In this section, 
shall always be the unit ball in 
. As
noted above, is a model of 
when supplied with the Bergman metric g:
where
is the Bergman kernel function. The distance function defined by g,
i.e. the hyperbolic metric, is denoted by 
. The group of
holomorphic isometries of is
Here, U(1,d) is the subgroup of 
leaving invariant the
hermitian form
It acts linearly on 
.
By definition, is the space of
complex lines in 
, which in inhomogeneous coordinates
corresponds to the ball .
Dividing by the center 
of
U(1,d), we get the quotient PU(1,d) = U(1,d)/Z, which acts
effectively on .
Alternatively, is the rank
one symmetric space G/K, where 
is the stabilizer of 
.
The symmetry at 0 is just 
.
The metric g can be defined at z=0 by the Killing form of the Lie
algebra of G and then be transported everywhere by elements of G.
Together with the natural complex structure J, complex hyperbolic space also carries a symplectic structure
i.e. 
, and is a
Kähler manifold.
The Heisenberg group arises as the `translation part' of the
stabilizer of the point 
of the action of G on
. The Iwasawa decomposition of this stabilizer is MAN,
where 
, 
, and N is the
Heisenberg group, see [13]. The group N acts simply
transitively on 
, hence its one-point
compactification 
can be identified with . In
suitable coordinates, the Heisenberg group N is
with the group law
The Heisenberg metric d is defined by
where
By definition it is left-invariant.
Via stereographic projection, d is conformally equivalent to the
metric 
on ,
see [13, section F,]. Thus, we say a homeomorphism of
is `quasiconformal with respect to the Heisenberg
metric' if and only if it is quasiconformal with respect to .
For two points 
, let
be the equidistant hypersurface separating them.
A 
-family of equidistant hypersurfaces 
is given by 
with 
of class
in t.
In order to deform fundamental domains for Schottky groups, we will
need the lemma below relating -families of equidistant
hypersurfaces to flows. Recall that a continuously time dependent
smooth vector field 
defined for 
on a manifold
M generates a two-parameter family of diffeomorphisms 
,
such that
provided M is compact or X does not grow too fast at infinity.
To be specific, 
is defined as y(t), where y is
the solution of 
with y(s)=x.
For simplicity we shall call the family the flow
generated by , even though it is not a flow in the usual sense.
Proof .
Let 
be the (real) geodesic through 
and 
.
There exist unique points 
such that
Because 
is bounded from below, the midpoint of the
geodesic arc 
and the endpoints of are in
t. Hence 
and 
are in t.
Assume 
satisfies
Then
Let 
be the subgroup fixing 
and 
and let 
be the canonical projection.
(By [4, Lemma 3.2.2,], H is a closed compact subgroup
isomorphic to 
.)
The projection 
is uniquely determined by the
condition (2.1).
Hence with any choice of 
satisfying
(2.1), is in t, because and
are .
Since H is a submanifold of G, it is easy to see in local
coordinates that we can choose a -lift of the curve
which is the identity at t=0.
We denote this lift again by 
.
Now we define the vector field on 
by
Because is an isometry and is biholomorphic, it is a
symplectomorphism of .
Thus, the inner product 
is closed:
(See for instance [10], formula (22.1).)
In other words, the vector field is locally Hamiltonian.
But is simply connected, so there exists a global Hamiltonian
for .
From the definition of the vector field we see that it is
continuous in time.
In the variable 
it is 
, because the action
, 
is smooth in both variables:
where 
is an
element of the Lie algebra of G, only depending on t.
Since the Hamiltonian is locally determined up to a constant by
, it can be chosen continuous in t and
in x.
We shall also need a lemma about extending Hamiltonian vector fields.
Again we begin with some notation.
A defining function (or Kähler potential) for is
, that is, 
.
The contact structure on is given by the contact
form 
and the Reeb vector field T is
defined by 
and 
.
The Levi form on the horizontal bundle 
is given by
and it is positive definite, i.e. is a strictly pseudoconvex
domain.
Finally, the horizontal gradient 
is defined by
for a differentiable function p on and 
.
Proof .
By [14, Proposition 1,], the flow of H extends to a contact
deformation on , i.e. the extension of the vector field
to equals 
for some smooth function 
.
The following calculation is the same as in the proof of
Proposition 3 in [14].
In a neighborhood of set 
and
supplement 
to an orthogonal basis 
of
(with respect to
g extended as a Hermitian metric on 
) normalized
such that 
for 
.
Then calculation yields
Note that we got rid of the factor 2/(d+1) in [14], because
of our different normalization of the symplectic structure:
instead of
.
(With the normalization used here, the holomorphic sectional
curvature is -2, cf. [7].)
Now, 
as one approaches the boundary
(cf. [14]).
It follows that 
.
Thus, 
is
on and the formula (2.2) with H replaced by
shows the extension of 
to
is given by 
.
