Hyperk\"ahler ambient metrics associated with twistor CR manifolds

Twistor CR manifolds, introduced by LeBrun, are Lorentzian (neutral) CR 5-manifolds defined as $\mathbb{P}^1$-bundles over 3-dimensional conformal manifolds. In this paper, we embed a real analytic twistor CR manifold into the twistor space of the anti self-dual Poincar\'e-Einstein metric whose conformal infinity is the base conformal 3-manifold, and construct the associated Fefferman ambient metric as a neutral hyperk\"ahler metric on the spinor bundle with the zero section removed. We also describe the structure of the Cheng--Yau type K\"ahler-Einstein metric which has the twistor CR manifold as the boundary at infinity.