseudo-differential operator of order zero in 
$b$, whose coefficients depend upon the coefficients of the 
metric $\o_{\psi}$ and its curvature tensor, all of which are continuous 
functions. For each $t$ on 
a time interval where all the metrics $\o_{\psi}$ are uniformly equivalent, 
we have that 
\begin{equation}
-\frac{1}{2}\Delta_\psi +P_0(\psi) :H^{2}(M) \rightarrow H^{0}(M)=L^2(M)
\label{ao}
\end{equation}
continuously. We shall consider the equation
\begin{equation}
\frac{d}{dt}b= -\frac{1}{2}\Delta_{\psi}b +P_0(\psi)b \, ,
\label{ae} 
\end{equation}
whose Cauchy problem will be studied in the next
section. We shall refer to it as the {\it approximate} linearized equation,
the reasons being ---we hope--- clear at this point.

We end this section with the following

\begin{proposition}
Let $\v_1$ be the Cauchy data for {\rm (\ref{ae})}. If
$b(t) \in C_{(s,0)}(I,{\mathcal Y})$ is a solution, then
$\d_t^r b \mid_{t=0}=\v_{r+1}$, $0\leq r \leq s$.
\end{proposition}

{\it Proof}. We have seen above that if $\v(t)$ satisfies (\ref{evol2}), then
$$\d_t^r\v=F_r(\v, \d_t \v, \ldots, \d_t^{r-1}\v) \, , \quad r\geq 2\, ,$$
where $F_r$ is some operator whose coefficients depend upon the 
coefficients of the metric $\o_\v=\o+i\ddb \v$, 
and whose restriction at $t=0$ depends only on the 
sequence $\v_0 , \v_1, \ldots, \v_{r-1}$. The approximate linearized equation
(\ref{ae}) is obtained from the linearization of (\ref{evol2}) given in
(\ref{lf}), when we replace the r\^ole played by $\v(t)$ by that of
$\psi(t)$. But $\psi(t)$ and $\v(t)$ have the same coefficients in their
Taylor expansions up to order $s+1$. Therefore, the solution $b(t)$ to the 
Cauchy problem of (\ref{ae}) with data $b(0)=\v_1$ will have necessarily a 
Taylor series of order $s$ that agrees with the Taylor series of the 
solution to the Cauchy problem of linearized equation (\ref{lf}). The 
conclusion follows by Proposition \ref{cd}. \qed

\section{Local solvability of the extremal flow equation}\label{loc}
In this section, we prove local time existence of solution to 
the extremal flow (\ref{evol2}). We do so by adapting to our situation
a method of T. Kato for the solvability of abstract differential equations
and 
non-linear problems \cite{ka}. The pseudo-differential nature of our
linearized equation (\ref{lf}) makes the task harder. But fortunately 
enough, the strictly pseudo-differential part of the equation is lower order,
and most of the analysis is based on that of the standard 
time-dependent heat equation.

\subsection{The Cauchy problem for the approximate linearized equation}
From now on, we take $s$ to be an integer such that $2s> n+2$ and
${\mathcal Y}_{j}=H^{2j}(M)$ as in the previous section.  Given 
Cauchy data $\v_0=0$ for (\ref{evol2}), Proposition \ref{cd} determines
the sequence $\{ \v_{j}\}_{j=0}^{s+1}$ that in turn allows us to define the
space $W(I)$ of (\ref{spa}). The interval $I=[0,T]$ will be determined later.
For $\psi \in W(I)$, we consider the metrics $\o_{\psi}=\o+i\ddb \psi$ and
the Cauchy problem of the approximate linearized equation (\ref{ae}),
$$\frac{d}{dt}b=-\frac{1}{2}\Delta_{\psi} b + P_0(\psi)b \, ,$$
where $P_0(\psi)$ is given by (\ref{lo}), a pseudo-differential operator of 
order zero whose coefficients depend non-linearly on the coefficients of the
metric $\o_{\psi}$ and its curvature tensor. 

Let $p(t,s)$ be the evolution operator of 
$$\frac{d}{dt}b=-\frac{1}{2}\Delta_{\psi} b \, .$$
Thus, $p(t,s)$ is a two-parameter family of strongly continuous operators
on ${\mathcal Y}_0$ and ${\mathcal Y}_1$, respectively, such that
$p(t,s)p(s,r) = p(t,r)$, $p(t,t)=1$, and for $b\in {\mathcal Y}_1$ we have
\begin{equation}
\begin{array}{rcl}
\d_t p(t,s)b & = & -\frac{1}{2}\Delta_{\psi(t)}p(t,s)b \, , \\
\d_s p(t,s)b & = & -\frac{1}{2}p(t,s)\Delta_{\psi(s)}b \, .
\end{array}
\label{eop}
\end{equation}
This family of operators exists for $0\leq s\leq t \leq T$, and their 
operator norm is bounded uniformly by a 
constant that only depends upon a bound on $I=[0,T]$ of the coefficients
of $\o_{\psi(t)}$. The function solving (\ref{ae}) with Cauchy data
$\be$ must satisfy the integral equation
\begin{equation}
b(t)=p(t,0)\be+\int_{0}^{t}p(t,s)P_{0}(\psi(s))b(s) d s \, .
\label{ie}
\end{equation}

Consider the set of functions $b(t)$ in $C_{(1,0)}(I;{\mathcal Y})=
C(I;{\mathcal Y}_0)\cap C^{1}(I;{\mathcal Y}_1)$ such that $b(0)=\beta$. The
right hand side of the expression above defines an operator in
this space,
$$P: b \mapsto p(t,0)\beta+\int_{0}^{t}p(t,s)P_{0}
(\psi(s))b(s) d s \, ,$$
and by the explicit form of the coefficients of $P_{0}(\psi)$ discussed 
above, combined with continuity of 
pseudo-differential operators on Sobolev spaces, we have that
 $$\| Pb -P\tilde{b}\| \leq C T \| b-\tilde{b}\| \, ,$$
where $C$ is a constant that depends upon the $L^{\infty}$-norm of the 
coefficients of $\o_{\psi(t)}$ and its curvature tensor on the time interval 
$I$. A fixed point argument now yields the following result:

\begin{theorem}\label{thae}
Consider the Cauchy problem for {\rm (\ref{ae})} with Cauchy data 
$b(t)\mid_{t=0}
\in {\mathcal Y}_1$. Then there exists $T$ such that this problem has a
unique solution in $C_{(1,0)}(I;{\mathcal Y})=C(I;{\mathcal Y}_0)\cap 
C^{1}(I;{\mathcal Y}_1)$. The value of $T$ only depends on
supremum norms of the coefficients of $\o_{\psi(t)}$ and its curvature
tensor. 
\end{theorem}

Of course, the regularity of the solution in the theorem above can be
improved if we start with a better initial condition. For that observe 
that the coefficients of the operator $\Delta_{\psi}$ are curves
in $C_{(s,0)}(I;{\mathcal Y})$, and consequently,
$$\Delta_{\psi(t)}: H^{2j}(M) \mapsto H^{2j-2}(M)\, , \; 1\leq j\leq s\, ,$$
continuously. While the metrics remain equivalent, we can choose a uniform
constant for the operator norm of these maps, and (\ref{eop}) holds for
$b\in H^{2j}(M)$ with $j$'s in this range. Then we have

\begin{corollary}
If the initial data $b(t)\mid_{t=0}=\v_1 \in {\mathcal Y}_{s}$, the 
solution to the Cauchy problem for {\rm (\ref{ae})} belongs to 
$C_{(s,0)}(I;{\mathcal Y})=C(I;{\mathcal Y}_s)\cap \cdots C^{s}(I,{\mathcal
Y}_0)$.
\end{corollary}
 
{\it Proof}. The arguments in the proof of the theorem and the remarks made 
above show that we now have a solution $b(t)$ to the Cauchy problem for
(\ref{ae}) that is in $C(I;{\mathcal Y}_s)\cap C^{1}(I;{\mathcal Y}_{s-1})$.
This solution satisfies (\ref{ie}) with $\be=\v_1$.

We can differentiate repeatedly the identity (\ref{ae}) in order to show
that the regularity of $b(t)$ with this initial condition can be improved.
Notice that the coefficients of the second order operators 
$d_t^r \Delta_{\psi(t)}$, 
$1\leq r \leq s-1$, are curves in $C(I;H^{2s-2r})$, and so we have
$d_t^r \Delta_{\psi(t)} \in C(I; {\mathcal L}({\mathcal Y}_{j+r+1}, 
{\mathcal Y}_{j})$ for $0\leq j \leq s-1-r$. Here, ${\mathcal L}(X,Y)$ is the
space of linear bounded operators from $X$ to $Y$, and the assertion follows
because in the stated range, $H^{2s-2r}\cdot H^{2j+2r}\subset H^{2j}$.
This suffices to conclude that the contributions to $\d_t^{k+1}b$ arising
from $\d_t^{k}\Delta_{\psi}b$ are in $H^{2s-2k-2}$ if we already 
know that $b\in C_{(s,k)}(I;{\mathcal Y})$.

The analysis of the contributions to $\d_t^{k+1}b$ arising from 
$d_t^k (P_0(\psi)b)$ is similar. This time, the coefficients of the
operators $d_t^r(P_{\psi(t)})$ are curves in $C(I;H^{2s-2r-2})$, one 
degree worse than those of $d_t^r \Delta_{\psi(t)}$, but the operators
are of pseudo-differentials of order zero. The desired improved regularity
follows by the same arguments as the ones in the previous paragraph. \qed

\subsection{An elliptic equation for $\gamma -F$}
Let us recall that $F(\v)=G_{\v}(s_{\v}-\pi_{\v}s_{\v})$ 
is the second order non-linear operator defined by the right side 
(\ref{evol2}). The derivative $L_{\psi}$ of this map at a general point 
$\psi$ in
${\mathcal Y}_{s+1}$ was computed in \S\ref{sl} and equals the
operator in the right side of (\ref{lf}):
\begin{equation}
L_{\psi}b= -\frac{1}{2}\Delta_\psi b -2G_\psi(\Pi_\psi \rho_\psi, 
i\ddb b)_\psi-G_\psi (\d_{\psi}^{\#}b,X_{\Omega})_{\psi}\, .
\label{lp}
\end{equation}
Since the top part of this
linearization is the negative operator $-\frac{1}{2}\Delta_{\psi}$,
while the lower order term is a pseudo-differential operator of order
zero, coercive estimates for this linearization imply that $\lambda -
L_{\psi}$ is an invertible operator as a map, say, from
${\mathcal Y}_{1}$ to ${\mathcal Y}_{0}$, for a sufficiently large 
constant $\lambda$. 

Let us then take a constant $\lambda$, and consider the non-linear
elliptic map
\begin{equation}
\begin{array}{rcl}
{\mathcal Y}_{s+1} & \longrightarrow & {\mathcal Y}_{s}\\
\v & \mapsto & \lambda \v - F(\v) \, .
\end{array}
\label{ne}
\end{equation}
We remind the reader here of the sequence $\{ \v_{r}\}$ given by 
Proposition \ref{cd}, whose first element is $\v_0=0$. 

\begin{proposition}
For $\lambda$ sufficiently large, there are neighborhoods ${\mathcal O}$ and
${\mathcal V}$ of $\v_0$ and $-\v_1$ in ${\mathcal Y}_{s+1}$ and 
${\mathcal Y}_{s}$, 
respectively, such that the restriction of {\rm (\ref{ne})} to ${\mathcal O}$
is an isomorphism onto ${\mathcal V}$.
\end{proposition}

{\it Proof}. This is a consequence of the Inverse Function Theorem. Indeed,
the linearization $\lambda -L_{0}$ is an invertible operator from
${\mathcal Y}_{1}$ to ${\mathcal Y}_{0}$. Hence, if $f\in {\mathcal Y}_{s}$,
there exists an element $b\in {\mathcal Y}_1$ that satisfies the equation
$$(\lambda-L_{0})b= f \, .$$
Thus, the image of $b$ under $L_{0}$ is in ${\mathcal Y}_{1}$, and by the
regularity properties of $\lambda -L_{0}$, we must have
$b\in {\mathcal Y}_2$. Iterating this argument, we conclude that
$b\in {\mathcal Y}_{s+1}$, and so, $b$ is an element of the tangent space
of ${\mathcal Y}_{s+1}$ at $0$. The desired result follows. \qed 

\begin{corollary}\label{coee}
Let $\psi \in {\mathcal Y}_s$ be sufficiently closed to $-\v_1$. Then, for
large $\lambda$, the equation 
$$\lambda \v - F(\v)= \psi $$
has a solution $\v \in {\mathcal Y}_{s+1}$. The solution is unique if
it is required to be closed enough to $\v_0=0$.
\end{corollary}

In the sequel, we let $D=D^{s}$ be the open neighborhood of $\v_0$ in
${\mathcal Y}_{s+1}$ where the operator $F(\v)$ is defined and smooth.

\subsection{The fixed point argument and the non-linear equation}
Proceeding in analogy with \cite{ka}, we define $E_{\v_0}(I)$ to be the 
set of curves $\psi(t)\in W^s(I)\subset C_{(s+1,0)}(I;{\mathcal Y})$ such that
$$\| \partial _{t}^{k}\psi(t)-\v_{k}\|_{s+1-k}\leq R,\; k=0,\ldots , s,
\; t\in I \, , $$ for some positive constant $R$. The value of $R$ is chosen
so the ball in ${\mathcal Y}_{s+1}$ with center $\v_0$ and radius
$R$ is contained in the domain $D$ of $F(\v)$. This space is not empty 
for some $R>0$ and some $I=[0,T]$. 

By the form (\ref{lp}) of the linearization of $F(\v)$ at $\psi$, 
we may conclude that if $\psi_{1}$ and $\psi_{2}$ are elements of 
${\mathcal Y}_{s+1}$, then the operator norm, as a map from ${\mathcal Y}_{s}$
to ${\mathcal Y}_{0}$, satisfies the estimate
$$\| L_{\psi_1}-L_{\psi_2}\| _{s,0} \leq C\| \psi_{1}-\psi_{2}\| _{1}
\, ,$$ for some constant $C$. Indeed, the top part of $L_{\psi}$ in
(\ref{lp}) is half of the 
Laplacian, and its lower order part is a zeroth order pseudo-differential 
operator with nicely behaved coefficients. Regularity of pseudo-differential
operators on Sobolev spaces yields to the assertion made.

We now define a key mapping in our proof of the local time
existence to the extremal flow. Let $\psi(t)$ be an element of 
$E_{\v_0}(I)$, and consider the solution $b(t)$ of $(\ref{ae})$ given in 
Theorem \ref{thae}, with initial data $\v_1$. We then solve the 
equation
\begin{equation}
\lambda \v - F(\v) = -b(t)+\lambda \left( \v_0+\int_{0}^{t} b(s) ds\right) 
\, . 
\label{abue}
\end{equation}
This is a stationary equation in $\v$ that is solved for each $t\in I$.
Here $\lambda$ is a real number such that $\lambda - L_{0}$ is an
isomorphism, where 
$L_0$ is the linearization (\ref{lp}) of $F(\v)$ at $\v=\v_0$.
Notice that if $t$ is sufficiently small, the right side of the equation 
lies in a 
neighborhood of $-\v_1$, and Corollary \ref{coee} applies to produce a 
solution $\v(t)$ in a neighborhood of $\v_0$.

The following two results are the versions of Proposition 7.4 and 
Proposition 7.6 in \cite{ka} adapted to our problem. We give 
proofs here for the sake of completeness.

\begin{proposition}
For sufficiently small $t$, {\rm (\ref{abue})} has a unique solution
$\v(t)$ in a neighborhood of $\v_0$ in  $D \subset {\mathcal Y}_{s+1}$,
with $\v(0)=\v_0 =0$. Furthermore, $\v(t)\in C_{(s+1,0)}(I;{\mathcal Y})$
and $\d_t^r \v(t) \mid_{t=0}=\v_r$, $0\leq r \leq s$ provided $T$ is chosen
sufficiently small, uniformly in $\psi \in E_{\v_0}(I)$. In that case,
$\v(t)\in E_{\v_0}(I)$.
\end{proposition}

{\it Proof}. The operator $\v \mapsto \lambda \v-F(\v)$ is a local 
diffeomorphism of
a neighborhood of $\psi (t)$ in ${\mathcal Y}_{s+1}$ into a 
neighborhood of $\lambda \psi -F(\psi)$ in ${\mathcal Y}_{s}$.
By Theorem \ref{thae}, the right side of (\ref{abue}) is a curve in
$C(I,{\mathcal Y}_s)$ that has value $-\v_1$ at $t=0$. By Corollary
\ref{coee}, we may solve the equation uniquely for $\v(t)$ in a 
${\mathcal Y}_{s+1}$ neighborhood of $\v_0$ and obtain that 
$\v(t)\in C(I,{\mathcal Y}_{s+1})$. This requires to choose $T$
sufficiently small but uniformly in $\psi \in E_{\v_0}(I)$. 

Formal differentiation of the equation solved by $\v(t)$ yields that
$$(\lambda -L_{\v(t)}) \d_t \v =\lambda b - \d_t b=
(\lambda -L_{\psi(t)})b(t) \, .$$
By the invertibility of the operator $\lambda -L_{\v(t)}$ and the known
regularity of the right side, it follows that
$\d_t \v \in C(I,{\mathcal Y}_{s})$ and has value $\v_1$ at $t=0$.
Iterated differentiation yields that $\v(t)\in C_{(s+1,0)}(I;{\mathcal Y})$
and has the desired coefficients in its Taylor series expansion up to 
order $s$. Moreover, the way the equation is solved, we have that
$$\| \d_t^k \v(t) -\v_r\|_{s+1-k} \leq R$$
for $t\in I$. This completes the proof. \qed
 
\begin{proposition}
For $\psi \in E_{\v_0}(I)$, let $\v(t)\in E_{\v_0}(I)$ be the solution 
curve given by the previous proposition. If $T$ is sufficiently small,
the mapping 
\begin{equation}
\begin{array}{c}
E_{\v_0}(I) \longrightarrow E_{\v_0}(I) \\
\psi(t) \longrightarrow \v(t) 
\end{array} \label{fawn} 
\end{equation}
is a contraction in the metric induced by the norm
$||| w |||_{1}=\sup_{t\in I}\| w(t)\| _{1}$, relative to
which, $E_{\v_0}(I)$ is complete.
\end{proposition}

{\it Proof}. Given a curve $b(t)$ in ${\mathcal Y}_k$, we define a norm
by $||| b|||_k=\sup_{t\in I}\| b(t)\|_{k}$. We shall only make use
of the $1$ and $0$ norm, respectively.

Let $\psi_1$ and $\psi_2$ be two elements of
$E_{\v_0}(I)$ and let $b_1$ and $b_2$ be the solutions to the 
corresponding approximate linearized equations with the same initial
condition $\v_1$. We the have that
$$b_1(t)=p_{\psi_1}(t,0)\v_1 \, , \quad b_2(t)=p_{\psi_2}(t,0)\v_1 \, ,$$
where $p_{\psi_1}(t,s)$ and $p_{\psi_2}(t,0)$ are the evolution operators
of the linear equations $\d_t v=L_{\psi_1(t)}v$ and 
 $\d_t v=L_{\psi_2(t)}v$, respectively. Consequently, 
$$b_2(t)-b_1(t)= (p_{\psi_2}(t,0)-p_{\psi_1}(t,0))\v_1 \, ,$$
and using the identity
$$p_{\psi_2}(t,0) \v -p_{\psi_1}(t,0)\v =-\int_{0}^t 
p_{\psi_2}(t,\tau)(L_{\psi_2(\tau)}-L_{\psi_1(\tau)}) p_{\psi_1}(\tau,0)
\v d\tau \, ,$$
we obtain the estimate
$$\| b_2 (t)-b_1(t)\|_{0} \leq C \| \v_1 \|_{s}\int_0^t \| L_{\psi_2(\tau)}-
L_{\psi_1(\tau)}\|_{s,0} d\tau$$
for some constant $C$. But we have observed that
$\| L_{\psi_2(\tau)}-L_{\psi_1(\tau)}\|_{s,0}$ is bounded by a constant
times $\| \psi_2(\tau)-\psi_1(\tau ) \|_{1}$. For small enough $R$, this 
last constant can be chosen uniformly. 
We then obtain that
$$||| b_2 -b_1|||_{0} \leq C T\| \v_1\|_{s} ||| \psi_2 -\psi_1 |||_{1}\, ,$$
showing that the map
$$\psi(t) \mapsto b(t) $$
is a contraction from the $1$-norm to the $0$-norm, with contraction factor 
arbitrarily small with $T$.

That the map $\psi(t) \mapsto \v(t)$ is a contraction now follows because
the map $b(t) \mapsto \v(t)$ is uniformly $C^1$ from the 
$0$-norm to the $1$-norm. This last map is simply the inverse of
$\v \mapsto \lambda \v - F(\v)$ from ${\mathcal Y}_{1}$ to 
${\mathcal Y}_{0}$, and we have that $\lambda - L_{\v(t)}$ is an 
isomorphism from ${\mathcal Y}_{1}$ to 
${\mathcal Y}_{0}$, uniformly in $\psi(t)$ when $\psi(t)$ is close
to $\v_0$. \qed


In view of the previous results, there exists a unique fixed point $\v(t)$
of the map (\ref{fawn}). Since $b(t)$ solves
(\ref{ae}) with initial data $\v_1$, differentiating with respect to $t$ 
in (\ref{abue}) we obtain:
$$(\lambda -L_{\v(t)})\d_{t}\v(t) = -\dot{b}(t) +\lambda b(t)=
(\lambda - L_{\v(t)})b (t)\, ,$$ and since 
$\lambda -L_{\v(t)}$ is injective, we must have that
$$b(t)=\partial _{t}\v(t)\, .$$
We may now use this fact in carrying the time integral in (\ref{abue}), and 
conclude that
$$\frac{d}{dt}\v(t)=F(\v(t)) \, .$$
Thus, the fixed point $\v(t)\in E_{\v_0}(I)$ is a solution to the 
initial value problem (\ref{evol2}). 

We thus arrive at the following 

\begin{theorem}
Let $(M,J,\Omega)$ be a polarized K\"ahler manifold and let $G$ be a maximal 
compact
subgroup of $Aut(M,J)$. The extremal flow equation 
$$\d _t \o _t  = -\rho_t+\Pi _t\rho_t $$
in ${\mathfrak M}_{\Omega,G}$ with a given initial data has a 
unique solution for a short time.
\end{theorem}

In fact, our proof carefully analyses how the time of existence depends 
upon the coefficients of the metric and its curvature tensor. 
Indeed, it shows that the local time of existence depends on
the $L^{\infty}$-norm of the coefficients of the initial metric and
its curvature operator. We can improve a bit the statement above
in relation to the lifespan of the extremal flow. 

\begin{corollary}
Given an initial condition $\o\in {\mathfrak M}_{\Omega,G}$, the extremal 
evolution equation has a unique solution on a maximal time interval 
$0\leq t < T \leq \infty$. If $T< \infty$, then the maximum of the 
pointwise norm of the curvature tensor blows-up as $t\rightarrow T$.
\end{corollary}

\section{Further remarks}
It is of course important to know if the extremal flow has solutions for
all time. Indeed, once the local time existence is known, the next problem to 
consider is the use of the flow to show the existence of extremal 
metrics representing a given cohomology class $\Omega$, task that could be 
accomplished if we manage to prove global time existence and convergence of 
the metrics as $t\rightarrow \infty$. 

This scheme could not possible work in all cases, as we
already know of examples of polarized K\"ahler manifolds without extremal
metrics. But as a testing ground of its usefulness, we have started its
analysis when in pursue of extremal metrics on polarized manifolds $(M,J,
\Omega)$ with $c_1<0$, or on polarized complex surfaces with $c_1>0$. The 
partial results obtained so far are quite encouraging.

We have two types of fairly strong reasons supporting our belief
that this approach will produce extremal metrics in the said cases.
The first of these reasons is directly related to the flow itself, 
while the other one involves some relation between this flow and the study 
of {\it families} of extremal problems as we vary the cohomology class 
$\Omega$. We discuss them briefly in this section.
 
The evolution equation (\ref{evol}) implies evolution equations 
for various metric tensors associated to the varying metrics. For instance,
the Ricci form evolves according to the equation
$$\frac{d}{dt}\rho=-\frac{1}{2}\Delta \rho + \frac{i\ddb (\pi s)}{2} \, ,$$
the scalar curvature evolves according to the equation
$$\frac{d}{dt}s=-\frac{1}{2}\Delta (s-\pi s) -2 (\rho, i\ddb G(s-\pi s)) \, ,$$
and the Ricci potential evolves according to the equation
$$\frac{d}{dt}\psi =-\frac{1}{2}\Delta \psi -
2G(\rho_H,i\ddb (\psi + G(\pi s))) -(\pi s - s_0) +
\frac{1}{2v}\int \psi (s-\pi s)d\mu \, .$$
Here $\rho_H$ is the harmonic component of $\rho$, and $v$ is the volume
of $M$ relative to $\o$. In particular, using the first of these
equations, we easily obtain the following result:

\begin{proposition}
Let $(M,J)$ be a complex manifold of K\"ahler type polarized by a 
K\"ahler class $\Omega$, all of whose non-trivial holomorphic vector fields
have no zeroes. If $\pm \rho\geq 0$ at $t=0$, then it remains so along
the flow {\rm (\ref{evol})} for $0\leq t \leq T$, $T$ the lifespan of
the solution.
\end{proposition}

{\it Proof}. Under the stated hypothesis, the Ricci form evolves according to
the heat equation. By Hamilton's maximum principle for tensors
(Theorem 9.1 in \cite{ha}), the result follows. \qed

This result applies directly to manifolds with no non-trivial holomorphic
vector fields, such as any complex manifold $(M,J)$ with negative first 
Chern class, or most complex surfaces with positive first Chern classes.
We have also verified the analogous result for the blow-up of 
${\mathbb C}{\mathbb P}^2$ at one point,  a complex surface that carries
non-trivial holomorphic vector fields. It is likely that this result holds
for any complex surface with positive $c_1$. 

We may also refine our earlier Theorem \ref{tc} when dealing with a 
complex surface of positive first Chern class. Indeed, we have the following
result, whose proof will be given elsewhere.

\begin{theorem}
Let $(M,J,\O)$ be a polarized complex surface of positive first Chern class.
Given any K\"ahler metric $g$ in ${\mathfrak M}_{\O,G}$, the image of the
holomorphy potential $\pi_g s_g$ is an interval contained in the set of 
positive real numbers, interval that only depends on $\O$ and not on $g$.
\end{theorem}

Thus, if for a given initial condition with positive Ricci
curvature the flow (\ref{evol}) would exist for all time and converge to
an extremal metric on this type of surfaces, the extremal metric so obtained
would have positive scalar curvature, which is what one expects.

The preservation of the sign of the Ricci tensor should have very strong 
implications on the global analysis of (\ref{evol}). This  property has 
been of utmost importance already in the work of Hamilton \cite{ha}, and 
should remain so in the general analysis of our flow equation as well. 
We venture the following two conjectures.

\begin{conjecture}
Let $(M,J)$ be a complex manifold of K\"ahler type polarized by a 
K\"ahler class $\Omega$. If $c_1(M,J)<0$, there exists an initial condition
to the extremal flow {\rm (\ref{evol2})} equation so that the solution 
exists on $[0,\infty)$ and, as $t\rightarrow \infty$, converges to a metric 
of constant negative scalar curvature representing $\Omega$. 
\end{conjecture}
\medskip

\begin{conjecture}
Let $(M,J)$ be a complex surface of positive first Chern class polarized by
a K\"ahler class $\Omega$. Then there exists an initial condition
to the extremal flow {\rm (\ref{evol2})} equation so that the solution 
exists on $[0,\infty)$ and, as $t\rightarrow \infty$, converges to an 
extremal metric of positive scalar curvature representing $\Omega$.
\end{conjecture}
\medskip

The initial condition we have in mind in these two cases is given by a
metric whose Ricci form is negative or positive, respectively.

These conjectures are further supported by the results in \cite{ss2}, that
we proceed to describe in brief detail. For a complex manifold
$(M,J)$ of complex dimension $n$, we denote by ${\mathfrak M}$ the space 
of K\"ahler metrics on $(M,J)$. As before, given a positive class 
$\Omega \in H^{1,1}(M,{\mathbb C})\cap 
H^{2}(M,{\mathbb R})$, we let ${\mathfrak M}_{\Omega}$ be the space of 
of K\"ahler metrics whose K\"ahler forms represent $\Omega$. We shall
also consider the space ${\mathfrak M}_1$ of 
K\"ahler metrics of volume one, and ${\mathcal K}_{1}$, the space of 
cohomology classes that can be represented by K\"ahler forms of metrics in
${\mathfrak M}_1$:
\begin{equation}
{\mathcal K}_{1}=\{ \Omega \in H^{1,1}(M,{\mathbb C}): \; \Omega= [\o]
\; {\rm for \; some\; }\o \in {\mathfrak M}_{1}\} \, .\label{cvo}
\end{equation}

Extremal metrics in ${\mathfrak M}_{\Omega}$ achieve the infimum of the 
functional $E_{\Omega}$ in (\ref{ene}), and we have the lower bound
(\ref{lb}):
$$E(\Omega)=\int (\pi _g s_g)^2 d\mu_g \, .$$

One approach to providing $(M,J)$ with a canonical shape would be to 
find critical points of the functional 
\begin{equation}
\begin{array}{rcl}
{\mathfrak M}_1 & \rightarrow & {\mathbb R} \\
\omega & \mapsto & {\displaystyle \int_M s_{\omega}^2 d\mu _{\omega }}\, .
\end{array} \label{fun1}
\end{equation}
A special metric $\o$ of this type must have the following properties:
\begin{enumerate}
\item[a)] $\o$ achieves the lower bound $E([\o])$, that is to say, $\o$ is
extremal relative to the polarization defined by the K\"ahler class $\Omega=
[\omega]$ that it represents; 
\item[b)] the K\"ahler class $\Omega=[\omega]$ is a critical point of 
$E(\Omega)$ 
as a functional defined over ${\mathcal K}_1$.
\end{enumerate}
Thus, the search for critical points of (\ref{fun1}) ---or {\it strongly
extremal metrics} \cite{si2}--- achieving an optimal lower bound involves the 
solution of back-to-back minimization problems: the
first solving for critical points of (\ref{ene}) within a fixed cohomology
class $\Omega$, and the second solving for those classes that minimize the
critical value $E(\Omega)$ as the class $\Omega$ varies within 
${\mathcal K}_1$. Naturally, we separate the two problems by, in addition
to (\ref{ene}), introducing the functional

\begin{equation}
\begin{array}{rcl}
{\mathcal K}_1 & \rightarrow & {\mathbb R} \\
\Omega & \mapsto & E(\Omega)=
{\displaystyle \int_M (\pi s)^2 d\mu }\, ,
\end{array} \label{cun1}
\end{equation}
where the geometric quantities in the right are those associated with any
$G$-invariant metric that represents $\Omega$, for $G$ a fixed maximal 
compact subgroup of the automorphism group of $(M,J)$. Its extremal 
points will be called either critical or canonical classes. We then have 
\cite{ss} the following 

\begin{theorem} \label{count}
Let $\Omega$ be a cohomology class that is represented by a K\"ahler
metric $g$, assumed to be invariant under the maximal compact 
subgroup $G$ of the biholomorphism group of $(M,J)$. Then $\Omega$ is 
critical class if and only if
$$\int_{M} (\pi_g s_g) (\Pi_g \rho , \a) d\mu_g =0 $$
for any trace-free harmonic $(1,1)$-form $\a$. In this expression, 
$\rho$ is the Ricci form of the metric $g$, 
$\pi$ is the $L^2$ projection {\rm (\ref{proj1})} onto the space
of holomorphy potentials, and $\Pi$ is its lift {\rm (\ref{proj})} at the 
level of {\rm (1,1)}-forms.
\end{theorem}

This theorem states that $\Omega$ is a critical class of 
(\ref{cun1}) if and only if
$$\int_{M} (\pi_g s_g) (\Pi_g \rho , \a) d\mu_g =0 $$
for any trace-free harmonic $(1,1)$-form $\a$. In other words, 
the form $\pi s \Pi \rho$ is $L^2$-perpendicular to the space of 
trace-free harmonic (1,1)-forms, and therefore, by Hodge decomposition, the
class must be such that
\begin{equation}
\pi s \Pi \rho = \lambda \omega +\partial G_{\partial}(\partial^{*}(\pi s\,
 \Pi \rho)) +\partial^{*}G_{\partial}(\partial(\pi s\, \Pi \rho))\, ,
\end{equation}
for $\lambda$ equal to the $L^2$-projection of $(\pi s)^2$ onto
the constants, divided by $2n$:
\begin{equation}
\lambda = \frac{1}{2n}\int (\pi s)^2 d\mu_g \, . \label{lam}
\end{equation}

In order to study the existence of critical classes, we may 
consider \cite{ss2} the evolution equation 
\begin{equation}
\frac{d\Omega}{dt} = \pi s \, \Pi \rho - \lambda \omega +
\partial G_{\partial}(\partial^{*}(\pi s\,
 \Pi \rho)) +\partial^{*}G_{\partial}(\partial(\pi s\, \Pi \rho))\, .
\label{hfe}
\end{equation}

The flow equation (\ref{hfe}) defines a dynamical system on
${\mathcal K}_1$ provided the solutions remain in ${\mathcal K}_1$ 
throughout time. Unfortunately, this is not true in general \cite{ss2}.

In the generic case where all non-trivial holomorphic vector fields of
$(M,J)$ have no zeroes, equation (\ref{hfe}) can be extended to a dynamical 
system on
$$\overline{{\mathcal K}}_1=\{ \Omega \in H^{1,1}(M,{\mathbb C})\cap
H^2(M,{\mathbb R}): \; \frac{\Omega^n}{n!}=1\} \, .$$
Indeed, given $\Omega \in 
\overline{{\mathcal K}}_1$, let us define the function
$$s_{\Omega} := 4\pi n \frac{c_1 \cdot \O ^{n-1}}{\O^n} 
\, .$$ 
If $\Omega$ were a K\"ahler class represented by a metric $g$, this function 
would be precisely the holomorphy potential $\pi_g s_g$.
The equation
\begin{equation}
\frac{d}{dt}\O= 2\pi s_{\Omega} c_1 - \frac{s_{\Omega}^2 }{2n}\O \, , 
\label{hfe2}
\end{equation}
extends (\ref{hfe}), which as such is defined only on ${\mathcal K}_1$,
all the way to a dynamical system on $\overline{{\mathcal K}}_1$.

Solutions to (\ref{hfe2}) with initial data
in $\overline{{\mathcal K}}_1$ remain in $\overline{{\mathcal K}}_1$. 
In fact, we have that \cite{ss2}

\begin{theorem}
Suppose that all non-trivial holomorphic vector fields of $(M,J)$ 
have no zeroes. 
Then solutions to {\rm (\ref{hfe2})} with initial data in 
$\overline{{\mathcal K}}_1$ converge, as $t\rightarrow \infty$, to a 
stationary point of the equation in the space $\overline{{\mathcal K}}_1$. 
\end{theorem}

It is then of natural interest to see if solutions to the equation with
Cauchy data given by a positive class, that is to say, an element of
${\mathcal K}_1$, remain positive thereafter. We already know \cite{ss2} of 
examples where this is not so, with solutions to the flow
equation that are initially in the K\"ahler cone but that, in converging to a 
critical point of the flow in $\overline{{\mathcal K}}_1$, must 
eventually leave the cone through its walls. 

In fact, this situation occurs already on complex surfaces, where the 
stability of 
${\mathcal K}_1$ under the flow
(\ref{hfe2}) can be analyzed using a criterion giving necessary
and sufficient for a cohomology class to be K\"ahler, criterion that
extends that of Nakai for integral classes.
Applied to our problem, if the Chern number $c_1^2 \neq 0$, we have that a 
path $\O_t$ solving (\ref{hfe2}) with initial condition in
${\mathcal K}_1$ stays there forever after if, and only if,
$$\O_0 \cdot [D] +8\pi^2 (c_1\cdot \O_0)( c_1\cdot[D])
\left( \frac{e^{c_1^2 t}-1}{c_1^2}\right) > 0$$
for all $t\geq 0$ and for all effective divisors $D$ in $(M,J)$.
When $c_1^2=0$ we still obtain a similar criterion, replacing the expression 
in parentheses above by its limit $t$ as $c_1^2 \rightarrow 0$.
This forward stability of the K\"ahler cone holds in very general 
situations, as can be seen by a run-down of the various cases in the 
Enriques-Kodaira classification of complex surfaces \cite{ss2}. In 
particular, it holds 
if the complex surface has a signed first Chern class $c_1$, 
condition under
which all solutions to the flow (\ref{hfe2}) that start in
${\mathcal K}_1$ stay there forever after, and as $t\rightarrow \infty$,
they either converge to the only critical class 
$\sqrt{2}({\rm sgn}\, c_1)c_1/c_1^2$ of (\ref{cun1}) if $c_1^2>0$, or all 
classes are critical and the flow is constant if $c_1=0$.

Notice that the positivity condition above involves the evaluation of $c_1$ 
over the divisor $D$, and only in the case when there are effective 
divisors $D$ for which $c_1\cdot[D]$ changes sign from one to another could 
the condition fail to hold. Merely fixing the sign of $c_1$ prevents this
from happening, but the counterpart to that is of great interest.
It shows that the existence of divisors on which $c_1$ achieves values
of opposite signs is in effect part of the reason why the
the K\"ahler cone might be poorly behaved in relation to 
the flow (\ref{hfe}).

When the surface in question has positive first Chern class and
carries non-trivial holomorphic fields, the forward stability of the 
K\"ahler cone under the flow (\ref{hfe}) seems to hold also, though we have
only verified that for the case of ${\mathbb C}{\mathbb P}^2$ blown-up
at one point.

In higher dimension and for manifolds $(M,J)$ where $c_1$ is either 
positive or negative, the space of K\"ahler classes is also forward stable 
under the flow (\ref{hfe2}). As a matter of fact, there is a positivity 
criterion that generalizes the one outlined above for surfaces, which 
guarantees forward stability of the K\"ahler cone under the flow. 
Manifolds with signed first Chern classes meet this criterion, though one
can give a direct argument to prove the flow stability of the cone in such 
a case.

All of these facts combined give further support to the conjectures
made earlier. We end up venturing a final one. 

\begin{conjecture}
Suppose the flow equation {\rm (\ref{hfe})} with initial data in the 
K\"ahler cone converges to a stationary point that is outside it.
Then the extremal K\"ahler cone is not a closed subset of the K\"ahler cone.
\end{conjecture}
\medskip

In other words, under the given hypothesis, there should exist cohomology 
classes in the K\"ahler cone that cannot be represented by extremal metrics.


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\end{document}

