Four-dimensional Einstein manifolds with Heisenberg symmetry

Abstract

We classify Einstein metrics on $$\mathbb {R}^4$$ R 4 invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. We consider metrics which are either Ricci-flat or of negative Ricci curvature. We show that all of the Ricci-flat metrics, including the simplest ones which are hyper-Kähler, are incomplete. By contrast, those of negative Ricci curvature contain precisely two complete examples: the complex hyperbolic metric and a metric of cohomogeneity one known as the one-loop deformed universal hypermultiplet.

Heisenberg-invariant self-dual Einstein manifolds

“…Remark 3.13. In the Riemannian case, it is possible to see that the solution (37) is completely equivalent to that obtained in [CS,proposition 4.1]. Using the subindex CS to make reference to quantities of that article, by performing the identifications e…”

Heisenberg-invariant self-dual Einstein manifolds

“…Remark 3.13. In the Riemannian case, it is possible to see that the solution ( 37) is completely equivalent to that obtained in [CS,Proposition 4.1]. Using the subindex CS to make reference to quantities of that article, by performing the identifications…”

Inhomogeneous deformations of Einstein solvmanifolds

“…Our main aim in this section is to show that there is a cohomogeneity one Einstein manifold interpolating between the Einstein solvmanifold (𝖲, g 𝑆 ) and real hyperbolic space (ℝ𝐻 𝑛+1 , g 𝐻 ). In the case that 𝖲 0 = 𝐻 3 (ℝ) is the Heisenberg group, this is the one-loop deformed universal hypermultiplet metric studied in [11]. In this case, 𝖲 = ℂ𝐻 2 .…”

“…These are asymptotic at one end to (a quotient of) an Einstein solvmanifold. Moreover, the Einstein manifold in [11] is asymptotic at its other end to $$\mathbb {R}H^4$$; the Kähler–Ricci solitons in [27] are asymptotic to a cone whose link is $$\Gamma \backslash \mathsf {S}_0$$ at their other end; and, the solitons in [24] are asymptotic to a cone whose link is the two‐torus.…”