ℋ -Space with a cosmological constant

Abstract

It is demonstrated that a real-analytic 3-manifold with Riemannian conformal metric is naturally the conformal infinity of a germ-unique real-analytic 4-manifold with real-analytic Riemannian metric satisfying the self-dual Einstein equations with cosmological constant - 1. Moreover, this result holds if 'Riemannian' is replaced in the first case by 'Lorentzian' (i.e. signature + - -) and in the second case by 'pseudo-Riemannian with signature + + - -', or if 'real-analytic' is replaced by 'complex-analytic' and 'Riemannian' is replaced by 'holomorphic'. This provides a cosmological-constant analogue of Newman's H-space construction (Newman 1976, 1977).