A construction of Poincar\'e-Einstein metrics of cohomogeneity one on the ball

We exhibit an explicit one-parameter smooth family of Poincar\'e-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ans\"atz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic metric at one of the ends, and at the other end the ball equipped with our metric collapses to a Poincar\'e-Einstein manifold of one lower dimension with an isolated conical singularity.