In this section, . As noted above, when supplied with the Bergman metric g:
[displaymath843]
where
[displaymath844]
is the Bergman kernel function. The distance function defined by g, i.e. the hyperbolic metric, is denoted by [tex2html_wrap_inline899]. The group of holomorphic isometries of is
Here, leaving invariant the hermitian form
It acts linearly on . By definition, [tex2html_wrap_inline909] is the space of complex lines in , which in inhomogeneous coordinates
corresponds to the ball . Dividing by the center of , which acts effectively on .
Alternatively, is the rank one symmetric space [tex2html_wrap_inline925], where [tex2html_wrap_inline927] is the stabilizer of [tex2html_wrap_inline929]. 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. 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 . 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 ,
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 is given by 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 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, , where y is the solution of . For simplicity we shall call the family the flow generated by , even though it is not a flow in the usual sense.
Proof. Let . There exist unique points such that
Because is bounded from below, the midpoint of the geodesic arc in t. Hence in t.
Assume satisfies
Then
Let
be the canonical projection.
(By [4, Lemma 3.2.2,], H is a closed compact subgroup
isomorphic to [tex2html_wrap_inline1093].)
The projection
is uniquely determined by the
condition (
).
Hence with any choice of
satisfying
(),
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 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 , 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 . The contact structure on is given by the contact form and the Reeb vector field T is defined by . The Levi form on the horizontal bundle is given by
[displaymath861]
and it is positive definite, i.e. [tex2html_wrap_inline1181] is a strictly pseudoconvex domain. Finally, the horizontal gradient is defined by
for a differentiable function p on .
Proof. By [14, Proposition 1,], the flow of H extends to a contact deformation on , i.e. the extension of the vector field [tex2html_wrap_inline1211] to [tex2html_wrap_inline1213] equals [tex2html_wrap_inline1215] for some smooth function [tex2html_wrap_inline1217]. The following calculation is the same as in the proof of Proposition 3 in [14]. In a neighborhood of and supplement of (with respect to g extended as a Hermitian metric on ) normalized such that . Then calculation yields
Note that we got rid of the factor 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,
on
and the formula () with H replaced by
to
.