Suppose that
is a strictly pseudoconvex CR manifold bounding a compact complex manifold
of complex dimension
. Under appropriate
geometric conditions on
,
the manifold
admits an approximate Kähler–Einstein metric
which makes
the interior of
a complete Riemannian manifold. We identify certain residues of the scattering operator
on
as conformally covariant differential operators on
and obtain the
CR
-curvature
of
from
the scattering operator as well. In order to construct the Kähler–Einstein metric on
, we
construct a global approximate solution of the complex Monge–Ampère equation on
,
using Fefferman’s local construction for pseudoconvex domains in
. Our
results for the scattering operator on a CR-manifold are the analogue in
CR-geometry of Graham and Zworski’s result on the scattering operator on a real
conformal manifold.
Keywords
CR geometry, Q curvature, geometric scattering theory