  A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal
metric and a trace-free second fundamental form. Conversely, such a data on the
3-sphere is the boundary of a unique selfdual conformal metric, defined in a
neighborhood of the sphere. In this paper we characterize the conformal metrics
and trace-free second fundamental forms on the 3-sphere (close to the standard
round metric) which are boundaries of selfdual conformal metrics on the whole
4-ball.
  When the data on the boundary is reduced to a conformal metric (the
trace-free part of the second fundamental form vanishes), one may hope to find
in the conformal class of the filling metric an Einstein metric, with a pole of
order 2 on the boundary. We determine which conformal metrics on the 3-sphere
are boundaries of such selfdual Einstein metrics on the 4-ball. In particular,
this implies the Positive Frequency Conjecture of LeBrun.
  The proof uses twistor theory, which enables to translate the problem in
terms of complex analysis; this leads us to prove a criterion for certain
integrable CR structures of signature (1,1) to be fillable by a complex domain.
  Finally, we solve an analogous, higher dimensional problem: selfdual Einstein
metrics are replaced by quaternionic-Kahler metrics, and conformal structures
on the boundary by quaternionic contact structures (previously introduced by
the author); in contrast with the 4-dimensional case, we prove that any small
deformation of the standard quaternionic contact structure on the (4m-1)-sphere
is the boundary of a quaternionic-Kahler metric on the 4m-ball.
