Hyperk\"ahler ambient metrics associated with twistor CR manifolds

TL;DR

This paper constructs hyperk"ahler ambient metrics linked to twistor CR manifolds, embedding them into twistor spaces of anti self-dual Poincaré-Einstein metrics and analyzing associated K"ahler-Einstein structures.

Contribution

It introduces a method to embed twistor CR manifolds into twistor spaces and constructs associated hyperk"ahler and K"ahler-Einstein metrics, advancing geometric understanding.

Findings

Embedded twistor CR manifolds into twistor spaces.

Constructed Fefferman ambient hyperk"ahler metrics.

Described structure of Cheng--Yau type K"ahler-Einstein metrics.

Abstract

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.