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\begin{document}

\title{Pseudoholomorphic mappings \\ and Kobayashi hyperbolicity}
\author{Boris~S.~Kruglikov and Marius Overholt}
\date{}
\maketitle

\begin{abstract}

We extend the definition of the Kobayashi pseudodistance to
almost complex manifolds and show that its familiar properties are for the
most part preserved. We also study the automorphism group
of an almost complex manifold. We give special consideration to almost
complex structures tamed by some symplectic form. The notions and
pseudoholomorphic curves involved are illustrated in some examples.

\end{abstract}

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\footnotetext{Keywords: Almost complex structure, Kobayashi distance}
\footnotetext{MS Classification: Primary 32H15, Secondary 53C15, 53C56}

\section*{Introduction}

\hspace{13.5pt}
The Poincar\'{e} metric on the open unit disk $\D$
in the complex plane $\C$ is a Riemannian metric
 $$
|v|=\frac{|v|_{euc}}{1-|z|^2}
 $$
conformal with the Euclidean metric $|\cdot|_{euc}$, that induces a
distance $d$ on $\D$ with the remarkable
property that every holomorphic mapping $f : \D \to \D$ is distance
nonincreasing in $d$. This fact, discovered in 1915 by Pick \cite{22} is
an invariant formulation of the Schwarz lemma. By means of
holomorphic mappings of the unit disk into a complex manifold $M$,
Kobayashi in 1967 used the distance $d$ to define a pseudodistance $d_M$
on $M$. This has the property that any holomorphic mapping from a complex
manifold $N$ to a complex manifold $M$ is distance nonincreasing with
respect to $d_N$ and $d_M$.  When this Kobayashi pseudodistance $d_M$ is
a distance, it can be used to obtain information about holomorphic
mappings to or from $M$; in this situation $M$ is said to be (Kobayashi-)
hyperbolic. Some references for Kobayashi hyperbolicity are
\cite{8,10,11,17,21}.

An almost complex manifold $(M^{2n},J)$ is a smooth manifold of dimension
$2n$ equipped with a smooth field $J$ of automorphisms of the
tangent spaces of $M^{2n}$, which satisfy $J^2 = -\1$. Such an
assignment of automorphisms $J$ is called an almost complex structure
on $M^{2n}$. A pseudoholomorphic mapping of the unit disk $\D$ into
$(M^{2n},J)$ is a smooth mapping $f : \D \rightarrow (M^{2n},J)$ which
satisfies $J{\circ}f_{\ast} = f_{\ast}{\circ}J_{0}$ where $J_{0}$ is
the almost complex structure on $\D$ induced by the standard complex
structure. Using such pseudoholomorphic disks, the definition of the
Kobayashi pseudodistance can be carried over to almost
complex manifolds. Many proofs of standard results about $d_{M}$
in the almost complex case are similar to known ones from the complex
situation; results of this kind will be merely noted in passing where
appropriate. However some facts need a treatment involving
new ideas. We will emphasise such results. To deal with them we
apply the Gromov pseudoholomorphic curve technique which was considerably
developed in symplectic geometry during the past decade (\cite{7,19}).

A symplectic manifold $(M^{2n},{\omega)}$ is a smooth manifold $M^{2n}$
together with a closed
and nondegenerate (${\omega}^n \neq 0$) 2-form ${\omega}$ on
$M^{2n}$. In the famous paper \cite{7} Gromov showed how to use
almost complex structures to prove new results in symplectic geometry.
He proved the existence, on each symplectic manifold, of almost
complex structures $J$ satisfying ${\omega}(v,Jv) > 0$ for any
$v \neq 0$ in $TM^{2n}$. Such an almost complex structure is called
tame, and $\omega$ is said to tame $J$. Pseudoholomorphic
mappings $f : S \rightarrow (M^{2n},J)$ from Riemann surfaces $S$
yield information about $(M^{2n},{\omega)}$ when $\omega$ tames $J$.

In the paper \cite{1} Bangert used a technique of hyperbolic
complex analysis called Brody reparametrization to prove that
the standard symplectic torus $(T^{2n},{\omega}_{0})$ contains a
entire $J$-holomorphic line for any $J$ taming ${\omega}_{0}$.
In the preliminary version of this paper \cite{12} we
independently treated the Brody criterion for the nonintegrable
almost complex case.
We consider the connection of our results with that of Bangert
and outline an alternative proof of his theorem in section 4.

The taming condition has some bearing on hyperbolicity
of almost complex manifolds. Roughly speaking, two conditions -- tameness
and hyperbolicity -- are opposite to each other in the large.
This makes possible connections with symplectic geometry rather
awkward as we discuss in the conclusion.

This paper was initiated during the visit of the first
author to the University of Tromsoe. He thanks professor V.\,Lychagin
and other mathematicians in the Department
for their hospitality and kind attention.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 1 %
\section{\hskip-20pt . \hskip2pt Definition of the pseudometric}

\hspace{13.5pt}
Let  $\C$ denote the complex plane with its standard complex structure,
$\D$ the open unit disk in it and $e=1\in T_0\D$.
Let $(M^{2n},J)$ be an almost complex manifold, which in this paper
will be taken to be connected and $C^\infty$.  This means that the smooth
field of automorphisms  $J\in T^*M \otimes TM$  satisfies  $J^2 = -\1$.
A Kobayashi chain joining two
points $p,q\in M$ is a sequence of pseudoholomorphic mappings
 $$
f_k : \D \rightarrow (M^{2n},J), \ \ \ k=1,...,m,
 $$
and points $z_k,w_k \in \D$ such that $f_1(z_1) = p, \ f_m(w_m) = q$
and $f_{k}(w_k) = f_{k+1}(z_{k+1})$.  The {\it Kobayashi pseudodistance\/}
from $p$ to $q$ is defined by
 $$
d_M(p,q) = \op{inf} \ \sum_{k=1}^{m}d(z_k,w_k)
 $$
where the infimum is taken over all Kobayashi chains joining $p$ to $q$,
if there exists some Kobayashi chain joining $p$ to $q$, and is defined to
be $+\infty$ otherwise.  It is well known that $d_{\C} \equiv 0$ and $d_{\D}
 \equiv d$. We record the very basic properties of $d_{M}$:

 \begin{prop}\po
{\rm\bf(i)}
The function $d_M : M \times M \to \R$ is nonnegative, symmetric
and satisfies the triangle inequality, i.e.\ it is a pseudodistance.
\newline
{\rm\bf(ii)}
Pseudoholomorphic mappings between almost complex manifolds are distance
nonincreasing with respect to the Kobayashi pseudodistance.
 \end{prop}

As in the case of complex manifolds, $d_M$ is finite on any almost complex
manifold, but this requires an existence theorem for pseudoholomorphic
disks due to Nijenhuis and Woolf \cite{20}.

 \begin{th}\po
{\rm\bf(i)} The Kobayashi pseudodistance $d_M$ is finite and continuous
on $M \times M$ for any almost complex manifold $(M^{2n},J)$.\newline
{\rm\bf(ii)} If $d_M$ is a distance, it induces the standard topology
on $M$.
 \end{th}

 \begin{proof}
We must first prove that any two points in $M$ can be joined by a
Kobayashi chain.  It is enough to prove that for any point
$p \in M$ there is a neighborhood $U_p$ of $p$ such that
any point $q$ in $U_p$ can be joined to $p$ by a single
pseudoholomorphic disk.  For then the set of points that can
be joined by a Kobayashi chain to some fixed point is open
and nonempty, and its complement is open,
so every point can be joined to the fixed point since $M$
is connected.

The problem is now local, so we consider $({\R}^{2n},J)$.
Let $v \in T_{0}{\R}^{2n} = {\R}^{2n}$.
By theorem III of \cite{20}, there is
some neighborhood $V$ of $0$ in ${\R}^{2n}$ such that
if $v \in V$, then there exists a pseudoholomorphic mapping
$f : {\D} \rightarrow ({\R}^{2n},J)$ with $f(0) = 0$ and
$f_{\ast}(0)e = v$; this mapping could be chosen canonically.
We write this mapping as $f(z;v)$ to show its
dependence on $v$. We denote its differential with respect to $z$
by $f_{\ast}(z;v)$. By 5.4a
of \cite{20}, $f : {\D} \times V \rightarrow
{\R}^{2n}$ is a $C^\infty$ mapping.  It satisfies
$f(0;v) = 0$ and $f_{*}(0;v)e = v$.  In addition, by 5.2a of \cite{20}
, it satisfies $f(z;0) = 0$ on the unit disk.  Since
 $$
\frac{\partial f_{*}(0;v)e}{\partial v} = I
 $$
there is some neighborhood $W$ of $0$ in $\D$ such that for every
$\zeta \in W$, the matrix
 $$
\left.\frac{\partial f_{*}(\zeta;v)e}{\partial v}\right|_{v=0}
 $$
is of full rank.  Now
 $$
\left.\frac{\partial f(\zeta;v)}{\partial v}\right|_{v=0} =
\left.{\zeta}\frac{\partial f_{*}(\zeta;v)e}{\partial v}\right|_{v=0} +
O(|\zeta|^2),
 $$
where multiplication is with respect to the complex structure on ${\R}^{2n}
\simeq \C^n$ induced by the linearization.
So there exists some $\zeta_0 \in W \backslash {\{}0{\}}$ such that
 $$
\left.\frac{\partial f(\zeta_0;v)}{\partial v}\right|_{v=0}
 $$
is a matrix of full rank.  Then $f(\zeta_0;V)$ contains an open
neighborhood of $0$ in ${\R}^{2n}$, because $f(\zeta_0;0) = 0$.  So if
$q$ is in this neighborhood, there is some $v \in V$ such that
$f(\cdot;v)$ is a pseudoholomorphic disk joining $0$ and $q$.

Since
 $$
|d_M(p,q) - d_M(p_0,q_0)| \leq d_M(p,p_0) + d_M(q,q_0)
 $$
by the triangle inequality, to prove continuity of $d_M$ it is enough to
prove
continuity of the mappings $d(\cdot,p_0)$ for all $p_0 \in M$.  The
reasoning
above shows that
for every $\ve > 0$ there exists some neighborhood $U$ of $p_0$ such that
for
some $\zeta_0 \in \D$ with $d(0,\zeta_0) = \ve$ the image $f(\zeta_0;V)$
contains  $U$  and thus  $d_M(p,p_0) \leq \ve$  for any  $p \in U$.
The proof that $d_M$ induces the standard
topology on $M$ if it is a distance can now be carried through by the
method of Barth \cite{2}.\qed
 \end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 2 %
\section{\hskip-20pt . \hskip2pt Hyperbolicity}

\hspace{13.5pt}
An almost complex manifold $(M^{2n},J)$ is {\it Kobayashi hyperbolic\/}
if the Kobayashi pseudodistance $d_M$ is a distance.  It is
{\it Brody hyperbolic\/} if any pseudoholomorphic mapping
$f : {\C} \to M$ is constant. It is clear that Kobayashi hyperbolicity
implies Brody hyperbolicity. Brody \cite{4} discovered that for complex
manifolds the
converse holds under the additional condition of compactness. As is
usual, we take the term hyperbolic to mean Kobayashi hyperbolic,
and denote other hyperbolic properties by a prefix.

We shall sometimes need to consider hyperbolicity of subsets of almost
complex manifolds; the definition is simply that the subset has a
hyperbolic open neighborhood. In particular we will encounter
hyperbolic manifolds with boundary, which for us will always be
contained in a larger almost complex manifold. Thus the term compact
manifold means possibly with boundary. Note that our terminology here
is not consistent with standard terminology in the theory of complex
manifolds. On the other hand, if a manifold is compact and has no
boundary, we use the term closed manifold.

The following is a version of Brody's theorem \cite{4} for almost complex
manifolds:

 \begin{th}\po
{\rm\bf(i)}
Let $(M^{2n},J)$ be an almost complex manifold and $|\cdot|$ a
continuous norm on $TM$. $M$ is hyperbolic if and only if
there exists a function $C: M\to \R$ (possibly a constant) bounded
on compact sets in $M$ and such that for all pseudoholomorphic mappings
$f : \D \to M$ we have:
 $$
|f_{\ast}(0)e| \leq C(f(0)).
 $$
{\flushleft
{\rm\bf(ii)}
}
A compact almost complex manifold is hyperbolic if and only if it
is Brody hyperbolic.
 \end{th}

 \begin{proof}
Suppose that a function $C: M\to \R$ exists with $|f_{\ast}(0)e| \leq
C(f(0))$ for all pseudoholomorphic mappings $f: \D \to M$ and that $C$
is bounded on compact sets. Then
 $$
|f_{\ast}(z)e| \leq \frac{C(f(0))}{1 - |z|^2}
 $$
by precomposing with suitable disk automorphisms.
Let $\rho$ be the distance associated to $|\cdot|$. Then for every
pseudoholomorphic mapping $f:\D\to M$ and for every pair $z_1, z_2\in
\D$ the following holds: $\rho(f(z_1),f(z_2))\le C(f(0))\cdot
d(z_1,z_2)$. Now it is easy to see
that for any two points $p\ne q$ with $R=\rho(p,q)$ sufficiently small, the
last
inequality implies $d_M(p,q)\ge \dfrac R {\mathstrut C_0}>0$,
$C_0=\op{sup}\limits_{p\in\bar B}C(p)$, where $\bar B$ is the ball
$\bar B_\rho(p;R)$ with center $p$ and radius $R$ (one can assume that
in the definition of a Kobayashi chain all the points $z_k=0$). We
note that it is sufficient to show that $d_M(p,q) > 0$ for $p$ and $q$
sufficiently close, by the triangle inequality. So the manifold $M$ is
hyperbolic.

Now assume that there exists no function $C(p)$ bounded on compact sets
such that
 $$
|f_{\ast}(0)e| \leq C(f(0))
 $$
for all pseudoholomorphic mappings $f : \D \to M$. Then there has to
be some compact set $K_{0}$ on which such an inequality does not hold.
For otherwise we can exhaust $M$ by an ascending sequence $K_{j}$ of
compact subsets such that $|f_{\ast}(0)e|$ has a finite upper bound
depending on $j$ alone, valid for all pseudoholomorphic mappings
$f : \D \to M$ with $f(0) \in K_{j}$. But clearly this yields a function
$C: M\to \R$ as above.

So let
$f_k : \D \to M$ be a sequence of pseudoholomorphic mappings with
 $$
|(f_k)_{\ast}(0)e| \to \infty,
 $$
and assume that all the points $f_k(0)$ lie in the compact set $K_{0}$.
We will show that $M$ is not hyperbolic.
We can extract a subsequence to assure that
$f_k(0) \to p \in M$.  Let $K$ be a compact set with $p$ an interior point
on which
$J$ is tamed by an exact symplectic form, and let $r_k$ be the supremum
of radii $r \leq 1$ such that $f_k(r\D) \subseteq K$.
Gromov's Schwarz lemma, see Corollary 4.1.4 of \cite{18}, implies that
$r_k \to 0$,
since $|(f_k)_{\ast}(0)e| \to \infty$.  It is clear that there exists
a sequence $z_k \in r_k\overline{\D}$ such that
$q_k = f(z_k) \in {\partial}K$.
Extract a subsequence to assure that $q_k \to q \in {\partial}K$.
Then $d_M(p,q_k) \to d_M(p,q)$ by the continuity of $d_M$ and
 $$
d_M(p,q_k) = d_M(f_k(0),f_k(z_k)) \leq d(0,z_k) \to 0
 $$
and so $d_M(p,q) = 0$, hence $M$ is not hyperbolic.

It is clear that a hyperbolic almost complex manifold is Brody hyperbolic.
Assume on the other hand that $M$ is a compact, but not hyperbolic,
almost complex manifold.  We shall prove that there exists a
nonconstant pseudoholomorphic mapping $g : \C \to M$.

We put a Riemannian metric $|\cdot|$ on $M$.  Since $M$ is not
hyperbolic, there is a sequence of pseudoholomorphic mappings
$f_k : \D \to M$ with $|(f_k)_{\ast}(0)e| \to \infty$.  The proof of
Brody's reparametrization lemma \cite{4} goes through unchanged for almost
complex manifolds.  We use it in the form that if $(M^{2n},J)$ is an almost
complex manifold, $|\cdot|$ a continuous norm on $TM$ and $f : r\D \to M$
a pseudoholomorphic mapping such that $|f_{\ast}(0)e| \geq c > 0$, then
there exists a pseudoholomorphic mapping $h : r\D \to M$ such that
$|h_{\ast}(0)e| = c/2$,
 $$
|h_{\ast}(z)e| \leq \frac{c}{2}\frac{r^2}{r^2 - |z|^2},
 $$
and $h(r\D) \subseteq f(r\D)$.  See p. 27 of \cite{21} for a proof that
can be adapted virtually unchanged.  We apply Brody's reparametrization
lemma to the sequence of pseudoholomorphic mappings
 $$
f_k\left(\frac{2z}{|(f_k)_{\ast}(0)e|}\right)
 $$
from $r_k\D$ to $M$, where $r_k = |(f_k)_{\ast}(0)e|/2 \to \infty$.
This gives a sequence of pseudoholomorphic mappings $g_k : r_k\D \to M$
with $|(g_k)_{\ast}(0)e| = 1$ and
 $$
|(g_k)_{\ast}(z)e| \leq \frac{r_k^2}{r_k^2 - |z|^2}.
 $$
Using the last inequality and the compactness theorem 4.1.3 of \cite{19},
we conclude that
$g_k$ has a subsequence which converges uniformly with all derivatives
on compact subsets of $\C$ to a pseudoholomorphic mapping $g : \C \to M$.
It is nonconstant because $|(g)_{\ast}(0)e| = 1$.\qed
\end{proof}

We remark that the proof of theorem~2 is applicable to compact
subsets of almost complex manifolds. Observe in particular that the proof
of theorem~2 shows that a compact subset of
an almost complex manifold is hyperbolic if and only if it is Brody
hyperbolic.

More general versions of Brody's theorem are known for complex manifolds.
See theorem 2.2 and 2.3 of Chap. III of \cite{17}.  They have analogs for
almost complex manifolds, which are easily proved by modifying the proofs in
\cite{17} along the lines of the above proof.
Then one can prove the following theorem in the same way as theorem 3.1
is proved in Chap. III of \cite{17}:

\begin{th}\po
Let $E$ and $B$ be subsets of almost complex manifolds and $\pi : E \to B$ a
proper (restriction of a) pseudoholomorphic mapping. If $B$ is hyperbolic
and each fiber is a hyperbolic subset of $E$, then $E$ is hyperbolic. If
the fiber above a point of $B$ is hyperbolic, this point has a neighborhood
such that the fibers above all points of this neighborhood are hyperbolic.
 \end{th}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3 %
\section{\hskip-20pt . \hskip2pt The automorphism group}

\hspace{13.5pt}
In contrast to the complex case, generic almost complex
structures admit no local automorphisms other than the identity
and so the group $\op{Aut}(M^{2n},J)$ is usually trivial.
The obstruction theory for local and formal automorphisms of almost
complex manifolds was developed in \cite{13}. However in practice
we usually need criteria for this generic property. Here we give
one connected with the notion of hyperbolicity.

It is a result of Kobayashi \cite{10} that the automorphism group
of a closed (usually called just compact in the literature but with
the assumption of empty boundary) hyperbolic complex manifold is finite.
This is also true in the almost complex case.
For closed almost complex manifolds, the group
of pseudoholomorphic diffeomorphisms of the manifold to itself is a Lie
transformation group, when equipped with the topology of
$\Sigma$-convergence, i.e.\ uniform convergence on compact sets
of the mappings and their derivatives through the third order,
as was proved by Boothby, Kobayashi and Wang \cite{3}.

  \begin{th}\po
The automorphism group  $G = \op{Aut}(M^{2n},J)$  of a closed hyperbolic
almost complex manifold $(M^{2n},J)$ is finite.
 \end{th}

 \begin{proof}
Since  $G$  is a Lie group we can consider its Lie algebra $\G$ consisting
of pseudoholomorphic vector fields; i.e.\ fields $\xi$ on $M^{2n}$
such that  $L_{\xi}J = 0$ or  $[\xi,J\eta] = J[\xi,\eta]$ for every vector
field $\eta$ on $M^{2n}$.  In particular, $[\xi,J\xi] = 0$.  This means
that if $0 \neq \xi \in \G$ then we have a nonconstant pseudoholomorphic
curve $f : \C \rightarrow M^{2n}$ through a point $p \in M^{2n}$ of the
form
 $$
z = x + iy \mapsto \exp(x\xi + y(J\xi))p.
 $$
The exponential mapping is globally defined since $M^{2n}$ was assumed
closed.  Now $(M^{2n},J)$ was assumed hyperbolic and so such a
nonconstant pseudoholomorphic curve cannot exist.  Hence $\G$ is trivial
and thus $G$ is discrete.

We must now show that $G$ is compact.  Since $M$ is compact and second
countable, $G$ is second countable.  So it is enough to prove that any
sequence $\vp_k \in G$ has a $\Sigma$-convergent subsequence.  In the
following, each time we extract a subsequence, we give it the same
notation as the original sequence.  Note that
$G$ is a subgroup of the isometry group $\I(M,d_M)$ of $M$,
where $d_M$ is the Kobayashi distance on $M$.
According to the theorem of van Dantzig and van der Waerden \cite{5},
see also pp. 46-50 of \cite{9}, $\I(M,d_M)$ is compact in the topology
induced by $d_M$.  Moreover this theorem easily generalizes to the
following statement which we also use:

 \begin{lem}
Let $(A,d_A)$ and $(B,d_B)$ be compact metric spaces.  Let $\I(A,B)$ be
the set of isometries $f : A \to B$, and topologize it by the
compact-open topology.  Then $\I(A,B)$ is compact.
 \end{lem}

Let $\vp_k \in G$ be a sequence, and extract a subsequence converging
in $\I(M,d_M)$ to some $\vp \in \I(M,d_M)$.  Fix a Riemannian metric on
$M$ with corresponding norm $|\cdot|$.  We use the boundness of
derivatives from theorem~2 to
deduce that the sequence $(\vp_k)_\ast$ of first derivatives must be
bounded.  Otherwise we can extract a subsequence to assure that
 $$
|(\vp_k)_\ast(p_k)v_k| \to \infty
 $$
where $|v_k|=1$ and $v_k \in T_{p_k}M$.  Now extract subsequences to
assure that $p_k \to p \in M$ and $v_k \to v \in T_pM$, $|v|=1$.
Let $\xi$ be a smooth vector field extending $v$ and such that $|\xi|=1$
on a neighborhood of $p$.  By 5.4a of \cite{20} and arguments from
\S 1, there exists a number $r > 0$ and
a smooth family $f_q : r\D \to M$
for $q \in U$, where $U$ is a neighborhood of $p$, such that
$f_q(0) = q$ and $(f_q)_\ast(0)e = \xi_q$.  Now we obtain a sequence
of pseudoholomorphic mappings $\vp_k \circ f_{p_k} : r\D \to M$
with the property that
 $$
|(\vp_k \circ f_{p_k})_\ast(0)e| \to \infty,
 $$
which contradicts hyperbolicity.

Note also that we might obtain the boundness of the sequence of
derivatives by the nonlinear Schwarz lemma of \cite{7} using the
compactness.  Such a Schwarz lemma also holds for higher derivatives
\cite{18}, which yields the desired convergence result.

Now consider the space $J_{PH}^1(M,M)$ of 1-jets of pseudoholomorphic
mappings of $M$ to itself, i.e.\ the set of points $(p,q,\Phi)$ with
$p,q\in M$, $\Phi\in T^*_pM\otimes T_qM$, satisfying the equality
$\Phi\c J_p=J_q\c\Phi$.
This space carries a canonical almost
complex structure \cite{6} for which the standard projection
$\pi : J_{PH}^1(M,M) \to M \times M$ is a pseudoholomorphic mapping.
Let $\sigma : M \times M \to M$ be the projection on the second factor.
Denote the composition by $\rho = \sigma \circ \pi$.
Let $J_{PH}^1(M,M)_r$ denote the set of points $(p,q,\Phi) \in J_{PH}^1$
such that the element $\Phi \in T_p^{\ast}M \otimes T_qM$
which also could be considered as a tangent vector to $J_{PH}^1$,
satisfies $|\Phi| \leq r$.
Thus we have pseudoholomorphic mappings $\rho_r : J_{PH}^1(M,M)_r\to M$
and $\pi_r : J_{PH}^1(M,M)_r\to M\times M$. The compact ball
$\pi_r^{-1}(p\times q)$ in the complex affine
space $\pi^{-1}(p\times q)$ carries the standard complex structure and
hence is hyperbolic for every $p,\,q \in M$ and $M$ is
also hyperbolic. The standard complex structure here means the
restriction of the standard complex structure from an open neighborhood.
We remind the reader that we call a
subset of an almost complex manifold hyperbolic if it has a hyperbolic
open neighborhood. Now by theorem~3
the compact set $J_{PH}^1(M,M)_r$ in $J_{PH}^1(M,M)$ is hyperbolic.
For $\pi_r : J_{PH}^1(M,M)_r\to M\times M$ is clearly a proper mapping.

By the arguments above for some $r$ we have the following commutative
diagram of pseudoholomorphic mappings:

\[
\begin{array}{ccc}
& \hsb                         & \hsp   J_{PH}^{1}(M,M)_{r}   \\
& \hsb ^{j^{1}\psi}\!\nearrow  & \hsp \downarrow\! ^{\rho_{r}} \\
M \!\!\!\!\!\!\!\!\!           & \hsb \ \ \ \ \ \ \ \
\stackrel{\psi}{\longrightarrow}  & \hsp M.
\end{array}
\quad
\]

For $p,q \in M$ and an isometry $\psi \in G$ we have
 $$
d_M(p,q) \geq d_{(J_{PH}^1)_r}(j^1\psi(p),j^1\psi(q))
\geq d_M(\psi(p),\psi(q)) = d_M(p,q),
 $$
and so for every $\psi \in G$, $j^1\psi$ is an isometry with respect
to $d_M$ and $d_{(J_{PH}^1)_r}$.  Using the generalization of the
result of van Dantzig and van der Waerden quoted above, we conclude that
we can extract a subsequence to assure that $\vp_k$ converges
uniformly in $J_{PH}^1$, i.e.\ the sequence of derivatives converges
uniformly.  Exactly the same arguments as above show that the derivatives
of the mappings $j^1\vp_k$ are bounded and we can consider the hyperbolic
almost complex manifold $J_{PH}^1(M,J_{PH}^1(M,M)_r)_r$ to extract a
subsequence for which we have uniform convergence in the $C^2$ sense.
Applying the argument one more time, we obtain a $\Sigma$-convergent
subsequence, thus $\vp_k \to \vp \in G$.\qed
 \end{proof}

 \begin{th}\po\label{ZZZ}
Let $(M^{2n},J)$ be a hyperbolic almost complex manifold.
\newline
{\rm\bf(i)}
$\op{dim}\op{Aut}(M^{2n},J)\le 2n+n^2$ and equality holds iff
there is an isomorphism with the standard ball
$(M^{2n},J)\simeq (B^{2n},J_0)$.
\newline
{\rm\bf(ii)}
The isotropy subgroup $\op{Aut}(M^{2n},J)_p$ is compact for every
$p\in M$.
\newline
{\rm\bf(iii)}
No almost complex Lie group of positive dimension acts effectively
as a pseudoholomorphic transformation group on $(M^{2n},J)$.
 \end{th}

 \begin{proof}
This is an analog of theorem~9.1 of~\cite{11}, see also~\cite{10}.
The idea is the same with slight changes:
we may modify the corresponding proof that $G$ is
a closed subgroup of $\I(M)$ using the ideas from the proof of
theorem~4; we can avoid using the lemma and
change $(J^1_{PH})_r$ to $(J^1_{PH})_C$, where $C: M\times M\to \R$ is
some function bounded on compact sets and the PH-jets $\Phi\in
T^*_pM\otimes T_qM$ satisfy $|\Phi|\le C(p,q)$. The third part is trivial:
if vector fields $\xi, J\xi \in\G$, then they are complete
and the exponential mapping is defined.\qed
 \end{proof}

It turns out that the third statement of theorem~\ref{ZZZ} is still
true for another wide class of almost complex manifolds. The following
is a weak version of the general position property from~\cite{13}.

 \begin{dfn}
An almost complex manifold $(M^{2n},J)$ is of slightly general position
if for a dense set of points $p \in M^{2n}$ the Nijenhuis tensor
$N_J$ at each of these points satisfies
 $$
\op{Ker}[(N_J)_p(\xi, \cdot )] \neq T_{p}M^{2n}
 $$
for all vectors $\xi \in T_{p}M \backslash {\{}0{\}}$.
 \end{dfn}

 \begin{prop}\po
Let an almost complex manifold $(M^{2n},J)$ be of slightly general position.
Then the Lie algebra $\G$ of the automorphism group
$G = \op{Aut}(M^{2n},J)$ has the property
$\G \cap (J\G) = 0$, i.e.\ the tangent space of $G$ contains no complex
lines: $\xi \in \G \Rightarrow J\xi \not\in \G$.
 \end{prop}

 \begin{proof}
If  $\xi \in \G = \op{aut}(M^{2n},J)$  then  $[\xi,J\eta] =
J[\xi,\eta]$ for any
vector field  $\eta$.  If in addition  $J\xi \in \G$  then
$[J\xi,J\eta] = J[J\xi,\eta]$.  These two equations give
$N_{J}(\xi,\eta) = 0$  at  $p$  for every  $\eta$ and hence $\xi_p=0$.
Since the set of such points $p$ is dense we have $\xi=0$.\qed
 \end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 4 %
\section{\hskip-20pt . \hskip2pt Examples}

\hspace{13.5pt}
Numerous examples of hyperbolic and nonhyperbolic complex manifolds are
considered in the literature.  Here we consider examples with
non-integ\-ra\-ble almost complex structures.

 \begin{ex} 1
We start with an example of a nonhyperbolic almost
complex manifold.
The Kobayashi pseudodistance on a homogeneous almost complex manifold
is invariant (for definitions and examples of homogeneous almost complex
manifolds see \cite{9,23}).
Consider $S^6 = {G_2}/SU(3)$ with its well-known nonintegrable almost
complex structure $J$ that is defined by means of the octonions
(or Cayley numbers).  The definition (see \cite{9} for the
full details) goes like this:  let
${\R}^7 = \op{Im} {\mbox{\boldmath $C\! a$}}$ be the purely
imaginary octonions and $S^6 \subset {\R}^7$ the unit
sphere.  On ${\R}^7$ there is defined a vector
product $\times$ and we define $J : T_{w}S^6 \rightarrow T_{w}S^6$ by
$\eta \mapsto {\eta}\times{w}$ where $\eta \in {\R}^7$
and $\eta \bot w$.

Now we show that $d_{S^6}\equiv0$. Let us take arbitrary $\eta\in T_wS^6$
such that $|w|=|\eta|=1$, $\eta\perp w$ and set $\nu=\eta\times w$.
The alternativity law implies that
$\R^3=\langle \eta,w,\nu\rangle\subset\Ca$
is a subspace invariant under the vector product $\times$ which is
isomorphic to the standard $(\R^3=\langle i,j,k\rangle,\times)$. Thus
$S^2=\R^3\cap S^6$ is $J$-invariant, i.e. pseudoholomorphic. Now the claim
follows since for every two points $w_1,w_2\in S^6$ we can find $\eta\in
T_{w_1}S^6$ such that the corresponding sphere $S^2$ goes through $w_2$.
 \end{ex}

 \begin{ex} 2
An almost complex structure  $J$  is called tame if
there exists a symplectic form  $\omega$  such that  $\omega(X,JX) > 0$
for any nonzero vector $X$, see \cite{7}.  For tame almost complex
structures on bounded domains in ${\R}^{2n}$ we have a
sufficient condition for hyperbolicity.

 \begin{th}\po\label{XXX}
Let $(D^{2n},J)$ be an almost complex domain with $J$ tame.  If there
exists a ball
$D' \,\,{\supset}\!{\supset} D$
such that $J$ is the
restriction of some tame almost complex structure on $D^{\prime}$, then
$(D^{2n},J)$ is hyperbolic.
 \end{th}

 \begin{proof}
It was shown by Lafontaine \cite{16} that if $D$ is a
bounded domain in $({\R}^{2n},J)$, where the almost
complex structure $J$ is tamed by a symplectic form with bounded
coefficients, then $(D^{2n},J)$ is Brody hyperbolic.
We can suppose the coefficients of the taming symplectic form
on $D'$ are bounded, shrinking this ball a little otherwise.
Now consider a diffeomorphism of $D^{\prime}$ onto
${\R}^{2n}$ which maps $D$ onto a bounded domain.
The coefficients of the transported symplectic form remain
bounded. Thus we have exhibited $D$ as a bounded domain in a tame
$({\R}^{2n},J^{\prime})$, since the diffeomorphism does
not disturb the taming condition. Then the closure of $D$ is Brody
hyperbolic since it is contained in a slightly bigger open ball.
Now apply the remark following the proof of
theorem~2 to the closure of $D$. \qed
 \end{proof}

The above result is the analog of the known sufficient condition for
a domain in ${\C}^n$ to be hyperbolic; that it be bounded.  For this can
be formulated as the statement that a bounded domain in
${\R}^{2n}$ with complex structure is
hyperbolic if the complex structure can be extended to the standard
complex structure on an ambient ball.  A statement of this kind is
however false for a general almost complex domain where some extra
condition such as tameness or standard integrability is not imposed.
For  $S^6 \backslash {\{}p_0{\}}$  is diffeomorphic to
${\R}^{6}$ so we can essentially impose on
${\R}^{6}$  the restriction of the almost complex
structure on  $S^6$  discussed above.  Choosing $p_0$ not to lie on the
pseudoholomorphic 2-sphere that we found, and removing a closed ball
around $p_0$, we obtain a bounded domain in  ${\R}^{6}$
(actually an open ball), such that the almost complex structure
extends to the whole of  ${\R}^{6}$. Yet this domain is not hyperbolic.

Note that the reasoning above shows that the almost complex structure
on  $S^6$  cannot be tamed even on the complement of a small closed
ball (globally it is evident since $S^6$ is not symplectic: $H^2(S^6)=0$).
 \end{ex}

 \begin{ex} 3
In Example~2 we saw that not every bounded domain in an almost complex
$\R^{2n}$ with $n>1$ is hyperbolic (except maybe for $\R^4$). However
theorem~\ref{XXX}
implies that every point possesses a neighborhood which is hyperbolic.
Thus it follows from theorem~3 that for every almost complex manifold
$M$ with finite dimensional ($C^\infty$-)parametrized almost complex
structure $J^{(\tau)}$, $\tau\in\R^p$, the property of
$(M,J^{(\tau)})$ being hyperbolic is open with respect to (the usual)
topology on $\R^p$. This can be generalized.

The following statement gives examples of non-integrable hyperbolic
almost complex structures on compact manifolds, for example
on the product of closed Riemann surfaces each of genus greater than one
or on a closed Hermitian manifold with negative holomorphic sectional
curvature.

 \begin{th}\po
Let $(M,J)$ be a compact hyperbolic almost complex manifold. Then for a
small neighborhood $\ok$ of $J$ in the $C^\infty$ topology every almost
complex manifold $(M,J')$ with $J'\in\ok$ is hyperbolic.
 \end{th}

 \begin{proof}
Assume that the statement is false. Then for every sequence of neighborhoods
$\ok_k$ of $J$ shrinking to $J$ there exists a sequence $J_k\in\ok_k$
of almost complex structures such that the manifold $(M,J_k)$ is not
hyperbolic. For example one can take a sequence of neighborhoods
$\ok^{(m)}_k$ to be the ball of radius $\dfrac1k$ around $J$ in
$C^m$-norm (or $W^{m,p}$-norm),
intersect it with $\ok$ and then take the diagonal subsequence. According
to theorem~2 for every $J_k$ there exists a nonconstant pseudoholomorphic
curve $f_k:\C\to(M,J_k)$. The sequence of almost complex structures $J_k$
tends to $J$ in the $C^\infty$ topology. As in the proof of theorem~2 we
can apply Brody's reparametrization lemma to obtain uniform boundness of
derivatives on compact sets and the condition $|(f_k)_*(0)e|=1$,
and thus due to the compactness theorem~B.4.2.
of~\cite{19} there exists a subsequence converging uniformly
with all derivatives on compact sets to some pseudoholomorphic
curve $f:\C\to(M,J)$. Since $|f_*(0)e|=1$ this curve is nonconstant which
contradicts the hyperbolicity of $(M,J)$. \qed
 \end{proof}
 \end{ex}

 \begin{ex} 4
In the paper~\cite{1} Bangert proved that almost complex tori $(T^{2n},J)$
with $J$ taming the standard symplectic structure $\oo_0$ admit
nonconstant pseudoholomorphic curves $f:\C\to T^{2n}$. Here we sketch
another proof of this fact.

Consider the universal covering $\R^{2n}$ of $T^{2n}$ and denote the
almost complex structure induced on $\R^{2n}$ via the covering
projection by $J$.
Compactify $\R^{2n}$ in a different way from Bangert's: change
the almost complex structure outside a big polydisk centered at the
origin to become integrable
and glue the neighborhood to the product
$M^{2n}= S^2 \times S^2 \times \dots \times S^2$
(cf.\ the construction from proposition~2 of \cite{14}). Denote the
constructed almost complex structure on $M^{2n}$ by $\tilde J$.

The structure $\tilde J$ can be perturbed to become a structure of general
position $J_\nu$. Consider the homology class $A_k$ of the
$k$-th sphere-factor of $M^{2n}= S^2 \times S^2 \times \dots \times S^2$.
By definition it is $\oo_M$-simple for the
standard symplectic structure $\oo_M$ on $M^{2n}$. Thus
calculating the evaluation map (see \cite{19}) we conclude that there
exists a
$J_\nu$-holomorphic $A_k$-sphere through every point of $M^{2n}$.
Choose such a sphere through the origin, and precompose with a
complex-affine mapping of $S^2$ onto itself, so that the
$J_{\nu}$-holomorphic map of the sphere into $(M^{2n},J_{\nu})$ is
normalized at the origin. So we have arranged that the origin goes to
the origin and the derivative at the origin has norm $1$.
Now let a sequence $J_\nu$ of almost complex structures of general
position tend to $\tilde J$. By Gromov's compactness theorem
\cite{7} the corresponding sequence of $J_\nu$-holomorphic
spheres $u_\nu:S^2\to M^{2n}$ in a fixed homology class $A_k$ has
a subsequence converging to some $\tilde J$-holomorphic sphere
$u:S^2\to M^{2n}$ in the same homology class. Since the polydisk from
which we glue $M^{2n}$ can be chosen as large as one desires the
construction yields normalized $J$-holomorphic disks
$f_r:r\D\to(\R^{2n},J)$ with $r$ as large as we please. Postcomposing
with the covering projection yields normalized $J$-holomorphic disks
$g_r:r\D\to(T^{2n},J)$, so $(T^{2n},J)$ is not hyperbolic. Since it is
compact, it cannot be Brody-hyperbolic, so there exists a nonconstant
pseudoholomorphic mapping $g:\C\to T^{2n}$.

Note that actually this method leads to a generalization of
Bangert's result giving $n$ distinct entire lines $g_k:\C\to T^{2n}$
through a given point.
Moreover a further generalization is connected with the fact that we can
take the
$C^\infty$-closure of the set of all tame almost complex structures. This
follows from theorem 7.

Another approach to get an almost complex structure of general position
on $M^{2n}$ is based on theorem~7: if we assume hyperbolicity of the almost
complex structure on the torus $T^{2n}$ it can be perturbed into
general position with preservation of the hyperbolicity. But then
we would need to control our construction in order that this general
position property not be destroyed under the gluing.

Note also that in dimension $2n=4$ Bangert's result follows for a subset of
all almost complex structures from the paper by Kuksin \cite{15}.
Actually he proved that generic almost complex structures are divided into
two classes: one has degree equal to an even nonzero number and the
other equal to zero. Here degree
means the index of the corresponding elliptic operator as in the Gromov
theory.  The nonvanishing of index is equivalent to the existence of at
least $\op{deg}=\op{ind}$ tori $T^2$ in a given homology class. Thus
for the first class the existence of entire lines follows. The
second case remains unclear with this approach.
 \end{ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 5 %
\section{\hskip-20pt . \hskip2pt Concluding remarks}

\hspace{13.5pt}
Note that the Kobayashi-Royden infinitesimal pseudometric  $F_M$  may
also be defined on almost complex manifolds using the results of Nijenhuis
and Woolf.  The definition is
 $$
F_M(p;v) = \inf \frac{1}{r}
 $$
where  $p \in M$, $v \in T_{p}M$  and the infimum is taken with respect to
all pseudoholomorphic mappings  $f : \D \rightarrow M$ with $f(0) = p$
and
$f_{\ast}(0)e = rv$.  This may be reformulated as another invariant
${\pi}F_M(p;v)^{-2} =\sup({\pi}r^2)$ which looks similar to the so-called
Gromov width from symplectic geometry, see \cite{7,19}. The
pseudometric $F_M$ and the corresponding geometry of pseudoholomorphic
disks are studied in~\cite{14}.

On a K\"ahler manifold one can consider further invariants coming from
the Kobayashi-Royden pseudometric and the Riemannian metric.
As follows from our theorem~1 this is also the case for almost
K\"ahler manifolds. For example we consider the following functions on
almost K\"ahler manifold $(M,g,\oo,j)$ with values from $[0,+\infty]$:
 $$
\rho_{M,j}^+(x)=\inf_{\x\in T_xM\setminus\{0\}}\dfrac{|\x|_g}{F_M(\x)}
,\ \ \
\rho_{M,j}^-(x)=\sup_{\x\in T_xM\setminus\{0\}}\dfrac{|\x|_g}{F_M(\x)}.
 $$

Note that the standard methods of generating symplectic invariants
via almost complex ones are not working here. Actually:
$\sup_{j\in\TT(\oo)} \rho^+_{M,j}(x)=+\infty$, $\inf_{j\in\TT(\oo)}
\rho^-_{M,j}(x)=0$.  Here $\TT(\oo)$ is the set of all almost complex
structures compatible with the symplectic form $\oo$ on $M$, i.e. such
structures $j$ that the manifold $M$ become almost K\"ahler with the
metric $g(\x,\e)=\oo(\x,j\e)$.

%\newpage
%
%
%
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%
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%
\begin{thebibliography}{ABCD}
%\addcontentsline{toc}{chapter}{\bf\quad \  Bibliography}

\addtocounter{f}{1}

\bib{1}
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{\it "Existence of a complex line in tame almost complex tori"\/},
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\bib{2}
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\bib{3}
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\bib{4}
R. Brody,
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\bib{5}
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\bib{6}
P. Gauduchon, \addtocounter{f}{1}
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\bib{8}
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\bib{9}
S. Kobayashi, K. Nomizu,
{\it "Foundations of Differential Geometry"\/},
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\bib{10}
S. Kobayashi,
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\bib{11}
S. Kobayashi,
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\bib{12}
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e-print: xxx.lanl.gov\slash abs\slash dg-ga\slash 9611004

\bib{14}
B. Kruglikov,
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S. B. Kuksin,
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J. Lafontaine,
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S. Lang,
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\bib{18}
M. P. Muller, \addtocounter{f}{1}
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D.~McDuff, D.~Salomon,
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A. Nijenhuis, W. Woolf,
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{\it Ann. Math.}, {\bf 77} (1963), 424--489

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J. Noguchi, T. Ochiai,
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K. Yang,
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World Scientific, (1987)


\end{thebibliography}

\ {\hbox to 12.5cm{ \hrulefill }}

\bigskip
{\footnotesize
\begin{itemize}

\item
Math.\ Modelling Chair, Moscow State Techn.\ Univ.\,n.\,a.\ Baumann,
Moscow, Russia

\item[]
kruglikov\verb"@"glasnet.ru

\item
Dept. of Mathematics and Statistics, Univ. of Tromso, N-9037,
Tromso, Norway

\item[]
marius\verb"@"math.uit.no

\end{itemize}
}

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