  We prove rigidity and gap theorems for self-dual and even Poincar\'e-Einstein
metrics in dimension four. As a corollary, we give an obstruction to the
existence of self-dual Poincar\'e-Einstein metrics in terms of conformal
invariants of the boundary and the topology of the bulk. As a by-product of our
proof we identify a new scalar conformal invariant of three-dimensional
Riemannian manifolds.
