Quarter-pinched Einstein metrics interpolating between real and complex hyperbolic metrics

Abstract

We show that the one-loop quantum deformation of the universal hypermultiplet provides a family of complete 1/4-pinched negatively curved quaternionic Kähler (i.e. half conformally flat Einstein) metrics g c , c ≥ 0, on R 4 . The metric g 0 is the complex hyperbolic metric whereas the family (g c ) c>0 is equivalent to a family of metrics (h b ) b>0 depending on b = 1/c and smoothly extending to b = 0 for which h 0 is the real hyperbolic metric. In this sense the one-loop deformation interpolates between the real and the complex hyperbolic metrics. We also determine the (singular) conformal structure at infinity for the above families.

Symmetries of quaternionic Kähler manifolds with S1‐symmetry

“…The one-loop deformed universal hypermultiplet is not symmetric, and in fact not even locally homogeneous. This follows from the results of [12], which we will review below. Thus, even the trivial PSK manifold yields an interesting quaternionic Kähler manifold.…”

“…Since the universal hypermultiplet is diffeomorphic to R >0 × R 3 , we may use global coordinates (ρ,φ, ζ,ζ), with respect to which the (one-loop deformed) Ferrara-Sabharwal metric takes the following form [12]:…”

“…Proof. An explicit eigenbasis for the (self-adjoint) curvature operator was constructed in [12]. With our normalization, the corresponding eigenvalues are…”

Examples of compact Einstein four-manifolds with negative curvature

“…It was recently observed by Cortés and Saha [12] that Pedersen's metrics are also negatively curved, even when far from the hyperbolic metric. Other explicit examples can also be found in [12].…”

“…In 4 dimensions, a 1parameter family of such deformations with an explicit formula was given by Pedersen [30]. It was recently observed by Cortés and Saha [12] that Pedersen's metrics are also negatively curved, even when far from the hyperbolic metric. Other explicit examples can also be found in [12].…”